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Area under the Curve-Optimized Synthesis of Prediction Models from a Meta-Analytical Perspective

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Title: Area under the Curve-Optimized Synthesis of Prediction Models from a Meta-Analytical Perspective
Language: English
Authors: Yoneoka, Daisuke (ORCID 0000-0002-3525-5092), Omae, Katsuhiro, Henmi, Masayuki, Eguchi, Shinto
Source: Research Synthesis Methods. Mar 2023 14(2):234-246.
Availability: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
Peer Reviewed: Y
Page Count: 13
Publication Date: 2023
Document Type: Journal Articles
Reports - Research
Descriptors: Prediction, Task Analysis, Medical Research, Outcomes of Treatment, Meta Analysis, Predictor Variables, Models
DOI: 10.1002/jrsm.1612
ISSN: 1759-2879
1759-2887
Abstract: The number of clinical prediction models sharing the same prediction task has increased in the medical literature. However, evidence synthesis methodologies that use the results of these prediction models have not been sufficiently studied, particularly in the context of meta-analysis settings where only summary statistics are available. In particular, we consider the following situation: we want to predict an outcome Y, that is not included in our current data, while the covariate data are fully available. In addition, the summary statistics from prior studies, which share the same prediction task (i.e., the prediction of Y), are available. This study introduces a new method for synthesizing the summary results of binary prediction models reported in the prior studies using a linear predictor under a distributional assumption between the current and prior studies. The method provides an integrated predictor combining all predictors reported in the prior studies with weights. The vector of the weights is designed to achieve the hypothetical improvement of area under the receiver operating characteristic curve (AUC) on the current available data under a practical situation where there are different sets of covariates in the prior studies. We observe a counterintuitive aspect in typical situations where a part of weight components in the proposed method becomes negative. It implies that flipping the sign of the prediction results reported in each individual study would improve the overall prediction performance. Finally, numerical and real-world data analysis were conducted and showed that our method outperformed conventional methods in terms of AUC.
Abstractor: As Provided
Notes: https://hbiostat.org/data
Entry Date: 2023
Accession Number: EJ1369258
Database: ERIC
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  Value: <anid>AN0162399693;[bdct]01mar.23;2023Mar15.08:32;v2.2.500</anid> <title id="AN0162399693-1">Area under the curve‐optimized synthesis of prediction models from a meta‐analytical perspective </title> <p>The number of clinical prediction models sharing the same prediction task has increased in the medical literature. However, evidence synthesis methodologies that use the results of these prediction models have not been sufficiently studied, particularly in the context of meta‐analysis settings where only summary statistics are available. In particular, we consider the following situation: we want to predict an outcome Y, that is not included in our current data, while the covariate data are fully available. In addition, the summary statistics from prior studies, which share the same prediction task (i.e., the prediction of Y), are available. This study introduces a new method for synthesizing the summary results of binary prediction models reported in the prior studies using a linear predictor under a distributional assumption between the current and prior studies. The method provides an integrated predictor combining all predictors reported in the prior studies with weights. The vector of the weights is designed to achieve the hypothetical improvement of area under the receiver operating characteristic curve (AUC) on the current available data under a practical situation where there are different sets of covariates in the prior studies. We observe a counterintuitive aspect in typical situations where a part of weight components in the proposed method becomes negative. It implies that flipping the sign of the prediction results reported in each individual study would improve the overall prediction performance. Finally, numerical and real‐world data analysis were conducted and showed that our method outperformed conventional methods in terms of AUC.</p> <p>Keywords: area under the curve; clinical prediction model; linear predictors; multivariate meta‐analysis; research synthesis</p> <hd id="AN0162399693-2">Highlights</hd> <p></p> <hd id="AN0162399693-3">What is already known?</hd> <p>The number of published studies that share the same prediction task or population has increased substantially. Previous studies have mainly aimed to synthesize regression coefficients from an estimation perspective to improve the (asymptotic) estimation efficiency of the average effect size.</p> <hd id="AN0162399693-4">What is new?</hd> <p>Our study focuses on synthesizing the regression coefficients from a prediction perspective to maximize prediction accuracy measured by the area under the receiver operating characteristic curve.</p> <hd id="AN0162399693-5">Potential impact for RSM readers outside the authors'' field</hd> <p>Our new method guarantees the meta‐analytical procedure that combines the regression coefficients reported from the prior studies to improve the prediction accuracy on the current dataset.</p> <hd id="AN0162399693-6">INTRODUCTION</hd> <p>The development of accurate clinical prediction models has been one of the main interests in the medical literature, and the number of published studies that share the same prediction task or population has increased substantially over the past few decades.[<reflink idref="bib1" id="ref1">1</reflink>] For example, 363 prediction models and 473 external validations are proposed for the risk of cardiovascular disease,[<reflink idref="bib1" id="ref2">1</reflink>] and 25 for the risk of Type 2 diabetes.[<reflink idref="bib2" id="ref3">2</reflink>] However, no single prediction model always has sufficient ability to identify the risk factors that can be generalized to larger and broader populations.[[<reflink idref="bib3" id="ref4">3</reflink>]] To address this, a new methodology of synthesizing clinical prediction models has received interest to achieve more accurate and generalizable predictions on larger datasets.[[<reflink idref="bib5" id="ref5">5</reflink>], [<reflink idref="bib7" id="ref6">7</reflink>]] Debray et al. (2012, 2013)[[<reflink idref="bib8" id="ref7">8</reflink>]] addressed this issue by proposing a multivariate meta‐analysis approach for regression coefficients using individual patient data (IPD). In contrast, Yoneoka et al. (2015, 2017)[[<reflink idref="bib6" id="ref8">6</reflink>], [<reflink idref="bib10" id="ref9">10</reflink>]] studied the synthesis of regression coefficients using only summary statistics reported in the published articles instead of assuming the availability of IPD. Especially they focused on cases where the included regression models in the meta‐analysis have different covariate sets and categorization schemes of continuous covariates. Moreover, recently, Debray et al. (2017)[<reflink idref="bib5" id="ref10">5</reflink>] and Ahmed et al. (2014)[<reflink idref="bib11" id="ref11">11</reflink>] comprehensively discussed meta‐analytical methods to synthesize and examine prediction performances on different settings and populations. Guidance for the systematic review of prediction models have been widely used in terms of how to formulate the review aim, search strategy, critical appraisal (CHARMS), and data extraction.[[<reflink idref="bib12" id="ref12">12</reflink>]]</p> <p>Previous studies have mainly aimed to synthesize regression coefficients of binary prediction models from an <emph>estimation</emph> perspective: the synthesis procedure aims to improve the (asymptotic) estimation efficiency of the average effect size. In contrast, our study focuses on synthesizing the regression coefficients from a <emph>prediction</emph> perspective: the synthesis procedure aims to hypothetically maximize prediction accuracy measured by the area under the receiver operating characteristic curve (AUC) in one available dataset. This study can be considered as an extension of the work by Su and Liu (1993)[<reflink idref="bib14" id="ref13">14</reflink>] to a meta‐analytical setting by using only summary statistics reported in previous articles. Similar attempts can be also found in Komori and Eguchi (2010)[<reflink idref="bib15" id="ref14">15</reflink>] and Takenouchi et al. (2012).[<reflink idref="bib16" id="ref15">16</reflink>] They proposed new boosting‐based algorithms to combine multiple predictors to maximize the AUC. While they worked on one IPD to maximize the AUC, we utilize only the summary information from several datasets to improve the prediction from a meta‐analytical perspective.</p> <p>Here, we are interested in the problem of prediction of a binary outcome by developing a class of linear predictors for the binary outcome in the context of meta‐analysis. In this study, we focus on the weighted‐sum of the linear predictors reported in prior studies. We then derive the combined linear predictor maximizing AUC under a distributional assumption, where the weights are estimated based on (i) observed covariates of the current available dataset, and (ii) summary statistics reported in several previous studies that are assumed to share the same prediction task and have the subset of the covariates in the current dataset. We show theoretical proof that the proposed weights achieve the optimal AUC in a specific situation, as well as simulation and application studies, which empirically illustrate the improved performance of our method compared with the standard approaches. In addition, we show that there are counterintuitive examples in Appendix: the weights, which measure the importance of the results of prior studies on the final AUC, become negative in some typical cases.</p> <p>The remainder of the article is organized as follows. In Section 2, we introduce our motivating example of the prediction of respiratory infection using a series of epidemiological studies. In Section 3, we then define a new method to synthesize the reported regression coefficients to maximize AUC. To demonstrate that the proposed approach outperforms the standard approaches in terms of AUC, the results of extensive simulation studies are presented in Section 4. To analyze the performance of the proposed method in a real‐world setting, the dataset introduced in Section 2 is examined in Section 5. Finally, Section 6 contains a discussion and our conclusions.</p> <hd id="AN0162399693-7">PRACTICAL EXAMPLE: WORLD HEALTH ORGANIZATION/ACUTE RESPIRATORY INFECTION MULTICENTRE STUDY</hd> <p>Annually, more than five million deaths are observed during the neonatal period. Approximately 97% of these deaths occur in developing countries, and more than 40% of those might be caused by infection.[<reflink idref="bib17" id="ref16">17</reflink>] Many researchers have focused on preventing and reducing the neonatal mortality. A popular approach has been to educate medical personnel to quickly recognize the clinical signs of infections at their local clinics, and administer the appropriate treatment as soon as possible.[<reflink idref="bib18" id="ref17">18</reflink>] In most developing countries, it is rare to diagnose an infection based on laboratory tests such as blood culture tests, and the only available tool is the observation of clinical signs. Therefore, in such a medical frontier, more accurate diagnostic models to predict infant illness using limited observational clinical signs could help improve clinical decisions for the referral of children to other well‐facilitated hospitals for additional diagnose and treatments. In this study, we use the count of white blood cells (WBCs) that is a simple and popular biomarker of systemic inflammation, and standardized and high‐precision tests are commonly available. Leukocytosis defined as the number of WBCs ≥11,000 cells per microliter of blood is generally caused by the response of the body to help fight an infection.[[<reflink idref="bib19" id="ref18">19</reflink>]]</p> <p>There are four studies conducted by the World Health Organization/Acute Respiratory Infection (WHO/ARI) Multicentre Study of Clinical Signs and Etiological Agents of Pneumonia, Sepsis, and Meningitis in Young Infants. The series of studies were conducted in Ethiopia, Gambia, Papua New Guinea, and the Philippines. Each study was designed and attempted to construct better screening rules for infants at high risk of serious infections. In each country, infants younger than 91 days with symptoms indicating potential infection are included in the analysis. More detailed information can be found elsewhere.