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Book Individual Treatment Effect Heterogeneity in Multiple Time Points Trials

Download or read book Individual Treatment Effect Heterogeneity in Multiple Time Points Trials written by and published by . This book was released on 2009 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In biomedical studies, the treatment main effect is often expressed in terms of an "average difference." A treatment that appears superior based on the average effect may not be superior for all subjects in a population if there is substantial "subject-treatment interaction." A parameter quantifying subject-treatment interaction is inestimable in two sample completely randomized designs. Crossover designs have been suggested as a way to estimate the variability in individual treatment effects since an "individual treatment effect" can be measured. However, variability in these observed individual effects may include variability due to the treatment plus inherent variability of a response over time. We use the "Neyman - Rubin Model of Causal Inference" (Neyman, 1923; Rubin, 1974) for analyses. This dissertation consists of two parts: The quantitative and qualitative response analyses. The quantitative part focuses on disentangling the variability due to treatment effects from variability due to time effects using suitable crossover designs. Next, we propose a variable that defines the variance of a true individual treatment effect in a two crossover designs and show that they are not directly estimable but the mean effect is estimable. Furthermore, we show the variance of individual treatment effects is biased under both designs. The bias depends on time effects. Under certain design considerations, linear combinations of time effects can be estimated, making it possible to separate the variability due to time from that due to treatment. The qualitative section involves a binary response and is centered on estimating the average treatment effect and bounding a probability of a negative effect, a parameter which relates to the individual treatment effect variability. Using a stated joint probability distribution of potential outcomes, we express the probability of the observed outcomes under a two treatment, two periods crossover design. Maximum likelihood estimates of these probabilities are found using an iterative numerical method. From these, we propose bounds for an inestimable probability of negative effect. Tighter bounds are obtained with information from subjects that receive the same treatments over the two periods. Finally, we simulate an example of observed count data to illustrate estimation of the bounds.

Book Developing a Protocol for Observational Comparative Effectiveness Research  A User s Guide

Download or read book Developing a Protocol for Observational Comparative Effectiveness Research A User s Guide written by Agency for Health Care Research and Quality (U.S.) and published by Government Printing Office. This book was released on 2013-02-21 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: This User’s Guide is a resource for investigators and stakeholders who develop and review observational comparative effectiveness research protocols. It explains how to (1) identify key considerations and best practices for research design; (2) build a protocol based on these standards and best practices; and (3) judge the adequacy and completeness of a protocol. Eleven chapters cover all aspects of research design, including: developing study objectives, defining and refining study questions, addressing the heterogeneity of treatment effect, characterizing exposure, selecting a comparator, defining and measuring outcomes, and identifying optimal data sources. Checklists of guidance and key considerations for protocols are provided at the end of each chapter. The User’s Guide was created by researchers affiliated with AHRQ’s Effective Health Care Program, particularly those who participated in AHRQ’s DEcIDE (Developing Evidence to Inform Decisions About Effectiveness) program. Chapters were subject to multiple internal and external independent reviews. More more information, please consult the Agency website: www.effectivehealthcare.ahrq.gov)

Book Small Clinical Trials

    Book Details:
  • Author : Institute of Medicine
  • Publisher : National Academies Press
  • Release : 2001-01-01
  • ISBN : 0309171148
  • Pages : 221 pages

Download or read book Small Clinical Trials written by Institute of Medicine and published by National Academies Press. This book was released on 2001-01-01 with total page 221 pages. Available in PDF, EPUB and Kindle. Book excerpt: Clinical trials are used to elucidate the most appropriate preventive, diagnostic, or treatment options for individuals with a given medical condition. Perhaps the most essential feature of a clinical trial is that it aims to use results based on a limited sample of research participants to see if the intervention is safe and effective or if it is comparable to a comparison treatment. Sample size is a crucial component of any clinical trial. A trial with a small number of research participants is more prone to variability and carries a considerable risk of failing to demonstrate the effectiveness of a given intervention when one really is present. This may occur in phase I (safety and pharmacologic profiles), II (pilot efficacy evaluation), and III (extensive assessment of safety and efficacy) trials. Although phase I and II studies may have smaller sample sizes, they usually have adequate statistical power, which is the committee's definition of a "large" trial. Sometimes a trial with eight participants may have adequate statistical power, statistical power being the probability of rejecting the null hypothesis when the hypothesis is false. Small Clinical Trials assesses the current methodologies and the appropriate situations for the conduct of clinical trials with small sample sizes. This report assesses the published literature on various strategies such as (1) meta-analysis to combine disparate information from several studies including Bayesian techniques as in the confidence profile method and (2) other alternatives such as assessing therapeutic results in a single treated population (e.g., astronauts) by sequentially measuring whether the intervention is falling above or below a preestablished probability outcome range and meeting predesigned specifications as opposed to incremental improvement.

