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Path Analysis for Binary Random Variables

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Title: Path Analysis for Binary Random Variables
Language: English
Authors: Raggi, Martina (ORCID 0000-0002-3309-9006), Stanghellini, Elena (ORCID 0000-0002-2503-8342), Doretti, Marco
Source: Sociological Methods & Research. 2023 52(4):1883-1915.
Availability: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
Peer Reviewed: Y
Page Count: 33
Publication Date: 2023
Document Type: Journal Articles
Reports - Evaluative
Descriptors: Path Analysis, Student Attitudes, Museums, Error of Measurement, Foreign Countries, Graphs, Regression (Statistics)
Geographic Terms: Italy
DOI: 10.1177/00491241211031260
ISSN: 0049-1241
1552-8294
Abstract: The decomposition of the overall effect of a treatment into direct and indirect effects is here investigated with reference to a recursive system of binary random variables. We show how, for the single mediator context, the marginal effect measured on the log odds scale can be written as the sum of the indirect and direct effects plus a residual term that vanishes under some specific conditions. We then extend our definitions to situations involving multiple mediators and address research questions concerning the decomposition of the total effect when some mediators on the pathway from the treatment to the outcome are marginalized over. Connections to the counterfactual definitions of the effects are also made. Data coming from an encouragement design on students' attitude to visit museums in Florence, Italy, are reanalyzed. The estimates of the defined quantities are reported together with their standard errors to compute p values and form confidence intervals.
Abstractor: As Provided
Entry Date: 2023
Accession Number: EJ1397548
Database: ERIC
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  Value: <anid>AN0173122300;som01nov.23;2023Oct25.06:20;v2.2.500</anid> <title id="AN0173122300-1">Path Analysis for Binary Random Variables </title> <p>The decomposition of the overall effect of a treatment into direct and indirect effects is here investigated with reference to a recursive system of binary random variables. We show how, for the single mediator context, the marginal effect measured on the log odds scale can be written as the sum of the indirect and direct effects plus a residual term that vanishes under some specific conditions. We then extend our definitions to situations involving multiple mediators and address research questions concerning the decomposition of the total effect when some mediators on the pathway from the treatment to the outcome are marginalized over. Connections to the counterfactual definitions of the effects are also made. Data coming from an encouragement design on students' attitude to visit museums in Florence, Italy, are reanalyzed. The estimates of the defined quantities are reported together with their standard errors to compute p values and form confidence intervals.</p> <p>Keywords: directed acyclic graph; logistic regression; recursive system; effect decomposition; multiple mediators</p> <hd id="AN0173122300-2">Introduction</hd> <p>The decomposition of the total effect of a treatment <emph>X</emph> on an outcome variable <emph>Y</emph> into direct and indirect effects is a central topic in empirical research. In linear models, the relationship between total, direct, and indirect effects is well understood ([<reflink idref="bib1" id="ref1">1</reflink>]; [<reflink idref="bib4" id="ref2">4</reflink>]; [<reflink idref="bib3" id="ref3">3</reflink>]; [<reflink idref="bib7" id="ref4">7</reflink>]) and a simple decomposition is available. Such a decomposition is based on the linearity of the marginal model for <emph>Y</emph> against <emph>X</emph>, where the coefficient of <emph>X</emph> is equal to the sum of the direct effect and the indirect effects. Outside the linear case, this simplicity is lost, as in the marginal model of <emph>Y</emph> against <emph>X</emph> only, either the effect of <emph>X</emph> on <emph>Y</emph> is a complex function of the original parameters or the error term does not possess nice properties or both.</p> <p>We here consider situations where the outcome <emph>Y</emph> is a binary random variable. Contributions have addressed the case of one continuous mediator (see, e.g., [<reflink idref="bib5" id="ref5">5</reflink>], [<reflink idref="bib6" id="ref6">6</reflink>]; [<reflink idref="bib16" id="ref7">16</reflink>]; [<reflink idref="bib20" id="ref8">20</reflink>]). Recent results concern the exact parametric form of the marginal effect of <emph>X</emph> on <emph>Y</emph> on the log odds ratio scale when the mediator is also binary ([<reflink idref="bib27" id="ref9">27</reflink>]). In this setting, when <emph>X</emph> is continuous the marginal model of <emph>Y</emph> against <emph>X</emph> is nonlinear unless some conditional independence assumptions hold, and a rather complex formula links the marginal and conditional effect of <emph>X</emph> on <emph>Y</emph>. Similarly, for a discrete <emph>X</emph>, the parameters of the conditional model combine in a nonlinear fashion to form the marginal effect. For analogous results on the log relative risk scale, see [<reflink idref="bib19" id="ref10">19</reflink>].</p> <p>Starting from the results in [<reflink idref="bib27" id="ref11">27</reflink>], we here elaborate a novel proposal for the direct and indirect effect definitions on the log odds scale for a treatment variable <emph>X</emph> either continuous or discrete. The postulated system can be represented by a directed acyclic graph (DAG); see [<reflink idref="bib18" id="ref12">18</reflink>], to which we refer for definitions, see also [<reflink idref="bib11" id="ref13">11</reflink>] for an account in the sociological context. Our proposal is based on zeroing the path-specific regression coefficients. Graphically, this corresponds to deleting one arrow in the associated DAG and thereby represents the analogue of the path analysis method.</p> <p>We initially focus on a single-mediator context and show that the marginal effect can be written as the sum of the indirect and direct effects plus a residual term that vanishes under some specific conditions. The proposed parametric relationship allows, for the specific setting under investigation, to solve the debate on which method should be used to disentangle the total effect, that is, the product method or the difference method ([<reflink idref="bib6" id="ref14">6</reflink>]). It also avoids fitting two nested models, thereby sidestepping the issue of unequal variance ([<reflink idref="bib33" id="ref15">33</reflink>]). We then extend our derivations to the case of multiple binary mediators, also modeled as a recursive system of univariate logistic regressions. In this context, additional path-specific effects can be defined and different research questions addressed. Although the paper draws from the derivation in [<reflink idref="bib27" id="ref16">27</reflink>], some novel results are also presented. With reference to a single mediator, a general formulation of functional form linking the log odds of the mediator and of the outcome to the covariates is considered. This is then extended to the multiple mediator context, for which a strategy for deriving the direct and indirect effects when marginalizing over an intermediate or outer mediator is also illustrated.</p> <p>Our approach is developed in a purely associational context that, in general, holds no interpretation for causal inference. However, if the recursive system of equation is structural ([<reflink idref="bib24" id="ref17">24</reflink>], chap. 7) and no unmeasured confounders exist, the total effect and some of its components can be endowed with a causal interpretation. Notice that a decomposition of the total effect based on counterfactual entities has been given by [<reflink idref="bib23" id="ref18">23</reflink>], [<reflink idref="bib25" id="ref19">25</reflink>]) and extended to the odds ratio scale by [<reflink idref="bib31" id="ref20">31</reflink>]. This parallelism is also addressed in this article.</p> <p>In the second section, we offer the general theory for the case of a single mediator. A case study concerning a randomized encouragement experiment on cultural consumption performed in Florence (Italy) is also presented as a guiding example. Results of a simulation study investigating maximum likelihood (ML) estimation of the effects and other related measures is also reported in the third section, while the extension to the multiple mediator setting is contained in the fourth section. In the fifth section, we address other complex issues concerning path-specific effects, whereas links with counterfactual definitions are explored in the sixth section. Finally, in the seventh section, we draw some conclusions.</p> <p>Graph: Figure 1. Data generating process when (A) no conditional independences hold, (B) X╨Y|W, (C) W╨Y|X, and (D) W╨X.</p> <hd id="AN0173122300-3">Effect Decomposition With a Single Mediator</hd> <p>We first focus on a very simple model for a binary outcome <emph>Y</emph>, a binary mediator <emph>W</emph>, and a treatment <emph>X</emph>, that can be either discrete or continuous (see Figure 1A for the corresponding DAG). Our aim is to decompose the total effect of <emph>X</emph> on <emph>Y</emph> on the log odds scale. Our postulated models are a logistic regression for <emph>Y</emph> given <emph>X</emph> and <emph>W</emph> and for <emph>W</emph> given <emph>X</emph>, that is,</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><mi>W</mi><mo>=</mo><mi>w</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><mi>W</mi><mo>=</mo><mi>w</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo>+</mo><msub><mi>β</mi><mi>w</mi></msub><mi>w</mi><mo>+</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mi>x</mi><mi>w</mi><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>and</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><msub><mi>γ</mi><mn>0</mn></msub><mo>+</mo><msub><mi>γ</mi><mi>x</mi></msub><mi>x</mi><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>Notice that we allow for the interaction between <emph>X</emph> and <emph>W</emph> in the outcome equation. In order to make the paper self-contained, the derivations in [<reflink idref="bib27" id="ref21">27</reflink>] to evaluate the total effect as a function of the parameters of models (<reflink idref="bib1" id="ref22">1</reflink>) and (<reflink idref="bib2" id="ref23">2</reflink>) are here reproduced, prior the introduction of its decomposition into direct, indirect, and residual effects.</p> <p>As a guiding example through this section, we consider an experiment aiming at identifying the best incentives to offer high school students in Florence to enhance cultural interest and increase art museum attendance. Three treatment levels are considered: A flyer given to the students with the main information about the Palazzo Vecchio museum constitutes the first level; a flyer and a presentation of the museum from an expert constitute the second level; a flyer, the presentation, and a reward in the form of extra-credit points for their final school grade constitute the third level. All students receive a free entry ticket to Palazzo Vecchio. The aim of the experiment is not only to assess the total effect of the treatment (<emph>X</emph>) on students museum's attendance (<emph>Y</emph>) but also to understand to what extent this effect could be stimulated by student's visit to Palazzo Vecchio (<emph>W</emph>); see [<reflink idref="bib13" id="ref24">13</reflink>]; [<reflink idref="bib17" id="ref25">17</reflink>].</p> <p>The interest is in the marginal model of <emph>Y</emph> against <emph>X</emph>, as a function of the parameters in equations (<reflink idref="bib1" id="ref26">1</reflink>) and (<reflink idref="bib2" id="ref27">2</reflink>). From first principles of probabilities, it follows that:</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mo>−</mo><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mi>w</mi><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mi>w</mi><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>W</mi><mo>=</mo><mi>w</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>W</mi><mo>=</mo><mi>w</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>The second term of the righthand side (RHS) of the above equality is given from model (<reflink idref="bib1" id="ref28">1</reflink>), while the parametric expression of the first term is not immediately derived from models (<reflink idref="bib1" id="ref29">1</reflink>) and (<reflink idref="bib2" id="ref30">2</reflink>), as it involves the probability of <emph>W</emph> after conditioning on <emph>X</emph> and <emph>Y</emph> (not after conditioning on <emph>X</emph> only). However, by repeated use of the previous relationship, we have:</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">|</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">|</mo><mi>W</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>Using equations (<reflink idref="bib1" id="ref31">1</reflink>) and (<reflink idref="bib2" id="ref32">2</reflink>), after some simplifications, we find:</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mi>y</mi><mo stretchy="false">(</mo><msub><mi>β</mi><mi>w</mi></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><mo stretchy="false">(</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><mo stretchy="false">(</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo>+</mo><msub><mi>β</mi><mi>w</mi></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><msub><mi>γ</mi><mn>0</mn></msub><mo>+</mo><msub><mi>γ</mi><mi>x</mi></msub><mi>x</mi><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>In what follows, we denote with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>g</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> the <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> </ephtml> , that is,</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>g</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo /></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>y</mi><mo stretchy="false">(</mo><msub><mi>β</mi><mi>w</mi></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><mo stretchy="false">(</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><mo stretchy="false">(</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo>+</mo><msub><mi>β</mi><mi>w</mi></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><msub><mi>γ</mi><mn>0</mn></msub><mo>+</mo><msub><mi>γ</mi><mi>x</mi></msub><mi>x</mi><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Graph</p> <p>Since <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> </ephtml> corresponds to <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> , substituting in equation (<reflink idref="bib3" id="ref33">3</reflink>) for <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>w</mi><mo>=</mo><mn>0</mn></mrow></math> </ephtml> , we find:</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>Denoting with</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mtext>RR</mtext></mrow><mrow><mi>W</mi><mo stretchy="false">|</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>the relative risk of <emph>W</emph> for varying <emph>Y</emph> in the distribution of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>=</mo><mi>x</mi></mrow></math> </ephtml> , we have</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mtext>RR</mtext></mrow><mrow><mi>W</mi><mo stretchy="false">|</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi></mrow></msub><mo>=</mo><mfrac><mrow><mtext>exp</mtext><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>{</mo><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo></mrow><mrow><mtext>exp</mtext><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>{</mo><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo></mrow></mfrac><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>Analogously, letting <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mi>W</mi><mo>¯</mo></mover><mo>=</mo><mn>1</mn><mo>−</mo><mi>W</mi></mrow></math> </ephtml> , then</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mtext>RR</mtext></mrow><mrow><mover accent="true"><mi>W</mi><mo>¯</mo></mover><mo stretchy="false">|</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>It then follows that equation (<reflink idref="bib6" id="ref34">6</reflink>) can be rewritten as</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo>−</mo><mtext>log</mtext><msub><mrow><mtext>RR</mtext></mrow><mrow><mover accent="true"><mi>W</mi><mo>¯</mo></mover><mo stretchy="false">|</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi></mrow></msub><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>or, alternatively, as</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo>+</mo><msub><mi>β</mi><mi>w</mi></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mi>x</mi><mo>−</mo><mtext>log</mtext><msub><mrow><mtext>RR</mtext></mrow><mrow><mi>W</mi><mo stretchy="false">|</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi></mrow></msub><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>Notice that conditioning on a set of covariates <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>C</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>C</mi><mn>1</mn></msub><mo>,</mo><mo>...