[[<reflink idref="bib18" id="ref19">18</reflink>], [<reflink idref="bib21" id="ref20">21</reflink>]] Logistic regression model is fitted to each data in Ethiopia, Gambia and Papua New Guinea to predict the binary outcome, the status of leukocytosis infection (i.e., WBCs ≥11,000), and only regression results such as the estimated coefficients from each model are available for the three countries. These summary results are regarded as the prior studies. In contrast, the IPD in the Philippines is partially available and regarded as the current study: i.e., the IPD of covariates is available, while the outcome is assumed not to be available due to the expensive cost of observation or lack of equipment in the medical frontier. Each study has the following set of covariates: weight‐for‐age z‐score (waz), respiratory rate (rr), age in days, temperature (temp), heart‐rate oximeter (hrat), total number of x‐ray readings (nxray), and pneumonia (pneu). However, shown in Table 1, the set of available covariates is assumed to differ by country. Note that the missing values with respect to "age" and "temp" were artificially created to induce the different covariate sets across countries. In this sense, this is a mixture of real and artificial motivating examples.</p> <p>1 TABLE Estimated regression coefficients (standard error) and AUC calculated on each country in WHO/ARI study</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left" /><th align="left">Individual studies</th></tr><tr><th align="left">Covariates</th><th align="left">Ethiopia</th><th align="left">Gambia</th><th align="left">Papua New Guinea</th></tr></thead><tbody valign="top"><tr><td>Intercept</td><td>−2.52 (3.45)</td><td>−2.72 (5.06)</td><td>−1.31 (0.65)</td></tr><tr><td>waz</td><td>−0.09 (0.09)</td><td>−0.07 (0.07)</td><td>−0.02 (0.08)</td></tr><tr><td>rr</td><td>0.01 (0.01)</td><td>0.01 (0.01)</td><td>0.00 (0.01)</td></tr><tr><td>age</td><td /><td>0.01 (0.00)</td><td>0.01 (0.00)</td></tr><tr><td>temp</td><td>0.04 (0.09)</td><td>0.03 (0.14)</td><td /></tr><tr><td>hrat</td><td>0.00 (0.00)</td><td>0.01 (0.01)</td><td>0.01 (0.00)</td></tr><tr><td>nxray</td><td /><td>0.08 (0.26)</td><td /></tr><tr><td>pneu</td><td>−0.37 (0.23)</td><td /><td /></tr><tr><td>AUC on individual country (95%CI)</td><td>0.604 (0.548–0.661)</td><td>0.551 (0.510–0.628)</td><td>0.594 (0.551–0.637)</td></tr></tbody></table> </ephtml> </p> <p>1 Abbreviations: age, age in days; hrat, heart‐rate oximeter; nxray, total number of x‐ray readings; pneu, pneumonia; rr, respiratory rate; temp, temperature; was, weight‐for‐age z‐score.</p> <p>Conventionally, the following methods have been used to analyze such a data: (M1) calculate the mean of the available regression coefficients, and then construct the linear predictor on the current dataset by plugging the mean coefficients into the predictor; (M2) separately plug the reported regression coefficients from each study into the linear predictor; and (M3) use the reported regression coefficients from the study with the best reported AUC in their article. In contrast, the aim of our study is to synthesize the reported prediction results from the three prior studies with weights that are designed to maximize the AUC on the current data (i.e., the IPD in the Philippines) in which the outcome is not available. We develop a new prediction rule that could help infants at risk of leukocytosis by using the limited information available in most developing countries, such as clinical vital signs and demographic information. In the following sections, we explain how the weights should be calculated and its preferable property.</p> <p>As other practical applications for our method, we believe that it can be used to assist in determining the design of studies and in pre‐treatment selection in clinical trials. In other words, when designing the clinical trial, our method can be used to predict how often an outcome will occur in advance (before the outcome is observed) by using the baseline covariates and prior information, and then use this information to design cohorts, calculate sample sizes, etc., or to calculate how much of a therapeutic agent to prepare in advance.</p> <hd id="AN0162399693-8">METHODS</hd> <p></p> <hd id="AN0162399693-9">Settings</hd> <p>We consider a binary classification problem, and thus the outcome of interest <emph>Y</emph> is coded as either 0 or 1. Let <emph>K</emph> be the number of the available prior studies, (<bold><emph>X</emph></bold>, <emph>Y</emph>) be a set of <emph>p</emph>‐covariate vector and the outcome of the current available study, and (<bold><emph>Z</emph></bold><subs><emph>k</emph></subs>, <emph>Y</emph><subs><emph>k</emph></subs>) a set of <emph>p</emph><subs><emph>k</emph></subs>‐covariate vector and outcome of the prior <emph>k</emph>‐th study, where <emph>p</emph><subs><emph>k</emph></subs> covariates of <bold><emph>Z</emph></bold><subs><emph>k</emph></subs> are included in the set of <emph>p</emph> covariates of <bold><emph>X</emph></bold> ( <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0001" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>...</mi><mo>,</mo><mi>K</mi></mrow></math> </ephtml> ). Our method does not require the availability of IPD in the prior studies, and requires only summary statistics as explained below. Note that, for simple explanation, we here use the notation <emph>Y</emph> tentatively, but it is supposed to be unavailable hereafter while the IPD of <bold><emph>X</emph></bold> (the current dataset) is available. This situation (i.e., the IPD of <emph>Y</emph> is not available) is the key point here and frequently occurs in clinical practice, where a researcher wants to predict a certain type of <emph>Y</emph> that is not included in the available dataset collected for other purposes. While <emph>Y</emph> is not available, we are considering the optimization of a hypothetical AUC on the current data (<bold><emph>X</emph></bold>, <emph>Y</emph>). What is meant by the hypothetical AUC here is that <emph>Y</emph> is not observed for some reason, but if it were, the AUC would be calculated based on <emph>Y</emph>.</p> <p>We introduce a basic assumption for the relationship between (<bold><emph>X</emph></bold>, <emph>Y</emph>) and <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0002" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mfenced open="{" close="}"><mfenced open="(" close=")" separators=","><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub><msub><mi mathvariant="bold">Y</mi><mi>k</mi></msub></mfenced></mfenced><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></msubsup></mrow></math> </ephtml> as follows:</p> <hd id="AN0162399693-10">1 Assumption</hd> <p>The conditional distribution of <bold><emph>Z</emph></bold><subs><emph>k</emph></subs> given <emph>Y</emph><subs><emph>k</emph></subs> is the same as the conditional distribution of <bold><emph>X</emph></bold><subs>(<emph>k</emph>)</subs> given <emph>Y</emph>, where <bold><emph>X</emph></bold><subs>(<emph>k</emph>)</subs> is a <emph>p</emph><subs><emph>k</emph></subs>‐subvector of <bold><emph>X</emph></bold> corresponding to <bold><emph>Z</emph></bold><subs><emph>k</emph></subs>. In addition, the set of covariates included in the prior studies is smaller than that in the current study: i.e., <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0003" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mfenced open="{" close="}"><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub></mfenced><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></msubsup></mrow></math> </ephtml> is a subset of <bold><emph>X</emph></bold>.</p> <p>The similar assumption has frequently been used in previous studies[[<reflink idref="bib10" id="ref21">10</reflink>], [<reflink idref="bib22" id="ref22">22</reflink>]] and the robustness of the results under the violation of the assumption is examined in the following sections. Under the assumption, we use a set of the following summary statistics of <bold><emph>X</emph></bold><subs>(<emph>k</emph>)</subs>: numbers of observations <emph>n</emph><subs>0</subs> and <emph>n</emph><subs>1</subs> of two classes, sample mean vectors <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0004" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mi mathvariant="bold">μ</mi><mo>̂</mo></mover><mrow><mi>k</mi><mn>0</mn></mrow></msub><mo>,</mo><msub><mover accent="true"><mi mathvariant="bold">μ</mi><mo>̂</mo></mover><mrow><mi>k</mi><mn>1</mn></mrow></msub></mrow></math> </ephtml> , the pooled sample variances <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0005" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mfenced open="{" close="}"><msubsup><mover accent="true"><mi>σ</mi><mo>̂</mo></mover><mi mathvariant="italic">kj</mi><mn>2</mn></msubsup></mfenced><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>p</mi><mi>k</mi></msub></msubsup></mrow></math> </ephtml> , and the estimated slope vector <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0006" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo>̂</mo></mover><mfenced open="(" close=")"><mi>k</mi></mfenced></msub></mrow></math> </ephtml> based on an unobservable sample from (<bold><emph>Z</emph></bold><subs><emph>k</emph></subs>, <emph>Y</emph><subs><emph>k</emph></subs>). We tentatively assume that each prior study used a logistic regression model to estimate <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0007" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo>̂</mo></mover><mfenced open="(" close=")"><mi>k</mi></mfenced></msub></mrow></math> </ephtml> , but it can be generalized to other models that assume a class of linear predictors. Our main objective is to propose an efficient prediction rule for a given new covariate vector <bold><emph>X</emph></bold> in the current dataset. In particular, we confine our interests to investigate an optimal form of new linear predictor defined by1 <ephtml> <math display="block" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0008" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>f</mi><mi mathvariant="bold">w</mi></msub><mfenced open="(" close=")"><mi mathvariant="bold">X</mi></mfenced><mo linebreak="goodbreak">=</mo><munderover><mo movablelimits="false">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></munderover><msub><mi>w</mi><mi>k</mi></msub><msub><mi mathvariant="bold">X</mi><mfenced open="(" close=")"><mi>k</mi></mfenced></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo>̂</mo></mover><mfenced open="(" close=")"><mi>k</mi></mfenced></msub><mo>,</mo></mrow></math> </ephtml> where <bold><emph>w</emph></bold> = (<emph>w</emph><subs>1</subs>, ⋯, <emph>w</emph><subs><emph>K</emph></subs>)<sups><emph>T</emph></sups> is a weight vector and <bold><emph>w</emph></bold> is of our interest. Therefore, the key task is to estimate the optimal weight to improve the prediction performance on the current dataset. In addition, we consider a practical situation for a user of our method: i.e., the user wants to predict unobserved <bold><emph>Y</emph></bold> using the (summary) information of <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0009" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mfenced open="{" close="}"><mfenced open="(" close=")" separators=","><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub><msub><mi mathvariant="bold">Y</mi><mi>k</mi></msub></mfenced></mfenced><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></msubsup></mrow></math> </ephtml> and <bold><emph>X</emph></bold>, which includes categorical covariates, and the number of covariates <emph>p</emph> is much greater than the number of studies <emph>K</emph>. For example, in the application section, we examine the <emph>p</emph> = 7 (including one categorical covariate) and <emph>K</emph> = 3 case. Accordingly, we add one more assumption to the current study:</p> <hd id="AN0162399693-11">2 Assumption</hd> <p>Let <bold><emph>∑</emph></bold><subs><emph>X</emph></subs> be the estimated covariance matrix of <bold><emph>X</emph></bold> in the current study. The estimation is unstable or impossible for several reasons including the small sample size of the current dataset, the high dimensional <emph>p</emph>‐covariate vector, or categorical variables in the covariate set (See Simulation and Application sections for examples). In contrast, one can (stably) calculate <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0010" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mover accent="true"><mi mathvariant="bold">B</mi><mo>̂</mo></mover><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mi mathvariant="bold">X</mi></msub><msub><mover accent="true"><mi mathvariant="bold">B</mi><mo>̂</mo></mover><mn>0</mn></msub></mrow></math> </ephtml> , where <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0011" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mi mathvariant="bold">B</mi><mo>̂</mo></mover><mn>0</mn></msub><mo>=</mo><mfenced open="[" close="]" separators=",,"><msubsup><mover accent="true"><mi mathvariant="bold">β</mi><mo>̂</mo></mover><mfenced open="(" close=")"><mn>1</mn></mfenced><mo>′</mo></msubsup><mi>...</mi><msubsup><mover accent="true"><mi mathvariant="bold">β</mi><mo>̂</mo></mover><mfenced open="(" close=")"><mi>K</mi></mfenced><mo>′</mo></msubsup></mfenced></mrow></math> </ephtml> and each <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0012" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mover accent="true"><mi mathvariant="bold">β</mi><mo>̂</mo></mover><mfenced open="(" close=")"><mi>k</mi></mfenced><mo>′</mo></msubsup></mrow></math> </ephtml> is the modified <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0013" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo>̂</mo></mover><mfenced open="(" close=")"><mi>k</mi></mfenced></msub></mrow></math> </ephtml> that is aligned to a <emph>p</emph>‐length vector by imputing 0 to the elements where the <emph>k</emph>th study does not report the coefficients.