Book Cochrane Handbook for Systematic Reviews of Interventions

Download or read book Cochrane Handbook for Systematic Reviews of Interventions written by Julian P. T. Higgins and published by Wiley. This book was released on 2008-11-24 with total page 672 pages. Available in PDF, EPUB and Kindle. Book excerpt: Healthcare providers, consumers, researchers and policy makers are inundated with unmanageable amounts of information, including evidence from healthcare research. It has become impossible for all to have the time and resources to find, appraise and interpret this evidence and incorporate it into healthcare decisions. Cochrane Reviews respond to this challenge by identifying, appraising and synthesizing research-based evidence and presenting it in a standardized format, published in The Cochrane Library (www.thecochranelibrary.com). The Cochrane Handbook for Systematic Reviews of Interventions contains methodological guidance for the preparation and maintenance of Cochrane intervention reviews. Written in a clear and accessible format, it is the essential manual for all those preparing, maintaining and reading Cochrane reviews. Many of the principles and methods described here are appropriate for systematic reviews applied to other types of research and to systematic reviews of interventions undertaken by others. It is hoped therefore that this book will be invaluable to all those who want to understand the role of systematic reviews, critically appraise published reviews or perform reviews themselves.

Book Statistical Methods for Studying Heterogeneous Treatment Effects with Instrumental Variables

Download or read book Statistical Methods for Studying Heterogeneous Treatment Effects with Instrumental Variables written by Michael William Johnson and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: There is a growing interest in estimating heterogeneous treatment effects in randomized and observational studies. However, most of the work relies on the assumption of ignorability, or no unmeasured confounding on the treatment effect. While instrumental variables (IV) are a popular technique to control for unmeasured confounding, there has been little research conducted to study heterogeneous treatment effects with the use of an IV. This dissertation introduces methods using an IV to discover novel subgroups, estimate their heterogeneous treatment effects, and identify individualized treatment rules (ITR) when ignorability is expected to be violated. In Chapter 2, we present a two-part algorithm to estimate heterogeneous treatment effects and detect novel subgroups using an IV with matching. The first part uses interpretable machine learning techniques, such as classification and regression trees, to discover potential effect modifiers. The second part uses closed testing to test for statistical significance of each effect modifier while strongly controlling the familywise error rate. We apply this method on the Oregon Health Insurance Experiment, estimating the effect of Medicaid on the number of days an individual's health does not impede their usual activities by using a randomized lottery as an instrument. In Chapter 3, we generalize methods to identify ITR using a binary IV to using multiple, discrete valued instruments, or equivalently, multilevel instruments. Several new problems arise when generalizing to multilevel instruments, requiring novel solutions. In particular, multilevel IV give rise to many latent subgroups that may experience heterogeneous treatment effects. Additionally, it may be unclear how to combine and compare the different levels of the IV to estimate treatment heterogeneity. We provide methods that use a prediction of the latent subgroup to identify optimal ITR, and methods to dynamically combine levels of the multilevel IV to estimate the heterogeneous treatment effects, effectively individualizing estimation of an ITR. Further, we provide and discuss necessary and sufficient conditions to identify an optimal ITR using a multilevel IV. We apply our methods to identify an ITR for two competing treatments, carotid endarterectomy and carotid artery stenting, on preventing stroke or death within 30 days of their index procedure.

