Matt Laidler, MPH, MA Acute and Communicable Disease Program

Oregon Health Authority

SOSUG, April 17, 2014

The conditional probability of being assigned to a particular treatment given a vector of observed covariates (Rosenbaum and Rubin, 1983)

The probability of treatment assignment conditional on observed characteristics (Austin, 2011)

When you want to estimate the effect (i.e., difference in means, difference in proportions, risk reduction) of a treatment/intervention/exposure on an outcome among individuals, and you [only] have observational data

Examples: • Does Medication X prevent Disease Y? • Did Program A reduce alcohol consumption among Population K?

In a nutshell: emulates randomized controlled trials (RCTs)

Observational studies vs. RCTs Causality Reduce or eliminate confounding (selection bias) Balance treatment groups on covariates Can allow direct comparison of effects between

treatment subjects (versus regression adjustment of measured covariates)

Propensity score-matched analysis (PSM) Stratified analysis (stratified on the PS) Covariate adjustment or regression models

Estimate the scores Match (treatment and non-treatment pair) Check for balance of treatment groups Calculate effect of treatment/intervention/exposure Check sensitivity of effect

Estimate a logistic regression model • Treatment is the response variable (treatment=1, non-treatment=0) • Measured covariates are the predictors

proc logistic data= treatments descend; class agecat race bmi; model treatment= agecat race sex bmi chronicdis1 chronicdis2 chronicdis3 chronicdis4 chronicdis5; output out=allscores prob=prob; run;

What covariates (predictor variables) to select? • All measured baseline covariates • Variables that might effect the outcome, treatment assignment, or

both • Interactions (?)

Model significance, diagnostics…? Outcome: a dataset with the estimated propensity

scores

proc univariate data= allscores plot; var prob; class treatment; histogram; run;

Match 1 treated subject to N non-treated subjects (i.e., 1:1, 1:M)

Matched subjects have similar values of estimated PS Greedy (nearest best match), or Optimal (minimizes

within-pair difference on PS) Replacement, or no replacement? Use macros…widely available

%gmatch and %vmatch (Mayo Clinic, http://www.mayo.edu/research/departments-divisions/department-health-sciences-research/division-biomedical-statistics-informatics/software/locally-written-sas-macros)

%OneToManyMTCH (Parsons, 2004 http://www2.sas.com/proceedings/sugi29/165-29.pdf)

%mahalanobis (Wuwei et al, 2006 http://www.lexjansen.com/pharmasug/2006/publichealthresearch/pr05.pdf)

%MACRO OneToManyMTCH ( Lib, /* Library Name */ Dataset, /* Data set of all your study subjects */ depend, /* Dependent variable that indicates Tx group, 1 for Tx, 0 for no Tx*/ SiteN, /* Site or Hospital ID */ PatientN, /* Patient ID */ matches, /* Output data set of matched pairs */ NoContrls); /* Number of controls to match to each case */ /* …just to show the macro parameters that are required… this macro assumes your propensity score variable is called “prob”, but you can change the code of course…*/ %OneToManyMTCH(work,allscores,treatment,site,patientn,Matches_1,1);

d= (ptreatment – pcontrol)/√((ptreatment(1- ptreatment) + pcontrol(1- pcontrol)) /2) ptreatment and pcontrol are the prevalence of the dichotomous

variable in the treated and untreated groups Compares prevalence of a baseline covariate between

treatment groups in the matched sample Standard difference < 0.1 suggests insignificant difference

in the prevalence of a covariate between treated and untreated groups (i.e., “Balance”)

Not influenced by sample size

Only dichotomous formula shown here. For continuous formula, see Austin, 2011 (full reference on last slide)

/*macro for standard diff for dichotomous variables*/ %macro binSD(var=); proc means mean data=matches_1 noprint; var &var; by treatment; output out=outmean (keep = treatment mean) mean = mean; run; data treatment 20; set outmean; if treatment = 0; mean_0 = mean; keep mean_0; run; data treatment 21; set outmean; if treatment = 1; mean_1 = mean; keep mean_1; run;

data balance; length label $ 30; merge treatment 20 treatment 21; d1 = (mean_1 - mean_0)/ sqrt((mean_1*(1-mean_1) + mean_0*(1-mean_0))/2); d2 = round(abs(d1),0.001); keep d2 ; run; proc append data= balance base=j force; run; %mend binSD; /*call the macro…do this for each dichotomous variable*/ %binSD(var=bmi); proc print data=j; title 'Standard differences, PS matched sample'; run;

Code adapted from: Analysis of Observational Health Care Data Using SAS. By Douglas E. Faries, Robert L. Obenchain, Josep Maria Haro, and Andrew C. Leon. 2010, SAS institute.