</p> <p>Note that <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0014" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mover accent="true"><mi mathvariant="bold">B</mi><mo>̂</mo></mover><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mi mathvariant="bold">X</mi></msub><msub><mover accent="true"><mi mathvariant="bold">B</mi><mo>̂</mo></mover><mn>0</mn></msub></mrow></math> </ephtml> , which is reduced to a <emph>K</emph> × <emph>K</emph> matrix and thus stably calculated compared with <bold><emph>∑</emph></bold><subs><emph>X</emph></subs>, will be used in Equation (<reflink idref="bib3" id="ref23">3</reflink>) in the following section. In practice, for example, we consider such a practical situation where the linear predictor <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0015" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">X</mi><msubsup><mover accent="true"><mi mathvariant="bold">β</mi><mo>̂</mo></mover><mfenced open="(" close=")"><mi>k</mi></mfenced><mo>′</mo></msubsup></mrow></math> </ephtml> can be regarded as continuous (thus, <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0016" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mover accent="true"><mi mathvariant="bold">B</mi><mo>̂</mo></mover><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mi mathvariant="bold">X</mi></msub><msub><mover accent="true"><mi mathvariant="bold">B</mi><mo>̂</mo></mover><mn>0</mn></msub></mrow></math> </ephtml> can be calculated), but <bold><emph>X</emph></bold> includes categorical variables (thus, <bold><emph>∑</emph></bold><subs><emph>X</emph></subs> cannot be calculated).</p> <hd id="AN0162399693-12">Maximum AUC</hd> <p>We briefly review a binary prediction theory under a normality assumption based on the work by Su and Liu (1993).[<reflink idref="bib14" id="ref24">14</reflink>] It is well‐known that the AUC for a predictor is maximized by the Fisher's linear discriminant function, which is equal to the log‐likelihood function up to a constant, under the normality assumption. Assume that the conditional random vector <bold><emph>X</emph></bold> given <emph>Y</emph> = <emph>y</emph> follows a <emph>p</emph>‐normal distribution <emph>N</emph><subs><emph>p</emph></subs>(<bold><emph>μ</emph></bold><subs><emph>y</emph></subs>, <bold><emph>∑</emph></bold>) for <emph>y</emph> = 0, 1. Define <bold><emph>δ</emph></bold> = <bold><emph>μ</emph></bold><subs>1</subs> − <bold><emph>μ</emph></bold><subs>0</subs>. Now, we consider a combination of linear predictors as2 <ephtml> <math display="block" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0017" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>g</mi><mi mathvariant="bold">w</mi></msub><mfenced open="(" close=")"><mi mathvariant="bold">X</mi></mfenced><mo linebreak="goodbreak">=</mo><munderover><mo movablelimits="false">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></munderover><msub><mi>w</mi><mi>k</mi></msub><msub><mi mathvariant="bold">Xβ</mi><mi>k</mi></msub><mo linebreak="goodbreak">=</mo><msub><mi mathvariant="bold">XB</mi><mn>0</mn></msub><mi mathvariant="bold">w</mi><mo>,</mo></mrow></math> </ephtml> where <bold><emph>B</emph></bold><subs>0</subs> = [<bold><emph>β</emph></bold><subs>1</subs>,⋯, <bold><emph>β</emph></bold><subs><emph>K</emph></subs>] and <bold><emph>β</emph></bold><subs><emph>k</emph></subs> are fixed <emph>p</emph>‐vectors and linearly independent. Then, <emph>g</emph><subs><emph>w</emph></subs>(<bold><emph>X</emph></bold>) is distributed as <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0018" xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced open="(" close=")" separators=","><msub><mi>μ</mi><mrow><mi>y</mi><mi mathvariant="bold">w</mi></mrow></msub><msubsup><mi>σ</mi><mi mathvariant="bold">w</mi><mn>2</mn></msubsup></mfenced></math> </ephtml> , where <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0019" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>μ</mi><mrow><mi>y</mi><mi mathvariant="bold">w</mi></mrow></msub><mo>=</mo><munderover><mo movablelimits="false">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></munderover><msub><mi>w</mi><mi>k</mi></msub><msubsup><mi mathvariant="bold">μ</mi><mi mathvariant="italic">yk</mi><mi>T</mi></msubsup><msub><mi mathvariant="bold">β</mi><mi>k</mi></msub></mrow></math> </ephtml> and <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0020" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>σ</mi><mi mathvariant="bold">w</mi><mn>2</mn></msubsup><mo>=</mo><munderover><mo movablelimits="false">∑</mo><mrow><mi mathvariant="normal">ℓ</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></munderover><munderover><mo movablelimits="false">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></munderover><msub><mi>w</mi><mi>k</mi></msub><msub><mi>w</mi><mi mathvariant="normal">ℓ</mi></msub><msubsup><mi mathvariant="bold">β</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑β</mi><mi mathvariant="normal">ℓ</mi></msub></mrow></math> </ephtml> . Note that, in the following section, when we substitute <bold><emph>β</emph></bold><subs><emph>k</emph></subs> with <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0021" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><msub><mi mathvariant="bold">β</mi><mi>k</mi></msub><mo>̂</mo></mover></mrow></math> </ephtml> , <emph>p</emph><subs><emph>k</emph></subs>‐subvector <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0022" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><msub><mi mathvariant="bold">β</mi><mi>k</mi></msub><mo>̂</mo></mover></mrow></math> </ephtml> is imputed with 0 to align to the length of <emph>p</emph>.</p> <p>We propose the following estimator of the optimal weight:3 <ephtml> <math display="block" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0023" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mi mathvariant="bold">w</mi><mo>̂</mo></mover><mi>opt</mi></msub><mo linebreak="goodbreak">=</mo><mover accent="true"><mi mathvariant="bold">δ</mi><mo>̂</mo></mover><msub><mover accent="true"><mi mathvariant="bold">B</mi><mo>̂</mo></mover><mn>0</mn></msub><msup><mfenced open="(" close=")"><mrow><msubsup><mover accent="true"><mi mathvariant="bold">B</mi><mo>̂</mo></mover><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mi mathvariant="bold">X</mi></msub><msub><mover accent="true"><mi mathvariant="bold">B</mi><mo>̂</mo></mover><mn>0</mn></msub></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>,</mo></mrow></math> </ephtml> where <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0024" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mi mathvariant="bold">δ</mi><mo>̂</mo></mover></mrow></math> </ephtml> is estimated by the standard meta‐analytical technique (such as a fixed‐effect model for mean difference) by using <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0025" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mfenced open="{" close="}" separators=","><msub><mover accent="true"><mi mathvariant="bold">μ</mi><mo>̂</mo></mover><mrow><mi>k</mi><mn>0</mn></mrow></msub><msub><mover accent="true"><mi mathvariant="bold">μ</mi><mo>̂</mo></mover><mrow><mi>k</mi><mn>1</mn></mrow></msub></mfenced><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></msubsup></mrow></math> </ephtml> and <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0026" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mfenced open="{" close="}"><msubsup><mover accent="true"><mi>σ</mi><mo>̂</mo></mover><mi mathvariant="italic">kj</mi><mn>2</mn></msubsup></mfenced><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>p</mi><mi>k</mi></msub></msubsup></mrow></math> </ephtml> . Detailed derivation and interesting examples are given in Appendix.</p> <p>In addition, it is known that, under the conditional normality assumption, the AUC for all predictors is maximized by <emph>g</emph><subs>opt</subs>(<bold><emph>X</emph></bold>) = <bold><emph>Xβ</emph></bold><subs>opt</subs>, where <bold><emph>β</emph></bold><subs>opt</subs> = <bold><emph>∑</emph></bold><sups>−1</sups><bold><emph>δ</emph></bold><sups><emph>T</emph></sups>. In accordance with this,4 <ephtml> <math display="block" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0027" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>AUC</mi><mfenced open="(" close=")"><msub><mi>g</mi><msub><mi mathvariant="bold">w</mi><mi>opt</mi></msub></msub></mfenced><mo>≤</mo><mi>AUC</mi><mfenced open="(" close=")"><msub><mi>g</mi><mi>opt</mi></msub></mfenced><mo>.</mo></mrow></math> </ephtml></p> <p>Since, based on the assumption 2, <bold><emph>∑</emph></bold> is not supposed to be available, <bold><emph>β</emph></bold><subs>opt</subs> is not estimable. Thus, we focus on the situation where the equality in (<reflink idref="bib4" id="ref25">4</reflink>) holds. Even under assumption 2, the following proposition 1 is obtained with the proof in the Appendix:</p> <hd id="AN0162399693-13">1 Proposition</hd> <p>Let <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0028" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>L</mi><mfenced open="(" close=")"><msub><mi mathvariant="bold">B</mi><mn>0</mn></msub></mfenced></mrow></math> </ephtml> be the linear hull of column vectors of <bold><emph>B</emph></bold><subs>0</subs>. Then, <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0029" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>AUC</mi><mfenced open="(" close=")"><msub><mi>g</mi><msub><mi mathvariant="bold">w</mi><mi>opt</mi></msub></msub></mfenced><mo>=</mo><mi>AUC</mi><mfenced open="(" close=")"><msub><mi>g</mi><mi>opt</mi></msub></mfenced></mrow></math> </ephtml> if and only if the difference <bold><emph>δ</emph></bold> between label conditional mean vectors is in <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0030" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>L</mi><mfenced open="(" close=")"><msub><mi mathvariant="bold">B</mi><mn>0</mn></msub></mfenced></mrow></math> </ephtml> .</p> <hd id="AN0162399693-14">SIMULATION EXPERIMENTS</hd> <p>We numerically examine the prediction performance of our method to maximize AUC in one available dataset, comparing it to conventional methods. In the simulations, we fix the following parameters: a <emph>p</emph>‐mean vector <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0031" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold">μ</mi><mn>0</mn></msub><mo>=</mo><mfenced open="(" close=")" separators=",,"><mn>0</mn><mi>...</mi><mn>0</mn></mfenced></mrow></math> </ephtml> , which corresponds to <bold><emph>δ</emph></bold> = <bold><emph>μ</emph></bold><subs>1</subs>, a covariance matrix <bold><emph>∑</emph></bold> = <bold><emph>V</emph></bold><sups>1/2</sups><bold><emph>RV</emph></bold><sups>1/2</sups>, where <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0032" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">V</mi><mo>=</mo><mi>diag</mi><mfenced open="(" close=")"><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>...</mi><mo>,</mo><mi>p</mi></mrow></mfenced></mrow></math> </ephtml> , <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0033" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>diag</mi><mfenced open="(" close=")" separators=",,"><msub><mi>a</mi><mn>1</mn></msub><msub><mi>a</mi><mn>2</mn></msub><mi>...</mi></mfenced></mrow></math> </ephtml> is a diagonal matrix with <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0034" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><mi>...</mi></mrow></math> </ephtml> as the diagonal elements, <bold><emph>R</emph></bold> is a <emph>p</emph> × <emph>p</emph>‐correlation matrix as <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0035" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="(" close=")">R1rR2rR3</mfenced></mrow></math> </ephtml> , and the sub‐correlation matrix <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0036" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>R</mi><mi>i</mi></msub><mo>=</mo><mfenced open="(" close=")">1ri⋱ri1</mfenced></mrow></math> </ephtml> for <emph>i</emph> = 1, 2, 3 is defined in the following scenarios. The varying parameters are <bold><emph>μ</emph></bold><subs>1</subs>, which is described below, <emph>K</emph> ∈ {2, 3, 6}, <emph>p</emph> ∈ {10, 20}, and <emph>r</emph> ∈ {−0.3, −0.1, 0.1, 0.3, 0.6} in <bold><emph>R</emph></bold>. Based on these parameter settings, the individual datasets for each prior study and one current dataset, which will be used for checking the prediction performance, are simulated as <bold><emph>X</emph></bold><subs><emph>i</emph></subs>|<emph>Y</emph> = 1 ∼ <emph>N</emph>(<bold><emph>μ</emph></bold><subs>1</subs>, <bold><emph>∑</emph></bold>) and <bold><emph>X</emph></bold><subs><emph>i</emph></subs>|<emph>Y</emph> = 0 ∼ <emph>N</emph>(<bold><emph>μ</emph></bold><subs>0</subs>, <bold><emph>∑</emph></bold>). The number of observations is set to (<emph>n</emph><subs>0</subs>, <emph>n</emph><subs>1</subs>) = (<reflink idref="bib500" id="ref26">500</reflink>, 500) for the prior and current datasets, respectively. Based on these settings, we generate IPD for each study. The IPD of <bold><emph>X</emph></bold> in the current study is fully stored while the IPD of <bold><emph>Y</emph></bold> in the current study is masked and used only when evaluating the performance. Finally, each simulated IPD for the prior studies is aggregated into summary statistics after fitting the logistic regressions, and then the statistics are stored. Comparison methods are as follows: (M1) use the mean of the available regression coefficients; (M2) use the reported regression coefficients from each study; and (M3) use the reported regression coefficients from the study with the best reported AUC. Monte Carlo simulations are iterated 1000 times.