Book Doing Meta Analysis with R

Download or read book Doing Meta Analysis with R written by Mathias Harrer and published by CRC Press. This book was released on 2021-09-15 with total page 500 pages. Available in PDF, EPUB and Kindle. Book excerpt: Doing Meta-Analysis with R: A Hands-On Guide serves as an accessible introduction on how meta-analyses can be conducted in R. Essential steps for meta-analysis are covered, including calculation and pooling of outcome measures, forest plots, heterogeneity diagnostics, subgroup analyses, meta-regression, methods to control for publication bias, risk of bias assessments and plotting tools. Advanced but highly relevant topics such as network meta-analysis, multi-three-level meta-analyses, Bayesian meta-analysis approaches and SEM meta-analysis are also covered. A companion R package, dmetar, is introduced at the beginning of the guide. It contains data sets and several helper functions for the meta and metafor package used in the guide. The programming and statistical background covered in the book are kept at a non-expert level, making the book widely accessible. Features • Contains two introductory chapters on how to set up an R environment and do basic imports/manipulations of meta-analysis data, including exercises • Describes statistical concepts clearly and concisely before applying them in R • Includes step-by-step guidance through the coding required to perform meta-analyses, and a companion R package for the book

Book Treatment Heterogeneity and Potential Outcomes in Linear Mixed Effects Models

Download or read book Treatment Heterogeneity and Potential Outcomes in Linear Mixed Effects Models written by Troy E. Richardson and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Studies commonly focus on estimating a mean treatment effect in a population. However, in some applications the variability of treatment effects across individual units may help to characterize the overall effect of a treatment across the population. Consider a set of treatments, {T, C}, where T denotes some treatment that might be applied to an experimental unit and C denotes a control. For each of N experimental units, the duplet {gamma[subscript]Ti, gamma[subscript]Ci}, i=1,2 ..., N, represents the potential response of the i[superscript]th experimental unit if treatment were applied and the response of the experimental unit if control were applied, respectively. The causal effect of T compared to C is the difference between the two potential responses, gamma[subscript]Ti- gamma[subscript]Ci. Much work has been done to elucidate the statistical properties of a causal effect, given a set of particular assumptions. Gadbury and others have reported on this for some simple designs and primarily focused on finite population randomization based inference. When designs become more complicated, the randomization based approach becomes increasingly difficult. Since linear mixed effects models are particularly useful for modeling data from complex designs, their role in modeling treatment heterogeneity is investigated. It is shown that an individual treatment effect can be conceptualized as a linear combination of fixed treatment effects and random effects. The random effects are assumed to have variance components specified in a mixed effects "potential outcomes" model when both potential outcomes, gamma[subscript]T, gamma[subscript]C, are variables in the model. The variance of the individual causal effect is used to quantify treatment heterogeneity. Post treatment assignment, however, only one of the two potential outcomes is observable for a unit. It is then shown that the variance component for treatment heterogeneity becomes non-estimable in an analysis of observed data. Furthermore, estimable variance components in the observed data model are demonstrated to arise from linear combinations of the non-estimable variance components in the potential outcomes model. Mixed effects models are considered in context of a particular design in an effort to illuminate the loss of information incurred when moving from a potential outcomes framework to an observed data analysis.

Book Targeted Learning in Data Science

Download or read book Targeted Learning in Data Science written by Mark J. van der Laan and published by Springer. This book was released on 2018-03-28 with total page 655 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook for graduate students in statistics, data science, and public health deals with the practical challenges that come with big, complex, and dynamic data. It presents a scientific roadmap to translate real-world data science applications into formal statistical estimation problems by using the general template of targeted maximum likelihood estimators. These targeted machine learning algorithms estimate quantities of interest while still providing valid inference. Targeted learning methods within data science area critical component for solving scientific problems in the modern age. The techniques can answer complex questions including optimal rules for assigning treatment based on longitudinal data with time-dependent confounding, as well as other estimands in dependent data structures, such as networks. Included in Targeted Learning in Data Science are demonstrations with soft ware packages and real data sets that present a case that targeted learning is crucial for the next generation of statisticians and data scientists. Th is book is a sequel to the first textbook on machine learning for causal inference, Targeted Learning, published in 2011. Mark van der Laan, PhD, is Jiann-Ping Hsu/Karl E. Peace Professor of Biostatistics and Statistics at UC Berkeley. His research interests include statistical methods in genomics, survival analysis, censored data, machine learning, semiparametric models, causal inference, and targeted learning. Dr. van der Laan received the 2004 Mortimer Spiegelman Award, the 2005 Van Dantzig Award, the 2005 COPSS Snedecor Award, the 2005 COPSS Presidential Award, and has graduated over 40 PhD students in biostatistics and statistics. Sherri Rose, PhD, is Associate Professor of Health Care Policy (Biostatistics) at Harvard Medical School. Her work is centered on developing and integrating innovative statistical approaches to advance human health. Dr. Rose’s methodological research focuses on nonparametric machine learning for causal inference and prediction. She co-leads the Health Policy Data Science Lab and currently serves as an associate editor for the Journal of the American Statistical Association and Biostatistics.

