Propensity Scores

1. Propensity Scores October 2012 Alexander M. Walker MD, DrPH Extensive parts of this presentation incorporate the work of John D. Seeger, PharmD, DrPH

2. “In mathematics, you don't understand things. You just get used to them.” - John von Neumann 1903-1957

3. Research Goal • Compare two treatments with respect to a health or economic outcome • “Counterfactual” ideal • If the same people had received B instead of A, how would their outcomes have differed? • What is achievable: “similar” not “same” • Comparable treatment groups • … insofar as you can tell!

4. Pictures for Confounding

5. Comparison of Heterogeneous Groups E1 E2

6. Internal Composition May Differ E1 E2

7. Risks that Depend on Subgroup Status E1 E2 15% 15% Affected individuals 50% 50%

8. Internal Risk Factor Heterogeneity Creates an Differences in Group Risk E1 E2 These differences in risk are due to the covariate structure of compared populations, not to the differential effects of E1 and E2

9. Propensity Scores to Create Populations with Similar Covariate Structure

10. Covariate Heterogeneity E1 E2 E1 has more Yellow E2has more Gray

11. Covariate Status as a Predictor of Treatment E2 Gray predicts E2 Yellow predicts E1 E2 E1 E1

12. Propensity Scores • PS is the predicted probability of treatment, given all the covariates • Matching on the PS creates study populations that have balance on the covariates • Perfect for a single, dichotomous covariate • Not perfect, but very good for multiple covariates

13. Propensity for Covariate Patterns Think of orange and green as two distinct covariate patterns that have the same predicted Pr(E1). E1 E2 Pr(E1)=x Pr(E1)=x

14. Conditioning on Propensity Permits Unconfounded Comparisons Gathering subjects with identical propensity puts all individuals with covariate patterns orange and green into the same stratum. E1 E2 At a given propensity level, there is no association between treatment and covariate patterns. Pr(E1)=x Pr(E1)=x

15. Formal Expression Propensity(x)  P(T=1|x) = E(T|x) The propensity associated with level x of the covariate X is the probability that treatment is present (equivalently, is “B” as opposed to “A”), given level x, and this is in turn equal to the expected value of treatment, given x. Note that the definition does not specify the parametric form of thePropensity(x) . The examples in this talk use a logistic function; others -- including nonparametric functions -- are also used. Notation. A single capital letter denotes a variable, a single lower case letter denotes a particular value for that variable.

16. Probability Calculus • Under propensity matching, how do x (covariate status) and t (treatment status) relate to one another? • Pr( x, t | p ) = Pr( x | p ) Pr( t | x, p ) • Probability Theory • Pr( t | x, p ) = Pr( t | p ) • p incorporates all information about t • that is in x •  Pr( x, t | p ) = Pr( x | p ) Pr( t | p )

17. Pr( x, t | p ) = Pr( x | p ) Pr( t | p ) Given a particular value of the propensity score variable, that is at P=p, the covariates X andTare uncorrelated. At particular levels of P individually and therefore collectively as well (“conditionally on P), Xcannot confound the association betweenTand any outcome.

18. Matching on Propensity Scores

19. Propensity Matching: Method • Identify candidate predictors of treatment B v A • Perform a logistic regression of B v A • Obtain from the regression a “predicted” probability of B v A • Sort all members of A and B according to this propensity • Match A patients to B patients on the propensity

20. Duragesic and Long-Acting Opioids

21. Straightforward Regression proc logistic data = mother.propensity2 descending; model DuragesicUser = DischCostIndex EncCostIndex RxCostIndex OtherCostIndex RxCostPrior1 OtherCostPrior1 AnyRx OneDisch TwoDisch ThreePlusDisch AnyICD443 AnyICD719 AnyICD724 AnyICD787 AnyICD789 q3_95_new q4_95 q1_96 q2_96 q3_96 q4_96 q1_97 q2_97 q3_97 q4_97 q1_98 q2_98 q3_98 q4_98 hmo men young old /rl; where enrbaseflag = 1 and validindex = 1 and sameday = 0 and medicare = 0 and malignant = 0; output out = mother.propensity3 p = score ; run;

22. Propensity Output Obs PATIENT score   1 2909 0.57475 2 3438 0.32899 3 3841 0.01324 4 5674 0.48411 5 5734 0.06892 6 8573 0.11780 7 10210 0.34692 8 13056 0.09737 9 13376 0.37350 10 16026 0.15635 11 16865 0.06106 12 16949 0.10568

23. Matching on Propensity Choose from E2 a sample that matches E1 in size. E1 E2(sample) E2 (residual) Pr(E1)=x Pr(E1)=x

24. Matching on Propensity E1 E2 At every level of propensity in the constructed cohorts, Pr(E1) = 0.5. Therefore, treatment is uncorrelated with propensity, and you can collapse all the propensity-matched groups together to form a cohort in which all covariate patterns are uncorrelated with treatment, and there will be no confounding bias. Pr(E1) = 0.5 Pr(E1) = 0.5