</p> <p>Given the settings above, we simulate the following six scenario settings, S1–S6. Table 2 shows the set of covariates in each study.</p> <p></p> <ulist> <item> <bold> S1: _I_p_i_  = 10, _HT_ <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0037" xmlns="<ulink href="http://www.w3.org/1998/Math/MathML">http://www.w3.org/1998/Math/MathML</ulink>"><mrow><msub><mi mathvariant="bold">μ</mi><mn>1</mn></msub><mo>=</mo><mfenced open="(" close=")" separators=","><munder><munder accentunder="true"><mrow><mn>1</mn><mo>,</mo><mi>...</mi><mo>,</mo><mn>1</mn></mrow><mo>⏟</mo></munder><mrow><mn>5</mn><mspace width="0.25em" /><mtext>times</mtext></mrow></munder><munder><munder accentunder="true"><mrow><mn>0.5</mn><mo>,</mo><mi>...</mi><mo>,</mo><mn>0.5</mn></mrow><mo>⏟</mo></munder><mrow><mn>5</mn><mspace width="0.25em" /><mtext>times</mtext></mrow></munder></mfenced></mrow></math> _ht_ , _B__I_R_i_</bold> <subs>1</subs> and <bold><emph>R</emph></bold><subs>2</subs> are 3 × 3 sub‐correlation matrices with <emph>r</emph><subs>1</subs>  = <emph>r</emph><subs>2</subs>  = 0.5, respectively, and <bold><emph>R</emph></bold><subs>3</subs> is 4 × 4 sub‐correlation matrix with <emph>r</emph><subs>3</subs>  = 0.5. In addition, there are two studies (<emph>K</emph>  = 2) with no shared covariate set, and the 4th and 9th variables are dichotomized with a cut‐off value of 1.</item> <p></p> <item> S2: The same parameter settings as S1, except that the 4, 5, 9, 10th variables are dichotomized with a cut‐off value of 1. Note that in S1 and S2, M1 corresponds to Example 1.</item> <p></p> <item> S3: The same parameter settings as S1, except that there are three studies (<emph>K</emph>  = 3) with three shared covariates (i.e., 1‐3rd variables); the 1st variable is dichotomized with a cut‐off value of 1 and <emph>r</emph><subs>1</subs>  = <emph>r</emph><subs>2</subs>  = <emph>r</emph><subs>3</subs>  = 0.8.</item> <p></p> <item> S4: The same parameter settings as S3, except that the 1st and 9th variables are dichotomized with a cut‐off value of 1.</item> <p></p> <item> <bold> S5: _I_p_i_  = 20, _HT_ <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0038" xmlns="<ulink href="http://www.w3.org/1998/Math/MathML">http://www.w3.org/1998/Math/MathML</ulink>"><mrow><msub><mi mathvariant="bold">μ</mi><mn>1</mn></msub><mo>=</mo><mfenced open="(" close=")" separators=",,"><munder><munder accentunder="true"><mrow><mn>2</mn><mo>,</mo><mi>...</mi><mo>,</mo><mn>2</mn></mrow><mo>⏟</mo></munder><mrow><mn>7</mn><mspace width="0.25em" /><mtext>times</mtext></mrow></munder><munder><munder accentunder="true"><mrow><mn>1</mn><mo>,</mo><mi>...</mi><mo>,</mo><mn>1</mn></mrow><mo>⏟</mo></munder><mrow><mn>7</mn><mspace width="0.25em" /><mtext>times</mtext></mrow></munder><munder><munder accentunder="true"><mrow><mn>0.5</mn><mo>,</mo><mi>...</mi><mo>,</mo><mn>0.5</mn></mrow><mo>⏟</mo></munder><mrow><mn>6</mn><mspace width="0.25em" /><mtext>times</mtext></mrow></munder></mfenced></mrow></math> _ht_ , _B__I_R_i_</bold> <subs>1</subs> and <bold><emph>R</emph></bold><subs>2</subs> are 7 × 7 sub‐correlation matrices with <emph>r</emph><subs>1</subs>  = <emph>r</emph><subs>2</subs>  = 0.8, respectively, and <bold><emph>R</emph></bold><subs>3</subs> is 6 × 6 sub‐correlation matrix with <emph>r</emph><subs>3</subs>  = 0.8. In addition, there are six studies with five shared covariates (i.e., 1‐5th variables), and the 1st variable is dichotomized with a cut‐off value of 1.</item> <p></p> <item> S6: The same parameter settings as S5, except that the 1, 9, 19th variables are dichotomized with a cut‐off value of 1.</item> <item>et of variables in six scenarios</item> </ulist> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left" /><th align="left" /><th align="left">Variables</th></tr><tr><th align="left">Scenario</th><th align="left">Study</th><th align="left">1</th><th align="left">2</th><th align="left">3</th><th align="left">4</th><th align="left">5</th><th align="left">6</th><th align="left">7</th><th align="left">8</th><th align="left">9</th><th align="left">10</th><th align="left">11</th><th align="left">12</th><th align="left">13</th><th align="left">14</th><th align="left">15</th><th align="left">16</th><th align="left">17</th><th align="left">18</th><th align="left">19</th><th align="left">20</th></tr></thead><tbody valign="top"><tr><td>S1 and S2</td><td>1</td><td>c</td><td>c</td><td>c</td><td>b</td><td>c/b</td><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /></tr><tr><td /><td>2</td><td /><td /><td /><td /><td /><td>c</td><td>c</td><td>c</td><td>b</td><td>c/b</td><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /></tr><tr><td>S3 and S4</td><td>1</td><td>b</td><td>c</td><td>c</td><td>c</td><td>c</td><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /></tr><tr><td /><td>2</td><td>b</td><td>c</td><td>c</td><td /><td /><td>c</td><td>c</td><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /></tr><tr><td /><td>3</td><td>b</td><td>c</td><td>c</td><td /><td /><td /><td /><td>c</td><td>c/b</td><td>c</td><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /></tr><tr><td>S5 and S6</td><td>1</td><td>b</td><td>c</td><td>c</td><td>c</td><td>c</td><td>c</td><td>c</td><td>c</td><td>c/b</td><td>c</td><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /></tr><tr><td /><td>2</td><td>b</td><td>c</td><td>c</td><td>c</td><td>c</td><td /><td>c</td><td>c</td><td>c/b</td><td>c</td><td>c</td><td /><td /><td /><td /><td /><td /><td /><td /><td /></tr><tr><td /><td>3</td><td>b</td><td>c</td><td>c</td><td>c</td><td>c</td><td /><td /><td /><td>c/b</td><td>c</td><td>c</td><td>c</td><td>c</td><td /><td /><td /><td /><td /><td /><td /></tr><tr><td /><td>4</td><td>b</td><td>c</td><td>c</td><td>c</td><td>c</td><td /><td /><td /><td /><td /><td>c</td><td>c</td><td>c</td><td>c</td><td>c</td><td /><td /><td /><td /><td /></tr><tr><td /><td>5</td><td>b</td><td>c</td><td>c</td><td>c</td><td>c</td><td /><td /><td /><td /><td /><td /><td /><td>c</td><td>c</td><td>c</td><td>c</td><td>c</td><td /><td /><td /></tr><tr><td /><td>6</td><td>b</td><td>c</td><td>c</td><td>c</td><td>c</td><td /><td /><td /><td /><td /><td /><td /><td /><td /><td>c</td><td>c</td><td>c</td><td>c</td><td>c/b</td><td>c</td></tr></tbody></table> </ephtml> </p> <p>2 Abbreviations: b, binary; c, continuous; c/b, depends on the scenarios (See Scenario details).</p> <p>Figure 1 shows the simulation results of our method, compared with (M1) mean, (M2) individual fitting and (M3) the best reported AUC methods. Our method performs better than M1, M2 and M3 in most settings: our method improves the AUC by 5.5% (max and min improvement: 12.7% and 1.1%), 8.1% (max and min improvement: 21.1% and 1.4%) and 4.5% (max and min improvement: 20.9% and 0.1%) on average across all settings, compared with M1, M2 and M3, respectively. Especially when the number of covariates increases, our method outperforms M1 even more (2.7% and 12.7% improvements of AUC on average in S3–S4 and S5–S6, respectively). Additionally, in S1 and S2, the AUC of Study 2 is poor while that of Study 1 is good. It implies that, under the M2 assumption, if researchers could "luckily" select Study 1 for fitting to the current data since there is no <emph>Y</emph> in the current dataset and thus it is impossible to know in advance which study to use for the higher AUC, they can obtain comparable AUC with ours. A similar discussion can be applied to S3–S6. However, we emphasize that since we cannot know in advance which study's results we should apply to obtain good AUC on the current data, the best result among M2 is not always achieved in practice. Regarding the correlations between the covariates, we observe that the higher the correlation, the greater the improvement of AUC, compared with M1, especially in S1–S4.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/BDCT/01mar23/jrsm1612-fig-0001.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jrsm1612-fig-0001.jpg" title="1 Simulation results of prediction performance based on AUC. our: our method using ŵopt in Equation (A1); mean: take an average of the available regression coefficients (M1); Studyi: separately use the regression result of the ith study (M2); Best prior AUC: use the reported regression coefficients from the study with the best reported AUC (M3) [Colour figure can be viewed at wileyonlinelibrary.com]" /> </p> <p></p> <hd id="AN0162399693-16">APPLICATION REVISITED: REAL WORLD DATA ANALYSIS</hd> <p>In this section, we apply the proposed method to a series of epidemiological studies that share prediction tasks for the occurrence of leukocytosis as explained in Section 2. Now we have four studies conducted by the WHO/ARI in Ethiopia, Gambia, Papua New Guinea, and the Philippines. The data from Ethiopia, Gambia, and Papua New Guinea are used as the datasets of prior studies in the context of conventional meta‐analysis and the IPD from the Philippines is used as the current dataset for calculating AUC. Table 3 shows the set of available covariates and the estimated regression coefficients. The sample size of each IPD is 430, 453, 659, and 586 for Ethiopia, Gambia, Papua New Guinea, and the Philippines, respectively.</p> <p>3 TABLE Estimated regression coefficients and AUC in the current study (Philippines) in WHO/ARI study</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left" /><th align="left">Results</th></tr><tr><th align="left">Covariates</th><th align="left">Our 1</th><th align="left">Our 2</th><th align="left">Our 3</th><th align="left">Our 4</th></tr></thead><tbody valign="top"><tr><td>Intercept</td><td>−3.34</td><td>−3.09</td><td>−1.87</td><td>−2.51</td></tr><tr><td>waz</td><td>−0.11</td><td>−0.08</td><td>−0.04</td><td>−0.07</td></tr><tr><td>rr</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.00</td></tr><tr><td>age</td><td>0.00</td><td>0.01</td><td>0.01</td><td>0.01</td></tr><tr><td>temp</td><td>0.05</td><td>0.03</td><td>0.02</td><td>0.02</td></tr><tr><td>hrat</td><td>0.00</td><td>0.01</td><td>0.01</td><td>0.01</td></tr><tr><td>nxray</td><td>0.04</td><td /><td>0.04</td><td>−0.03</td></tr><tr><td>pneu</td><td>−0.29</td><td>−0.27</td><td /><td>−0.32</td></tr><tr><td>AUC on Philippines</td><td>0.542 (0.495–0.590)</td><td>0.561 (0.514–0.609)</td><td>0.552 (0.505–0.600)</td><td>0.563 (0.516–0.611)</td></tr></tbody></table> </ephtml> </p> <ulist> <item>3 <emph>Note</emph>: Our 1: combination of Ethiopia and Gambia; Our 2: combination of Ethiopia and Papua New Guinea; Our 3: combination of Gambia and Papua New Guinea; Our 4: combination of Ethiopia, Gambia and Papua New Guinea.</item> <item>4 Abbreviations: age, age in days; hrat, heart‐rate oximeter; nxray, total number of x‐ray readings; pneu, pneumonia; rr, respiratory rate; temp, temperature; was, weight‐for‐age z‐score.</item> </ulist> <p>A new prediction model is then constructed based on the estimated weights defined in Equation (<reflink idref="bib3" id="ref27">3</reflink>), and the performance measured by AUC is compared with that of M2 (individual fitting) and M3 (best reported AUC) defined in Section 4. The 95% confidence intervals (CIs) of AUC are calculated based on DeLong's method with 2000 bootstrap iterations.[<reflink idref="bib23" id="ref28">23</reflink>] Finally, to check the sensitivity of the results, we prepare four sets of combinations of datasets from prior studies: (Our 1) a combination of Ethiopia and Gambia, (Our 2) a combination of Ethiopia and Papua New Guinea, (Our 3) a combination of Gambia and Papua New Guinea, and (Our 4) a combination of Ethiopia, Gambia, and Papua New Guinea.</p> <p>Figure 2 shows the kernel density plots with the Gaussian kernel to check whether assumption 1 is plausible. Table 3 shows the results of our methods (stratified by the combination of datasets for prior studies). The results demonstrate that while the 95% CIs of AUCs are overlapped each other, which implicates our results may have limited clinical significance, our method provides higher AUC in the sense of the point estimation, especially in Our 4: the AUC of our method ranges from 0.542 (95% CI: 0.495–0.590) (Our 1) to 0.563 (0.516–0.611) (Our 4), the AUC of M2 is 0.542 (average), and AUC of M2 in Ethiopia (corresponds to M3), Gambia and Papua New Guinea are 0.522 (0.475–0.570), 0.545 (0.497–0.593) and 0.559 (0.511–0.607), respectively. For reader's reference, we report that the AUC that is calculated based on the model fitted by using the available data (Philippines) is 0.581 (0.534–0.628). The estimated weights are shown in Table 4. It is interesting to note that <emph>w</emph><subs><emph>Gambia</emph></subs> in Our 4 is a negative value, implying that the regression results in Gambia should be used after flipping the sign of the linear predictor in order to improve the AUC in the current dataset. It is notable that, in Our 1 and 3, the AUCs of our method are relatively poorer than the AUCs of M2 (i.e., Ethiopia and Gambia for Our 1, and Gambia and Papua New Guinea for Our 3), while the AUCs of our method are better than the average AUC of M2: the average AUC of M2 is 0.534 and 0.552 for Our 1 and 3, respectively. However, again, we emphasize that since we cannot know in advance which study's results we should apply to obtain good AUC on the current dataset, the best result among M2 is not always achieved in practice.