Book Design and Analysis of Cluster Randomization Trials in Health Research

Download or read book Design and Analysis of Cluster Randomization Trials in Health Research written by Allan Donner and published by Wiley. This book was released on 2010-05-17 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: A cluster randomization trial is one in which intact social units, or clusters of individuals, are randomized to different intervention groups. Trials randomizing clusters have become particularly widespread in the evaluation of non-therapeutic interventions, including lifestyle modification, educational programmes and innovations in the provision of health care. The increasing popularity of this design among health researchers over the past two decades has led to an extensive body of methodology on the subject. This is the first book to present a systematic and united treatment of this topic; it contains distinctive chapters on the history of cluster randomized trials, ethical issues and reporting guidelines.

Book Theory of U Statistics

    Book Details:
  • Author : Vladimir S. Korolyuk
  • Publisher : Springer Science & Business Media
  • Release : 2013-03-09
  • ISBN : 9401735158
  • Pages : 558 pages

Download or read book Theory of U Statistics written by Vladimir S. Korolyuk and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 558 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of U-statistics goes back to the fundamental work of Hoeffding [1], in which he proved the central limit theorem. During last forty years the interest to this class of random variables has been permanently increasing, and thus, the new intensively developing branch of probability theory has been formed. The U-statistics are one of the universal objects of the modem probability theory of summation. On the one hand, they are more complicated "algebraically" than sums of independent random variables and vectors, and on the other hand, they contain essential elements of dependence which display themselves in the martingale properties. In addition, the U -statistics as an object of mathematical statistics occupy one of the central places in statistical problems. The development of the theory of U-statistics is stipulated by the influence of the classical theory of summation of independent random variables: The law of large num bers, central limit theorem, invariance principle, and the law of the iterated logarithm we re proved, the estimates of convergence rate were obtained, etc.

Book Encyclopedia of Biopharmaceutical Statistics   Four Volume Set

Download or read book Encyclopedia of Biopharmaceutical Statistics Four Volume Set written by Shein-Chung Chow and published by CRC Press. This book was released on 2018-09-03 with total page 4031 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the publication of the first edition in 2000, there has been an explosive growth of literature in biopharmaceutical research and development of new medicines. This encyclopedia (1) provides a comprehensive and unified presentation of designs and analyses used at different stages of the drug development process, (2) gives a well-balanced summary of current regulatory requirements, and (3) describes recently developed statistical methods in the pharmaceutical sciences. Features of the Fourth Edition: 1. 78 new and revised entries have been added for a total of 308 chapters and a fourth volume has been added to encompass the increased number of chapters. 2. Revised and updated entries reflect changes and recent developments in regulatory requirements for the drug review/approval process and statistical designs and methodologies. 3. Additional topics include multiple-stage adaptive trial design in clinical research, translational medicine, design and analysis of biosimilar drug development, big data analytics, and real world evidence for clinical research and development. 4. A table of contents organized by stages of biopharmaceutical development provides easy access to relevant topics. About the Editor: Shein-Chung Chow, Ph.D. is currently an Associate Director, Office of Biostatistics, U.S. Food and Drug Administration (FDA). Dr. Chow is an Adjunct Professor at Duke University School of Medicine, as well as Adjunct Professor at Duke-NUS, Singapore and North Carolina State University. Dr. Chow is the Editor-in-Chief of the Journal of Biopharmaceutical Statistics and the Chapman & Hall/CRC Biostatistics Book Series and the author of 28 books and over 300 methodology papers. He was elected Fellow of the American Statistical Association in 1995.

