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  pages = {773--783},
  issn = {0022-006X},
  doi = {10.1037/a0036515},
  abstract = {Objective: This study provides guidance on how propensity score methods can be combined with moderation analyses (i.e., effect modification) to examine subgroup differences in potential causal effects in nonexperimental studies. As a motivating example, we focus on how depression may affect subsequent substance use differently for men and women. Method: Using data from a longitudinal community cohort study (N = 952) of urban African Americans with assessments in childhood, adolescence, young adulthood, and midlife, we estimate the influence of depression by young adulthood on substance use outcomes in midlife, and whether that influence varies by gender. We illustrate and compare 5 different techniques for estimating subgroup effects using propensity score methods, including separate propensity score models and matching for men and women, a joint propensity score model for men and women with matching separately and together by gender, and a joint male/female propensity score model that includes theoretically important gender interactions with matching separately and together by gender. Results: Analyses showed that estimating separate models for men and women yielded the best balance and, therefore, is a preferred technique when subgroup analyses are of interest, at least in this data. Results also showed substance use consequences of depression but no significant gender differences. Conclusions: It is critical to prespecify subgroup effects before the estimation of propensity scores and to check balance within subgroups regardless of the type of propensity score model used. Results also suggest that depression may affect multiple substance use outcomes in midlife for both men and women relatively equally. (PsycINFO Database Record (c) 2016 APA, all rights reserved). (journal abstract)},
  file = {/Users/NoahGreifer/Zotero/storage/FUTHAJER/Green and Stuart - 2014 - Examining moderation analyses in propensity score .pdf},
  journal = {Journal of Consulting and Clinical Psychology},
  keywords = {Blacks,causal effects,causality,depression,Drug Abuse,effect modification,Gender differences,Human Sex Differences,Major Depression,nonexperimental study,Observation Methods,observational data,substance abuse},
  number = {5},
  series = {Advances in {{Data Analytic Methods}}}
}





@article{kreifMethodsEstimatingSubgroup2012,
  title = {Methods for Estimating Subgroup Effects in Cost-Effectiveness Analyses That Use Observational Data},
  author = {Kreif, Noemi and Grieve, Richard and Radice, Rosalba and Sadique, Zia and Ramsahai, Roland and Sekhon, Jasjeet S.},
  year = {2012},
  month = nov,
  volume = {32},
  pages = {750--763},
  issn = {0272-989X},
  doi = {10.1177/0272989X12448929},
  abstract = {Decision makers require cost-effectiveness estimates for patient subgroups. In nonrandomized studies, propensity score (PS) matching and inverse probability of treatment weighting (IPTW) can address overt selection bias, but only if they balance observed covariates between treatment groups. Genetic matching (GM) matches on the PS and individual covariates using an automated search algorithm to directly balance baseline covariates. This article compares these methods for estimating subgroup effects in cost-effectiveness analyses (CEA). The motivating case study is a CEA of a pharmaceutical intervention, drotrecogin alfa (DrotAA), for patient subgroups with severe sepsis (n = 2726). Here, GM reported better covariate balance than PS matching and IPTW. For the subgroup at a high level of baseline risk, the probability that DrotAA was cost-effective ranged from 30\% (IPTW) to 90\% (PS matching and GM), at a threshold of \textsterling 20 000 per quality-adjusted life-year. We then compared the methods in a simulation study, in which initially the PS was correctly specified and then misspecified, for example, by ignoring the subgroup-specific treatment assignment. Relative performance was assessed as bias and root mean squared error (RMSE) in the estimated incremental net benefits. When the PS was correctly specified and inverse probability weights were stable, each method performed well; IPTW reported the lowest RMSE. When the subgroup-specific treatment assignment was ignored, PS matching and IPTW reported covariate imbalance and bias; GM reported better balance, less bias, and more precise estimates. We conclude that if the PS is correctly specified and the weights for IPTW are stable, each method can provide unbiased cost-effectiveness estimates. However, unlike IPTW and PS matching, GM is relatively robust to PS misspecification.},
  file = {/Users/NoahGreifer/Zotero/storage/2DX775MH/Web Appendix.pdf;/Users/NoahGreifer/Zotero/storage/IUAWWYQP/Kreif et al. - 2012 - Methods for Estimating Subgroup Effects in Cost-Ef.pdf},
  journal = {Medical Decision Making},
  language = {en},
  number = {6}
}