</mo><msub><mi>C</mi><mi>p</mi></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> does not strongly alter the structure of equations (<reflink idref="bib6" id="ref35">6</reflink>) and (<reflink idref="bib8" id="ref36">8</reflink>). We here offer an example for <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>p</mi><mo>=</mo><mn>1</mn></mrow></math> </ephtml> , with both additive and interaction effects up to the second order in both models for <emph>Y</emph> and <emph>W</emph>. After the marginalization over <emph>W</emph>, we obtain the marginal model for <emph>Y</emph> given <emph>X</emph> and <emph>C</emph> as:</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><mi>C</mi><mo>=</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><mi>C</mi><mo>=</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo>+</mo><msub><mi>β</mi><mi>c</mi></msub><mi>c</mi><mo>+</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>c</mi></mrow></msub><mi>x</mi><mi>c</mi><mo>−</mo><mtext>log</mtext><msub><mrow><mtext>RR</mtext></mrow><mrow><mover accent="true"><mi>W</mi><mo>¯</mo></mover><mo stretchy="false">|</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><mi>C</mi><mo>=</mo><mi>c</mi></mrow></msub></mrow></math> </ephtml> </p> <p>Graph</p> <p>with</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mtext>RR</mtext></mrow><mrow><mover accent="true"><mi>W</mi><mo>¯</mo></mover><mo stretchy="false">|</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><mi>C</mi><mo>=</mo><mi>c</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>and</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>g</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>y</mi><mfenced><mrow><msub><mi>β</mi><mi>w</mi></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mi>x</mi><mo>+</mo><msub><mi>β</mi><mrow><mi>c</mi><mi>w</mi></mrow></msub><mi>c</mi></mrow></mfenced></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo /></mrow></mtd><mtd columnalign="left"><mrow><mo>+</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><mo stretchy="false">(</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo>+</mo><msub><mi>β</mi><mi>c</mi></msub><mi>c</mi><mo>+</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>c</mi></mrow></msub><mi>x</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><mo stretchy="false">(</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo>+</mo><msub><mi>β</mi><mi>c</mi></msub><mi>c</mi><mo>+</mo><msub><mi>β</mi><mi>w</mi></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mi>x</mi><mo>+</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>c</mi></mrow></msub><mi>x</mi><mi>c</mi><mo>+</mo><msub><mi>β</mi><mrow><mi>c</mi><mi>w</mi></mrow></msub><mi>c</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo /></mrow></mtd><mtd columnalign="left"><mrow><mo>+</mo><msub><mi>γ</mi><mn>0</mn></msub><mo>+</mo><msub><mi>γ</mi><mi>x</mi></msub><mi>x</mi><mo>+</mo><msub><mi>γ</mi><mi>c</mi></msub><mi>c</mi><mo>+</mo><msub><mi>γ</mi><mrow><mi>x</mi><mi>c</mi></mrow></msub><mi>x</mi><mi>c</mi><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Graph</p> <p>See Online Appendix A (Supplementary material for this article is available online) for a general formulation that includes more covariates and possibly nonlinear link functions.</p> <p>With reference to the cultural consumption data, 15 classes for a total of 294 students, all aged between 15 and 18, from three different schools, were randomly assigned at baseline (March/April 2014) to the three treatment levels (<emph>X</emph>). At the second occasion, after two months, researchers collected the entry tickets to record student visits (<emph>W</emph>). Finally, after six months, they collected the concluding questionnaire with general information on visits to other museums (<emph>Y</emph>). A questionnaire with information on background characteristics of the students and their families was also administered. Among all the covariates, only one appears to be relevant in the model for the outcome, that is, the binary variable <emph>C</emph> taking value 1 for students considering themselves mainly interested in mathematics/science and 0 if they are mainly interested in humanities. At follow-up, 28 students were absent, so the final sample included 266 students. Data are reported in Table 1 and are publicly available at https://<ulink href="http://www.tandfonline.com/doi/abs/10.1080/07350015.2019.1647843">www.tandfonline.com/doi/abs/10.1080/07350015.2019.1647843</ulink> as supplementary material of [<reflink idref="bib13" id="ref37">13</reflink>].</p> <p>Graph</p> <p>Table 1. Contingency Tables for ( <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Y</mi><mo>,</mo><mi>X</mi><mo>,</mo><mi>W</mi><mo>,</mo><mi>C</mi></mrow></math> </ephtml> ) for the Cultural Consumption Experiment.</p> <p> <ephtml> <table><thead><tr><th /><th /><th><italic>Y</italic></th><th /><th /><th /><th><italic>Y</italic></th><th /></tr></thead><tbody><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>W</mi><mo>=</mo><mn>0</mn></mrow></math></p></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>C</mi><mo>=</mo><mn>0</mn></mrow></math></p></td><td>0</td><td>1</td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>W</mi><mo>=</mo><mn>0</mn></mrow></math></p></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>C</mi><mo>=</mo><mn>1</mn></mrow></math></p></td><td>0</td><td>1</td></tr><tr><td><italic>X</italic></td><td>1</td><td>19</td><td>2</td><td><italic>X</italic></td><td>1</td><td>48</td><td>17</td></tr><tr><td /><td>2</td><td>14</td><td>21</td><td /><td>2</td><td>14</td><td>28</td></tr><tr><td /><td>3</td><td>3</td><td>3</td><td /><td>3</td><td>23</td><td>21</td></tr><tr><td /><td /><td>Y</td><td /><td /><td /><td><italic>Y</italic></td><td /></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>W</mi><mo>=</mo><mn>1</mn></mrow></math></p></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>C</mi><mo>=</mo><mn>0</mn></mrow></math></p></td><td>0</td><td>1</td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>W</mi><mo>=</mo><mn>1</mn></mrow></math></p></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>C</mi><mo>=</mo><mn>1</mn></mrow></math></p></td><td>0</td><td>1</td></tr><tr><td><italic>X</italic></td><td>1</td><td>0</td><td>0</td><td><italic>X</italic></td><td>1</td><td>1</td><td>2</td></tr><tr><td /><td>2</td><td>1</td><td>0</td><td /><td>2</td><td>6</td><td>3</td></tr><tr><td /><td>3</td><td>1</td><td>9</td><td /><td>3</td><td>19</td><td>11</td></tr></tbody></table> </ephtml> </p> <p>Table 2 contains the output of the ML estimation of the logistic regression models for the outcome and for the mediator. We use the subscript <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></math> </ephtml> to denote the contrast of level 2 (Flyer + Presentation) versus level 1 (Flyer) and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></math> </ephtml> for the contrast of level 3 (Flyer + Presentation + Reward) versus level 1. Notice that the interaction terms <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>β</mi><mrow><msub><mi>x</mi><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub><mi>w</mi></mrow></msub></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>β</mi><mrow><msub><mi>x</mi><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub><mi>w</mi></mrow></msub></mrow></math> </ephtml> in the outcome equation are significant.</p> <p>Graph</p> <p>Table 2. Maximum Likelihood Estimates of the Two Logistic Models for Y and W for the Cultural Consumption Experiment.</p> <p> <ephtml> <table><thead><tr><th>Parameter</th><th>Estimate</th><th>Standard Error</th><th align="center" colspan="2">95 Percent Confidence Interval</th><th><italic>p</italic> Value</th></tr></thead><tbody><tr><td /><td align="center" colspan="5"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>Y</mi><mo>∼</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>X</mi><mo>+</mo><msub><mi>β</mi><mi>c</mi></msub><mi>C</mi><mo>+</mo><msub><mi>β</mi><mi>w</mi></msub><mi>W</mi><mo>+</mo><msub><mi>β</mi><mrow><mi>c</mi><mi>w</mi></mrow></msub><mi>C</mi><mi>W</mi></mrow></math></p></td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>β</mi><mn>0</mn></msub></mrow></math></p></td><td>−1.6186</td><td>0.3857</td><td>−2.3746</td><td>−0.8626</td><td>.0000</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>β</mi><mrow><msub><mi>x</mi><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></msub></mrow></math></p></td><td>1.9345</td><td>0.3676</td><td>1.2139</td><td>2.6550</td><td>.0000</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>β</mi><mrow><msub><mi>x</mi><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></msub></mrow></math></p></td><td>1.1329</td><td>0.3865</td><td>0.3754</td><td>1.8904</td><td>.0034</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>β</mi><mi>c</mi></msub></mrow></math></p></td><td>0.4597</td><td>0.3540</td><td>−0.2342</td><td>1.1536</td><td>.1941</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>β</mi><mi>w</mi></msub></mrow></math></p></td><td>4.3290</td><td>1.5427</td><td>1.3053</td><td>7.3527</td><td>.0050</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>β</mi><mrow><msub><mi>x</mi><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub><mi>w</mi></mrow></msub></mrow></math></p></td><td>−3.7077</td><td>1.4725</td><td>−6.5937</td><td>−0.8217</td><td>.0118</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>β</mi><mrow><msub><mi>x</mi><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub><mi>w</mi></mrow></msub></mrow></math></p></td><td>−2.2708</td><td>1.3365</td><td>−4.8903</td><td>0.3488</td><td>.0893</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>β</mi><mrow><mi>c</mi><mi>w</mi></mrow></msub></mrow></math></p></td><td>−2.4770</td><td>0.9255</td><td>−4.2910</td><td>−0.6630</td><td>.0074</td></tr><tr><td align="center" colspan="6"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>W</mi><mo>∼</mo><msub><mi>γ</mi><mn>0</mn></msub><mo>+</mo><msub><mi>γ</mi><mi>x</mi></msub><mi>X</mi></mrow></math></p></td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>γ</mi><mn>0</mn></msub></mrow></math></p></td><td>−3.3557</td><td>0.5873</td><td>−4.5069</td><td>−2.2046</td><td>.0000</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>γ</mi><mrow><msub><mi>x</mi><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></msub></mrow></math></p></td><td>1.3145</td><td>0.6767</td><td>−0.0118</td><td>2.6409</td><td>.0521</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>γ</mi><mrow><msub><mi>x</mi><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></msub></mrow></math></p></td><td>3.1326</td><td>0.6245</td><td>1.9087</td><td>4.3565</td><td>.0000</td></tr></tbody></table> </ephtml> </p> <p>Ignoring for now the sampling errors, we see that the <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>g</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> function can be formed by plugging in equation (<reflink idref="bib10" id="ref38">10</reflink>) the ML estimates of the parameters. The function expresses the log odds ratio of <emph>W</emph> after conditioning on <emph>X</emph>, <emph>Y</emph> and the covariate <emph>C</emph>.</p> <p>We now present a definition for the total, direct, and indirect effects in the situation with no covariates. When covariates <emph>C</emph> are present, the parametric formula of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> and its decomposition, for both continuous and discrete <emph>X</emph>, vary with the level of <emph>C</emph>. Notice that the direct and indirect effects so defined do not sum to the total effect, but a residual term remains. This term is zero only under some specific conditions that we are going to discuss.</p> <p> <emph>Total effect</emph>: Let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> be the effect of <emph>X</emph> on <emph>Y</emph>, on the log odds scale, in the distribution of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">|</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> obtained after marginalization on <emph>W</emph>. For <emph>X</emph> continuous and differentiable, the total effect is defined as the derivative of equation (<reflink idref="bib6" id="ref39">6</reflink>) with respect to <emph>x</emph>. For <emph>X</emph> discrete, the total effect is defined as the difference between equation (<reflink idref="bib6" id="ref40">6</reflink>) evaluated at two different levels of <emph>X</emph>.</p> <p> <emph>Indirect effect</emph>: Let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> be the indirect effect of <emph>X</emph> on <emph>Y</emph> on the log odds scale. The indirect effect is defined as the part of the total effect of <emph>X</emph> on <emph>Y</emph> through <emph>W</emph> only. It is evaluated after imposing, in the total effect, the coefficients of <emph>X</emph> in the model for <emph>Y</emph> equal to zero, that is, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>β</mi><mi>x</mi></msub><mo>=</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> , that is, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mi>β</mi><mi>x</mi></msub><mo>=</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub></mrow></math> </ephtml> , so that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>╨</mo><mi>Y</mi><mo stretchy="false">|</mo><mi>W</mi></mrow></math> </ephtml> and the effect of <emph>X</emph> on <emph>Y</emph> is mediated by <emph>W</emph> (see Figure 1B).</p> <p> <emph>Direct effect</emph>: Let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> be the direct effect of <emph>X</emph> on <emph>Y</emph> on the log odds scale. The direct effect is defined as the part of the total effect due to <emph>X</emph> only. It is obtained after imposing, in the total effect, the coefficients of <emph>W</emph> in the model for <emph>Y</emph> equal to zero, that is, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>β</mi><mi>w</mi></msub><mo>=</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> so that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>W</mi><mo>╨</mo><mi>Y</mi><mo stretchy="false">|</mo><mi>X</mi></mrow></math> </ephtml> (see Figure 1C). In other words, we have <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mi>β</mi><mi>w</mi></msub><mo>=</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub></mrow></math> </ephtml> . This definition is aligned with the collapsibility of odds ratio, as explained by [<reflink idref="bib35" id="ref41">35</reflink>], Corollary 3). It is important to notice that the direct effect can also be seen as the effect of <emph>X</emph> on <emph>Y</emph> keeping <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>W</mi><mo>=</mo><mn>0</mn></mrow></math> </ephtml> .</p> <p> <emph>Residual effect:</emph> Let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> be the residual effect of <emph>X</emph> on <emph>Y</emph> on the log odds scale defined as <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> . Clearly, by construction</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>Notice that this residual term is always null in linear models. In this context, the total effect can be decomposed into the sum of the direct and indirect effect. Provided that these two are positive, it is therefore meaningful to look at the ratio between the indirect effect and the total effect, as it gives an indication of the proportion of total effect due to the mediator <emph>W</emph> (i.e., the proportion mediated). When a residual effect is present, the ratio between the indirect and total effect can still provide information on the weight of the indirect effect on the total effect, though with a less clear interpretation (see Continuous Case and Discrete Case subsections).</p> <p>In what follows, we study in detail the decomposition of the total effect for the simple case without covariates, where <emph>X</emph> can be either continuous or discrete. Addition of covariates can be done in a straightforward manner.