Matching estimators for average treatment effects are widely used in the binary treatment setting, in which missing potential outcomes are imputed as the average of observed outcomes of all matches for each unit. With more than two treatment groups, however, estimation using matching requires additional techniques. In this paper, we propose a nearest-neighbors matching estimator for use with multiple, nominal treatments, and use simulations to show that this method is precise and has coverage levels that are close to nominal. In addition, we implement the proposed inference methods to examine the effects of different medication regimens on long-term pain for patients experiencing motor vehicle collision.
RoystonPParmarMKQianW. Novel designs for multi-arm clinical trials with survival outcomes with an application in ovarian cancer. Stat Med2003; 22: 2239–2256.
2.
JakiT. Multi-arm clinical trials with treatment selection: what can be gained and at what price?Clinical Invest2015; 5: 393–399.
3.
Dunnett CW. Selection of the best treatment in comparison to a control with an application to a medical trial. In: Santner TJ and Tamhane AC (eds) Design of experiments: ranking and selection. New York: Marcel Dekker, 1984, pp.47–66.
4.
D’AgostinoRBHeerenTC. Multiple comparisons in over-the-counter drug clinical trials with both positive and placebo controls. Stat Med1991; 10: 1–6.
5.
RosnerB. Fundamentals of biostatistics, 6 ed. Pacific Grove, CA: Duxbury Press, 2005.
RaoCVSwarupchandU. Multiple comparison procedures-a note and a bibliography. J Stat2009; 16: 66–109.
8.
NiskaRBhuiyaFXuJ. National hospital ambulatory medical care survey: 2007 emergency department summary. National Health Stat Rep2010; 26: 1–31.
9.
BeaudoinFL, et al.Persistent pain after motor vehicle collision: comparative effectiveness of opioids vs nonsteroidal antiinflammatory drugs prescribed from the emergency department–a propensity matched analysis. Pain2017; 158: 289–295.
10.
VroomenJMBosmansJEEekhoutI, et al.The cost-effectiveness of two forms of case management compared to a control group for persons with dementia and their informal caregivers from a societal perspective. PloS One2016; 11: e0160908–e0160908.
11.
PuriRNichollsSJShaoM, et al.Impact of statins on serial coronary calcification during atheroma progression and regression. J Am College Cardiol2015; 65: 1273–1282.
12.
CochranWGRubinDB. Controlling bias in observational studies: a review. Sankhya Ser A1973; 35: 417–446.
13.
RubinDB. Matching to remove bias in observational studies. Biometrics1973; 29: 159–183.
14.
RubinDB. The use of matched sampling and regression adjustment to remove bias in observational studies. Biometrics1973; 29: 185–203.
15.
RosenbaumPRRubinDB. The central role of the propensity score in observational studies for causal effects. Biometrika1983; 70: 41–55.
16.
ImaiKvan DykDA. Causal inference with general treatment regimes: generalizing the propensity score. J Am Stat Assoc2004; 99: 854–866.
17.
StuartEA. Matching methods for causal inference: a review and a look forward. Stat Sci2010; 25: 1–21.
18.
LopezMJGutmanR. Estimation of causal effects with multiple treatments: a review and new ideas. Stat Sci2017; 32: 432–454.
19.
LechnerM. Identification and estimation of causal effects of multiple treatments under the conditional independence assumption. Econometric Eval Labour Market Policies2001; 13: 43–58.
20.
LechnerM. Program heterogeneity and propensity score matching: an application to the evaluation of active labor market policies. Rev Econom Stat2002; 84: 205–220.
21.
RassenJASolomonDHGlynnRJ, et al.Simultaneously assessing intended and unintended treatment effects of multiple treatment options: a pragmatic matrix design. Pharmacoepidemiol Drug Safety2011; 20: 675–683.
22.
RosenbaumPR. Optimal matching for observational studies. J Am Stat Assoc1989; 84: 1024–1032.
23.
ZubizarretaJR. Using mixed integer programming for matching in an observational study of kidney failure after surgery. J Am Stat Assoc2012; 107: 1360–1371.
24.
Karp RM. Reducibility among combinatorial problems. In: Miller RE and Thatcher JW (eds) Complexity of computer computations. New York, NY: Plenum, 1972, pp.85–103.
25.
AbadieAImbensGW. Large sample properties of matching estimators for average treatment effects. Econometrica2006; 74: 235–267.