@article{wangRelativePerformancePropensity2018,
  title = {Relative {{Performance}} of {{Propensity Score Matching Strategies}} for {{Subgroup Analyses}}},
  author = {Wang, Shirley V. and Jin, Yinzhu and Fireman, Bruce and Gruber, Susan and He, Mengdong and Wyss, Richard and Shin, HoJin and Ma, Yong and Keeton, Stephine and Karami, Sara and Major, Jacqueline M. and Schneeweiss, Sebastian and Gagne, Joshua J.},
  year = {2018},
  month = aug,
  volume = {187},
  pages = {1799--1807},
  issn = {0002-9262},
  doi = {10.1093/aje/kwy049},
  abstract = {Abstract.  Postapproval drug safety studies often use propensity scores (PSs) to adjust for a large number of baseline confounders. These studies may involve ex},
  file = {/Users/NoahGreifer/Zotero/storage/PFIYDGIG/Wang et al. - 2018 - Relative Performance of Propensity Score Matching .pdf},
  journal = {American Journal of Epidemiology},
  language = {en},
  number = {8}
}


@article{zubizarretaMatchingBalancePairing2014,
  title = {Matching for Balance, Pairing for Heterogeneity in an Observational Study of the Effectiveness of for-Profit and Not-for-Profit High Schools in {{Chile}}},
  author = {Zubizarreta, Jos{\'e} R. and Paredes, Ricardo D. and Rosenbaum, Paul R.},
  year = {2014},
  month = mar,
  volume = {8},
  pages = {204--231},
  issn = {1932-6157},
  doi = {10.1214/13-AOAS713},
  file = {/Users/NoahGreifer/Zotero/storage/JACNSVBA/Zubizarreta et al. - 2014 - Matching for balance, pairing for heterogeneity in.pdf},
  journal = {The Annals of Applied Statistics},
  language = {en},
  number = {1}
}





@article{bennettBuildingRepresentativeMatched2020,
  title = {Building {{Representative Matched Samples With Multi}}-{{Valued Treatments}} in {{Large Observational Studies}}},
  author = {Bennett, Magdalena and Vielma, Juan Pablo and Zubizarreta, Jos{\'e} R.},
  year = {2020},
  month = may,
  pages = {1--29},
  issn = {1061-8600, 1537-2715},
  doi = {10.1080/10618600.2020.1753532},
  abstract = {In this article, we present a new way of matching in observational studies that overcomes three limitations of existing matching approaches. First, it directly balances covariates with multi-valued treatments without explicitly estimating the generalized propensity score. Second, it builds self-weighted matched samples that are representative of a target population by design. Third, it can handle large datasets, with hundreds of thousands of observations, in a couple of minutes. The key insights of this new approach to matching are balancing the treatment groups relative to a target population and positing a linear-sized mixed integer formulation of the matching problem. We formally show that this formulation is more effective than alternative quadratic-sized formulations, as its reduction in size does not affect its strength from the standpoint of its linear programming relaxation. We also show that this formulation can be used for matching with distributional covariate balance in polynomial time under certain assumptions on the covariates and that it can handle large datasets in practice even when the assumptions are not satisfied. This algorithmic characterization is key to handling large datasets. We illustrate this new approach to matching in both a simulation study and an observational study of the impact of an earthquake on educational attainment. With this approach, the results after matching can be visualized with simple and transparent graphical displays: while increasing levels of exposure to the earthquake have a negative impact on school attendance, there is no effect on college admission test scores. Supplementary materials for this article are available online.},
  file = {/Users/NoahGreifer/Zotero/storage/FPDB8NKK/Bennett et al. - 2020 - Building Representative Matched Samples With Multi.pdf},
  journal = {Journal of Computational and Graphical Statistics},
  language = {en}
}





