Statistical Modeling And Application For Multi-treatment Bayesian Propensity Score Analysis

Background:For more than two decades, propensity score has been proved to be a more effective meansof dealing with large numbers of confounding covariates in observational studies. Themethod includes treatment model in the first step and outcome model throughsubclassification, regression, weighting and matching in the second step. However,propensity score did not take the uncertainty of the estimated propensity score intoconsideration and could not contain prior information in treatment effect estimation.Introducing Bayesian approach in propensity score analysis could deal with the aboveproblems. There had been several studies applying Bayesian approach in propensity scoreanalysis since2008. However, most of the studies were only focused on binary treatment.Bayesian approach in multi-treatment propensity score analysis remains unclear.Aim:This thesis aims to explore alternative methods applying Bayesian approach inmulti-treatment propensity score analysis, including both nominal and ordinal categoricaltreatments, and find out the optimal method for controlling the confounding factors ofmultiple treatment groups.Methods:1Statistical modeling:(1) We proposed an “intermediate Bayesian generalized propensityscore analysis” with Bayesian treatment model combined with conventional outcomemodel via regression, subclassification or weighting methods. When treatment wasnominal categorical variable, we used Bayesian multinomial logistic regression to estimatepropensity score in the first step with a conventional multiple linear regression model viaregression on the generalized propensity score in the second step to obtain the treatmenteffect. When treatment was ordinal categorical treatment variable, we used Bayesianordinal logistic regression to estimate propensity score in the first step and employed apropensity score subclassification together with weighting and conventional multiple linearregression model in the second step.(2) We also proposed a “two-step Bayesiangeneralized propensity score analysis” with Bayesian treatment model in the first stepcombined with Bayesian regression as the Bayesian outcome model in the second step. Themethods of estimating generalized propensity score were the same as above.2Data simulation: Based on the basic data frame of observational studies, we simulated 1000data sets including treatments, confounding factors and outcome variables, with threedifferent sample sizes N=100,250and500.When treatment variable was three-levelnominal categorical variable, we considered two sets of different true treatment effects1=-0.4,2=0.3and1=-1.5,2=2.5and four different prior precisions of treatment modelB t=0,1,10,100. When treatment was four-level ordinal categorical variable, we considered twotrue treatment effects=-0.4and=2.5, three different prior precisions of treatment modelB=1,10,100and three different applications (regression,subclassification andweighting).For two-step methods, we also considered0,1,10,100as four different priorprecisions of outcome model, and non-informative and true parameter values as twodifferent prior means.3Case study: Data came from the “Gastrointestinal diseses epidemiological survey inmainland, China” carried out by Department of Health Statistics, Second Military MedicalUniversity.We evaluated the effect of self-rated working pressure and marital status oneight scales of Health related qualtiy of life (HRQoL) using conventional and intermediatebayesian generalized propensity score,and compared with the results of traditional multplelinear regression.Results:1Results of simulation: Bias, the absolute difference of estimated and true value, wasused to evaluate the accuracy of treatment effect estimation. The smaller the bias, the moreaccurate the results. The corresponding95%confidence interval was used to determine thestatistical significance of treatment effect estimates. With respect to bias in treatment effectestimates, conventional generalized propensity score with1000replication offers moreprecise estimates than one replication. When N increased to500, Bayesian andconventional methods obtained similar results, with the bias around0.01or0.02. Forstandard error, Bayesian methods generated larger standard error than conventionalmethods.(1)Results of Intermediate Bayesian generalized propensity score:a) Treatment as nominal categorical variable: When1=-1.5,2=2.5and N=100,Bayesian methods withB t=0got more accurate treatment effect estimates thanconventional methods (For bayesian methods:bias of1and2were0.05and0.11respectively; for conventional methods:0.21and0.25). b) Treatment as ordinalcategorical variable: When=-0.4, after employing regression, results of Bayesianmethod with any prior precision and conventional methods were similar at N=100, with all biases smaller than0.01.After employing weighting, Bayesian method with prior precisionB t=100performed better than other prior precisions(bias was0.06whenB t=1,0.05whenB t=10,and0.01whenB t=100), and the result was closer to the true treatmenteffect than conventional method(bias=0.03).According to corresponding95%confidenceintervals, all the treatment effect estimation were statistically significance. After employingsubclassification, Bayesian method with prior precisionB t=10performed better thanother prior precisions, and the result was similar with conventional method, but withoutany statistically significance. When increased to2.5, the pattern of estimates for bothBayesian and conventional method was same with=-0.4.(2)Results of Two-step Bayesian generalized propensity score:In our study, the results were similar when using non-informative and true parameter valueas prior information. a) Treatment as nominal categorical variable: When1=-1.5,2=2.5and N=100, Bayesian methods withB t=0andB t=0got more accurate treatmenteffect estimates than conventional methods (For bayesian methods:bias of1and2were0.04and0.11respectively; for conventional methods:0.21and0.25). All theestimates were statistically significance. b) Treatment as ordinal categorical variable:With respect to bias of treatment effect estimates, Bayesian method with0as priorprecision in outcome model performed similar with conventional method under theconditions of all sample sizes and all kinds of prior precisions in treatment model whenemploying regression and subclassificaiton.All the biases were around0.01.However, theweighting approach in Bayesian method got less accurate estimates than conventionalmethod, with the larger the prior precision in outcome model, the larger the bias.2Results of case study:(1) Effect of Self rated working pressure on HRQoL: Theresults were similar among the three methods. Self rated working pressure had impact onVT and MH scales. When self-rated working pressure increased one degree, the VT scorewould drop1.32points and MH score would drop2points.(2) Effect of marital status onHRQoL: The result of multiple linear regression indicated that marital status would affectsuch five scales as PF,VT,SF,RE and MH.However, after controlling all the confoundingfactors, Bayesian and conventional generalized propensity score analysis provided adifferent results compared with multiple linear regression without any adjustment. Theresults indicated that marital status mainly affect four scales:GH,VT,RE and MH.Compared with those married, those divorced or separated or widowed would drop3points in GH,4points in VT, even6points in RE and5points in MH. Discussions:(1)Bayesian method showed a slight advantage in small sample size.(2) After consideringthe uncertainty of estimated generalized propensity score, the standard errors of Bayesianmethod were larger than conventional method.(3) When treatment was nominalcategorical variable, on one hand, Bayesian method with0as prior precision could obtainmore accurate estimates than conventional method when there was big difference amongdifferent true treatment effects; on the other hand, Bayesian method could greatly reducedbiases when there was large bias in the estimates of conventional method.(4) Whentreatment was ordinal categorical variable, we recommended intermediate Bayesiangeneralized propensity score employing regression or weighting under higher priorprecision, or two-step Bayesian generalized propensity score employing regression orsubclassification both under0as prior precision in outcome model.(5) The results ofintermediate Bayesian generalized propensity score and two-step Bayesian generalizedpropensity score were similar in our study. In practice, appropriate method withcorresponding optimal conditions could be chosen according to the real situation.