1. Observational Studies Based on Rosenbaum (2002) David Madigan Rosenbaum, P.R. (2002). Observational Studies (2 nd edition). Springer

2. Introduction A empirical study in which: Examples: smoking and heart disease vitamin C and cancer survival DES and vaginal cancer “The objective is to elucidate cause-and-effect relationships in which it is not feasible to use controlled experimentation” aspirin and mortality cocaine and birthweight diet and mortality

3. Asthma Study Have data on 2,000 kids What is the effect of tobacco experimentation on asthma? Sex Male Female Ethnicity African American Asian Hispanic Other White Smoking at Home Yes No Tobacco Experimentation Yes No Asthma Self-Diagnosis Yes No Asthma ISAAC Yes No

5. Cameron and Pauling Vitamin C Gave Vitamin C to 100 terminally ill cancer patients For each patient found 10 controls matched for age, gender, cancer site, and tumor type Vitamin C patients survived four times longer than controls Later randomized study found no effect of vitamin C Turns out the control group was formed from patients already dead… LESSONS: - observational studies are tricky - randomized study is the gold standard why?

6. Why does randomization work?

8. The two groups are comparable at baseline Could do a better job manually matching patients on 18 characteristics listed, but no guarantees for other characteristics Randomization did a good job without being told what the 18 characteristics were Chance assignment could create some imbalances but the statistical methods account for this properly

9. The Hypothesis of No Treatment Effect In a randomized experiment, can test this hypothesis essentially without making any assumptions at all “no effect” formally means for each patient the outcome would have been the same regardless of treatment assignment Test statistic, e.g., proportion (D|TT)-proportion(D|PCI) TT D D PCI L L TT D PCI D TT L PCI L TT D PCI D L TT L PCI D TT D L PCI L D TT D PCI L TT L PCI D D TT L L P=1/6 observed

10. Estimates, etc. Note: the probability distribution needed for the test is known, not assumed or modeled Randomized experiment provides unbiased estimator of the average treatment effect Internal versus external validity Confidence intervals by inverting tests Partially ordered outcomes, censoring, multivariate outcomes, etc.

11. Overt Bias in Observational Studies “An observational study is biased if treatment and control groups differ prior to treatment in ways that matter for the outcome under study” Overt bias: a bias that can be seen in the data Hidden bias: involves factors not in the data Can adjust for overt bias…

12. Overt Bias M units, j=1,…,M covariate vector treatment (assume binary 0 or 1).  j =Pr(Z j =1) unknown An OS is free of hidden bias if the  j ’s are known to depend only on the ’s (i.e., ) (so two units with same x have same prob of getting the treatment) unknown

13. Stratifying on x Suppose can group units into strata with identical x’s. Then: Conditional on all ’s are equally likely…just like in a uniform randomized experiment

14. Stratifying on the Propensity Score Obviously exact matching not always possible Idea: form strata comprising units with the same ’s ( i.e. could have ) Problem: don’t know the ’s Solution: estimate them (logistic regression, SVM, decision tree, etc.) Form strata containing units with “similar” probability of treatment

15. Matched Analysis Using a model with 29 covariates to predict VHA use, we were able to obtain an accuracy of 88 percent (receiver-operating-characteristic curve, 0.88) and to match 2265 (91.1 percent) of the VHA patients to Medicare patients. Before matching, 16 of the 29 covariates had a standardized difference larger than 10 percent, whereas after matching, all standardized differences were less than 5 percent

16. Conclusions VHA patients had more coexisting conditions than Medicare patients. Nevertheless, we found no significant difference in mortality between VHA and Medicare patients, a result that suggests a similar quality of care for acute myocardial infarction.

19. What about hidden bias? Sensitivity analysis! Consider two units j and k with the same x. hidden bias  they may not have the same  Consider this inequality: Sensitivity analysis will consider various  ’s

20. An equivalent latent variable model for two units j and k with the same x: between –1 and 1 so the model implies the previous inequality with (implication goes the other way too)

21. Matched Pairs Strata of size 2, one gets the treatment, one doesn’t If  =0, every unit has the same chance of treatment Standard test statistic for matched pairs is: rank of sum of the ranks for pairs in which treated unit > control unit Wilcoxon rank sum test

22. More on Matched Pairs No hidden bias => know the null distribution of T because sth pair contributes d s with prob ½ and 0 with prob ½ with hidden bias, the sth pair contributes d s with prob: and zero with prob 1-p s so null distribution of T is unknown…

23. Even More on Matched Pairs The P-value we are after is Lower bound on P-value: where T - is the sum of S quantities, the sth one being d s with prob and 0 otherwise Upper bound likewise using This directly provides bounds on P-values for fixed  easy to see that:

24. Smoking & Lung Cancer Example Hammond (1964) paired 36,975 heavy smokers to non-smokers. Matched on age, race, plus 16 other factors  MinimumMaximum 1< 0.0001 2 3 4 0.0036 5< 0.00010.03 6< 0.00010.1

25. Asthma Study Need a  of three to make the effect of tobacco experimentation on asthma become non-significant