</p> <hd id="AN0173122300-4">Continuous Case</hd> <p>We first look at the case of <emph>X</emph> continuous and differentiable. Let</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>It is possible to show that</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msub><mi>β</mi><mi>x</mi></msub><mo>{</mo><mn>1</mn><mo>−</mo><msub><mo>Δ</mo><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo /></mrow></mtd><mtd columnalign="left"><mrow><mo>+</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>{</mo><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo /></mrow></mtd><mtd columnalign="left"><mrow><mo>+</mo><msub><mi>γ</mi><mi>x</mi></msub><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Graph</p> <p>where</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mo>Δ</mo><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>W</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo /></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mrow><mtext>exp</mtext><mo stretchy="false">(</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo>+</mo><msub><mi>β</mi><mi>w</mi></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><mo stretchy="false">(</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo>+</mo><msub><mi>β</mi><mi>w</mi></msub><mo>+</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mfrac><mrow><mtext>exp</mtext><mo stretchy="false">(</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><mo stretchy="false">(</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr></mtable><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>and</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo /></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mrow><mtext>exp</mtext><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mfrac><mrow><mtext>exp</mtext><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Graph</p> <p>with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>g</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> as in equation (<reflink idref="bib5" id="ref42">5</reflink>). Equation (<reflink idref="bib12" id="ref43">12</reflink>) confirms the well-known fact that the marginal logistic model is nonlinear in <emph>x</emph>, also providing the explicit expression of it. Notice that, as shown in [<reflink idref="bib27" id="ref44">27</reflink>], all terms in curly bracket are bounded between 0 and 1, while <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is bounded between −1 and 1. Notice further that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>Δ</mo><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> share the same sign, and they are both zero whenever <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>W</mi><mo>╨</mo><mi>Y</mi><mo stretchy="false">|</mo><mi>X</mi></mrow></math> </ephtml> .</p> <p> <emph>Indirect effect</emph>: Following the definition, we evaluate the total effect assuming <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>β</mi><mi>x</mi></msub><mo>=</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> , that is,</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mi>β</mi><mi>x</mi></msub><mo>=</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><msub><mi>γ</mi><mi>x</mi></msub><msubsup><mo>Δ</mo><mi>w</mi><mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>where <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mo>Δ</mo><mi>w</mi><mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> evaluated at <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>β</mi><mi>x</mi></msub><mo>=</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> . The indirect effect of <emph>X</emph> on <emph>Y</emph> through <emph>W</emph> depends on the value of <emph>x</emph> and is null if either <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mi>x</mi></msub></mrow></math> </ephtml> or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>β</mi><mi>w</mi></msub></mrow></math> </ephtml> are zero. It can be shown that, for all <emph>x</emph>, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mo>Δ</mo><mi>w</mi><mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>β</mi><mi>w</mi></msub></mrow></math> </ephtml> share the same sign and, therefore, the indirect effect is concordant with the <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mi>x</mi></msub><msub><mi>β</mi><mi>w</mi></msub></mrow></math> </ephtml> product (see Online Appendix B, Supplementary material for this article is available online). However, the magnitude of the effect varies with <emph>x</emph>.</p> <p> <emph>Direct effect</emph>: Following the definition, we evaluate the direct effect after assuming <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>β</mi><mi>w</mi></msub><mo>=</mo><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> , that is,</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mi>w</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><msub><mo>β</mo><mi>x</mi></msub><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Graph</p> <p>Notice that equation (<reflink idref="bib14" id="ref45">14</reflink>) follows as <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>Δ</mo><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> are zero when <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>w</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> .</p> <p> <emph>Residual effect</emph>: Finally, the residual effect is given by difference as follows</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo /></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mo>−</mo><msub><mo>β</mo><mi>x</mi></msub><mo>{</mo><msub><mo>Δ</mo><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo /></mrow></mtd><mtd columnalign="left"><mrow><mo>+</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>{</mo><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo /></mrow></mtd><mtd columnalign="left"><mrow><mo>+</mo><msub><mi>γ</mi><mi>x</mi></msub><mo>{</mo><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><msubsup><mo>Δ</mo><mi>w</mi><mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Graph</p> <p>It is therefore apparent that the effect above vanishes whenever <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>w</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> . As a matter of fact, in the former case, we have <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>Δ</mo><mi>w</mi><mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> , whereas in the latter case, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>Δ</mo><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></math> </ephtml> . Notice that the latter case coincides with the condition of collapsibility of odds ratio (see [<reflink idref="bib35" id="ref46">35</reflink>], Corollary 3). Since all terms in curly brackets are bounded, expression (<reflink idref="bib15" id="ref47">15</reflink>) also highlights that the sign of the residual effect depends on the relative magnitude of the coefficients and is not linked to the logistic regression coefficients in a clear way. As a matter of fact, even if the direct and indirect effects share the same sign, the sign of the residual effect may be either positive or negative. Thus, as mentioned in the second section, for a fixed level of <emph>x</emph>, the ratio between the indirect effect and the total effect may provide an indication of the relative strength of the indirect effect, though with a less clear interpretation.</p> <p> <emph>Some cases of interest</emph>: Reformulating the total effect by definition as in equation (<reflink idref="bib11" id="ref48">11</reflink>), we study in detail the decomposition of the total effect into indirect (equation [<reflink idref="bib13" id="ref49">13</reflink>]), direct (equation [<reflink idref="bib14" id="ref50">14</reflink>]), and residual (equation [<reflink idref="bib15" id="ref51">15</reflink>]) effects for some cases of interest.</p> <p></p> <ulist> <item> <emph>Case</emph> (<emph>i</emph>) When the recursive logistic models can be depicted as in Figure 1B, that is, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>╨</mo><mi>Y</mi><mo stretchy="false">|</mo><mi>W</mi></mrow></math> </ephtml> , it follows from the definition above that</item> </ulist> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>and both <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> are zero.</p> <p></p> <ulist> <item> <emph>Case</emph> (<emph>ii</emph>) When the recursive logistic models can be depicted as in Figure 1C, that is, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>W</mi><mo>╨</mo><mi>Y</mi><mo stretchy="false">|</mo><mi>X</mi></mrow></math> </ephtml> , it follows from the definition above that</item> </ulist> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mi>w</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>and both <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> are zero.</p> <p></p> <ulist> <item> <emph>Case</emph> (<emph>iii</emph>) A noticeable situation arises after imposing <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> . In this case, the total effect is</item> </ulist> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>where</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mo>−</mo><msub><mo>β</mo><mi>x</mi></msub><mo>{</mo><msub><mo>Δ</mo><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>+</mo><msub><mi>γ</mi><mi>x</mi></msub><mo>{</mo><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><msubsup><mo>Δ</mo><mi>w</mi><mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>Notice that this assumption does not itself reflect into any conditional independence. If we further assume <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> , then some simplifications arise. Thus,</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>where</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mo>−</mo><msub><mo>β</mo><mi>x</mi></msub><mo>{</mo><msub><mo>Δ</mo><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>It is possible to see that under this condition, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">|</mo><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>≤</mo><mo stretchy="false">|</mo><msub><mo>β</mo><mi>x</mi></msub><mo stretchy="false">|</mo></mrow></math> </ephtml> , in line with results obtained by [<reflink idref="bib21" id="ref52">21</reflink>] in a more general context.</p> <p></p> <ulist> <item> <emph>Case</emph> (<emph>iv</emph>) When the recursive logistic models can be depicted as in Figure 1D, that is, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>W</mi><mo>╨</mo><mi>X</mi></mrow></math> </ephtml> , it follows from the definition above that</item> </ulist> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>where</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mo>−</mo><msub><mo>β</mo><mi>x</mi></msub><msub><mo>Δ</mo><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>{</mo><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo><msub><mo stretchy="false">|</mo><mrow><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo /></mrow></mtd><mtd columnalign="left"><mrow><mo>+</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>{</mo><mi>P</mi><mo stretchy="false">(</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mo>Δ</mo><mi>w</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>W</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo><msub><mo stretchy="false">|</mo><mrow><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Graph</p> <p>In this case, there is an effect modification due to conditioning of an additional variable, in line with well-known results on noncollapsibility of parameters of logistic regression models (see [<reflink idref="bib35" id="ref53">35</reflink>]). In addition, we notice that even in this simple case, the linearity of <emph>X</emph> in the marginal model is lost. Furthermore, if <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>x</mi></msub></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub></mrow></math> </ephtml> are both positive (negative), the marginal effect is also positive (negative), thereby recovering the finding in [<reflink idref="bib8" id="ref54">8</reflink>] on the condition to avoid the effect reversal (i.e., the marginal and conditional effects having opposite signs).</p> <hd id="AN0173122300-5">Discrete Case</hd> <p>Without loss of generality, we here assume that <emph>X</emph> is binary. The total effect of <emph>X</emph> on <emph>Y</emph> can be derived by taking the first difference of, equivalently, equations (<reflink idref="bib7" id="ref55">7</reflink>) or (<reflink idref="bib8" id="ref56">8</reflink>). We here opt for differentiating equation (<reflink idref="bib7" id="ref57">7</reflink>). Then,</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>β</mo><mi>x</mi></msub><mo>+</mo><mtext>log</mtext><msub><mrow><mtext>RR</mtext></mrow><mrow><mover accent="true"><mi>W</mi><mo>¯</mo></mover><mo stretchy="false">|</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mn>0</mn></mrow></msub><mo>−</mo><mtext>log</mtext><msub><mrow><mtext>RR</mtext></mrow><mrow><mover accent="true"><mi>W</mi><mo>¯</mo></mover><mo stretchy="false">|</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mn>1</mn></mrow></msub><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>which explicitly becomes</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>β</mo><mi>x</mi></msub><mo>+</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>g</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> as in equation (<reflink idref="bib5" id="ref58">5</reflink>). Notice that, in order to make the extension to <emph>X</emph> discrete straightforward, we maintain the <emph>x</emph> notation. Obviously, in the case of a binary <emph>X</emph>, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mtext /><mi>c</mi><mi>p</mi><mi>r</mi><mtext /><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> , a constant term corresponding to the cross product ratio of the marginal table for <emph>Y</emph> and <emph>X</emph>.</p> <p> <emph>Indirect effect:</emph> Following the definition above, we evaluate the total effect assuming <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> , that is,</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>1</mn><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>0</mn><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>where <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>g</mi><mi>y</mi><mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>g</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> evaluated at <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> . Notice that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>g</mi><mi>y</mi><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>g</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> , see equation (<reflink idref="bib5" id="ref59">5</reflink>). In parallel with the continuous case, the indirect effect is null if either <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>w</mi></msub></mrow></math> </ephtml> or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mi>x</mi></msub></mrow></math> </ephtml> are zero. It can be shown with some algebra that it is concordant with the product of the two coefficients.</p> <p> <emph>Direct effect:</emph> Following the definition of the direct effect, we evaluate the direct effect after assuming in the total effect <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>w</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> , that is,</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mi>w</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><msub><mo>β</mo><mi>x</mi></msub><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p> <emph>Residual effect</emph>: By definition, the remaining effect is evaluated by difference, such as</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>1</mn><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>0</mn><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>g</mi><mi>y</mi><mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> as above. It can easily be seen that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is zero as soon as <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> , leading to <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>g</mi><mi>y</mi><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>g</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> , or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>w</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> , leading to <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>g</mi><mn>1</mn><mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>g</mi><mn>0</mn><mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> (see equation [<reflink idref="bib5" id="ref60">5</reflink>]). The latter situation coincides with the condition of collapsibility of odds ratio (see [<reflink idref="bib35" id="ref61">35</reflink>], Corollary 3). However, like in the continuous case, there is no clear relationship between the sign of this effect and the logistic regression coefficients.