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/BDCT/01mar23/jrsm1612-fig-0002.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="jrsm1612-fig-0002.jpg" title="2 Kernel density plots of each (continuous) covariates, stratified by country [Colour figure can be viewed at wileyonlinelibrary.com]" /> </p> <p></p> <p>4 TABLE Estimated weights in WHO/ARI study</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left" /><th align="left"><italic>w</italic><sub><italic>Ethiopia</italic></sub></th><th align="left"><italic>w</italic><sub><italic>Gambia</italic></sub></th><th align="left"><italic>w</italic><sub><italic>Papua</italic></sub></th></tr></thead><tbody valign="top"><tr><td>Our 1: Ethiopia and Gambia</td><td>0.79</td><td>0.50</td><td /></tr><tr><td>Our 2: Ethiopia and Papua New Guinea</td><td>0.73</td><td /><td>0.95</td></tr><tr><td>Our 3: Gambia and Papua New Guinea</td><td /><td>0.55</td><td>0.28</td></tr><tr><td>Our 4: Ethiopia, Gambia and Papua New Guinea</td><td>0.87</td><td>−0.39</td><td>1.05</td></tr></tbody></table> </ephtml> </p> <hd id="AN0162399693-18">DISCUSSION</hd> <p>Along with increasing attention being paid to prediction models, there is a higher demand for meta‐analytical approaches to synthesize regression results. However, these methodologies have not been well developed due to many difficulties caused by the different study settings, and further research is still needed,[[<reflink idref="bib6" id="ref29">6</reflink>], [<reflink idref="bib10" id="ref30">10</reflink>]] in particular compared well‐established conventional meta‐analysis methods including the synthesis of mean differences, correlations, and so on.[<reflink idref="bib24" id="ref31">24</reflink>] This study demonstrated a new method to combine regression coefficients to improve the prediction accuracy measured by AUC on a new dataset in which the outcome of interest is unobserved. Based on the proposed weights for combining linear predictors, we showed interesting and counterintuitive examples in Appendix in which some of the weights could take negative values, suggesting that flipping the sign of the prediction results reported in each individual study would improve the overall prediction performance. From a different perspective, when we consider that the proposed weight would measure the importance of each prior study for the prediction performance in the available data, the negative weight implies that the results from the study with the negative weight is negatively important. However, it should be noted that we need to carefully examine the validity of the negative importance. In addition, we must be careful that developing a new model increases the burden of new research in the prediction modeling literature: i.e., the new model must be carefully examined and the other performance metrics such as calibration must be also carefully considered. We would like to emphasize that it is also important to validate existing models with new datasets and compare their performance head‐to‐head to identify the best performing existing models.</p> <p>The prediction performance was examined by numerical and real‐world data, and our method is shown to outperform conventional methods including the methods that (M1) take an average of the available regression coefficients, (M2) fit the regression coefficients of each study separately to the available data or (M3) use the reported regression coefficients from the study with the best reported AUC. We emphasize that our method does not aim to improve the estimation accuracy of the post‐integration coefficients. Therefore, while it is shown to be useful for improving prediction performance, it requires further examination to clarify whether it can be used in cases where we are interested in combining the coefficient estimates that would discover useful risk factors in the context of conventional observational studies, such as case–control/cross‐sectional studies. One practical feature of our new method is that we can naturally accept the different sets of covariates among the prior studies as explained in the simulation and application section. Such a discrepancy is typical in large‐scale multicenter cohort studies.[<reflink idref="bib10" id="ref32">10</reflink>] Another important aspect is the calibration performance of the model in addition to the discriminant performance measured by AUC. Since our method uses only the reported linear predictors from the prior studies and provides one synthesized linear predictor, any (link) function that maps from the linear predictor to individual risk can be used for both the individual risk prediction and its calibration. In other words, the user is free to determine the functional form to estimate the individual risk. The choice of this functional form to predict the individual risk should be based on previous studies and the guidelines for prediction modeling studies such as TRIPOD (Transparent Reporting of a multivariable prediction model for Individual Prognosis Or Diagnosis).[[<reflink idref="bib25" id="ref33">25</reflink>], [<reflink idref="bib27" id="ref34">27</reflink>]]</p> <p>Our method makes the following assumptions: (A1) the joint distribution of covariates are similar among all included studies, and (A2) the covariance matrix is not estimable. The first assumption is required to derive the weights in Equation (<reflink idref="bib3" id="ref35">3</reflink>), and the second assumption is required in Equation (<reflink idref="bib5" id="ref36">5</reflink>). Although the similar assumption of A1 has frequently been used in previous studies,[[<reflink idref="bib10" id="ref37">10</reflink>], [<reflink idref="bib22" id="ref38">22</reflink>]] it may sometimes be implausible in practice.[[<reflink idref="bib28" id="ref39">28</reflink>]] In conjunction with that, it is possible that the regression parameters underlying the prior study are highly heterogeneous across other prior studies. One simple idea is to run some sensitivity analysis by removing several prior studies and then to examine the robustness of the result. Another possibility for dealing with the heterogeneity of the distribution of <bold><emph>X</emph></bold> would be to extend <emph>w</emph><subs><emph>k</emph></subs> to the more flexible weight that depends on the discrepancy of the mean and standard deviation vectors of covariates between the current and prior observation. We need further research about how to adjust for the discrepancy, and it is our ongoing research aim to develop a statistical method to check the validity of the assumption of A1 by applying the technique of sensitivity analysis to the parameters for the distribution. Furthermore, our real‐world data analysis shows that our method is still likely to work well even if the assumption of A1 is violated to some extent. The second assumption is relatively plausible in such a practical situation where <bold><emph>X</emph></bold><subs>(<emph>k</emph>)</subs> contains the categorical covariates, and thus making it difficult to calculate the covariance matrix, but the linear predictor <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:17592879:media:jrsm1612:jrsm1612-math-0040" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold">X</mi><mfenced open="(" close=")"><mi>k</mi></mfenced></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo>̂</mo></mover><mfenced open="(" close=")"><mi>k</mi></mfenced></msub></mrow></math> </ephtml> can be regarded as continuous, and thus it is possible to calculate the variance.</p> <p>As a limitation of this study, our method was examined in only one practical dataset, while conducting comprehensive numerical simulations to check the prediction performance. However, the data includes four different countries with potential heterogeneity among sample populations, which implies that our method would work preferably even when the assumption of A1 is not plausible in practice. In addition, the results showed that the 95% CIs of AUCs were overlapped each other. It is quite difficult to define what would be a "clinically important" difference in AUC, which highly depends on clinical conditions. In these cases, careful consultation with clinical experts and the use of additional measures such as net‐benefit should be considered to allow us to judge the clinical significance. From these perspectives, we welcome the re‐evaluation of our method in other practical cases. Another potential limitation is that although it works in conjunction with the limitation due to the assumption of A1, our method incorporates the mean, standard deviation and regression coefficient information of prior studies, but ignores other potentially informative statistics such as prediction performance measures. It would also be possible to incorporate other prior information by extending the weights or by adding a shrinkage factor for the weights to improve the external validity of our method, and it is our ongoing research aim to investigate this. The last limitation is that we assume the typical situation where the set of covariates included in the prior studies is smaller than that in the current study. However, in practice, the opposite might be possible (i.e., the set of covariates included in the prior studies is larger than that in the current study.) Our method tries to restore the missing correlation information in the prior studies by using the current data. Therefore, it would be difficult to apply our method in such a case.</p> <hd id="AN0162399693-19">AUTHOR CONTRIBUTIONS</hd> <p>D.Y. led the study. All the authors conceived and designed the study and analysed the data. All authors took responsibility for the integrity of the data and the accuracy of the data analysis. All the authors made critical revisions to the manuscript for important intellectual content and gave final approval of the manuscript.</p> <hd id="AN0162399693-20">FUNDING INFORMATION</hd> <p>This research was partially supported by research grants from JST, PRESTO Japan (JPMJPR21RC), and the Ministry of Education, Culture, Sports, Science and Technology of Japan (21K17292).</p> <hd id="AN0162399693-21">CONFLICT OF INTEREST</hd> <p>The author declares there is no potential conflict of interest.</p> <hd id="AN0162399693-22">DATA AVAILABILITY STATEMENT</hd> <p>The data that support the findings of this study are openly available in https://hbiostat.org/data/ (Dataset name: WHO ARI Multicentre Study of clinical signs and etiologic agents). The R program to replicate the result is available in https://github.com/kingqwert/R/blob/master/Combine_LP/combine_lp_case1.R.</p> <hd id="AN0162399693-23">APPENDIX</hd> <p></p> <hd id="AN0162399693-24">Derivation of optimal weights, the equation. The AUC of g w (X) is given by</hd> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>AUC</mi><mfenced open="(" close=")"><msub><mi>g</mi><mi mathvariant="bold">w</mi></msub></mfenced><mo linebreak="goodbreak">=</mo><mi mathvariant="normal">Φ</mi><mfenced open="(" close=")"><mrow><msub><mi mathvariant="bold">δB</mi><mn>0</mn></msub><mi mathvariant="bold">w</mi></mrow><msqrt><mrow><msup><mi mathvariant="bold">w</mi><mi>T</mi></msup><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub><mi mathvariant="bold">w</mi></mrow></msqrt></mfenced><mo>.