Book Comparative Effectiveness Review Methods

Download or read book Comparative Effectiveness Review Methods written by U. S. Department of Health and Human Services and published by Createspace Independent Pub. This book was released on 2013-05-17 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Agency for Healthcare Research and Quality (AHRQ) commissioned the RTI International–University of North Carolina at Chapel Hill (RTI-UNC) Evidence-based Practice Center (EPC) to explore how systematic review groups have dealt with clinical heterogeneity and to seek out best practices for addressing clinical heterogeneity in systematic reviews (SRs) and comparative effectiveness reviews (CERs). Such best practices, to the extent they exist, may enable AHRQ's EPCs to address critiques from patients, clinicians, policymakers, and other proponents of health care about the extent to which “average” estimates of the benefits and harms of health care interventions apply to individual patients or to small groups of patients sharing similar characteristics. Such users of reviews often assert that EPC reviews typically focus on broad populations and, as a result, often lack information relevant to patient subgroups that are of particular concern to them. More important, even when EPCs evaluate literature on homogeneous groups, there may be varying individual treatment for no apparent reason, indicating that average treatment effect does not point to the best treatment for any given individual. Thus, the health care community is looking for better ways to develop information that may foster better medical care at a “personal” or “individual” level. To address our charge for this methods project, the EPC set out to answer six key questions (KQ). Key questions for methods report on clinical heterogeneity include: 1. What is clinical heterogeneity? a. How has it been defined by various groups? b. How is it distinct from statistical heterogeneity? c. How does it fit with other issues that have been addressed by the AHRQ Methods Manual for CERs? 2. How have systematic reviews dealt with clinical heterogeneity in the key questions? a. What questions have been asked? b. How have they pre-identified population subgroups with common clinical characteristics that modify their intervention-outcome association? c. What are best practices in key questions and how these subgroups have been identified? 3. How have systematic reviews dealt with clinical heterogeneity in the review process? a. What do guidance documents of various systematic review groups recommend? b. How have EPCs handled clinical heterogeneity in their reviews? c. What are best practices in searching for and interpreting results for particular subgroups with common clinical characteristics that may modify their intervention-outcome association? 4. What are critiques in how systematic reviews handle clinical heterogeneity? a. What are critiques from specific reviews (peer and public) on how EPCs handled clinical heterogeneity? b. What general critiques (in the literature) have been made against how systematic reviews handle clinical heterogeneity? 5. What evidence is there to support how to best address clinical heterogeneity in a systematic review? 6. What questions should an EPC work group on clinical heterogeneity address? Heterogeneity (of any type) in EPC reviews is important because its appearance suggests that included studies differed on one or more dimensions such as patient demographics, study designs, coexisting conditions, or other factors. EPCs then need to clarify for clinical and other audiences, collectively referred to as stakeholders, what are the potential causes of the heterogeneity in their results. This will allow the stakeholders to understand whether and to what degree they can apply this information to their own patients or constituents. Of greatest importance for this project was clinical heterogeneity, which we define as the variation in study population characteristics, coexisting conditions, cointerventions, and outcomes evaluated across studies included in an SR or CER that may influence or modify the magnitude of the intervention measure of effect (e.g., odds ratio, risk ratio, risk difference).

Book Statistical Issues in Drug Development

Download or read book Statistical Issues in Drug Development written by Stephen S. Senn and published by John Wiley & Sons. This book was released on 2008-02-28 with total page 523 pages. Available in PDF, EPUB and Kindle. Book excerpt: Drug development is the process of finding and producingtherapeutically useful pharmaceuticals, turning them into safe andeffective medicine, and producing reliable information regardingthe appropriate dosage and dosing intervals. With regulatoryauthorities demanding increasingly higher standards in suchdevelopments, statistics has become an intrinsic and criticalelement in the design and conduct of drug development programmes. Statistical Issues in Drug Development presents anessential and thought provoking guide to the statistical issues andcontroversies involved in drug development. This highly readable second edition has been updated toinclude: Comprehensive coverage of the design and interpretation ofclinical trials. Expanded sections on missing data, equivalence, meta-analysisand dose finding. An examination of both Bayesian and frequentist methods. A new chapter on pharmacogenomics and expanded coverage ofpharmaco-epidemiology and pharmaco-economics. Coverage of the ICH guidelines, in particular ICH E9,Statistical Principles for Clinical Trials. It is hoped that the book will stimulate dialogue betweenstatisticians and life scientists working within the pharmaceuticalindustry. The accessible and wide-ranging coverage make itessential reading for both statisticians and non-statisticiansworking in the pharmaceutical industry, regulatory bodies andmedical research institutes. There is also much to benefitundergraduate and postgraduate students whose courses include amedical statistics component.