</p> <p> <emph>Some cases of interest</emph>: Following the definition, we reformulate the total effect of a binary <emph>X</emph> on a binary <emph>Y</emph> as in equation (<reflink idref="bib11" id="ref62">11</reflink>) and we study in detail the decomposition of the total effect into the indirect (equation [<reflink idref="bib17" id="ref63">17</reflink>]), direct (equation [<reflink idref="bib18" id="ref64">18</reflink>]), and residual (equation [<reflink idref="bib19" id="ref65">19</reflink>]) effects for some cases of interest.</p> <p></p> <ulist> <item> <emph>Case</emph> (<emph>i</emph>) When the recursive logistic models can be depicted as in Figure 1B, that is, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>╨</mo><mi>Y</mi><mo stretchy="false">|</mo><mi>W</mi></mrow></math> </ephtml> , it follows from the definition above that,</item> </ulist> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>and both <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> are zero.</p> <p></p> <ulist> <item> <emph>Case</emph> (<emph>ii</emph>) When the recursive logistic models can be depicted as in Figure 1C, that is, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>W</mi><mo>╨</mo><mi>Y</mi><mo stretchy="false">|</mo><mi>X</mi></mrow></math> </ephtml> , it follows from the definition above that</item> </ulist> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mi>w</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>as all other terms are zero.</p> <p></p> <ulist> <item> <emph>Case</emph> (<emph>iii</emph>) After imposing <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> , the total effect is</item> </ulist> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>in which</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>−</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>1</mn><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>0</mn><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>Notice that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>g</mi><mi>y</mi><mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>g</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> evaluated at <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> . If we further assume <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> , after some algebra, it is possible to show that under this condition <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">|</mo><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>≤</mo><mo stretchy="false">|</mo><msub><mo>β</mo><mi>x</mi></msub><mo stretchy="false">|</mo></mrow></math> </ephtml> in line with results obtained by [<reflink idref="bib21" id="ref66">21</reflink>].</p> <p></p> <ulist> <item> <emph>Case</emph> (<emph>iv</emph>) When the recursive logistic models can be depicted as in Figure 1D, that is, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>W</mi><mo>╨</mo><mi>X</mi></mrow></math> </ephtml> , it follows from the case above</item> </ulist> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub></mrow></math> </ephtml> </p> <p>Graph</p> <p>where</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><msub><mo stretchy="false">|</mo><mrow><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>−</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>1</mn><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>0</mn><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><msub><mo stretchy="false">|</mo><mrow><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>After some algebra, it is possible to show that this condition is sufficient to avoid effect reversal as proved by [<reflink idref="bib8" id="ref67">8</reflink>] in a more general context.</p> <p>With reference to the cultural consumption data, in Table 3, the decomposition of the total effect of moving from level 1 to the other levels of <emph>X</emph> is reported. Notice that this decomposition is based on the estimated parameters of Table 2 and does not require to estimate the marginal model of <emph>Y</emph> against <emph>X</emph> and <emph>C</emph> only, thereby avoiding the issue of comparing parameters coming from two logistic models with unequal variance. The 95 percent confidence intervals and p-values are calculated using the approximated standard errors evaluated via the delta method ([<reflink idref="bib22" id="ref68">22</reflink>]).</p> <p>Graph</p> <p>Table 3. Estimates (Est.), Standard Errors (SEs), 95 Percent Confidence Intervals (CIs), and p Values of the Effects for the Cultural Consumption Experiment.</p> <p> <ephtml> <table><thead><tr><th /><th colspan="5"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>C</mi><mo>=</mo><mn>0</mn></mrow></math></p></th><th colspan="5"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>C</mi><mo>=</mo><mn>1</mn></mrow></math></p></th></tr><tr><th>Effect</th><th>Est.</th><th><italic>SE</italic></th><th align="center" colspan="2">95 Percent CI</th><th><italic>p</italic> Value</th><th>Est.</th><th><italic>SE</italic></th><th align="center" colspan="2">95 Percent CI</th><th><italic>p</italic> Value</th></tr></thead><tbody><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>DE</mtext></mrow><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>1.934</td><td>.368</td><td>1.214</td><td>2.655</td><td>.000</td><td>1.934</td><td>.368</td><td>1.214</td><td>2.655</td><td>.000</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>IE</mtext></mrow><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>0.364</td><td>.192</td><td>−0.011</td><td>0.740</td><td>.057</td><td>0.176</td><td>.139</td><td>−0.096</td><td>0.449</td><td>.205</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>RES</mtext></mrow><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>−0.476</td><td>.197</td><td>−0.862</td><td>−0.089</td><td>.016</td><td>−0.475</td><td>.227</td><td>−0.919</td><td>−0.031</td><td>.036</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>TE</mtext></mrow><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>1.822</td><td>.348</td><td>1.141</td><td>2.506</td><td>.000</td><td>1.635</td><td>.341</td><td>0.968</td><td>2.303</td><td>.000</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>DE</mtext></mrow><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>1.133</td><td>.386</td><td>0.375</td><td>1.890</td><td>.003</td><td>1.133</td><td>.386</td><td>0.375</td><td>1.890</td><td>.003</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>IE</mtext></mrow><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>1.475</td><td>.316</td><td>0.856</td><td>2.094</td><td>.000</td><td>0.795</td><td>.477</td><td>−0.141</td><td>1.731</td><td>.096</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>RES</mtext></mrow><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>−0.846</td><td>.300</td><td>−1.435</td><td>−0.257</td><td>.005</td><td>−1.057</td><td>.567</td><td>−2.168</td><td>0.054</td><td>.062</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>TE</mtext></mrow><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>1.762</td><td>.369</td><td>1.038</td><td>2.486</td><td>.000</td><td>0.871</td><td>.340</td><td>0.205</td><td>1.538</td><td>.010</td></tr></tbody></table> </ephtml> </p> <p>In the upper part of Table 3, the decomposition for the <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></math> </ephtml> contrast is reported. For both levels of <emph>C</emph>, the total and direct effects are positive and statistically significant, while the indirect effect, also positive, is moderately significant in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>C</mi><mo>=</mo><mn>0</mn></mrow></math> </ephtml> (<emph>p</emph> value =.057) and nonsignificant for <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>C</mi><mo>=</mo><mn>1</mn></mrow></math> </ephtml> (<emph>p</emph> value =.205). As for the <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></math> </ephtml> contrast, in the lower part of Table 3, we see instead that all the direct, indirect, and total effects are positive and statistically significant for both <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>C</mi><mo>=</mo><mn>0</mn></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>C</mi><mo>=</mo><mn>1</mn></mrow></math> </ephtml> (though the indirect effect for <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>C</mi><mo>=</mo><mn>1</mn></mrow></math> </ephtml> is only moderately significant). Although, for each contrast, the total, direct, and indirect effects are all positive, their interpretation in terms of proportion mediated is not possible given the presence of a negative residual effect (see the second section). Such an effect is rather large in magnitude, possibly due to a large interaction coefficient (see Table 2).</p> <p>In summary, the direct and total effects of moving from level 1 to level 2 of <emph>X</emph> are positive and statistically significant in all groups of students, while the indirect effect, also positive, is significant only for students mainly interested in humanities. When moving from level 1 to level 3 of <emph>X</emph>, all effects are positive and significant. We believe that this is an important message on how to design incentives to increase museums attendance of high school students which cannot be easily derived by simply looking at the estimated coefficients in Table 2.</p> <p>Our results are aligned with the ones in the original studies. [<reflink idref="bib17" id="ref69">17</reflink>] estimated an average causal effect based on the mean difference and the difference in difference methods, marginally with respect to <emph>W</emph>. Instead, in the study of [<reflink idref="bib13" id="ref70">13</reflink>], the authors performed a decomposition of the total effect based on counterfactual entities using the principal stratification method.</p> <hd id="AN0173122300-6">Effect decomposition on the probability scale</hd> <p>So far, we have considered effect decompositions operating on the logistic scale. However, sociologists and econometricians are quite often concerned with effects on the probability scale (also called partial effects; see [<reflink idref="bib34" id="ref71">34</reflink>], chap. 15), for which effect decompositions in specific contexts, typically on the additive scale, have also been proposed ([<reflink idref="bib5" id="ref72">5</reflink>]; [<reflink idref="bib16" id="ref73">16</reflink>]). We here denote with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>η</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> the RHS of equation (<reflink idref="bib6" id="ref74">6</reflink>), that is,</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>log</mi><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mo>η</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>log</mi><mfrac><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><msub><mo>β</mo><mn>0</mn></msub><mo>+</mo><msub><mo>β</mo><mi>x</mi></msub><mi>x</mi><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>For the continuous <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> case, the probability effect is defined as the derivative with respect to <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> of the probability function. The total probability effect ( <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mi>P</mi><mi>E</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> ) is therefore so defined:</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TPE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>{</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>where <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> corresponds to the total effect on the logistic scale as defined in equation (<reflink idref="bib12" id="ref75">12</reflink>). The result follows after taking the derivative of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>expit</mtext><mo>{</mo><mi>η</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo><mo>=</mo><mtext>exp</mtext><mi>η</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>/</mo><mo>{</mo><mn>1</mn><mo>+</mo><mtext>exp</mtext><mi>η</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo></mrow></math> </ephtml> with respect to its argument <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>η</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> .</p> <p>On the other hand, for <emph>X</emph> binary or discrete, the total effect on the probability scale can be defined by simply taking the difference across levels of <emph>X</emph> of the marginal probability. For the binary <emph>X</emph>, this becomes:</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TPE</mtext><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>P</mi><mrow><mo>(</mo><mrow><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mi>P</mi><mrow><mo>(</mo><mrow><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mn>0</mn></mrow><mo>)</mo></mrow><mo>=</mo><mtext>expits </mtext><mo>η</mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>−</mo><mtext>expit </mtext><mo>η</mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>.</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>In analogy with the approach of the second section, the direct probability effect <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>(DPE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></math> </ephtml> and indirect probability effect <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>(IPE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></math> </ephtml> are defined by zeroing the corresponding coefficients in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TPE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> . To obtain an additive decomposition, we also define the residual probability effect (RPE(<emph>x</emph>)) by difference as:</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RPE</mtext><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mtext>TPE</mtext><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo /><mo>−</mo><mo /><mtext>DPE</mtext><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo /><mo>−</mo><mtext> IPE</mtext><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>.</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>With particular reference to the continuous case, this amounts to:</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>DPE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mtext>TPE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mi>w</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mtext>expit</mtext><mo stretchy="false">(</mo><msub><mo>β</mo><mn>0</mn></msub><mo>+</mo><msub><mo>β</mo><mi>x</mi></msub><mi>x</mi><mo stretchy="false">)</mo><mo>{</mo><mn>1</mn><mo>−</mo><mtext>expit</mtext><mo stretchy="false">(</mo><msub><mo>β</mo><mn>0</mn></msub><mo>+</mo><msub><mo>β</mo><mi>x</mi></msub><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo><msub><mo>β</mo><mi>x</mi></msub><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>IPE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mtext>TPE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mtext>expit</mtext><mo>{</mo><msup><mo>η</mo><mo>*</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><mtext>expit</mtext><mo>{</mo><msup><mo>η</mo><mo>*</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>}</mo><mo stretchy="false">]</mo><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Graph</p> <p>where <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>IE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is as in equation (<reflink idref="bib13" id="ref76">13</reflink>) and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mo>η</mo><mo>*</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>η</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> evaluated at <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> . Like the effects on the logistic scale, all these probability effects are local measures, since they depend on the specific value <emph>x.</emph> Global measures can be defined by averaging the aforementioned local quantities. For instance, given a population of <emph>N</emph> units (<emph>i</emph> = 1,..., <emph>N</emph>), one could define the average total probability effect (ATPE) as:</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo /><mtext>ATPE</mtext><mo /><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover></mstyle><mo /><mtext>TPE </mtext><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>with the average direct probability effect (ADPE) and the average indirect probability effect (AIPE) defined analogously. However, these average effects should be taken with caution when there is a strong variation across values of <emph>x</emph>.