</mo></mrow></math> </ephtml> </p> <p>The Cauchy‐Schwarz inequality suggests</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable groupalign="{right left}" displaystyle="true"><mtr><mtd><msup><mfenced open="(" close=")"><mrow><msub><mi mathvariant="bold">δB</mi><mn>0</mn></msub><mi mathvariant="bold">w</mi></mrow></mfenced><mn>2</mn></msup><mo linebreak="goodbreak">=</mo><msup><mfenced open="[" close="]"><mrow><msup><mfenced open="{" close="}"><mrow><msup><mfenced open="{" close="}"><msup><mfenced open="(" close=")"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mrow></mfenced><mrow><mo>−</mo><mn>1</mn><mn>2</mn></mrow></msup></mfenced><mi>T</mi></msup><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msup><mi mathvariant="bold">δ</mi><mi>T</mi></msup></mrow></mfenced><mi>T</mi></msup><msup><mfenced open="(" close=")"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mrow></mfenced><mn>1</mn><mn>2</mn></msup><mi mathvariant="bold">w</mi></mrow></mfenced><mn>2</mn></msup></mtd></mtr><mtr><mtd><mo>≤</mo><mfenced open="{" close="}"><mrow><msup><mfenced open="(" close=")"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msup><mi mathvariant="bold">δ</mi><mi>T</mi></msup></mrow></mfenced><mi>T</mi></msup><msup><mfenced open="(" close=")"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msup><mi mathvariant="bold">δ</mi><mi>T</mi></msup></mrow></mfenced><mfenced open="{" close="}"><mrow><msup><mi mathvariant="bold">w</mi><mi>T</mi></msup><mfenced open="(" close=")"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mrow></mfenced><mi mathvariant="bold">w</mi></mrow></mfenced></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>or equivalently</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mfenced open="(" close=")"><mrow><msub><mi mathvariant="bold">δB</mi><mn>0</mn></msub><mi mathvariant="bold">w</mi></mrow></mfenced><mn>2</mn></msup><mrow><msup><mi mathvariant="bold">w</mi><mi>T</mi></msup><mfenced open="(" close=")"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mrow></mfenced><mi mathvariant="bold">w</mi></mrow><mo>≤</mo><msup><mfenced open="(" close=")"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msup><mi mathvariant="bold">δ</mi><mi>T</mi></msup></mrow></mfenced><mi>T</mi></msup><msup><mfenced open="(" close=")"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msup><mi mathvariant="bold">δ</mi><mi>T</mi></msup></mrow></math> </ephtml> </p> <p>with equality if and only if <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">w</mi><mo>∝</mo><msub><mi mathvariant="bold">δB</mi><mn>0</mn></msub><msup><mfenced open="(" close=")"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> </ephtml> . Therefore, the maximum of AUC for <emph>g</emph><subs><bold><emph>w</emph></bold></subs>(<bold><emph>X</emph></bold>) is the standard normal quantile of the length of the projection of <bold><emph>δ</emph></bold> onto the linear subspace hulled by <bold><emph>β</emph></bold><subs><emph>k</emph></subs>'s given as</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Φ</mi><mfenced open="(" close=")"><msqrt><mrow><msub><mi mathvariant="bold">δB</mi><mn>0</mn></msub><msup><mfenced open="(" close=")"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msup><mi mathvariant="bold">δ</mi><mi>T</mi></msup></mrow></msqrt></mfenced><mo>,</mo></mrow></math> </ephtml> </p> <p>which is attained at</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold">w</mi><mi>opt</mi></msub><mo linebreak="goodbreak">=</mo><msub><mi mathvariant="bold">δB</mi><mn>0</mn></msub><msup><mfenced open="(" close=")"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>.</mo></mrow></math> </ephtml> </p> <p>We propose the following estimator of the optimal weight for Equation (<reflink idref="bib1" id="ref40">1</reflink>):</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mi mathvariant="bold">w</mi><mo stretchy="true">̂</mo></mover><mi>opt</mi></msub><mo linebreak="goodbreak">=</mo><mover accent="true"><mi mathvariant="bold">δ</mi><mo stretchy="true">̂</mo></mover><msub><mover accent="true"><mi mathvariant="bold">B</mi><mo stretchy="true">̂</mo></mover><mn>0</mn></msub><msup><mfenced open="(" close=")"><mrow><msubsup><mover accent="true"><mi mathvariant="bold">B</mi><mo stretchy="true">̂</mo></mover><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mi mathvariant="bold">X</mi></msub><msub><mover accent="true"><mi mathvariant="bold">B</mi><mo stretchy="true">̂</mo></mover><mn>0</mn></msub></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>,</mo></mrow></math> </ephtml> </p> <p>It is worth noting that</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><mi>max</mi><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>K</mi></mrow></munder><mi>AUC</mi><mfenced open="(" close=")"><msubsup><mi mathvariant="bold">Xβ</mi><mi>k</mi><mi>T</mi></msubsup></mfenced><mo>≤</mo><mi>AUC</mi><mfenced open="(" close=")"><msub><mi>g</mi><msub><mi mathvariant="bold">w</mi><mi>opt</mi></msub></msub></mfenced><mo>,</mo></mrow></math> </ephtml> </p> <p>which suggests that the AUC in the current dataset using each study's coefficients is less than that using the optimal weights.</p> <p>Proof of proposition 1Proof. Let <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup></mrow></math> </ephtml> be a <emph>p</emph> × (<emph>p</emph> − <emph>K</emph>) matrix such that</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msubsup><mi mathvariant="bold">∑B</mi><mn>0</mn><mo>⊥</mo></msubsup><mo linebreak="goodbreak">=</mo><mi mathvariant="bold">O</mi><mo>,</mo></mrow></math> </ephtml> </p> <p>where <bold>O</bold> is a zero matrix of size <emph>K</emph> × (<emph>p</emph> − <emph>K</emph>). We note that all the column vectors of <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup></mrow></math> </ephtml> is in the orthogonal complement space of the linear hull of the column vectors of <bold><emph>B</emph></bold><subs>0</subs>, so that the matrix <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="[" close="]" separators=","><msub><mi mathvariant="bold">B</mi><mn>0</mn></msub><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup></mfenced></mrow></math> </ephtml> is non‐singular. If <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Z</mi><mo>=</mo><mi mathvariant="bold">X</mi><msup><mfenced open="[" close="]" separators=","><msub><mi mathvariant="bold">B</mi><mn>0</mn></msub><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup></mfenced><mi>T</mi></msup></mrow></math> </ephtml> , then <bold><emph>Z</emph></bold> given <emph>Y</emph> = <emph>y</emph> is conditionally distributed as</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>N</mi><mi>p</mi></msub><mfenced open="(" close=")" separators=","><msup><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">μ</mi><mi>y</mi></msub></mtd></mtr><mtr><mtd><msup><mfenced open="(" close=")"><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup></mfenced><mi>T</mi></msup><msub><mi mathvariant="bold">μ</mi><mi>y</mi></msub></mtd></mtr></mtable></mfenced><mi>T</mi></msup><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msup><msub><mi mathvariant="bold">B</mi><mn>0</mn></msub><mi>T</mi></msup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mtd><mtd><mi mathvariant="bold">O</mi></mtd></mtr><mtr><mtd><msup><mi mathvariant="bold">O</mi><mi>T</mi></msup></mtd><mtd><msup><mfenced open="(" close=")"><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup></mfenced><mi>T</mi></msup><msubsup><mi mathvariant="bold">∑B</mi><mn>0</mn><mo>⊥</mo></msubsup></mtd></mtr></mtable></mfenced></mfenced><mo>.</mo></mrow></math> </ephtml> </p> <p>If <emph>g</emph><subs><bold><emph>w</emph></bold>, <bold><emph>v</emph></bold></subs>(<bold><emph>Z</emph></bold>) = <bold><emph>Z</emph></bold>(<bold><emph>w</emph></bold><sups><emph>T</emph></sups>, <bold><emph>v</emph></bold><sups><emph>T</emph></sups>)<sups><emph>T</emph></sups>, then</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>AUC</mi><mfenced open="(" close=")"><msub><mi>g</mi><mrow><mi mathvariant="bold">w</mi><mo>,</mo><mi mathvariant="bold">v</mi></mrow></msub></mfenced><mo linebreak="goodbreak">=</mo><mi mathvariant="normal">Φ</mi><mfenced open="(" close=")"><mrow><msub><mi mathvariant="bold">δB</mi><mn>0</mn></msub><mi mathvariant="bold">w</mi><mo>+</mo><msubsup><mi mathvariant="bold">δB</mi><mn>0</mn><mo>⊥</mo></msubsup><mi mathvariant="bold">v</mi></mrow><msqrt><mrow><msup><mi mathvariant="bold">w</mi><mi>T</mi></msup><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub><mi mathvariant="bold">w</mi><mo>+</mo><msup><mi mathvariant="bold">v</mi><mi>T</mi></msup><msup><mfenced open="(" close=")"><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup></mfenced><mi>T</mi></msup><msubsup><mi mathvariant="bold">∑B</mi><mn>0</mn><mo>⊥</mo></msubsup><mi mathvariant="bold">v</mi></mrow></msqrt></mfenced><mo>.</mo></mrow></math> </ephtml> </p> <p>Similarly, AUC(<emph>g</emph><subs><bold><emph>w</emph></bold>, <bold><emph>v</emph></bold></subs>) ≤ AUC(<emph>g</emph><subs>opt</subs>) with equality if and only if (<bold><emph>w</emph></bold>, <bold><emph>v</emph></bold>) = (<bold><emph>w</emph></bold><subs>opt</subs>, <bold><emph>v</emph></bold><subs>opt</subs>), where</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi mathvariant="bold">w</mi><mi>opt</mi></msub></mtd></mtr><mtr><mtd><msub><mi mathvariant="bold">v</mi><mi>opt</mi></msub></mtd></mtr></mtable></mfenced><mo linebreak="goodbreak">=</mo><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi mathvariant="bold">δB</mi><mn>0</mn></msub><msup><mfenced open="(" close=")"><mrow><msup><msub><mi mathvariant="bold">B</mi><mn>0</mn></msub><mi>T</mi></msup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr><mtr><mtd><msubsup><mi mathvariant="bold">δB</mi><mn>0</mn><mo>⊥</mo></msubsup><msup><mfenced open="{" close="}"><mrow><msup><mfenced open="(" close=")"><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup></mfenced><mi>T</mi></msup><msubsup><mi mathvariant="bold">∑B</mi><mn>0</mn><mo>⊥</mo></msubsup></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr></mtable></mfenced><mo>.</mo></mrow></math> </ephtml> </p> <p>Therefore, if we assume that <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi mathvariant="bold">δB</mi><mn>0</mn><mo>⊥</mo></msubsup><mo>=</mo><mn mathvariant="bold">0</mn></mrow></math> </ephtml> , or equivalently <bold><emph>v</emph></bold><sups><emph>T</emph></sups> = <bold>0</bold>, then</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="right center left" columnspacing="3pt" displaystyle="true"><mtr><mtd><msub><mi>g</mi><mi mathvariant="normal">opt</mi></msub><mfenced><mi mathvariant="bold">X</mi></mfenced></mtd><mtd><mo linebreak="goodbreak">=</mo></mtd><mtd><msup><mi mathvariant="bold">X∑</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="bold">δ</mi><mi>T</mi></msup></mtd></mtr><mtr><mtd><mrow /></mtd><mtd><mo linebreak="goodbreak">=</mo></mtd><mtd><mi mathvariant="bold">X</mi><mfenced open="[" close="]"><mrow><msub><mi mathvariant="bold">B</mi><mn>0</mn></msub><msup><mfenced><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><mo linebreak="goodbreak">+</mo><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup><msup><mfenced open="{" close="}"><mrow><msup><mfenced><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup></mfenced><mi>T</mi></msup><msubsup><mi mathvariant="bold">∑B</mi><mn>0</mn><mo>⊥</mo></msubsup></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mfenced><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup></mfenced><mi>T</mi></msup></mrow></mfenced><msup><mi mathvariant="bold">δ</mi><mi>T</mi></msup></mtd></mtr><mtr><mtd><mrow /></mtd><mtd><mo linebreak="goodbreak">=</mo></mtd><mtd><msub><mi mathvariant="bold">XB</mi><mn>0</mn></msub><msup><mfenced><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msup><mi mathvariant="bold">δ</mi><mi>T</mi></msup><mo linebreak="goodbreak">=</mo><msub><mi>g</mi><msub><mi mathvariant="bold">w</mi><mi mathvariant="normal">opt</mi></msub></msub><mfenced><mi mathvariant="bold">X</mi></mfenced><mo>.</mo></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>Because it follows from the decomposition theorem:</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">I</mi><mo linebreak="goodbreak">=</mo><msup><mi mathvariant="bold">∑</mi><mn>1</mn><mn>2</mn></msup><msub><mi mathvariant="bold">B</mi><mn>0</mn></msub><msup><mfenced open="(" close=")"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msup><mi mathvariant="bold">∑</mi><mn>1</mn><mn>2</mn></msup><mo linebreak="goodbreak">+</mo><msup><mi mathvariant="bold">∑</mi><mn>1</mn><mn>2</mn></msup><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup><msup><mfenced open="{" close="}"><mrow><msup><mfenced open="(" close=")"><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup></mfenced><mi>T</mi></msup><msubsup><mi mathvariant="bold">∑B</mi><mn>0</mn><mo>⊥</mo></msubsup></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mfenced open="(" close=")"><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup></mfenced><mi>T</mi></msup><msup><mi mathvariant="bold">∑</mi><mn>1</mn><mn>2</mn></msup></math> </ephtml> </p> <p>that</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi mathvariant="bold">∑</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo linebreak="goodbreak">=</mo><msub><mi mathvariant="bold">B</mi><mn>0</mn></msub><msup><mfenced open="(" close=")"><mrow><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑B</mi><mn>0</mn></msub></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mi>T</mi></msubsup><mo linebreak="goodbreak">+</mo><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup><msup><mfenced open="{" close="}"><mrow><msup><mfenced open="(" close=")"><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup></mfenced><mi>T</mi></msup><msubsup><mi mathvariant="bold">∑B</mi><mn>0</mn><mo>⊥</mo></msubsup></mrow></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mfenced open="(" close=")"><msubsup><mi mathvariant="bold">B</mi><mn>0</mn><mo>⊥</mo></msubsup></mfenced><mi>T</mi></msup><mo>.</mo></mrow></math> </ephtml> </p> <p>This immediately implies that AUC for <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>g</mi><msub><mi mathvariant="bold">w</mi><mi>opt</mi></msub></msub></mrow></math> </ephtml> (<bold><emph>X</emph></bold>) attains the global maximum AUC(<emph>g</emph><subs>opt</subs>). Then proposition 1 can be obtained. □.