1. Advanced Statistics for Interventional Cardiologists

2. What you will learn Introduction Basics of multivariable statistical modeling Advanced linear regression methods Logistic regression and generalized linear model Multifactor analysis of variance Cox proportional hazards analysis Propensity analysis Bayesian methods Resampling methods Meta-analysis Most popular statistical packages Conclusions and take home messages

3. Background

4. In randomized trials… All patients have a specified chance of receiving each treatment Treatments are concurrent Data collection is concurrent, uniform, and high quality All patient covariates, measured or unmeasured, are expected to be balanced between the two treatment groups

5. In randomized trials… Statistical assumptions underlying comparison tests are met Data collection is concurrent, uniform, and high quality The two groups are comparable and observed treatment difference is an unbiased estimate of true treatment difference

6. However… The above advantages are not guaranteed for small, poorly designed or poorly conducted randomized trials Large randomized trials take a long time and great cost to generate answers (analysis of existing data may be more timely, yet acceptably accurate) Randomized trials are not always feasible, e.g. when variables cannot be manipulated (smoke…)

7. Non-randomized studies None of advantages provided by randomized trials is available in non- randomized studies A potential problem: two treatment groups are not comparable before the start of treatment, i.e. not comparable due to imbalanced covariates between two treatment groups So, direct treatment comparisons are invalid

8. Adjustments for covariates Three common methods of adjusting for confounding covariates: – Matching – Stratification – Regression adjustment

9. Question: When there are many confounding covariates needed to adjust for: –Matching: based on many covariates is not practical –Stratification: is difficult, as the number of covariates increases, the number of strata grows exponentially: 1 covariate: 2 strata  5 covariates: 32 (2 5 ) strata –Regression adjustment: may not be possible: potential problem: over-fitting

10. Propensity score Replace the collection of confounding covariates with one scalar function of these covariates Age Gender Ejection fraction Risk factors Lesion characteristics … 1 composite covariate: Propensity Score Balancing score

11. Propensity score: conditional probability of receiving Treatment A rather than Treatment B, given a collection of observed covariates Purpose: simultaneously balance many covariates in the two treatment groups and thus reduce the bias Propensity score

12. What you will learn Propensity analysis –Building a propensity score –Exploiting a propensity score

13. Statistical modeling of the relationship between treatment membership and covariates –Statistical method: multiple logistic regression –Outcome: actual treatment membership –Predictor variables: all measured covariates, and even some interaction terms: e.g. age, gender, ejection fraction, risk factors, previous history, lesion characteristics… Propensity score construction

14. –Outcome variable of interest (e.g. MACE or TLR or restenosis) is NOT involved in the modeling –No concerns regarding over-fitting –A propensity score model is obtained: it is a mathematical equation: PS = f (age, gender, risk factors, …) –Calculate, through this equation, estimated propensity scores for all patients Propensity score construction

15. A group of patients with the same propensity score are equally likely to have been assigned to treatment A Within a group of patients with the same propensity score, some patients actually got treatment A and some got treatment B, just as they had been “randomly” allocated to whichever treatment they actually received Propensity score properties

16. (Sub-)Group with the same propensity score Treatment ATreatment B Propensity score properties

17. –When the propensity scores are balanced across two treatment groups, the distribution of all the covariates are balanced in expectation across the two groups –Use the propensity scores as a diagnostic tool to measure treatment group comparability –If the two treatment groups overlap well enough in terms of the propensity scores, we compare the two treatment groups adjusting for the PS

18. Comparability No comparison possible…

19. Cosgrave et al, AJC 2005

21. Beta is a regression coefficient

22. Y = C + B 1 *X 1 + B 2 *X 2 + B 3 *X 3 + … PS = -1.242 + (-0.206*sex [0,1]) + (0.01*age) +… + (0.618*IVUS [0,1]) We obtain a parameter, specific for each patient, and expressed in logit units To transform it in probability (included between 0 and 1) p = 1 / (1 + e -PS ) This is the probability (propensity) to receive stent A vs. stent B

23. What you will learn Propensity analysis –Building a propensity score –Exploiting a propensity score

24. Propensity score methods Fit propensity score model using all measured covariates Estimate propensity score for all patients using propensity model Compare treatments adjusting for propensity scores

25. Compare treatments with propensity score Three common methods of using the propensity score to adjust results: – Matching – Stratification – Regression adjustment

26. PS A vs. B PS 1 PS 2 PS m Matching Compare treatments based on matched pairs Problem: may exclude unmatched patients

27. Mauri et al, NEJM 2008

30. –All patients are sorted by propensity scores –Divide into equal-sized subclasses –Compare two treatments within each subclass, as in a randomized trial; then estimate overall treatment effect as weighted average –It is intended to use all patients –But, if trial size is small, some subclass may contain patients from only one treatment group PS 12 ……... 5 Stratification

31. Brener et al, Circulation 2004

34. Treatment effect estimation model fitting: the relationship of clinical outcome and treatment Outcome: Clinical outcome, e.g., MACE, TLR, … Predictor variables: treatment received, propensity score, (a subset of important covariates) Statistical method: e.g., logistical regression, Cox proportional hazards analysis Regression adjustment

35. Brener et al, Circulation 2004

37. Practical Issues Issues in propensity score estimation –How to handle missing baseline covariate values –What terms of covariates should be included –Evaluation of treatment group comparability Issues in treatment comparison –Which method: matching, stratification, regression Issues in study design with propensity analysis –Pre-specified vs. post hoc propensity analysis –Pre-specify the covariates needed to collect in the study and then included in propensity score estimation –Sample size estimation adjusting for the propensity scores

38. Limitations Propensity score methods can only adjust for observed confounding covariates and not for unobserved ones Propensity score is seriously degraded when important variables influencing selection have not been collected Propensity score may not eliminate all selection bias

39. Propensity score methods work better in larger samples Propensity score methods lack the discipline and rigor of randomized trials Randomized trials remain the highest level of evidence for comparison Limitations

40. Conclusions Propensity score is a technique that allows the creation of a single confounding covariate that permits simultaneous adjustment for many covariates thus reducing bias Propensity score methodology is an addition to, not a substitute of traditional covariate adjustment methods Randomized studies are still preferred and strongly encouraged whenever possible!

41. References Blackstone, EH, Comparing apples and oranges. Journal of Thoracic and Cardiovascular Surgery 2002;1:8-15. Rubin, DB, Estimating casual effects from large data sets using propensity scores. Annals of Internal Medicine 1997;127:757-763. D’agostino, RB, Jr., Propensity score methods for bias reduction in the comparison of a treatment to a non-randomized control group. Statistics in medicine 1998;7:2265-2281.

42. Questions?

43. For further slides on these topics please feel free to visit the metcardio.org website: http://www.metcardio.org/slides.html http://www.metcardio.org/slides.html