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

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

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

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

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

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

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 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 Logistic Regression Models

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 = p 0 = 18,784/2,897,205 =

14 Infant Mortality and Marital Status: Various Models

15 Logistic Regression: Body Weight and CHD

16 CHD and Body Weight: Various Models

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

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

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

21 Interpretation of Slope Coefficients with Dummy Variables CHD Event Dnot D Body Wt (lbs) >

22 Indicator (Dummy) Variables for Discrete Exposures ModelParameterEstimateOR a b a b1b b2b b3b b4b

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