</p> <hd id="AN0162399693-25">Interesting counterintuitive examples: Negative weight of w k</hd> <p>We begin with a simple and significant example of a two‐study case, where there is a pattern in two covariate vectors <bold><emph>X</emph></bold><subs>(<reflink idref="bib1" id="ref41">1</reflink>)</subs> and <bold><emph>X</emph></bold><subs>(<reflink idref="bib2" id="ref42">2</reflink>)</subs> such that</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold">X</mi><mfenced open="(" close=")"><mn>1</mn></mfenced></msub><mo linebreak="goodbreak">=</mo><mfenced open="(" close=")"><mrow><msub><mi mathvariant="bold">X</mi><mn>0</mn></msub><mspace width="0.5em" /><msub><mi mathvariant="bold">X</mi><mn>1</mn></msub></mrow></mfenced><mo linebreak="goodbreak">=</mo><mi mathvariant="bold">x</mi><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi mathvariant="bold">I</mi><mn>0</mn></msub></mtd><mtd><mn mathvariant="bold">0</mn></mtd><mtd><mn mathvariant="bold">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="bold">0</mn></mtd><mtd><msub><mi mathvariant="bold">I</mi><mn>1</mn></msub></mtd><mtd><mn mathvariant="bold">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="bold">0</mn></mtd><mtd><mn mathvariant="bold">0</mn></mtd><mtd><mn mathvariant="bold">0</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mspace width="1em" /><msub><mi mathvariant="bold">X</mi><mfenced open="(" close=")"><mn>2</mn></mfenced></msub><mo linebreak="goodbreak">=</mo><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi mathvariant="bold">X</mi><mn>0</mn></msub></mtd><mtd><msub><mi mathvariant="bold">X</mi><mn>2</mn></msub></mtd></mtr></mtable></mfenced><mo linebreak="goodbreak">=</mo><mi mathvariant="bold">x</mi><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi mathvariant="bold">I</mi><mn>0</mn></msub></mtd><mtd><mn mathvariant="bold">0</mn></mtd><mtd><mn mathvariant="bold">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="bold">0</mn></mtd><mtd><mn mathvariant="bold">0</mn></mtd><mtd><mn mathvariant="bold">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="bold">0</mn></mtd><mtd><mn mathvariant="bold">0</mn></mtd><mtd><msub><mi mathvariant="bold">I</mi><mn>2</mn></msub></mtd></mtr></mtable></mfenced><mo>,</mo></math> </ephtml> </p> <p>where <bold><emph>x</emph></bold> = (<bold><emph>X</emph></bold><subs>0</subs>, <bold><emph>X</emph></bold><subs>1</subs>, <bold><emph>X</emph></bold><subs>2</subs>). In this way, <bold><emph>X</emph></bold><subs>(<reflink idref="bib1" id="ref43">1</reflink>)</subs> and <bold><emph>X</emph></bold><subs>(<reflink idref="bib2" id="ref44">2</reflink>)</subs> have the common covariate <bold><emph>X</emph></bold><subs>0</subs> and the separate covariate vectors <bold><emph>X</emph></bold><subs>1</subs> and <bold><emph>X</emph></bold><subs>2</subs>, respectively. The predictor combined <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold">X</mi><mfenced open="(" close=")"><mn>1</mn></mfenced></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mfenced open="(" close=")"><mn>1</mn></mfenced></msub></mrow></math> </ephtml> with <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold">X</mi><mfenced open="(" close=")"><mn>2</mn></mfenced></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mfenced open="(" close=")"><mn>2</mn></mfenced></msub></mrow></math> </ephtml> is given by</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable groupalign="{right left}" displaystyle="true"><mtr><mtd><msub><mi>f</mi><mi mathvariant="bold">w</mi></msub><mfenced open="(" close=")"><mi mathvariant="bold">X</mi></mfenced><mo linebreak="goodbreak">=</mo><msub><mi>w</mi><mn>1</mn></msub><msub><mi mathvariant="bold">X</mi><mfenced open="(" close=")"><mn>1</mn></mfenced></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mfenced open="(" close=")"><mn>1</mn></mfenced></msub><mo linebreak="goodbreak">+</mo><msub><mi>w</mi><mn>2</mn></msub><msub><mi mathvariant="bold">X</mi><mfenced open="(" close=")"><mn>2</mn></mfenced></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mfenced open="(" close=")"><mn>2</mn></mfenced></msub></mtd></mtr><mtr><mtd><mo linebreak="newline">=</mo><msub><mi>w</mi><mn>1</mn></msub><mfenced open="(" close=")"><mrow><msub><mi mathvariant="bold">X</mi><mn>0</mn></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>10</mn></msub><mo linebreak="goodbreak">+</mo><msub><mi mathvariant="bold">X</mi><mn>1</mn></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>11</mn></msub></mrow></mfenced><mo linebreak="goodbreak">+</mo><msub><mi>w</mi><mn>2</mn></msub><mfenced open="(" close=")"><mrow><msub><mi mathvariant="bold">X</mi><mn>0</mn></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>20</mn></msub><mo linebreak="goodbreak">+</mo><msub><mi mathvariant="bold">X</mi><mn>2</mn></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>21</mn></msub></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></mrow></math> </ephtml> </p> <p>The covariance matrix of <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="(" close=")" separators=","><mrow><msub><mi mathvariant="bold">X</mi><mfenced open="(" close=")"><mn>1</mn></mfenced></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mfenced open="(" close=")"><mn>1</mn></mfenced></msub></mrow><mrow><msub><mi mathvariant="bold">X</mi><mfenced open="(" close=")"><mn>2</mn></mfenced></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mfenced open="(" close=")"><mn>2</mn></mfenced></msub></mrow></mfenced></mrow></math> </ephtml> is given by</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msubsup><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>10</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mn>00</mn></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>10</mn></msub><mo linebreak="goodbreak">+</mo><mn>2</mn><msubsup><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>10</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mn>01</mn></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>11</mn></msub><mo linebreak="goodbreak">+</mo><msubsup><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>11</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mn>11</mn></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>11</mn></msub></mtd><mtd><msup><mrow /><mo>*</mo></msup></mtd></mtr><mtr><mtd><msup><mrow /><mo>*</mo></msup></mtd><mtd><msubsup><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>20</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mn>00</mn></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>20</mn></msub><mo linebreak="goodbreak">+</mo><mn>2</mn><msubsup><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>20</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mn>02</mn></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>21</mn></msub><mo linebreak="goodbreak">+</mo><msubsup><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>21</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mn>22</mn></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>21</mn></msub></mtd></mtr></mtable></mfenced><mo>,</mo></math> </ephtml> </p> <p>where * is <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>10</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mn>00</mn></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>20</mn></msub><mo>+</mo><msubsup><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>10</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mn>02</mn></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>21</mn></msub><mo>+</mo><msubsup><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>11</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mn>10</mn></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>20</mn></msub><mo>+</mo><msubsup><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>11</mn><mi>T</mi></msubsup><msub><mi mathvariant="bold">∑</mi><mn>12</mn></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>21</mn></msub></mrow></math> </ephtml> . We denote the matrix as <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi>b</mi><mn>11</mn></msub></mtd><mtd><msub><mi>b</mi><mn>12</mn></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mn>12</mn></msub></mtd><mtd><msub><mi>b</mi><mn>22</mn></msub></mtd></mtr></mtable></mfenced></mrow></math> </ephtml> . It is easily estimated from the current sample with a size of more than two; however, in general, to estimate <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">∑</mi><mo>=</mo><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi mathvariant="bold">∑</mi><mn>00</mn></msub></mtd><mtd><msub><mi mathvariant="bold">∑</mi><mn>01</mn></msub></mtd><mtd><msub><mi mathvariant="bold">∑</mi><mn>02</mn></msub></mtd></mtr><mtr><mtd><msub><mi mathvariant="bold">∑</mi><mn>10</mn></msub></mtd><mtd><msub><mi mathvariant="bold">∑</mi><mn>11</mn></msub></mtd><mtd><msub><mi mathvariant="bold">∑</mi><mn>12</mn></msub></mtd></mtr><mtr><mtd><msub><mi mathvariant="bold">∑</mi><mn>20</mn></msub></mtd><mtd><msub><mi mathvariant="bold">∑</mi><mn>21</mn></msub></mtd><mtd><msub><mi mathvariant="bold">∑</mi><mn>22</mn></msub></mtd></mtr></mtable></mfenced></mrow></math> </ephtml> , we need more observations.</p> <p>Based on the discussion in Section 3.2, the two predictors <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold">X</mi><mfenced open="(" close=")"><mn>1</mn></mfenced></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>1</mn></msub></mrow></math> </ephtml> and <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold">X</mi><mfenced open="(" close=")"><mn>2</mn></mfenced></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>2</mn></msub></mrow></math> </ephtml> have Φ(<emph>A</emph><subs>1</subs>) and Φ(<emph>A</emph><subs>2</subs>) as AUC, respectively, where <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>A</mi><mi>k</mi></msub><mo>=</mo><msub><mi>d</mi><mi>k</mi></msub><mo>/</mo><msqrt><msub><mi>b</mi><mi mathvariant="italic">kk</mi></msub></msqrt></mrow></math> </ephtml> for <emph>k</emph> = 1, 2 with <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>d</mi><mi>k</mi></msub><mo>=</mo><msub><mi mathvariant="bold">δ</mi><mfenced open="(" close=")"><mi>k</mi></mfenced></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mfenced open="(" close=")"><mi>k</mi></mfenced></msub></mrow></math> </ephtml> . Then, the optimal weight vector is given by</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold">w</mi><mi>opt</mi></msub><mo linebreak="goodbreak">=</mo><msup><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi>b</mi><mn>11</mn></msub></mtd><mtd><msub><mi>b</mi><mn>12</mn></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mn>12</mn></msub></mtd><mtd><msub><mi>b</mi><mn>22</mn></msub></mtd></mtr></mtable></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi>d</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></mfenced><mo>∝</mo><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi>b</mi><mn>22</mn></msub></mtd><mtd><mo linebreak="goodbreak">−</mo><msub><mi>b</mi><mn>12</mn></msub></mtd></mtr><mtr><mtd><mo linebreak="goodbreak">−</mo><msub><mi>b</mi><mn>12</mn></msub></mtd><mtd><msub><mi>b</mi><mn>11</mn></msub></mtd></mtr></mtable></mfenced><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi>d</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></mfenced><mo>∝</mo><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msqrt><msub><mi>b</mi><mn>22</mn></msub></msqrt><mfenced open="(" close=")"><mrow><msub><mi>A</mi><mn>1</mn></msub><mo linebreak="goodbreak">−</mo><mi>ρ</mi><msub><mi>A</mi><mn>2</mn></msub></mrow></mfenced></mtd></mtr><mtr><mtd><msqrt><msub><mi>b</mi><mn>11</mn></msub></msqrt><mfenced open="(" close=")"><mrow><msub><mi>A</mi><mn>2</mn></msub><mo linebreak="goodbreak">−</mo><mi>ρ</mi><msub><mi>A</mi><mn>1</mn></msub></mrow></mfenced></mtd></mtr></mtable></mfenced><mo>,</mo></math> </ephtml> </p> <p>where <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ρ</mi><mo>=</mo><msub><mi>b</mi><mn>12</mn></msub><mo>/</mo><msqrt><mrow><msub><mi>b</mi><mn>11</mn></msub><msub><mi>b</mi><mn>22</mn></msub></mrow></msqrt></mrow></math> </ephtml> . The maximum AUC is given by Φ(<emph>A</emph><subs>opt</subs>), where</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>opt</mi></msub><mo linebreak="goodbreak">=</mo><msqrt><mrow><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi>d</mi><mn>1</mn></msub></mtd><mtd><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></mfenced><msup><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi>b</mi><mn>11</mn></msub></mtd><mtd><msub><mi>b</mi><mn>12</mn></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mn>12</mn></msub></mtd><mtd><msub><mi>b</mi><mn>22</mn></msub></mtd></mtr></mtable></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi>d</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></mfenced></mrow></msqrt><mo linebreak="goodbreak">=</mo><msqrt><mrow><msubsup><mi>A</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>A</mi><mn>2</mn><mn>2</mn></msubsup><mo>−</mo><mn>2</mn><mi>ρ</mi><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub></mrow><mrow><mn>1</mn><mo>−</mo><msup><mi>ρ</mi><mn>2</mn></msup></mrow></msqrt><mo>.