@article{delosangelesresaDirectStableWeight2020,
  title = {Direct and Stable Weight Adjustment in Non-Experimental Studies with Multivalued Treatments: Analysis of the Effect of an Earthquake on Post-Traumatic Stress},
  author = {{de los Angeles Resa}, Mar{\'i}a and Zubizarreta, Jos{\'e} R.},
  year = {2020},
  month = apr,
  volume = {n/a},
  publisher = {{John Wiley \& Sons, Ltd}},
  issn = {0964-1998},
  doi = {10.1111/rssa.12561},
  abstract = {Summary In February 2010, a massive earthquake struck Chile, causing devastation in certain parts of the country, affecting other areas, and leaving territories untouched. 2 months after the earthquake, Chile's Ministry of Social Development reinterviewed a representative subsample of its National Socioeconomic Characterization Survey, which had been completed 2 months before the earthquake, thereby creating a prospective longitudinal survey with detailed information of the same individuals before and after the earthquake. We use a new weighting method for non-experimental studies with multivalued treatments to estimate the effect of levels of exposure to the earthquake on post-traumatic stress. Unlike common weighting approaches for multivalued treatments, this new method does not require explicit modelling of the generalized propensity score and instead focuses on directly balancing the covariates across the multivalued treatments with weights that have minimum variance. As a result, the weighting estimator is stable and approximately unbiased. Furthermore, the weights are constrained to avoid model extrapolation. We illustrate this new method in a simulation study, with both categorical and continuous treatments. The results show that directly targeting balance instead of explicitly modelling the treatment assignment probabilities tends to provide the best results in terms of bias and root-mean-square error. Using this method, we estimate the effect of the intensity of the earthquake on post-traumatic stress. We implement this method in the new package msbw for R.},
  file = {/Users/NoahGreifer/Zotero/storage/RN4P9UNF/de los Angeles Resa and Zubizarreta - 2020 - Direct and stable weight adjustment in non-experim.pdf},
  journal = {Journal of the Royal Statistical Society: Series A (Statistics in Society)},
  keywords = {Causal inference,Inverse probability weights,Observational studies,Propensity score},
  number = {n/a}
}




@article{visconti2018,
	title = {Handling Limited Overlap in Observational Studies with Cardinality Matching},
	author = {{Visconti}, {Giancarlo} and {Zubizarreta}, {Jos{\'e} R.}},
	year = {2018},
	date = {2018},
	journal = {Observational Studies},
	pages = {217--249},
	volume = {4},
	number = {1},
	doi = {10.1353/obs.2018.0012},
	url = {https://doi.org/10.1353/obs.2018.0012},
	langid = {en}
}


@article{greiferChoosingEstimandWhen2021,
  title = {Choosing the {{Estimand When Matching}} or {{Weighting}} in {{Observational Studies}}},
  author = {Greifer, Noah and Stuart, Elizabeth A.},
  year = {2021},
  month = jun,
  journal = {arXiv:2106.10577 [stat]},
  eprint = {2106.10577},
  eprinttype = {arxiv},
  primaryclass = {stat},
  abstract = {Matching and weighting methods for observational studies require the choice of an estimand, the causal effect with reference to a specific target population. Commonly used estimands include the average treatment effect in the treated (ATT), the average treatment effect in the untreated (ATU), the average treatment effect in the population (ATE), and the average treatment effect in the overlap (i.e., equipoise population; ATO). Each estimand has its own assumptions, interpretation, and statistical methods that can be used to estimate it. This article provides guidance on selecting and interpreting an estimand to help medical researchers correctly implement statistical methods used to estimate causal effects in observational studies and to help audiences correctly interpret the results and limitations of these studies. The interpretations of the estimands resulting from regression and instrumental variable analyses are also discussed. Choosing an estimand carefully is essential for making valid inferences from the analysis of observational data and ensuring results are replicable and useful for practitioners.},
  archiveprefix = {arXiv},
  url = {https://arxiv.org/abs/2106.10577},
  keywords = {Statistics - Methodology},
}