</p> <p>For the cultural consumption data, the effects on the probability scale are summarized in Table 4, which has the same structure of Table 3. As expected, results are in line with the ones on the log odds scale. Notice that averaging the effects may be not appropriate in applications with a strong variation across levels of <emph>x</emph>, as in this case, especially with reference to the indirect effects.</p> <p>Graph</p> <p>Table 4. Estimates (Est.) of the Effects on the Probability Scale for the Cultural Consumption Experiment, with Standard Errors (SEs), 95 Percent Confidence Intervals (CIs), and p Values.</p> <p> <ephtml> <table><thead><tr><th /><th colspan="5"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>C</mi><mo>=</mo><mn>0</mn></mrow></math></p></th><th colspan="5"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>C</mi><mo>=</mo><mn>1</mn></mrow></math></p></th></tr><tr><th>Effect</th><th>Est.</th><th><italic>SE</italic></th><th colspan="2">95 Percent CI</th><th><italic>p</italic> Value</th><th>Est.</th><th><italic>SE</italic></th><th colspan="2">95 Percent CI</th><th><italic>p</italic> Value</th></tr></thead><tbody><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>DPE</mtext></mrow><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>.413</td><td>.069</td><td>.279</td><td>.547</td><td>.000</td><td>.446</td><td>.074</td><td>.301</td><td>.591</td><td>.000</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>IPE</mtext></mrow><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>.063</td><td>.031</td><td>.001</td><td>.124</td><td>.046</td><td>.035</td><td>.028</td><td>−.020</td><td>.090</td><td>.215</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>RPE</mtext></mrow><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>−.073</td><td>.032</td><td>−.135</td><td>−.010</td><td>.023</td><td>−.099</td><td>.047</td><td>−.190</td><td>−.008</td><td>.034</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>TPE</mtext></mrow><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>.403</td><td>.068</td><td>.269</td><td>.537</td><td>.000</td><td>.382</td><td>.072</td><td>.240</td><td>.523</td><td>.000</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>DPE</mtext></mrow><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>.216</td><td>.067</td><td>.084</td><td>.347</td><td>.001</td><td>.255</td><td>.082</td><td>.094</td><td>.415</td><td>.002</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>IPE</mtext></mrow><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>.317</td><td>.078</td><td>.164</td><td>.470</td><td>.000</td><td>.176</td><td>.119</td><td>−.058</td><td>.410</td><td>.141</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>RPE</mtext></mrow><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>−.144</td><td>.074</td><td>−.289</td><td>.000</td><td>.049</td><td>−.236</td><td>.154</td><td>−.539</td><td>.067</td><td>.127</td></tr><tr><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mtext>TPE</mtext></mrow><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></msub></mrow></math></p></td><td>.388</td><td>.091</td><td>.210</td><td>.566</td><td>.000</td><td>.194</td><td>.098</td><td>.003</td><td>.386</td><td>.046</td></tr></tbody></table> </ephtml> </p> <hd id="AN0173122300-7">Simulation Study</hd> <p>In this section, we present results of a simulation study to investigate how well the relative amount of indirect effect is recovered, also in relation with already existing methods. In particular, [<reflink idref="bib16" id="ref77">16</reflink>] and [<reflink idref="bib5" id="ref78">5</reflink>] derive a decomposition of the total effect that can be applied to the case of a continuous mediator <emph>W</emph>, when the response model for <emph>Y</emph> can either be a logistic or a probit one with no interaction terms. In this context, the total effect, as measured by the marginal coefficient of <emph>X</emph> on <emph>Y</emph>, is the sum of the direct and indirect effects as in linear models. When all effects are positive, it is therefore meaningful to compute the proportion mediated as the ratio between the indirect and total effect (see Equation [<reflink idref="bib21" id="ref79">21</reflink>] in [<reflink idref="bib5" id="ref80">5</reflink>]). The authors present a method to sidestep the well-known issue of unequal variances, known as the Karlson Holm and Breen (KHB) method. They also propose to adapt it to the binary mediator case, by postulating a linear probability model for <emph>W</emph>.</p> <p>We here postulate a logistic model for <emph>W</emph> and analyze, through simulations, the behavior in finite samples of the KHB method and of the proposed method. For <emph>X</emph> binary, we compare the KHB method with the ratio between the estimates of the effects IE(<emph>x</emph>) and TE(<emph>x</emph>) in the second section, obtained by plugging-in the ML estimates of the parameters in the corresponding expression. For <emph>X</emph> continuous, we notice that the KHB measure should be interpreted as the proportion mediated on the probability scale in the same fashion as discussed in Effect Decomposition on the Probability Scale subsection. For this reason, we proceed as follows. Given a sample of <emph>n</emph> units, an estimate of the corresponding effect is formed by averaging across units the corresponding entities. As an instance, the estimated ATPE is so formed:</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mtext>ATPE</mtext></mrow><mo stretchy="true">^</mo></mover><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><mover accent="true"><mrow><mtext>TPE</mtext></mrow><mo stretchy="true">^</mo></mover><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>where <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mtext>TPE</mtext></mrow><mo stretchy="true">^</mo></mover><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> is the estimated total probability effect of unit <emph>i</emph>, obtained by plugging-in the ML estimates in the corresponding expression. The estimated <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>AIPE</mtext></mrow></math> </ephtml> is formed accordingly and the ratio between the two entities is then taken. We recall from the second section that, due to the presence of the residual effect, this measure should not be interpreted as proportion mediated in the usual way. The above measure is then compared with the KHB measure.</p> <p>We consider a basic setting with no covariates, where the outcome <emph>Y</emph> and the mediator <emph>W</emph> are generated according to Equations (<reflink idref="bib1" id="ref81">1</reflink>) and (<reflink idref="bib2" id="ref82">2</reflink>) respectively. Though our method can accommodate for the treatment–mediator interaction <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub></mrow></math> </ephtml> in the outcome equation, for a fair comparison, it is here posed to zero. The remaining parameters are set to: <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mn>0</mn></msub><mo>=</mo><mo>−</mo><mn>2</mn><mo>=</mo><msub><mi>γ</mi><mn>0</mn></msub></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>w</mi></msub><mo>=</mo><mn>2</mn><mo>=</mo><msub><mi>γ</mi><mi>x</mi></msub></mrow></math> </ephtml> , while <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>x</mi></msub></mrow></math> </ephtml> is varying in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>{</mo><mn>0.4</mn><mo>,</mo><mn>0.9</mn><mo>,</mo><mn>1.8</mn><mo>}</mo></mrow></math> </ephtml> in order to explore different relative amounts of indirect effect.</p> <p>We define three sample sizes, that is, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>∈</mo><mo>{</mo><mn>250</mn><mo>,</mo><mn>500</mn><mo>,</mo><mn>1000</mn><mo>}</mo></mrow></math> </ephtml> . In the binary treatment case, the <emph>X</emph> variate is sampled from a Bernoulli distribution with probability equal to 0.5. In the continuous treatment case, we first generate a large pseudo-population of size <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>N</mi><mrow><mtext>pop</mtext></mrow></msub><mo>=</mo><mn>150</mn><mo>,</mo><mn>000</mn></mrow></math> </ephtml> from a Normal distribution with null mean and variance equal to 2 and then create <emph>X</emph> by extracting a random sample of size <emph>n</emph> from it. In this way, the true value of the <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>AIPE</mtext><mo>/</mo><mtext>ATPE</mtext></mrow></math> </ephtml> ratio is computed on the pseudo-population and does not vary with the sample size. Once <emph>X</emph> is obtained, for each <emph>n</emph>, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>N</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>000</mn></mrow></math> </ephtml> replications of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> are drawn and estimation is performed.</p> <p>Table 5 summarizes the simulation results. Notice that for <emph>X</emph> binary, the dependence of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>IE</mtext></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext></mrow></math> </ephtml> on <emph>x</emph> is removed. As expected, the proposed estimators always approach the true values as the sample size grows, with smaller root mean squared error (RMSE) in all scenarios considered. Conversely, the KHB estimator is biased, with RMSE increasing with the value of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>x</mi></msub></mrow></math> </ephtml> . Furthermore, if for the binary case the bias seems reasonable, it becomes consistent in the continuous case.</p> <p>Graph</p> <p>Table 5. True Value and Simulation Average, Variance, and Root Mean Squared Error (RMSE) for the KHB Method and the Proposed Method (RSD).</p> <p> <ephtml> <table><thead><tr><th colspan="3"><italic>n</italic></th><th>250</th><th>500</th><th>1,000</th><th>250</th><th>500</th><th>1,000</th><th>250</th><th>500</th><th>1,000</th></tr><tr><th><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mo>β</mo><mi>x</mi></msub></mrow></math></p></th><th>True Value</th><th>Method</th><th colspan="3">Average</th><th colspan="3">Variance</th><th colspan="3">RMSE</th></tr></thead><tbody><tr><td /><td align="center" colspan="11"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mtext>IE</mtext><mo>/</mo><mtext>TE</mtext></mrow></math></p> (<italic>X</italic> binary)</td></tr><tr><td rowspan="2">0.4</td><td rowspan="2">.716</td><td>KHB</td><td>.732</td><td>.683</td><td>.669</td><td>.144</td><td>.033</td><td>.014</td><td>.379</td><td>.184</td><td>.129</td></tr><tr><td>RSD</td><td>.757</td><td>.732</td><td>.724</td><td>.064</td><td>.024</td><td>.011</td><td>.256</td><td>.154</td><td>.107</td></tr><tr><td rowspan="2">0.9</td><td rowspan="2">.532</td><td>KHB</td><td>.475</td><td>.468</td><td>.462</td><td>.020</td><td>.008</td><td>.004</td><td>.153</td><td>.111</td><td>.092</td></tr><tr><td>RSD</td><td>.544</td><td>.539</td><td>.535</td><td>.020</td><td>.009</td><td>.004</td><td>.141</td><td>.094</td><td>.064</td></tr><tr><td>1.8</td><td>.364</td><td>KHB</td><td>.301</td><td>.300</td><td>.297</td><td>.004</td><td>.002</td><td>.001</td><td>.092</td><td>.079</td><td>.074</td></tr><tr><td /><td /><td>RSD</td><td>.367</td><td>.367</td><td>.364</td><td>.007</td><td>.003</td><td>.002</td><td>.082</td><td>.058</td><td>.040</td></tr><tr><td /><td align="center" colspan="11"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mtext>AIPE</mtext><mo>/</mo><mtext>ATPE</mtext></mrow></math></p> (<italic>X</italic> continuous)</td></tr><tr><td rowspan="2">0.4</td><td rowspan="2">.590</td><td>KHB</td><td>.531</td><td>.521</td><td>.513</td><td>.028</td><td>.013</td><td>.006</td><td>.178</td><td>.133</td><td>.108</td></tr><tr><td>RSD</td><td>.561</td><td>.589</td><td>.580</td><td>.010</td><td>.004</td><td>.002</td><td>.103</td><td>.064</td><td>.042</td></tr><tr><td rowspan="2">0.9</td><td rowspan="2">.437</td><td>KHB</td><td>.316</td><td>.317</td><td>.310</td><td>.008</td><td>.004</td><td>.002</td><td>.149</td><td>.135</td><td>.134</td></tr><tr><td>RSD</td><td>.411</td><td>.458</td><td>.440</td><td>.002</td><td>.002</td><td>.001</td><td>.055</td><td>.046</td><td>.026</td></tr><tr><td rowspan="2">1.8</td><td rowspan="2">.351</td><td>KHB</td><td>.180</td><td>.178</td><td>.180</td><td>.002</td><td>.001</td><td>.001</td><td>.178</td><td>.177</td><td>.173</td></tr><tr><td>RSD</td><td>.328</td><td>.343</td><td>.352</td><td>.001</td><td>.001</td><td>.000</td><td>.042</td><td>.026</td><td>.017</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note</emph>: AIPE = average indirect probability effect; ATPE = average total probability effect; RSD = Raggi Stanghellini Doretti.</p> <hd id="AN0173122300-8">Extension to Multiple Binary Mediators</hd> <p>The proposed definitions of direct, indirect, and residual effects, together with their parametric formulations, extend nicely to the situation where multiple binary mediators are present. Suppose there are <emph>k</emph> mediators and that a full ordering among the variables <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo>,</mo><msub><mi>W</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mi>W</mi><mi>k</mi></msub><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is available such that each variable is a potential response variable for the subsequent ones. The system can be represented via a DAG (see Figure 2). We assume that each response model is a hierarchical logistic model. In the following, if we impose the regression coefficient of one covariate to be zero, all higher order interaction terms involving this covariate are implicitly imposed to zero. For brevity, we denote with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>W</mi><mrow><mo>></mo><mi>j</mi></mrow></msub></mrow></math> </ephtml> the set of all <emph>W<subs>r</subs></emph> such that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>r</mi><mo>></mo><mi>j</mi></mrow></math> </ephtml> . The coefficients of <emph>X</emph> and of <emph>W<subs>j</subs></emph> and of their interactions in the logistic regression of <emph>Y</emph> are denoted with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>β</mo></math> </ephtml> in a self-explaining fashion.</p> <p>Graph: Figure 2. Directed acyclic graph with k mediators.</p> <p>In analogy with equation (<reflink idref="bib7" id="ref83">7</reflink>), the logistic model for <emph>Y</emph> given <emph>X</emph>, obtained after marginalization upon the <emph>k</emph> mediators, is</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msub><mo>β</mo><mn>0</mn></msub><mo>+</mo><msub><mo>β</mo><mi>x</mi></msub><mi>x</mi><mo>−</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover></mstyle><mtext>log</mtext><msub><mrow><mtext>RR</mtext></mrow><mrow><msub><mover accent="true"><mi>W</mi><mo>¯</mo></mover><mi>j</mi></msub><mo stretchy="false">|</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><msub><mi>W</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub></mrow></mtd></mtr></mtable><mo>,</mo></mrow></math> </ephtml> </p> <p>Graph</p> <p>where</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mtext>RR</mtext></mrow><mrow><msub><mover accent="true"><mi>W</mi><mo>¯</mo></mover><mi>j</mi></msub><mo stretchy="false">|</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><msub><mi>W</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>w</mi><mrow><mo>></mo><mi>j</mi></mrow></msub></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><msub><mi>w</mi><mrow><mo><</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>w</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><msub><mi>w</mi><mrow><mo><</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>w</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>In Online Appendix C (Supplementary material for this article is available online), the relevant expressions are given.</p> <p>In line with what done for the simple case, we here offer a generalization of the definitions for the total, direct, indirect, and residual effects under the situation of <emph>k</emph> mediators.