</mo></math> </ephtml> </p> <p>Example 1. Let us consider a naive weight vector <bold><emph>w</emph></bold><subs>naive</subs> = (<reflink idref="bib1" id="ref45">1</reflink>, 1) that leads to <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>f</mi><msub><mi mathvariant="bold">w</mi><mtext>naive</mtext></msub></msub><mfenced open="(" close=")"><mi mathvariant="bold">X</mi></mfenced><mo>=</mo><msub><mi mathvariant="bold">X</mi><mfenced open="(" close=")"><mn>1</mn></mfenced></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>1</mn></msub><mo>+</mo><msub><mi mathvariant="bold">X</mi><mfenced open="(" close=")"><mn>2</mn></mfenced></msub><msub><mover accent="true"><mi mathvariant="bold">β</mi><mo stretchy="true">̂</mo></mover><mn>2</mn></msub></mrow></math> </ephtml> . Then the AUC is given by Φ(<emph>A</emph><subs>naive</subs>), where</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>A</mi><mtext>naive</mtext></msub><mo linebreak="goodbreak">=</mo><mrow><msub><mi>A</mi><mn>1</mn></msub><msqrt><msub><mi>b</mi><mn>11</mn></msub></msqrt><mo>+</mo><msub><mi>A</mi><mn>2</mn></msub><msqrt><msub><mi>b</mi><mn>22</mn></msub></msqrt></mrow><msqrt><mrow><mn>1</mn><msub><mi>b</mi><mn>11</mn></msub><mo>+</mo><mn>1</mn><msub><mi>b</mi><mn>22</mn></msub><mo>+</mo><mrow><mn>2</mn><mi>ρ</mi></mrow><msqrt><mrow><msub><mi>b</mi><mn>11</mn></msub><msub><mi>b</mi><mn>22</mn></msub></mrow></msqrt></mrow></msqrt><mo>.</mo></mrow></math> </ephtml> </p> <p>Here, we observe the improvement of AUC for <bold><emph>w</emph></bold><subs>opt</subs> to <bold><emph>w</emph></bold><subs>naive</subs> given by</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><msub><mi>A</mi><mi>opt</mi></msub><mn>2</mn></msup><mo linebreak="goodbreak">−</mo><msup><msub><mi>A</mi><mtext>naive</mtext></msub><mn>2</mn></msup><mo linebreak="goodbreak">=</mo><msup><mfenced open="{" close="}"><mrow><msub><mi>A</mi><mn>1</mn></msub><mfenced open="(" close=")"><mrow><mi>ρ</mi><msub><mi>b</mi><mn>22</mn></msub><mo linebreak="goodbreak">+</mo><mn>1</mn><msub><mi>b</mi><mn>11</mn></msub></mrow></mfenced><mo linebreak="goodbreak">−</mo><msub><mi>A</mi><mn>2</mn></msub><mfenced open="(" close=")"><mrow><mi>ρ</mi><msub><mi>b</mi><mn>11</mn></msub><mo linebreak="goodbreak">+</mo><mn>1</mn><msub><mi>b</mi><mn>22</mn></msub></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>.</mo></mrow></math> </ephtml> </p> <p>Consider a counterintuitive aspect of <bold><emph>w</emph></bold><subs>opt</subs>. Without the loss of generality, we assume that <emph>A</emph><subs>1</subs> ≥ <emph>A</emph><subs>2</subs>. If <emph>A</emph><subs>2</subs> = <emph>ρA</emph><subs>1</subs>, then <bold><emph>w</emph></bold><subs>opt</subs> ∝ (<reflink idref="bib1" id="ref46">1</reflink>, 0) and <emph>A</emph><subs>opt</subs> = <emph>A</emph><subs>1</subs>. If <emph>A</emph><subs>2</subs> < <emph>ρA</emph><subs>1</subs>, then the second component of <bold><emph>w</emph></bold><subs>opt</subs> becomes negative. It seems counterintuitive to support negative weights. This result is related with the observation in Demler et al. (2013). [?] This is because we pose no requirement for two predictors. On the other hand, when two predictors are independent or equivalently <emph>ρ</emph> = 0, <ephtml> <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold">w</mi><mi>opt</mi></msub><mo>∝</mo><mfenced open="(" close=")" separators=","><mrow><msqrt><msub><mi>b</mi><mn>22</mn></msub></msqrt><msub><mi>A</mi><mn>1</mn></msub></mrow><msqrt><mrow><msub><mi>b</mi><mn>11</mn></msub><msub><mi>A</mi><mn>2</mn></msub></mrow></msqrt></mfenced></mrow></math> </ephtml> , both of the weights become positive.</p> <p>Example 2. Consider a simple numerical case as follows:</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi>b</mi><mn>11</mn></msub></mtd><mtd><msub><mi>b</mi><mn>12</mn></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mn>12</mn></msub></mtd><mtd><msub><mi>b</mi><mn>22</mn></msub></mtd></mtr></mtable></mfenced><mo linebreak="goodbreak">=</mo><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><mn>1</mn></mtd><mtd><mi>ρ</mi></mtd></mtr><mtr><mtd><mi>ρ</mi></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mspace width="1.5em" /><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi>d</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></mfenced><mo linebreak="goodbreak">=</mo><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mi>δ</mi></mtd></mtr></mtable></mfenced><mo>.</mo></mrow></math> </ephtml> </p> <p>Thus, we get that <emph>A</emph><subs>1</subs> = 1, <emph>A</emph><subs>2</subs> = <emph>δ</emph>, so</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold">w</mi><mi>opt</mi></msub><mo>∝</mo><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi>A</mi><mn>1</mn></msub><mo linebreak="goodbreak">−</mo><mi>ρ</mi><msub><mi>A</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>2</mn></msub><mo linebreak="goodbreak">−</mo><mi>ρ</mi><msub><mi>A</mi><mn>1</mn></msub></mtd></mtr></mtable></mfenced><mo linebreak="goodbreak">=</mo><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><mn>1</mn><mo linebreak="goodbreak">−</mo><mi mathvariant="italic">ρδ</mi></mtd></mtr><mtr><mtd><mi>δ</mi><mo linebreak="goodbreak">−</mo><mi>ρ</mi></mtd></mtr></mtable></mfenced><mo>.</mo></mrow></math> </ephtml> </p> <p>For example, if <emph>ρ</emph> = 0.8, <emph>δ</emph> = 0.5, then we have negative weight <bold><emph>w</emph></bold><subs>opt</subs> ∝ (<reflink idref="bib2" id="ref47">2</reflink>, −1) and the values of AUC are given by Φ(<emph>A</emph><subs>1</subs>) = 0.841, Φ(<emph>A</emph><subs>2</subs>) = 0.691 and Φ(<emph>A</emph><subs>opt</subs>) = 0.894.</p> <p>Example 3. A similar argument can be applied to a case of <emph>K</emph> = 3, where we obtain</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi>b</mi><mn>11</mn></msub></mtd><mtd><msub><mi>b</mi><mn>12</mn></msub></mtd><mtd><msub><mi>b</mi><mn>13</mn></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mn>12</mn></msub></mtd><mtd><msub><mi>b</mi><mn>22</mn></msub></mtd><mtd><msub><mi>b</mi><mn>23</mn></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mn>13</mn></msub></mtd><mtd><msub><mi>b</mi><mn>23</mn></msub></mtd><mtd><msub><mi>b</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mo linebreak="goodbreak">=</mo><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><mn>1</mn></mtd><mtd><mn>0.3</mn></mtd><mtd><mn>0.8</mn></mtd></mtr><mtr><mtd><mn>0.3</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0.5</mn></mtd></mtr><mtr><mtd><mn>0.8</mn></mtd><mtd><mn>0.5</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mspace width="1em" /><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><msub><mi>d</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>d</mi><mn>3</mn></msub></mtd></mtr></mtable></mfenced><mo linebreak="goodbreak">=</mo><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0.8</mn></mtd></mtr><mtr><mtd><mn>0.5</mn></mtd></mtr></mtable></mfenced><mo>.</mo></mrow></math> </ephtml> </p> <p>Then, we have the optimal weight vector as</p> <p> <ephtml> <math display="block" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold">w</mi><mi>opt</mi></msub><mo linebreak="goodbreak">=</mo><msup><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><mn>1</mn></mtd><mtd><mn>0.3</mn></mtd><mtd><mn>0.8</mn></mtd></mtr><mtr><mtd><mn>0.3</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0.5</mn></mtd></mtr><mtr><mtd><mn>0.8</mn></mtd><mtd><mn>0.5</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0.8</mn></mtd></mtr><mtr><mtd><mn>0.5</mn></mtd></mtr></mtable></mfenced><mo linebreak="goodbreak">=</mo><mfenced open="(" close=")"><mtable columnalign="center"><mtr><mtd><mn>1.94</mn></mtd></mtr><mtr><mtd><mn>0.992</mn></mtd></mtr><mtr><mtd><mo linebreak="goodbreak">−</mo><mn>1.55</mn></mtd></mtr></mtable></mfenced><mo>.</mo></mrow></math> </ephtml> </p> <p>The values of AUC are given as Φ(<emph>A</emph><subs>1</subs>) = 0.841, Φ(<emph>A</emph><subs>2</subs>) = 0.788, Φ(<emph>A</emph><subs>3</subs>) = 0.691 and Φ(<emph>A</emph><subs>opt</subs>) = 0.919.</p> <ref id="AN0162399693-26"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Funding information Japan Society for the Promotion of Science, Grant/Award Number: Grant‐in‐Aid for Early‐Career Scientists (21K17292)</bibtext> </blist> </ref> <ref id="AN0162399693-27"> <title> REFERENCES </title> <blist> <bibtext> Damen JA, Hooft L, Schuit E, et al. 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  Data: Area under the Curve-Optimized Synthesis of Prediction Models from a Meta-Analytical Perspective
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  Data: <searchLink fieldCode="AR" term="%22Yoneoka%2C+Daisuke%22">Yoneoka, Daisuke</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-3525-5092">0000-0002-3525-5092</externalLink>)<br /><searchLink fieldCode="AR" term="%22Omae%2C+Katsuhiro%22">Omae, Katsuhiro</searchLink><br /><searchLink fieldCode="AR" term="%22Henmi%2C+Masayuki%22">Henmi, Masayuki</searchLink><br /><searchLink fieldCode="AR" term="%22Eguchi%2C+Shinto%22">Eguchi, Shinto</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22Research+Synthesis+Methods%22"><i>Research Synthesis Methods</i></searchLink>. Mar 2023 14(2):234-246.
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  Data: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
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  Data: <searchLink fieldCode="DE" term="%22Prediction%22">Prediction</searchLink><br /><searchLink fieldCode="DE" term="%22Task+Analysis%22">Task Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Medical+Research%22">Medical Research</searchLink><br /><searchLink fieldCode="DE" term="%22Outcomes+of+Treatment%22">Outcomes of Treatment</searchLink><br /><searchLink fieldCode="DE" term="%22Meta+Analysis%22">Meta Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Predictor+Variables%22">Predictor Variables</searchLink><br /><searchLink fieldCode="DE" term="%22Models%22">Models</searchLink>
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  Data: 10.1002/jrsm.1612
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  Data: 1759-2879<br />1759-2887
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  Label: Abstract
  Group: Ab
  Data: The number of clinical prediction models sharing the same prediction task has increased in the medical literature. However, evidence synthesis methodologies that use the results of these prediction models have not been sufficiently studied, particularly in the context of meta-analysis settings where only summary statistics are available. In particular, we consider the following situation: we want to predict an outcome Y, that is not included in our current data, while the covariate data are fully available. In addition, the summary statistics from prior studies, which share the same prediction task (i.e., the prediction of Y), are available. This study introduces a new method for synthesizing the summary results of binary prediction models reported in the prior studies using a linear predictor under a distributional assumption between the current and prior studies. The method provides an integrated predictor combining all predictors reported in the prior studies with weights. The vector of the weights is designed to achieve the hypothetical improvement of area under the receiver operating characteristic curve (AUC) on the current available data under a practical situation where there are different sets of covariates in the prior studies. We observe a counterintuitive aspect in typical situations where a part of weight components in the proposed method becomes negative. It implies that flipping the sign of the prediction results reported in each individual study would improve the overall prediction performance. Finally, numerical and real-world data analysis were conducted and showed that our method outperformed conventional methods in terms of AUC.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: Note
  Label: Notes
  Group: Note
  Data: https://hbiostat.org/data
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2023
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1369258
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1369258
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1002/jrsm.1612
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 13
        StartPage: 234
    Subjects:
      – SubjectFull: Prediction
        Type: general
      – SubjectFull: Task Analysis
        Type: general
      – SubjectFull: Medical Research
        Type: general
      – SubjectFull: Outcomes of Treatment
        Type: general
      – SubjectFull: Meta Analysis
        Type: general
      – SubjectFull: Predictor Variables
        Type: general
      – SubjectFull: Models
        Type: general
    Titles:
      – TitleFull: Area under the Curve-Optimized Synthesis of Prediction Models from a Meta-Analytical Perspective
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Yoneoka, Daisuke
      – PersonEntity:
          Name:
            NameFull: Omae, Katsuhiro
      – PersonEntity:
          Name:
            NameFull: Henmi, Masayuki
      – PersonEntity:
          Name:
            NameFull: Eguchi, Shinto
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 03
              Type: published
              Y: 2023
          Identifiers:
            – Type: issn-print
              Value: 1759-2879
            – Type: issn-electronic
              Value: 1759-2887
          Numbering:
            – Type: volume
              Value: 14
            – Type: issue
              Value: 2
          Titles:
            – TitleFull: Research Synthesis Methods
              Type: main
ResultId 1