1. FINAL REVIEW BIOST/EPI 536 December 14, 2009

2. Outline Before the midterm: Interpretation of model parameters (Cohort vs case-control studies) Hypothesis testing Adjusting for confounding Exposure modeling

3. Outline Since the midterm: Interaction DAGs Omitted covariates Assessing model fit (influential observations: notes p370-389) Advanced coding Conditional logistic regression Conditional vs Marginal adjustment Conditional vs Marginal causal models Marginal structural models (briefly) Prediction (briefly)

4. Interaction Many possible forms Confirmatory: include as spec’d in prior hypothesis Exploratory: include interactions that have a priori rationale and find best-fitting, succinct form Rules 1.Interaction and main effect can have different forms 2.Interaction should be nested within main effect (HW 10 key, notes p260-300)

5. DAGs DAGs help us visualize confounding Confounding occurs when there are unblocked backdoor paths between E and D Backdoor paths may be blocked by: Colliders Controlling for variables Sufficient: a set of variables that control for confounding Minimally sufficient: smallest sufficient set (HW 11 key, notes p301-342) ED F G B C

6. Omitted covariates: take-home message 1.Adding an additional variable X to a logistic model will ALWAYS change the interpretation of the coefficient for the exposure of interest. 2.Why? OR compares two values with the same value of X (conditional OR) rather than randomly chosen individuals (marginal OR). 3.Marginal OR != Conditional OR 4.When X is strongly associated with E or D, controlling for X can reduce the precision to estimate the log OR and reduce the power of the test for the exposure parameter

7. Advanced Coding Many examples: Genetics (HW 14) Nutritional epi (HW 13, pop quiz) Notes p357-369 Take-home message: Interpretation depends on the coding of both the variable of interest and other variables in the model. “Other variables held constant” doesn’t always make sense. When in doubt, write out logits.

8. Conditional Logistic Regression Method of estimation (conditional likelihood) Estimates are interpreted as in normal logistic regression Use to reduce bias when you have many nuisance parameters (example: matched sets) Stratum-specific parameters aren’t estimated, but stratum variables are in the model Can only estimate coefficients for covariates that vary w/in at least one stratum (HW 15-16, notes p423-449)

9. Conditional vs. Marginal Causal Models Goal: Control confounding. Method depends on parameter of interest. Subject-specific “How much more likely is a person to get the disease if they are exposed?” Regression can be useful here Population average Compare average risk of disease in the population if everyone is exposed, vs. if no one was exposed Example: causal effect of population interventions Regression not useful here (Notes p398-422)

10. Conditional vs. Marginal Causal Models Homogeneous pop/subpop: pop/subpop parameters have subject-specific interpretation In general: if estimating P(disease|exp+) – P(disease|exp-), can estimate avg causal effect for pop/subpop In case-control studies: can’t estimate this difference! Can estimate pop/subpop ORs. With strong assumptions, we can say subpop ORs are avg subject-specific ORs.

11. Conditional vs. Marginal Causal Models Logistic regression Covariates define subpopulations Estimate pop/subpop level causal effects when there is no confounding Can’t control subpop confounding reliably Control confounding for average subject-specific causal effects when subpops defined by strata are homogeneous (strong assumption) Include variables related to disease (not necessarily confounders)

12. Conditional vs. Marginal Adjustment Conditional: compare similar individuals Common OR using logistic regression/M-H We’ve done this a lot! Marginal: compare individuals randomly selected from a population Reweight exposure probabilities so cases and controls have same confounder distribution, then calculate OR. 1.Standardization 2.Marginal structural models Conditional and marginal ORs will be different (Notes p390-397)

13. Crude OR uses confounded P(Exposed) for cases and controls: SUM[ P(C=c i ) x P(Exposed|C=c i ) ] Exposure probabilities are weighted differently b/c of different confounder distributions for cases and controls Conditional OR: Same exposure OR within confounder strata ControlsCases P(C = c 1 )P(Exposed|C=c 1 )P(C = c 1 )P(Exposed|C=c 1 ) P(C = c 2 )P(Exposed|C=c 2 )P(C = c 2 )P(Exposed|C=c 2 ) Conditional vs. Marginal Adjustment

14. Marginal OR uses adjusted P(Exposed) for cases and controls: SUM[ P(control, C=c i ) x P(Exposed|C=c i ) ] Weighting distribution of C in cases to match distribution in controls Marginal structural models also standardize pop-level ORs Weights: Proportion of observations that would be in the sample for C=c and E=e if C and E weren’t related (MSM: Notes p450-465) ControlsCases P(C = c 1 )P(Exposed|C=c 1 ) P(C = c 2 )P(Exposed|C=c 2 ) Conditional vs. Marginal Adjustment

15. Marginal Could standardize to any population Estimate is not conditional on value of confounding variable Interpretation of OR depends on population chosen Look at the association of interest comparing populations with different exposure levels, but each with the same distribution of the confounding variable. Conditional Better for subpopulation-level ORs Subpop ORs can sometimes be interpreted as subject-level Look at the association of interest comparing individuals with different exposure levels but with the same level of the confounding variable.

16. Prediction Goal: Develop a rule to predict outcomes Method Using training data, develop predictive models and obtain parameter estimates. Using validation data, pick the best model. Using test data, estimate prediction error associated with the best model. (We can fudge this a little but this is the best method.)

17. Questions?