1. 1 PH 240A: Chapter 12 Mark van der Laan University of California Berkeley (Slides by Nick Jewell)

2. 2 Regression Models: Motivation  Stratification methods break down (wrt precision) with large number of strata and moderate sample sizes  Stratification leads to high degree of freedom tests for interaction, for example, and therefore low power  Refined measures of exposure lead to high degree of freedom tests for association and therefore low power  Want parsimonious descriptions of patterns of risk Regression Models (how to diet wrt degrees of freedom)

3. 3 Regression Models: Linking Effects x x x x X, numerical measure of Exposure E P(D|E=x) x1x1 x2x2 x4x4 x3x3

4. 4 Linear Models (think CHD and body weight)  Form of model:  Interpretation of model parameters a and b:  b = ER associated with unit increase in X  b(x * -x) = ER assoc. with increase in X from x * to x Choose X = 0 value carefully! Choose the scale ofX carefully!

5. 5 Linear Models  Pros  Good modeling Excess Risk  Cons  Can’t be applied to case-control data  Can predict probabilities 1

6. 6 Log Linear Models  Form of model:  Interpretation of model parameters a and b: Choose X = 0 value carefully!

7. 7 Log Linear Models  Interpretation of model parameters a and b:  e b = RR associated with unit increase in X  e b(x*-x) = RR assoc. with increase in X from x * to x Choose the scale ofX carefully!

8. 8 Log Linear Models  Pros  Good modeling Relative Risk  Cons  Can’t be applied to case-control data  Can predict probabilities > 1

9. 9 Logistic Regression Models  Form of model:  Interpretation of model parameters a and b: Choose X = 0 value carefully!

10. 10 Logistic Regression Models  Interpretation of model parameters a and b:  e b = OR associated with unit increase in X  e b(x*-x) = OR assoc. with increase in X from x * to x Choose the scale ofX carefully!

11. 11 Logistic Regression Models  Pros  Good modeling Odds Ratio  Can be applied to case-control data  Predicted probabilities always lie between 0 and 1  Cons  Harder to interpret

12. 12 Logistic Regression Models

13. 13 1991 US Infant Mortality Mother’s Marital Status Infant Mortality Unmarried (X = 1) Married (X = 0) Total Death16,71218,78435,496 Live at 1 Year 1,197,1422,878,4214,075,563 Total1,213,8542,897,2054,111,059 p 1 = 16,712/1,213,854 = 0.0138 p 0 = 18,784/2,897,205 = 0.0065

14. 14 Infant Mortality and Marital Status: Various Models

15. 15 Logistic Regression: Body Weight and CHD

16. 16 CHD and Body Weight: Various Models

17. 17 Multiple Logistic Regression Models  Form of model: Think, e.g., D = CHD, X 1 = Behavior type, X 2 = Body weight

18. 18 Multiple Logistic Regression Models  Interpretation of model parameters a, b 1,..., b k : Choose 0 covariate values carefully! Choose covariate scales carefully!

19. 19 Indicator (Dummy) Variables for Discrete Exposures  Goal: to model exposures with several discrete levels without assuming dose response  Body weight (dose response)  Body weight (no assumed pattern)

20. 20 Indicator (Dummy) Variables for Discrete Exposures  Model: No assumed pattern!

21. 21 Interpretation of Slope Coefficients with Dummy Variables CHD Event Dnot D Body Wt (lbs) 15032558 150-16031505 160-17050594 170-18066501 >18078739

22. 22 Indicator (Dummy) Variables for Discrete Exposures ModelParameterEstimateOR a-2.839 b0.1801.20 a-2.859 b1b1 0.0681.07 b2b2 0.3841.47 b3b3 0.8322.30 b4b4 0.6101.84

23. 23 Logistic Regression: Body Weight and CHD Dose response (linear) X No pattern (dummy vars.) X 1,..., X 4