</p> <p> <emph>Total effect:</emph> Let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> be the total effect of <emph>X</emph> on <emph>Y</emph> on the log odds scale, after marginalization on <emph>k</emph> binary mediators. For <emph>X</emph> continuous and differentiable, the total effect is defined as the derivative of equation (<reflink idref="bib21" id="ref84">21</reflink>) with respect to <emph>x</emph>. It follows that</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msub><mo>β</mo><mi>x</mi></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover></mstyle><mfrac><mrow><mtext>exp</mtext><msubsup><mi>g</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><msub><mi>w</mi><mrow><mo><</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>w</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><msub><mi>w</mi><mrow><mo><</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>w</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mi>g</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><msub><mi>w</mi><mrow><mo><</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>w</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo /></mrow></mtd><mtd columnalign="left"><mrow><mo>−</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover></mstyle><mfrac><mrow><mtext>exp</mtext><msubsup><mi>g</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><msub><mi>w</mi><mrow><mo><</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>w</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><msub><mi>w</mi><mrow><mo><</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>w</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mi>g</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><msub><mi>w</mi><mrow><mo><</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>w</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Graph</p> <p>For <emph>X</emph> discrete, the total effect is defined as the difference between equation (<reflink idref="bib21" id="ref85">21</reflink>) evaluated at two different levels of <emph>X</emph>. Without loss of generality, we here assume <emph>X</emph> binary and take the difference for <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math> </ephtml> . Then,</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mo>=</mo><msub><mo>β</mo><mi>x</mi></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover></mstyle><mtext>log</mtext><mfrac><mrow><msub><mrow><mtext>RR</mtext></mrow><mrow><msub><mover accent="true"><mi>W</mi><mo>¯</mo></mover><mi>j</mi></msub><mo stretchy="false">|</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mn>0</mn><mo>,</mo><msub><mi>W</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub></mrow><mrow><msub><mrow><mtext>RR</mtext></mrow><mrow><msub><mover accent="true"><mi>W</mi><mo>¯</mo></mover><mi>j</mi></msub><mo stretchy="false">|</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>=</mo><mn>1</mn><mo>,</mo><msub><mi>W</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msub></mrow></mfrac><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Graph</p> <p>Equation (<reflink idref="bib23" id="ref86">23</reflink>) can also be written as follows</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msub><mo>β</mo><mi>x</mi></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover></mstyle><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><msub><mi>w</mi><mrow><mo><</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><msub><mi>w</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><msub><mi>w</mi><mrow><mo><</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><msub><mi>w</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover></mstyle><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><msub><mi>w</mi><mrow><mo><</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msub><mi>w</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msubsup><mi>g</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><msub><mi>w</mi><mrow><mo><</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msub><mi>w</mi><mrow><mo>></mo><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Graph</p> <p> <emph>Direct effect:</emph> Let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> be the direct effect of <emph>X</emph> on <emph>Y</emph> on the log odds scale. Let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mo>β</mo><mo>¯</mo></mover><mi>W</mi></msub></mrow></math> </ephtml> be the set of all <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>β</mo></math> </ephtml> regression coefficients of each mediator in the model for <emph>Y</emph>, including also the interaction terms both between mediators and between mediators and <emph>X</emph>. The direct effect is evaluated in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> after imposing <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mo>β</mo><mo>¯</mo></mover><mi>W</mi></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> , thereby <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Y</mi><mo>╨</mo><msub><mi>W</mi><mn>1</mn></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mi>W</mi><mi>k</mi></msub><mo stretchy="false">|</mo><mi>X</mi></mrow></math> </ephtml> , that is,</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mo>=</mo><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mover accent="true"><mo>β</mo><mo>¯</mo></mover><mi>W</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><msub><mo>β</mo><mi>x</mi></msub><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Graph</p> <p> <emph>Global indirect effect:</emph> The global indirect effect <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>GIE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> can be defined in analogy with the previous definitions, as the total effect evaluated when the direct effect of <emph>X</emph>, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>x</mi></msub></mrow></math> </ephtml> , is zero. Since we only deal with hierarchical models, this implies that all interaction terms between <emph>X</emph> and the mediators are also zero. Therefore,</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>GIE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p> <emph>Residual effect:</emph> Let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> be the residual effect evaluated by difference, that is,</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mtext>GIE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> </ephtml> </p> <p>Graph</p> <p>From previous derivations, nonzero residual effects are induced by graphs having more than one arrow pointing to <emph>Y</emph>. Therefore, we can state that the residual effect is zero whenever one of the two following graphical conditions holds: (i) there is no direct path from <emph>X</emph> to <emph>Y</emph> or (ii) there is the direct path from <emph>X</emph> to <emph>Y</emph> and no other arrow is pointing to <emph>Y</emph>. As an instance, the model corresponding to the DAG in Figure 3A has a nonzero <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> as there is a direct arrow form <emph>X</emph> to <emph>Y</emph> and other two arrows are pointing to <emph>Y</emph>, while models corresponding to DAGs as in Figure 3B and C are such that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>GIE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>RES</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></math> </ephtml> .</p> <p>Graph: Figure 3. Directed acyclic graphs with k=2 mediators when (A) no conditional independences hold; (B) X╨Y|{W1,W2}, W2╨W1|X, and X╨W2; and (C) Y╨{X,W2}|W1.</p> <p>In a setting with multiple mediators, one is also interested in a path-specific indirect effect, that is, the effect that is due to some mediators only, and is null whenever one arrow along the pathway is deleted. Notice that, in this setting, also other research questions are of interest, such as the path-specific indirect effects when some mediators are marginalized over. They are addressed in the fifth section.</p> <p> <emph>Path-specific indirect effect:</emph> Let <emph>A</emph> be one of the <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> </ephtml> ordered subsets of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mi>W</mi><mn>1</mn></msub><mo>,</mo><msub><mi>W</mi><mn>2</mn></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mi>W</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> containing at least one element of <emph>W</emph>. Let <emph>i<subs>A</subs></emph> be the ordered set of indices <emph>j</emph> such that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>W</mi><mi>j</mi></msub><mo>∈</mo><mi>A</mi></mrow></math> </ephtml> . The path-specific indirect effect <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mtext>PSIE</mtext></mrow><mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is obtained from the total effect after imposing that:</p> <p></p> <ulist> <item> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> ;</item> <p></p> <item> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mrow><msub><mi>w</mi><mi>j</mi></msub></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>j</mi><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>k</mi><mo>}</mo><mo>,</mo><mi>j</mi><mo>≠</mo><mtext> min </mtext><mo>{</mo><msub><mi>i</mi><mi>A</mi></msub><mo>}</mo></mrow></math> </ephtml> the smallest index in <emph>i<subs>A</subs></emph>;</item> <p></p> <item> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mrow><mi>r</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>W</mi><mi>r</mi></msub><mo>∈</mo><mi>A</mi></mrow></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>j</mi><mo>></mo><mi>r</mi><mo>,</mo><mi>j</mi><mo>≠</mo><msub><mi>ℓ</mi><mi>r</mi></msub></mrow></math> </ephtml> , where <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>ℓ</mi><mi>r</mi></msub></mrow></math> </ephtml> is the index following <emph>r</emph> in <emph>i<subs>A</subs></emph>; and</item> <p></p> <item> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mrow><mi>r</mi><mo>,</mo><mi>x</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>W</mi><mi>r</mi></msub><mo>∈</mo><mi>A</mi></mrow></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>r</mi><mo>≠ </mo><mtext>max </mtext><mo>{</mo><msub><mi>i</mi><mi>A</mi></msub><mo>}</mo></mrow></math> </ephtml> the largest index in <emph>i<subs>A</subs></emph>.</item> </ulist> <p>In this way, each path-specific indirect effect contains only the parameters pertaining to the path (including the intercepts). It then follows that the indirect effect <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mtext>PSIE</mtext></mrow><mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> is null whenever one of the following conditions holds:</p> <p></p> <ulist> <item> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mrow><msub><mi>w</mi><mi>j</mi></msub></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>j</mi><mo>= </mo><mtext>min </mtext><mo>{</mo><msub><mi>i</mi><mi>A</mi></msub><mo>}</mo></mrow></math> </ephtml> ;</item> <p></p> <item> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mrow><mi>r</mi><mo>,</mo><msub><mi>ℓ</mi><mi>r</mi></msub></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>W</mi><mi>r</mi></msub><mo>∈</mo><mi>A</mi></mrow></math> </ephtml> ; and</item> <p></p> <item> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mrow><mi>r</mi><mo>,</mo><mi>x</mi></mrow></msub></mrow></math> </ephtml> with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>r</mi><mo>= </mo><mtext>max</mtext><mo> {</mo><msub><mi>i</mi><mi>A</mi></msub><mo>}</mo></mrow></math> </ephtml> .</item> </ulist> <p>Notice that each of the conditions above implies deleting one arrow in the DAG corresponding to the model of interest.</p> <p>As an instance, let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>k</mi><mo>=</mo><mn>6</mn></mrow></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>i</mi><mi>A</mi></msub><mo>=</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>}</mo></mrow></math> </ephtml> . The path-specific indirect effect is obtained from <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>TE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> after imposing that:</p> <p></p> <ulist> <item> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><msub><mi>w</mi><mn>1</mn></msub></mrow></msub><mo>=</mo><mo>...</mo><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><msub><mi>w</mi><mn>6</mn></msub></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> ;</item> <p></p> <item> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mrow><msub><mi>w</mi><mn>1</mn></msub></mrow></msub><mo>=</mo><msub><mo>β</mo><mrow><msub><mi>w</mi><mn>3</mn></msub></mrow></msub><mo>=</mo><msub><mo>β</mo><mrow><msub><mi>w</mi><mn>4</mn></msub></mrow></msub><mo>=</mo><msub><mo>β</mo><mrow><msub><mi>w</mi><mn>5</mn></msub></mrow></msub><mo>=</mo><msub><mo>β</mo><mrow><msub><mi>w</mi><mn>6</mn></msub></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> ;</item> <p></p> <item> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mrow><mn>2</mn><mo>,</mo><mn>4</mn></mrow></msub><mo>=</mo><msub><mi>γ</mi><mrow><mn>2</mn><mo>,</mo><mn>5</mn></mrow></msub><mo>=</mo><msub><mi>γ</mi><mrow><mn>2</mn><mo>,</mo><mn>6</mn></mrow></msub><mo>=</mo><msub><mi>γ</mi><mrow><mn>3</mn><mo>,</mo><mn>4</mn></mrow></msub><mo>=</mo><msub><mi>γ</mi><mrow><mn>3</mn><mo>,</mo><mn>6</mn></mrow></msub><mo>=</mo><msub><mi>γ</mi><mrow><mn>5</mn><mo>,</mo><mn>6</mn></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> ; and</item> <p></p> <item> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mrow><mn>2</mn><mo>,</mo><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>γ</mi><mrow><mn>3</mn><mo>,</mo><mi>x</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> </item> </ulist> <p>where <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mrow><mi>j</mi><mo>,</mo><mi>x</mi></mrow></msub></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mrow><mi>j</mi><mo>,</mo><mi>i</mi></mrow></msub></mrow></math> </ephtml> are, in order, the coefficients of <emph>X</emph> and <emph>W<subs>i</subs></emph> in the equation of <emph>W<subs>j</subs></emph> against its parent nodes in the corresponding DAG. The above definition allows for only one path from <emph>X</emph> to <emph>Y</emph>, which is <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>→</mo><msub><mi>W</mi><mn>5</mn></msub><mo>→</mo><msub><mi>W</mi><mn>3</mn></msub><mo>→</mo><msub><mi>W</mi><mn>2</mn></msub><mo>→</mo><mi>Y</mi></mrow></math> </ephtml> . It is null whenever <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mrow><mn>5</mn><mo>,</mo><mi>x</mi></mrow></msub></mrow></math> </ephtml> or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mrow><mn>3</mn><mo>,</mo><mn>5</mn></mrow></msub></mrow></math> </ephtml> or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mrow><mn>2</mn><mo>,</mo><mn>3</mn></mrow></msub></mrow></math> </ephtml> or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mrow><msub><mi>w</mi><mn>2</mn></msub></mrow></msub></mrow></math> </ephtml> are zero.</p> <p>Notice that the definition of path-specific indirect effects allows only for direction preserving paths, that is, paths with all arrows pointing to the same direction. As a matter of fact, only ordered subsets of <emph>W</emph> are allowed to form <emph>A</emph>. This choice is justified by the fact that these are the only subsets with a nonzero path-specific indirect effect. To clarify the issue, see the graph in Figure 4. The path <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>→</mo><msub><mi>W</mi><mn>3</mn></msub><mo>→</mo><msub><mi>W</mi><mn>1</mn></msub><mo>←</mo><msub><mi>W</mi><mn>2</mn></msub><mo>→</mo><mi>Y</mi></mrow></math> </ephtml> is not admitted as it gives rise to <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mi>W</mi><mn>3</mn></msub><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo>,</mo><msub><mi>W</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> , not an ordered subset of <emph>W</emph>. However, as <emph>W</emph><subs>1</subs> is a collider node, the path between <emph>X</emph> and <emph>Y</emph> is blocked by <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mi>W</mi><mn>3</mn></msub><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo>,</mo><msub><mi>W</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> and the corresponding path-specific effect is zero (see [<reflink idref="bib24" id="ref87">24</reflink>], chaps. 1 and 3).</p> <p>Graph: Figure 4. Directed acyclic graph with W 1 acting as a collider node in the path X→W3→W1←W2→Y.</p> <p>It is also important to notice that, in parallel to the single-mediator case, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>DE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> coincides with the effect of <emph>X</emph> on <emph>Y</emph> keeping <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>W</mi><mn>1</mn></msub><mo>=</mo><msub><mi>W</mi><mn>2</mn></msub><mo>=</mo><mo>...