Treatment (N & (%)) No treatment (N & (%)) Standard Difference (matched)

StandardDifference (unmatched)

Age categories < 35 5 (0.75) 6 (0.90) 0.017 0.539 35-44 19 (2.84) 11 (1.64) 0.081 0.3 45-54 54 (8.06) 56 (8.36) 0.011 0.227 55-64 91(13.58) 93 (13.88) 0.009 0.078 ≥ 65 501 (74.78) 504 (75.22) 0.01 0.549 Sex Female 349 (47.91) 380 (56.72) - - Male 321(52.09) 290 (43.28) 0.093 0.157 Race White 535 (79.85) 535 (79.85) 0 0.549 Black 107 (15.97) 106 (15.82) 0.004 0.214 Other 29 (4.33) 30 (4.48) 0.007 0.019 Etc., etc.,….

Matching: matched pairs are more like each other than randomly selected subjects….outcomes should not be considered independent within matched pairs

Methods that assume independence should probably be avoided

Dichotomous outcome variable: McNemar’s test, Kaplan-Meier log rank test, Cox regression for matched data (if time to event data), relative risk (Agresti and Min, 2004), number needed to treat (NNT)

Continuous: Paired t-test, Wilcoxon Signed Rank test (nonparametric)

*Restructure your data first!; data treat1 treat2; set matches_1; if treatment = 1 then output treat1; if treatment = 0 then output treat2; run; /*then sort these datasets (code not shown) */ data comparison_matched; merge treat1(rename = (hospitalized= hospitalized1 died=died1/* etc., do this for each outcome variable in the dataset*/) treat2(rename = (hospitalized= hospitalized2 died=died2 /* etc., do this for each outcome variable in the dataset*/) by match_1; run;

proc freq data=comparison_matched; exact agree; tables died1*died2 /agree ; title "McNemar's test for comparing outcomes among matched pairs"; run;

Answers: could an unmeasured covariate bias the results of the study?

“Gamma,” (Rosenbaum, 2005) If there was an unmeasured

variable that increased the odds of treatment (Gamma) by X%, could this variable potentially account for the treatment effect?

%let a= /*sum of the discordant pairs*/; %let b= /*number of pairs where untreated had

outcome (e.g. death), treated did not*/; data g; do gamma_init = 0 to 50; gamma = 1 + gamma_init/20; p_plus = gamma/(1 + gamma); p_neg = 1/(1 + gamma); p_upper = 2*(1 - probbnml(p_plus,&a, &b) ); p_lower = 2*(1 - probbnml(p_neg,&a, &b) ); output; end; run; proc print data=g noobs; var gamma p_lower p_upper; title "Sensitivity analysis for McNemar's test"; run;

Adapted from: Analysis of Observational Health Care Data Using SAS. By Douglas E. Faries, Robert L. Obenchain, Josep Maria Haro, and Andrew C. Leon. 2010, SAS institute.

Sensitivity Analysis in Observational Studies. By Paul R. Rosenbaum. In Encyclopedia of Statistics in Behavioral Science, 2005, John Wiley & Sons, LTD

You can still adjust for covariates (but, do you need to?) Expectation of balance between measured and

unmeasured covariates (like RCTs) using PS methods….but doesn’t guarantee all bias is eliminated.

Unmeasured covariates…a reason to check sensitivity. PS methods assume that all covariates that effect treatment assignment have been measured (however, regression-based approaches also assume this…).

Austin PC. An Introduction to Propensity Score Methods for Reducing the Effects of Confounding in Observational Studies. Multivariate Behav Res. 2011 May;46(3):399-424.

Analysis of Observational Health Care Data Using SAS. By Douglas E. Faries, Robert L. Obenchain, Josep Maria Haro, and Andrew C. Leon. 2010, SAS institute.

Parsons LS, 2004. Performing a 1:N Case-Control Match on Propensity Score. SUGI 29, paper 165-29.

Questions? Matthew.Laidler@state.or.us