Book Applied Mixed Model Analysis

Download or read book Applied Mixed Model Analysis written by Jos W. R. Twisk and published by Cambridge University Press. This book was released on 2019-04-18 with total page 249 pages. Available in PDF, EPUB and Kindle. Book excerpt: Emphasizing interpretation of results, this hands-on guide explains why, when, and how to use mixed models with your data.

Book Using Multisite Instrumental Variables to Estimate Treatment Effects and Treatment Effect Heterogeneity

Download or read book Using Multisite Instrumental Variables to Estimate Treatment Effects and Treatment Effect Heterogeneity written by Christopher Ryan Runyon and published by . This book was released on 2020 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multisite randomized trials (MSTs) are an attractive research design to test the efficacy of an educational program at scale. Population models examining data from MSTs can provide information on the range of possible treatment effects that sites (such as schools) can expect from an educational program, even for those sites not included in the study. However, when some individuals at a site do not comply with their treatment assignment, conventional multilevel and meta-analytic estimation methods do not provide information on the effect of actually participating in the educational program. Instrumental variables (IV) is a method that can produce consistent estimates of the causal effect of participating in an educational program for those individuals that comply with their treatment assignment, an estimand called the complier-average treatment effect (CATE). IV methods for single-site trials are well understood and widely-used. Recently multisite IV models have been proposed to estimate the CATE and CATE heterogeneity across a population of sites, but the performance of these estimators has not been examined in a simulation study. Using Monte Carlo simulation, the current study examines the performance of three IV estimators and two conventional estimators in recovering the CATE and CATE heterogeneity under simulation conditions that resemble multisite trials of well-known educational programs

Book Treatment Heterogeneity and Individual Qualitative Interaction

Download or read book Treatment Heterogeneity and Individual Qualitative Interaction written by Robert S. Poulson and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The potential for high variability in treatment effects across individuals has been recognized as an important consideration in clinical studies. Surprisingly, little attention has been given to evaluating this variability in design of clinical trials or analyses of resulting data. High variation in a treatment's efficacy or safety across individuals (referred to herein as treatment heterogeneity) may have important consequences because the optimal treatment choice for an individual may be different from that suggested by a study of average effects. We call this an individual qualitative interaction (IQI), borrowing terminology from earlier work - referring to a qualitative interaction (QI) being present when the optimal treatment varies across 'groups' of individuals. At least three techniques have been proposed to investigate treatment heterogeneity: techniques to detect a QI, use of measures such as the density overlap of two outcome variables under different treatments, and use of cross-over designs to observe 'individual effects.' Connections, limitations, and the required assumptions are compared among these techniques through a quantity frequently referred to as subject-treatment (S-T) interaction, but shown here to be the probability of an IQI (PIQI). Their association is studied utilizing a potential outcomes framework that can add insights to results from usual data analyses and to study design features to more directly assess treatment heterogeneity. Particular attention is given to the density overlap of two outcome variables, each representing an individual's 'potential' response under a different treatment. Connections are made between the overlap quantified as the proportion of similar responses (PSR) and the PIQI. Given a bivariate normal model, the maximum PIQI is shown to be an upper bound for 1/2 the PSR. Additionally, the characterization of a conditional PSR allows for the PIQI boundaries to be developed within subgroups defined over observable covariates so that the subset contribution to treatment heterogeneity may be identified. The possibility of similar boundaries is explored outside the normal model using the skew normal distribution. Furthermore, a bivariate PIQI is developed along with its PSR counterpart to help characterize treatment heterogeneity resulting from a bivariate response such as the efficacy and safety of a treatment.