</mo><mo>=</mo><msub><mi>W</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> . However, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mtext>PSIE</mtext></mrow><mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> does not in general coincide with the indirect effect after keeping the mediators not in <emph>A</emph> equal to zero. To see this, notice that in Figure 4, the path-specific indirect effect for <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>W</mi><mn>3</mn></msub><mo>,</mo><msub><mi>W</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> is evaluated after imposing that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><msub><mi>w</mi><mn>1</mn></msub></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> . This effect does not coincide with the one obtained after conditioning on <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>W</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> (see [<reflink idref="bib12" id="ref88">12</reflink>]).</p> <p>Notice that the sum of all the path-specific indirect effects in general is not equal to the global indirect effect. This is true even when there is just one path from <emph>X</emph> to <emph>Y</emph>. This is due to the different ways to deal with the effects induced in noncollapsible subgraphs. These are subgraphs involving three random variables, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mi>W</mi><mi>i</mi></msub><mo>,</mo><msub><mi>W</mi><mi>j</mi></msub><mo>,</mo><msub><mi>W</mi><mi>r</mi></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> , or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mi>W</mi><mi>i</mi></msub><mo>,</mo><msub><mi>W</mi><mi>j</mi></msub><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>></mo><mi>j</mi><mo>></mo><mi>r</mi></mrow></math> </ephtml> , such that there are two arrows pointing to the inner node, that is, <emph>W<subs>r</subs></emph> or <emph>Y</emph>. In this case, <emph>W<subs>j</subs></emph> acts as a mediator between <emph>W<subs>i</subs></emph> and <emph>W<subs>r</subs></emph>, or <emph>Y</emph>, and there is a nonzero residual effect (see the second section). Specifically, the global indirect effect includes all residual effects, whereas path-specific indirect effects do not.</p> <p>As an example, consider the models with DAG as in Figure 5A and B. In both DAGs, there is just one indirect path leading from <emph>X</emph> to <emph>Y</emph>, that is, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>→</mo><msub><mi>W</mi><mn>3</mn></msub><mo>→</mo><msub><mi>W</mi><mn>1</mn></msub><mo>→</mo><mi>Y</mi></mrow></math> </ephtml> , with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>W</mi><mn>3</mn></msub><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> . Both models have <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>GIE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≠</mo><msub><mrow><mtext>PSIE</mtext></mrow><mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> . The model corresponding to Figure 5A has two noncollapsible subgraphs, namely, the ones induced by <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mi>W</mi><mn>3</mn></msub><mo>,</mo><msub><mi>W</mi><mn>2</mn></msub><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mi>W</mi><mn>2</mn></msub><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> , while the model corresponding to Figure 5B has one noncollapsible subgraph, namely the one induced by <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mi>W</mi><mn>2</mn></msub><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> . Notice the different meaning of the parameters attached to the arrow <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>W</mi><mn>3</mn></msub><mo>→</mo><msub><mi>W</mi><mn>1</mn></msub></mrow></math> </ephtml> in the two effects: In the <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>GIE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> , it is the total effect of <emph>W</emph><subs>3</subs> on <emph>W</emph><subs>1</subs>, while in the path-specific <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mtext>PSIE</mtext></mrow><mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> , it is the direct effect.</p> <p>Graph: Figure 5. Directed acyclic graph with k=3 mediators (A) Y╨{X,W3}|{W1,W2}, W1╨X|{W2,W3}, and W2╨X and (B) Y╨{X,W3}|{W1,W2}, W1╨{X,W2}|W3, and W2╨X.</p> <hd id="AN0173122300-9">Other Path-specific Indirect Effects of Interest</hd> <p>Suppose now that the research question involves path-specific effects in the model obtained after marginalization over some mediators while others are kept in the model. First of all, the parameters of the marginal model of interest should be obtained and then the path-specific indirect effects can be evaluated. Two different situations may arise. The first one involves marginalization over an inner mediator, and therefore equation (<reflink idref="bib21" id="ref89">21</reflink>) can be used in a straightforward manner. The second one involves marginalization over one intermediate/outer node and more technicalities are necessary. We here present an instance of both situations.</p> <p>Suppose that the research question involves investigation of the path-specific indirect effects in the model obtained after marginalization over <emph>W</emph><subs>1</subs> of a model corresponding to the DAG in Figure 6A. In Figure 6B, the DAG corresponding to the marginal model of interest is presented, with the red arrows corresponding to parameters that change due to the marginalization over <emph>W</emph><subs>1</subs>. The expressions for these parameters is reported in Online Appendix D (Supplementary material for this article is available online). The only nonzero path-specific indirect effects are for <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>A</mi><mo>=</mo><mo>{</mo><msub><mi>W</mi><mn>2</mn></msub><mo>}</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>A</mi><mo>=</mo><mo>{</mo><msub><mi>W</mi><mn>3</mn></msub><mo>}</mo></mrow></math> </ephtml> . They can be evaluated by making use of the parameters of the marginal model. Notice that, as expected, the <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>GIE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> in the marginal model is equal to the <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>GIE</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> in the original model.</p> <p>Graph: Figure 6. (A) Marginalization over the inner mediator W 1 and (B) quantifying the parameters (in red the parameters that change).</p> <p>Quantification of effects in models obtained after marginalization over intermediate or outer nodes involves repeated use of the derivations here presented. We here detail the steps to be followed for the case with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></math> </ephtml> mediators. Generalizations to more complex models can be derived after repeatedly applying the procedure here proposed.</p> <p>Suppose that we wish to evaluate the indirect effect in the model with <emph>W</emph><subs>1</subs> as unique mediator, that is, the model obtained after marginalization over <emph>W</emph><subs>2</subs> (see Figure 7). This implies deriving the parametric formulation of</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo>=</mo><msub><mi>w</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo>=</mo><msub><mi>w</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo>=</mo><msub><mi>w</mi><mn>1</mn></msub><mo>,</mo><msub><mi>W</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo>=</mo><msub><mi>w</mi><mn>1</mn></msub><mo>,</mo><msub><mi>W</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo /></mrow></mtd><mtd columnalign="left"><mrow><mo>+</mo><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>W</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo>=</mo><msub><mi>w</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>W</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo>=</mo><msub><mi>w</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Graph</p> <p>in which the second term of the RHS of the equation is to be determined. From repeated use of the derivations in Online Appendix A (Supplementary material for this article is available online), we have</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left" equalcolumns="true" equalrows="true"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>log</mtext><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>1</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo>=</mo><msub><mi>w</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><msub><mi>W</mi><mn>1</mn></msub><mo>=</mo><msub><mi>w</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msub><mo>β</mo><mn>0</mn></msub><mo>+</mo><msub><mo>β</mo><mi>x</mi></msub><mi>x</mi><mo>+</mo><msub><mo>β</mo><mrow><msub><mi>w</mi><mn>1</mn></msub></mrow></msub><msub><mi>w</mi><mn>1</mn></msub><mo>+</mo><msub><mo>β</mo><mrow><mi>x</mi><msub><mi>w</mi><mn>1</mn></msub></mrow></msub><mi>x</mi><msub><mi>w</mi><mn>1</mn></msub><mo>+</mo><mtext>log</mtext><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>h</mi><mrow><mn>1</mn><mo>,</mo><msub><mi>w</mi><mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><msub><mi>h</mi><mrow><mn>0</mn><mo>,</mo><msub><mi>w</mi><mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Graph</p> <p>with the expression of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>h</mi><mrow><mi>y</mi><mo>,</mo><msub><mi>w</mi><mn>1</mn></msub></mrow></msub></mrow></math> </ephtml> in Online Appendix E (Supplementary material for this article is available online). The values of the marginal parameters are straightforward (see, e.g., Online Appendix D, Supplementary material for this article is available online).</p> <p>Graph: Figure 7. (A) Marginalization over the outer mediator W 2 and (B) quantifying the parameters (in red the parameters that change).</p> <hd id="AN0173122300-10">Causal Interpretation of Total, Direct, and Indirect Effects</hd> <p>In the counterfactual framework, many approaches exist to mediation analysis, and a review is in [<reflink idref="bib14" id="ref90">14</reflink>]. In a single-mediator context, [<reflink idref="bib31" id="ref91">31</reflink>] define the counterfactual notion of direct and indirect effects when the outcome is binary, thereby focusing on the log odds scale. Within a regression analysis context with a continuous mediator, the authors present an approximated parametric formulation of the effects that holds under the rare outcome assumption of <emph>Y</emph>. [<reflink idref="bib30" id="ref92">30</reflink>] address the same problem when also the mediator is binary, again modeled under the rare outcome assumption. It is therefore worth to explore the links existing between the effects introduced here and these causal effects defined in a formal counterfactual framework. Since the latter are contrasts expressed, possibly after a logarithmic transformation, by a difference, this parallel holds for <emph>X</emph> binary. Notice that, differently from the above cited approaches, we here present a decomposition based on the exact formulation of the effects on the log odds scale.</p> <p>Under the assumption that the recursive system of equation is structural in the sense of [<reflink idref="bib24" id="ref93">24</reflink>], chap. 7), one can give the total effect and some of its components a causal interpretation. To say that the recursive system of equations is structural implies that the DAG is a causal graph that satisfies a set of axioms, namely, composition, effectiveness, and reversibility (see also [<reflink idref="bib29" id="ref94">29</reflink>]).</p> <p>With a single binary mediator, a parallelism between the structural definition of a DAG and the sequential ignorability assumption of [<reflink idref="bib15" id="ref95">15</reflink>] exists (see [<reflink idref="bib25" id="ref96">25</reflink>]; [<reflink idref="bib26" id="ref97">26</reflink>]). Under the assumption of no unmeasured counfounder of the treatment–outcome relationship, possibly after conditioning on a set of pretreatment covariates <emph>C</emph>, the total effect of <emph>X</emph> on <emph>Y</emph> here presented corresponds to the <emph>total causal effect</emph> as defined by [<reflink idref="bib31" id="ref98">31</reflink>]. Similarly, assuming that there are no unobserved confounders of the treatment–outcome relationship, possibly after conditioning on a set of pretreatment covariates <emph>C</emph>, and no unmeasured confounders of the mediator-outcome relationship, after conditioning on the treatment <emph>X</emph> and possibly some pretreatment covariates <emph>C</emph>, the direct effect can be seen as the <emph>controlled direct effect</emph> ([<reflink idref="bib31" id="ref99">31</reflink>]) after an external intervention to fix <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>W</mi><mo>=</mo><mn>0</mn></mrow></math> </ephtml> is performed (see Discrete Case subsection, Case (ii)). Less obvious are the parallelisms in terms of natural effects of [<reflink idref="bib31" id="ref100">31</reflink>]. It is possible to show that the <emph>pure natural indirect effect</emph> can be seen as the total effect after assuming <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mo>β</mo><mi>x</mi></msub><mo>=</mo><msub><mo>β</mo><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> (i.e., the indirect effect; see Discrete Case subsection, Case (i)) and that the <emph>pure natural direct effect</emph> corresponds to the total effect after assuming <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mi>x</mi></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> , that is, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>╨</mo><mi>W</mi></mrow></math> </ephtml> (see Discrete Case subsection, Case (iv)). More details are in [<reflink idref="bib10" id="ref101">10</reflink>].</p> <p>When multiple causally ordered mediators are present, several possible effects are of interest (see [<reflink idref="bib9" id="ref102">9</reflink>]; [<reflink idref="bib28" id="ref103">28</reflink>]). However, in this case, the definition of natural direct and indirect effects is more cumbersome, and sometimes effects of interest are nonidentifiable (see, e.g., the situation described in [<reflink idref="bib2" id="ref104">2</reflink>]]). Again, if no unobserved confounders exist, some parallelisms continue to hold. As an instance, the direct effect can be seen as the controlled direct effect of <emph>X</emph> on <emph>Y</emph> after an external intervention to fix the mediators <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>W</mi><mn>1</mn></msub><mo>=</mo><msub><mi>W</mi><mn>2</mn></msub><mo>=</mo><mo>...</mo><mo>=</mo><msub><mi>W</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> is performed (see also [<reflink idref="bib32" id="ref105">32</reflink>]). Our approach allows further to appreciate the total and controlled direct effect of <emph>X</emph> on <emph>Y</emph> also when some mediators are marginalized over, while others are kept in the system. We believe that this is an important research question in many applied studies.</p> <hd id="AN0173122300-11">Conclusions</hd> <p>Logistic regression is by far the most used model for a binary response. Further than in a mediation context, the models here proposed may arise in longitudinal studies with a binary outcome measured at different occasions.</p> <p>With reference to a single mediator, we have proposed a novel decomposition of the total effect into direct and indirect effects that is more appropriate for the nonlinear case and that, under certain conditions, reduces to the classical definition in the linear case. Additionally, this decomposition overcomes the issue of unequal variance when fitting two nested models. As an illustrative example, we have reanalyzed data based on an encouragement program to stimulate students' attitude to visit museums. We have shown how the decomposition of the total effect could avoid erroneous conclusions on the direct and indirect effects and provides additional information that cannot be found by just looking at the results of the separate regressions analysis. Although the total, direct, and indirect effects are all positive, a substantial residual effect, possibly due to a large interaction coefficient, hinders the interpretation in terms of proportion mediated.</p> <p>Additional important results concern the extension of the definitions to the multiple mediator context. Repeated use of the decomposition of the total effect allows to address complex issues like quantifying the total, direct, and indirect effects when a subset of mediators are marginalized over. Links to the causal effects have also been established.