25. Stratum I II III IV V

26. Stratum I II III IV V

27. Duragesic and Long-Acting Opioids

28. Propensity-Matched Cohorts

29. Pharmacoepidemiol Drug Saf. 2005 Jul;14(7):465-76.

30. Do Statins Affect Risk of AMI? • The purpose of the study was to assess whether Statins affect the risk of risk of acute myocardial infarction (AMI) • Strong predictors for statin use that affect risk of AMI • How to design an observational study? • Note: we would not ordinarily use observational data for efficacy questions, but this serves as a suitable test case because there is a known gold standard

31. Good Clinical Practice Creates Confounding Risk Category LDL to initiate LDL Goal of drug Tx drug Tx ³ No CHD and <160 190 <2 Risk Factors ³ No CHD and <130 160 ³ 2 Risk Factors £ With CHD >130 100 +Risk Factors: age (45M, 55F), diabetes, smoking, HTN, low HDL, family history of premature CHD -Risk Factor: high HDL NCEP ATP II guidelines (1993)

32. Gold Standard for the Effect of Statins CARE Trial Results Sacks FM, et al N Engl J Med. 1996;335:1001-9

33. Data Source • Fallon Community Health Plan • Central Massachusetts HMO • ~200,000 members • Claims Data available on: • Enrollment (age, sex, date) • Ambulatory care visits • Hospitalization • Pharmacy dispensings (drug & quantity) • Laboratory tests (tests & results)

34. Patient Entry, Analytic Sequence All Fallon members with any LDL > 130 mg/dl 1994 1995 1996 1997 1998 1999 1993 ~35,000 Members • Apply eligibility criteria • FCHP member for at least 1 year • At least one physician visit in last year • LDL, HDL, TG levels in last 6 months • At least one physician visit in cohort accrual block • No PAD diagnosis before index date • Not current statin user • 2) Estimate propensity score (statin initiation) • 3) Match statin initiators with non-initiators • 4) Repeat for all blocks of time • 5) Follow matched groups for diagnosis of MI Require 1 year Enrollment 1 of 9 Blocks 2nd/94

35. Month of 1/1/94 Total subjects in cohort (36,050) Non Statin Users, Not Eligible (24,799) Non Statin Users, Eligible (9,639) Propensity Score Matching Statin Initiators, Eligible (77) Statin Initiators, Not Eligible (34) Current Statin Users (1501)

36. “Typical” Statin Initiator and Non-Initiator

37. MI Outcome (Unmatched) 111% (46%-204%) Risk Increase HR=2.11 (1.46-3.04) Statin Initiators Cumulative Incidence Statin Non-Initiators Months of Follow-Up

38. Calculate Propensity Score • Predict Treatment • Statin Initiation vs Not • In Each 6-month Period of Cohort Accrual • Using Baseline Covariates • Obtain Fitted Value From Regression • Fitted Value is the Propensity Score

39. Construct Rich Model • More than 8 events per covariate leads to unbiased estimates • Many more persons exposed to drug of interest than study outcomes • In Drug Safety studies, usually the outcome is rare • Therefore can control for more covariates when exposure is dependent variable than when outcome is Cepeda S, et al. Am J Epidemiol 2003;158:280-287.

40. *build model for 9501; proclogisticdescendingdata=new1; model statin = male smokobes age9501 ang9501 usa9501 chf9501 isch9501 ath9501 cva9501 usa9501 mi9501 olmi9501 htn9501 tia9501 afib9501 ascv9501 hth9501 ost9501 cvs9501 htdx9501 circ9501 cond9501 rvsc9501 hhd9501 dysr9501 hrt9501 ns9501 ins9501 diab9501 skca9501 depr9501 adj9501 schz9501 deb9501 rheu9501 days9501 lres9501 tres9501 hres9501 hbac9501 cvhp9501 ekg9501 cvrx9501 cvvs9501 llab9501 lab9501 cvdg9501 hosp9501 rx9501 vist9501 diag9501 ; outputout=psmodelpred=PROPSCORE; run;

41. Propensity Regression Parameter Estimates

42. Output File – Propensity Scores Obs ID STATIN PROPSCORE   1 1909 0 0.57475 2 2438 0 0.33899 3 3841 0 0.01324 4 4674 0 0.48411 5 4734 0 0.06892 6 5573 0 0.11780 7 6210 1 0.34692 8 7056 1 0.09737 9 7376 1 0.37350 10 8026 1 0.15635 11 8865 1 0.06106 12 9949 1 0.10568 . . .

45. Output File – Propensity Scores Obs ID STATIN PROPSCOR   1 1909 0 0.57475 2 2438 0 0.33899 3 3841 0 0.01324 4 4674 0 0.48411 5 4734 0 0.06892 6 5573 0 0.11780 7 6210 1 0.34692 8 7056 1 0.09737 9 7376 1 0.37350 10 8026 1 0.15635 11 8865 1 0.06106 12 9949 1 0.10568 . . .

47. Balance Achieved by Matching Only 1 of 52 variables sig. different at P<0.05

48. MI Outcome (After Matching) 31% (7%-48%) Risk Reduction HR=0.69 (0.52-0.93) Cumulative Incidence Statin Non-Initiators Statin Initiators Months of Follow-Up

49. Interpreting Propensity Coefficients