@book{rosenbaumDesignObservationalStudies2010,
  title = {Design of Observational Studies},
  author = {Rosenbaum, Paul R.},
  year = {2010},
  series = {Springer Series in Statistics},
  publisher = {{Springer}},
  address = {{New York}},
  isbn = {978-1-4419-1212-1 978-1-4419-1213-8},
  langid = {english},
  lccn = {QA279 .R669 2010},
  keywords = {Analysis of variance,Beobachtungsstudie,Experimental design},
  annotation = {OCLC: ocn444428720},
  file = {/Users/NoahGreifer/Zotero/storage/WQDICIVJ/Rosenbaum - 2010 - Design of observational studies.pdf}
}





@article{rubinBiasReductionUsing1980,
  ids = {rubinBiasReductionUsing1980a,rubinBiasReductionUsing1980b},
  title = {Bias {{Reduction Using Mahalanobis-Metric Matching}}},
  author = {Rubin, Donald B.},
  year = {1980},
  journal = {Biometrics},
  volume = {36},
  number = {2},
  pages = {293--298},
  publisher = {{[Wiley, International Biometric Society]}},
  issn = {0006-341X},
  doi = {10.2307/2529981},
  abstract = {Monte Carlo methods are used to study the ability of nearest-available, Mahalanobis-metric matching to make the means of matching variables more similar in matched samples than in random samples.},
  file = {/Users/NoahGreifer/Zotero/storage/G3ES5W6C/Rubin - 1980 - Bias Reduction Using Mahalanobis-Metric Matching.pdf}
}





@article{ripolloneImplicationsPropensityScore2018,
  title = {Implications of the {{Propensity Score Matching Paradox}} in {{Pharmacoepidemiology}}},
  author = {Ripollone, John E. and Huybrechts, Krista F. and Rothman, Kenneth J. and Ferguson, Ryan E. and Franklin, Jessica M.},
  year = {2018},
  month = sep,
  journal = {American Journal of Epidemiology},
  volume = {187},
  number = {9},
  pages = {1951--1961},
  issn = {0002-9262},
  doi = {10.1093/aje/kwy078},
  abstract = {Abstract.  Recent work has demonstrated that propensity score matching may lead to increased covariate imbalance, even with the corresponding decrease in propen},
  langid = {english},
  file = {/Users/NoahGreifer/Zotero/storage/G2B49Z3J/Ripollone et al. - 2018 - Implications of the Propensity Score Matching Para.pdf;/Users/NoahGreifer/Zotero/storage/9SYCKYA5/4994700.html}
}




@article{savjeGeneralizedFullMatching2021,
	title = {Generalized Full Matching},
	author = {{Sävje}, Fredrik and Higgins, Michael J. and Sekhon, Jasjeet S.},
	year = {2021},
	month = {10},
	date = {2021-10},
	journal = {Political Analysis},
	pages = {423--447},
	volume = {29},
	number = {4},
	doi = {10.1017/pan.2020.32},
	url = {http://www.cambridge.org/core/journals/political-analysis/article/generalized-full-matching/3DA71D8BEDA6F02B5D36457E114C79B6},
	note = {Publisher: Cambridge University Press},
	langid = {en}
}

@book{savjeQuickmatchQuickGeneralized2018,
	title = {quickmatch: Quick generalized full matching},
	author = {Savje, Fredrik and Sekhon, Jasjeet and Higgins, Michael},
	year = {2018},
	date = {2018},
	url = {https://CRAN.R-project.org/package=quickmatch}
}


@article{austinDoublePropensityscoreAdjustment2017,
	title = {Double propensity-score adjustment: A solution to design bias or bias due to incomplete matching},
	author = {Austin, Peter C.},
	year = {2017},
	month = {02},
	date = {2017-02-01},
	journal = {Statistical Methods in Medical Research},
	pages = {201--222},
	volume = {26},
	number = {1},
	doi = {10.1177/0962280214543508},
	url = {https://doi.org/10.1177/0962280214543508},
	note = {Publisher: SAGE Publications Ltd STM}
}