</p> <hd id="AN0173122300-12">Supplemental Material</hd> <p>Graph: Supplemental Material, sj-docx-1-smr-10.1177_00491241211031260 for Path Analysis for Binary Random Variables by Martina Raggi, Elena Stanghellini and Marco Doretti in Sociological Methods & Research</p> <hd id="AN0173122300-13">Supplemental Material</hd> <p>Graph: Supplemental Material, sj-r-1-smr-10.1177_00491241211031260 for Path Analysis for Binary Random Variables by Martina Raggi, Elena Stanghellini and Marco Doretti in Sociological Methods & Research</p> <hd id="AN0173122300-14">Supplemental Material</hd> <p>Graph: Supplemental Material, sj-r-2-smr-10.1177_00491241211031260 for Path Analysis for Binary Random Variables by Martina Raggi, Elena Stanghellini and Marco Doretti in Sociological Methods & Research</p> <ref id="AN0173122300-15"> <title> References </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Alwin D. F., Hauser R. M. 1975. " The Decomposition of Effects in Path Analysis." American Sociological Review. 40(1):37–47.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref23" type="bt">2</bibl> <bibtext> Avin C., Shpitser I., Pearl J. 2005. "Identifiability of Path-specific Effects." Pp. 357–63 in Proceedings of International Joint Conference on Artificial Intelligence. Edinburgh, Schotland.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref3" type="bt">3</bibl> <bibtext> Baron R. M., Kenny D. A. 1986. " The Moderator–Mediator Variable Distinction in Social Psychological Research: Conceptual, Strategic, and Statistical Considerations." Journal of Personality and Social Psychology. 51(6):1173.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref2" type="bt">4</bibl> <bibtext> Bollen K. A.1987. " Total, Direct, and Indirect Effects in Structural Equation Models ". Sociological Methodology. 17:37–69.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref5" type="bt">5</bibl> <bibtext> Breen R., Karlson K. B., Holm A. 2013. " Total, Direct, and Indirect Effects in Logit and Probit Models." Sociological Methods & Research. 42(2):164–91.</bibtext> </blist> <blist> <bibl id="bib6" idref="ref6" type="bt">6</bibl> <bibtext> Breen R., Karlson K. B., Holm A. 2018. " A Note on a Reformulation of the KHB Method." Sociological Methods & Research. (https://doi.org/10.1177/0049124118789717). Accessed October 1, 2020.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref4" type="bt">7</bibl> <bibtext> Cochran W. G.1938. " The Omission or Addition of an Independent Variate in Multiple Linear Regression." Supplement to the Journal of the Royal Statistical Society. 5(2):171–6.</bibtext> </blist> <blist> <bibl id="bib8" idref="ref36" type="bt">8</bibl> <bibtext> Cox D., Wermuth N. 2003. " A General Condition for Avoiding Effect Reversal after Marginalization." Journal of the Royal Statistical Society: Series B (Statistical Methodology). 65(4):937–41.</bibtext> </blist> <blist> <bibl id="bib9" idref="ref102" type="bt">9</bibl> <bibtext> Daniel R., De Stavola B., Cousens S., Vansteelandt S. 2015. " Causal Mediation Analysis with Multiple Mediators." Biometrics. 71(1):1–14.</bibtext> </blist> <blist> <bibtext> Doretti M., Raggi M., Stanghellini E. 2021. " Exact Parametric Causal Mediation Analysis for a Binary Outcome with a Binary Mediator." Statistical Methods & Applications. doi: 10.1007/s10260-021-00562-w</bibtext> </blist> <blist> <bibtext> Elwert F.2013. "Graphical Causal Models." Pp. 245–73 in Handbook of Causal Analysis for Social Research, edited by Morgan S. L. Berlin, Germany: Springer.</bibtext> </blist> <blist> <bibtext> Elwert F., Winship C. 2014. " Endogenous Selection Bias: The Problem of Conditioning on a Collider Variable." Annual Review of Sociology. 40:31–53.</bibtext> </blist> <blist> <bibtext> Forastiere L., Lattarulo P., Mariani M., Mealli F., Razzolini L. 2019. " Exploring Encouragement, Treatment, and Spillover Effects using Principal Stratification, with Application to a Field Experiment on Teens' Museum Attendance." Journal of Business & Economic Statistics. 39(1): 244-258.</bibtext> </blist> <blist> <bibtext> Huber M.2019. " A Review of Causal Mediation Analysis for Assessing Direct and Indirect Treatment Effects." Technical report, Université de Fribourg, Fribourg.</bibtext> </blist> <blist> <bibtext> Imai K., Keele L., Yamamoto T. 2010. " Identification, Inference and Sensitivity Analysis for Causal Mediation Effects." Statistical Science. 25(1):51–71.</bibtext> </blist> <blist> <bibtext> Karlson K. B., Holm A., Breen R. 2012. " Comparing Regression Coefficients between same-Sample Nested Models Using Logit and Probit: A New Method." Sociological Methodology. 42(1):286–313.</bibtext> </blist> <blist> <bibtext> Lattarulo P., Mariani M., Razzolini L. 2017. " Nudging Museums Attendance: A Field Experiment with High School Teens." Journal of Cultural Economics. 41(3):259–77.</bibtext> </blist> <blist> <bibtext> Lauritzen S. L.1996. Graphical Models, Volume 17. Oxford, England: Clarendon Press.</bibtext> </blist> <blist> <bibtext> Lupparelli M.2019. " Conditional and Marginal Relative Risk Parameters for a Class of Recursive Regression Graph Models." Statistical Methods in Medical Research. 28(10-11):3466–86.</bibtext> </blist> <blist> <bibtext> MacKinnon D. P., Lockwood C. M., Brown C. H., Wang W., Hoffman J. M. 2007. " The Intermediate Endpoint Effect in Logistic and Probit Regression." Clinical Trials. 4(5):499–513.</bibtext> </blist> <blist> <bibtext> Neuhaus J. M., Jewell N. P. 1993. " A Geometric Approach to Assess Bias Due to Omitted Covariates in Generalized Linear Models." Biometrika. 80(4):807–15.</bibtext> </blist> <blist> <bibtext> Oehlert G. W.1992. " A Note on the Delta Method." The American Statistician. 46(1):27–9.</bibtext> </blist> <blist> <bibtext> Pearl J. 2001. "Direct and Indirect Effects." Pp. 411–20 in Proceedings of the 17th International Conference on Uncertainty in Artificial Intelligence, UAI'01, edited by Breese Jack, Koller Daphne. San Francisco, CA: Morgan Kaufmann.</bibtext> </blist> <blist> <bibtext> Pearl J.2009. Causality: Models, Reasoning, and Inference. 2nd ed. New York: Cambridge University Press.</bibtext> </blist> <blist> <bibtext> Pearl J.2012. The Mediation Formula: A Guide to the Assessment of Causal Pathways in Nonlinear Models. Hoboken, NJ: Wiley Online Library.</bibtext> </blist> <blist> <bibtext> Shpitser I., VanderWeele T. J. 2011. " A Complete Graphical Criterion for the Adjustment Formula in Mediation Analysis." The International Journal of Biostatistics. 7(1):16.</bibtext> </blist> <blist> <bibtext> Stanghellini E., Doretti M. 2019. " On Marginal and Conditional Parameters in Logistic Regression Models." Biometrika. 106(3):732–39.</bibtext> </blist> <blist> <bibtext> Steen J., Loeys T., Moerkerke B., Vansteelandt S. 2017. " Flexible Mediation Analysis with Multiple Mediators." American Journal of Epidemiology. 186(2):184–93.</bibtext> </blist> <blist> <bibtext> Steen J., Vansteelandt S. 2018. "Mediation Analysis." Pp. 423–56 in Handbook of Graphical Models, edited by Maathuis M., Wainwright M., Drton M., Lauritzen S. Boca Raton, FL: CRC Press.</bibtext> </blist> <blist> <bibtext> Valeri L., VanderWeele T. J. 2013. " Mediation Analysis allowing for Exposure–Mediator Interactions and Causal Interpretation: Theoretical Assumptions and Implementation with SAS and SPSS Macros." Psychological Methods. 18(2), 137–50.</bibtext> </blist> <blist> <bibtext> VanderWeele T. J., Vansteelandt S. 2010. " Odds Ratios for Mediation Analysis for a Dichotomous Outcome." American Journal of Epidemiology. 172(12):1339–48.</bibtext> </blist> <blist> <bibtext> VanderWeele T. J., Vansteelandt S. 2014. " Mediation Analysis with Multiple Mediators." Epidemiologic Methods. 2(1):95–115.</bibtext> </blist> <blist> <bibtext> Winship C., Mare R. D. 1983. " Structural Equations and Path Analysis for Discrete Data." American Journal of Sociology. 89(1):54–110.</bibtext> </blist> <blist> <bibtext> Wooldridge J. M.2010. Econometric Analysis of Cross Section and Panel Data. 2nd ed. Cambridge, MA: MIT Press.</bibtext> </blist> <blist> <bibtext> Xie X., Ma Z., Geng Z. 2008. " Some Association Measures and Their Collapsibility." Statistica Sinica. 18(3):1165–183.</bibtext> </blist> </ref> <ref id="AN0173122300-16"> <title> Footnotes </title> <blist> <bibtext> The author Martina Raggi is also affiliated with Université de Paris, INSERM U1153, Epidemiology of Ageing and Neurodegenerative diseases Paris, France</bibtext> </blist> <blist> <bibtext> The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibtext> The author(s) received no financial support for the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibtext> Martina Raggi https://orcid.org/0000-0002-3309-9006 Elena Stanghellini https://orcid.org/0000-0002-2503-8342</bibtext> </blist> <blist> <bibtext> The supplemental material for this article is available online.</bibtext> </blist> </ref> <aug> <p>By Martina Raggi; Elena Stanghellini and Marco Doretti</p> <p>Reported by Author; Author; Author</p> <p></p> <p>Martina Raggi has recently obtained a PhD in Statistics at the University of Neuchâtel, Switzerland. She is currently working as postdoctoral researcher in Biostatistics in the Epidemiology of Ageing and Neurodegenerative diseases (EpiAgeing) team at the INSERM, France. Her research interests mainly concern mediation analysis and causal inference as well as competing risk methods for time-to-event data, with a particular focus on epidemiological, medical and social science applications. She has recently published the following paper: "Exact parametric causal mediation analysis for a binary outcome with a binary mediator" (Statistical Methods & Applications 2021, with Marco Doretti and Elena Stanghellini).</p> <p>Elena Stanghellini is Professor of Statistics at the Department of Economics of the University of Perugia, Italy. Her research focusses mainly on Graphical Markov Models, both from the theoretical and applied point of view. She has been awarded international grants and fellowships. From 2015 to 2019 she has also been Affiliated Professor of the Umeå University, Sweden. She has recently published the following papers: "On marginal and conditional parameters in logistic regression models" (Biometrika, 2019, with Marco Doretti), and "Identification of Principal Causal Effects Using Secondary Outcomes in Concentration Graphs" (Journal of Educational and Behavioral Statistics, 2016, with Fabrizia Mealli and Barbara Pacini).</p> <p>Marco Doretti is Assistant Professor in Statistics at the Department of Political Sciences of the University of Perugia, Italy. His research interests mainly concern parametric methods for causal inference as well as statistical policy evaluation systems and latent variable models for the analysis of longitudinal data. He recently published the following papers: "Model-based two-way clustering of second-level units in ordinal multilevel latent Markov models" (Advances in Data Analysis and Classification, 2021, with Giorgio E. Montanari and Maria Francesca Marino), "Ranking nursing homes' performances through a latent Markov model with fixed and random effects" (Social Indicators Research, 2019, with Giorgio E. Montanari), and "Missing data: A unified taxonomy guided by conditional independence" (International Statistical Review, 2018, with Sara Geneletti and Elena Stanghellini).</p> </aug> <nolink nlid="nl1" bibid="bib16" firstref="ref7"></nolink> <nolink nlid="nl2" bibid="bib20" firstref="ref8"></nolink> <nolink nlid="nl3" bibid="bib27" firstref="ref9"></nolink> <nolink nlid="nl4" bibid="bib19" firstref="ref10"></nolink> <nolink nlid="nl5" bibid="bib18" firstref="ref12"></nolink> <nolink nlid="nl6" bibid="bib11" firstref="ref13"></nolink> <nolink nlid="nl7" bibid="bib33" firstref="ref15"></nolink> <nolink nlid="nl8" bibid="bib24" firstref="ref17"></nolink> <nolink nlid="nl9" bibid="bib23" firstref="ref18"></nolink> <nolink nlid="nl10" bibid="bib25" firstref="ref19"></nolink> <nolink nlid="nl11" bibid="bib31" firstref="ref20"></nolink> <nolink nlid="nl12" bibid="bib13" firstref="ref24"></nolink> <nolink nlid="nl13" bibid="bib17" firstref="ref25"></nolink> <nolink nlid="nl14" bibid="bib10" firstref="ref38"></nolink> <nolink nlid="nl15" bibid="bib35" firstref="ref41"></nolink> <nolink nlid="nl16" bibid="bib12" firstref="ref43"></nolink> <nolink nlid="nl17" bibid="bib14" firstref="ref45"></nolink> <nolink nlid="nl18" bibid="bib15" firstref="ref47"></nolink> <nolink nlid="nl19" bibid="bib21" firstref="ref52"></nolink> <nolink nlid="nl20" bibid="bib22" firstref="ref68"></nolink> <nolink nlid="nl21" bibid="bib34" firstref="ref71"></nolink> <nolink nlid="nl22" bibid="bib30" firstref="ref92"></nolink> <nolink nlid="nl23" bibid="bib29" firstref="ref94"></nolink> <nolink nlid="nl24" bibid="bib26" firstref="ref97"></nolink> <nolink nlid="nl25" bibid="bib28" firstref="ref103"></nolink> <nolink nlid="nl26" bibid="bib32" firstref="ref105"></nolink>
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  Data: Path Analysis for Binary Random Variables
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  Data: <searchLink fieldCode="AR" term="%22Raggi%2C+Martina%22">Raggi, Martina</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-3309-9006">0000-0002-3309-9006</externalLink>)<br /><searchLink fieldCode="AR" term="%22Stanghellini%2C+Elena%22">Stanghellini, Elena</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-2503-8342">0000-0002-2503-8342</externalLink>)<br /><searchLink fieldCode="AR" term="%22Doretti%2C+Marco%22">Doretti, Marco</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22Sociological+Methods+%26+Research%22"><i>Sociological Methods & Research</i></searchLink>. 2023 52(4):1883-1915.
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  Data: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
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  Data: 33
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  Data: 2023
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  Data: Journal Articles<br />Reports - Evaluative
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  Data: <searchLink fieldCode="DE" term="%22Path+Analysis%22">Path Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Student+Attitudes%22">Student Attitudes</searchLink><br /><searchLink fieldCode="DE" term="%22Museums%22">Museums</searchLink><br /><searchLink fieldCode="DE" term="%22Error+of+Measurement%22">Error of Measurement</searchLink><br /><searchLink fieldCode="DE" term="%22Foreign+Countries%22">Foreign Countries</searchLink><br /><searchLink fieldCode="DE" term="%22Graphs%22">Graphs</searchLink><br /><searchLink fieldCode="DE" term="%22Regression+%28Statistics%29%22">Regression (Statistics)</searchLink>
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  Data: 10.1177/00491241211031260
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  Data: 0049-1241<br />1552-8294
– Name: Abstract
  Label: Abstract
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  Data: The decomposition of the overall effect of a treatment into direct and indirect effects is here investigated with reference to a recursive system of binary random variables. We show how, for the single mediator context, the marginal effect measured on the log odds scale can be written as the sum of the indirect and direct effects plus a residual term that vanishes under some specific conditions. We then extend our definitions to situations involving multiple mediators and address research questions concerning the decomposition of the total effect when some mediators on the pathway from the treatment to the outcome are marginalized over. Connections to the counterfactual definitions of the effects are also made. Data coming from an encouragement design on students' attitude to visit museums in Florence, Italy, are reanalyzed. The estimates of the defined quantities are reported together with their standard errors to compute p values and form confidence intervals.
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  Data: 2023
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  Data: EJ1397548
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        PageCount: 33
        StartPage: 1883
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      – SubjectFull: Path Analysis
        Type: general
      – SubjectFull: Student Attitudes
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      – SubjectFull: Museums
        Type: general
      – SubjectFull: Error of Measurement
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      – SubjectFull: Foreign Countries
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      – SubjectFull: Graphs
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      – SubjectFull: Regression (Statistics)
        Type: general
      – SubjectFull: Italy
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      – TitleFull: Path Analysis for Binary Random Variables
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              Type: published
              Y: 2023
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            – TitleFull: Sociological Methods & Research
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