26.
HillJReiterJ. Interval estimation for treatment effects using propensity score matching. Stat Med2006; 25: 2230–2256.
27.
GutmanRRubinDB. Estimation of causal effects of binary treatments in unconfounded studies with one continuous covariate. Stat Meth Med Res2017; 26: 1199–1215.
28.
AbadieAImbensGW. On the failure of the bootstrap for matching estimators. Econometrica2008; 76: 1537–1557.
29.
AustinPCSmallDS. The use of bootstrapping when using propensity-score matching without replacement: a simulation study. Stat Med2014; 33: 4306–4319.
30.
McCaffreyDFGriffinBAAlmirallD, et al.A tutorial on propensity score estimation for multiple treatments using generalized boosted models. Stat Med2013; 32: 3388–3414.
31.
GutmanRRubinDB. Estimation of causal effects of binary treatments in unconfounded studies. Stat Med2015; 34: 3381–3398.
32.
YangSImbensGWCuiZ, et al.Propensity score matching and subclassification in observational studies with multi-level treatment. Biometrics2016; 72: 1055–1065.
33.
AbadieAImbensGW. Bias-corrected matching estimators for average treatment effects. J Business Economic Stat2011; 29: 1–11.
34.
RubinDB. Estimating causal effects of treatments in randomized and nonrandomized studies. J Educational Psychol1974; 56: 688–701.
35.
RubinDB. Bayesian inference for causal effects: the role of randomization. Ann Stat1978; 6: 34–58.
36.
RubinDB. Randomization analysis of experimental data: the Fisher randomization test comment. J Am Stat Assoc1980; 75: 591–593.
37.
ImbensGWRubinDB. Causal inference for statistics, social, and biomedical sciences: an introduction, Cambridge: Cambridge University Press, 2015.
38.
SiddiqueAASchnitzerMEBahamyirouA, et al.Causal inference with multiple concurrent medications: A comparison of methods and an application in multidrug-resistant tuberculosis. Stat Meth Med Res. Epub ahead of print 2018. DOI: 10.1177/0962280218808817.
39.
FengPZhouXZouQ, et al.Generalized propenstiy score for estimating the average treatment effect of multiple treatments. Stat Med2011; 31: 681–697.
40.
ImbensGW. The role of the propensity score in estimating dose-response functions. Biometrika2000; 87: 706–710.
41.
KangJDYSchaferJL. Demystifying double robustness: a comparison of alternative strategies for estimating a population mean from incomplete data. Stat Sci2007; 22: 523–539.
42.
ScotinaADGutmanR. Matching algorithms for causal inference with multiple treatments. Stat Med2019, pp. 1–29. DOI: 10.1002/sim.8147–1–29. DOI: 10.1002/sim.8147.
43.
SekhonJ. Multivariate and propensity score matching software with automated balance optimization: the Matching package for R. J Stat Software2011; 42: 1–52.
44.
R Studio Team. Rstudio: integrated development environment for R, 2015, http://www.rstudio.com/.
45.
AustinPC. Type I error rates, coverage of confidence intervals, and variance estimation in propensity-score matched analyses. Int J Biostat2009; 5: 1–21.
46.
DunnOJ. Multiple comparisons among means. J Am Stat Assoc1961; 56: 52–64.
47.
RubinDB. Using multivariate matched sampling and regression adjustment to control bias in observational studies. J Am Stat Assoc1979; 74: 318–328.
48.
CangulMZChretienYRGutmanR, et al.Testing treatment effects in unconfounded studies under model misspecification: logistic regression, discretization, and their combination. Stat Med2009; 28: 2531–2551.
49.
BeaudoinFLHutmanRZhaiW, et al.Racial differences in presentations and predictors of acute pain after motor vehicle collision. Pain2018; 159: 1056–1063.
50.
van BuurenS. Multiple impiutation of discrete and continuous data by fully conditional specification. Stat Meth Med Res2007; 16: 219–242.
51.
RubinDB. Multiple imputation for nonresponse in surveys2004; Vol. 81Hoboken, NJ: John Wiley & Sons.
52.
SuYA G, HillJ, et al.Multiple imputation with diagnostics (mi) in R: opening windows into the black box. J Stat Software2011; 45: 1–31.
53.
EndersCK. Applied missing data analysis, New York, NY: Guilford Press, 2010.
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