@article{snowdenImplementationGComputationSimulated2011,
	title = {Implementation of G-Computation on a Simulated Data Set: Demonstration of a Causal Inference Technique},
	author = {Snowden, Jonathan M. and Rose, Sherri and Mortimer, Kathleen M.},
	year = {2011},
	month = {04},
	date = {2011-04-01},
	journal = {American Journal of Epidemiology},
	pages = {731--738},
	volume = {173},
	number = {7},
	doi = {10.1093/aje/kwq472},
	url = {https://doi.org/10.1093/aje/kwq472},
	langid = {en}
}

@article{schaferAverageCausalEffects2008,
	title = {Average causal effects from nonrandomized studies: A practical guide and simulated example},
	author = {Schafer, Joseph L. and Kang, Joseph},
	year = {2008},
	month = {12},
	date = {2008-12},
	journal = {Psychological Methods},
	pages = {279--313},
	volume = {13},
	number = {4},
	doi = {10.1037/a0014268},
	url = {https://doi.org/10.1037/a0014268}
}

@article{bodoryFiniteSamplePerformance2020,
	title = {The Finite Sample Performance of Inference Methods for Propensity Score Matching and Weighting Estimators},
	author = {Bodory, Hugo and Camponovo, Lorenzo and Huber, Martin and Lechner, Michael},
	year = {2020},
	month = {01},
	date = {2020-01-02},
	journal = {Journal of Business & Economic Statistics},
	pages = {183--200},
	volume = {38},
	number = {1},
	doi = {10.1080/07350015.2018.1476247},
	url = {https://doi.org/10.1080/07350015.2018.1476247},
	langid = {en}
}

@article{abadieFailureBootstrapMatching2008,
	title = {On the Failure of the Bootstrap for Matching Estimators},
	author = {Abadie, Alberto and Imbens, Guido W.},
	year = {2008},
	date = {2008},
	journal = {Econometrica},
	pages = {1537--1557},
	volume = {76},
	number = {6},
	url = {https://www.jstor.org/stable/40056514},
	note = {Publisher: [Wiley, Econometric Society]}
}

@article{cohnProfileMatchingGeneralization2021,
  title = {Profile {{Matching}} for the {{Generalization}} and {{Personalization}} of {{Causal Inferences}}},
  author = {Cohn, Eric R. and Zubizarreta, Jose R.},
  year = {2021},
  month = may,
  journal = {arXiv:2105.10060 [stat]},
  eprint = {2105.10060},
  eprinttype = {arxiv},
  primaryclass = {stat},
  abstract = {We introduce profile matching, a multivariate matching method for randomized experiments and observational studies that finds the largest possible self-weighted samples across multiple treatment groups that are balanced relative to a covariate profile. This covariate profile can represent a specific population or a target individual, facilitating the tasks of generalization and personalization of causal inferences. For generalization, because the profile often amounts to summary statistics for a target population, profile matching does not require accessing individual-level data, which may be unavailable for confidentiality reasons. For personalization, the profile can characterize a single patient. Profile matching achieves covariate balance by construction, but unlike existing approaches to matching, it does not require specifying a matching ratio, as this is implicitly optimized for the data. The method can also be used for the selection of units for study follow-up, and it readily applies to multi-valued treatments with many treatment categories. We evaluate the performance of profile matching in a simulation study of generalization of a randomized trial to a target population. We further illustrate this method in an exploratory observational study of the relationship between opioid use treatment and mental health outcomes. We analyze these relationships for three covariate profiles representing: (i) sexual minorities, (ii) the Appalachian United States, and (iii) a hypothetical vulnerable patient. We provide R code with step-by-step explanations to implement the methods in the paper in the Supplementary Materials.},
  archiveprefix = {arXiv},
  langid = {english},
  keywords = {Statistics - Methodology},
  file = {/Users/NoahGreifer/Zotero/storage/PFFUW2AF/Cohn and Zubizarreta - 2021 - Profile Matching for the Generalization and Person.pdf;/Users/NoahGreifer/Zotero/storage/QX8CY5HV/Cohn and Zubizarreta - 2021 - Profile Matching for the Generalization and Person.pdf}
}

