1. April 11 Logistic Regression –Modeling interactions –Analysis of case-control studies –Data presentation

2. Subgroup Analyses Journal Tables Treatment ATreatment BOR Overall 1001500.67 Men4090x.xx Women6060x.xx Age <502530x.xx Age 51-603550x.xx Age 60 +4070x.xx SBP < 1604070x.xx SBP ≥ 1606080x.xx Is there any evidence that the effect of treatment differs among subgroups

3. TOMHS Example Question: Does the effect of active BP treatment on CVD differ for young versus older persons? Looking at an interaction effect (effect modification) Compare –Odds CVD (treatment/placebo) in younger patients –Odds CVD (treatment/placebo) in older patients

4. Logistic regression equation Model log odds of outcome as a linear function of one or more variables X i = predictors, independent variables  is increase in log odds of 1-unit increase in X e   is relative odds of a 1-unit increase in X The model is:

5. Logistic Model For Interaction X1 = 1 for active treatment and 0 for placebo X2 = 1 for age ≥ 55 and 0 for age < 55 X3 = X1 * X2

6. Logistic Model For Interaction X1 = 1 for active treatment and 0 for placebo X2 = 1 for age ≥ 55 and 0 for age < 55 X3 = X1 * X2 Log Odds (placebo, young) =  0 Log Odds (active, young) =  0 +  1 Log Odds (placebo, old) =  0 +  2 Log Odds (active, old)=  0 +  1 +  2 +  3 Dif =  1 ; exp(  1 ) is odds (A v P) for young Dif =  1 +   ; exp(  1 +  3 ) is odds (A v P) for old

7. Log Odds (placebo, young) =  0 Log Odds (active, young) =  0 +  1 Log Odds (placebo, old) =  0 +  2 Log Odds (active, old)=  0 +  1 +  2 +  3 exp(  1 ) is odds (A v P) for young exp(  1 +  3 ) is odds (A v P) for old What does  3 Mean? = Odds (A v P) for Oldexp(  1 +  3 ) Odds (A v P) for Youngexp (  1 ) exp (  3 ) = A ratio of ratios!!

8. Interaction Hypothesis Ho:  3 = 0 Ha:  3 ≠ 0 Test in SAS just like any other coefficient

9. TOMHS: Overall Effect of Active Treatment PROC MEANS DATA=temp N MEAN SUM; CLASS active; VAR cvd; RUN; Analysis Variable : cvd N active Obs N Mean Sum ============================================================ 0 234 234 0.1623932 38.0000000 1 668 668 0.1107784 74.0000000 ============================================================ Active: 38/234 or 11.1% Placebo: 74/668 or 16.2% RR = 0.68 (32% lower rate of CVD with active treatment)

10. OVERALL (ACTIVE VERSUS PLACEBO) The LOGISTIC Procedure Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.6405 0.1773 85.6626 <.0001 active 1 -0.4423 0.2159 4.1964 0.0405 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits active 0.643 0.421 0.981 Active group at 36% lower risk of CVD compared to placebo.

11. Reading DATA and Creating indicator variables and interaction variable LIBNAME tomhs 'C:/'; DATA temp; SET tomhs.bpstudy; cvd = second; if group = 6 then active = 0; else active = 1; if age < 55 then old = 0; else old =1; *compute interaction term (x3); active_old = active*old;

12. * Get simple counts and proportions first; PROC MEANS DATA=temp N MEAN SUM; CLASS old active; VAR cvd; RUN; The MEANS Procedure Analysis Variable : cvd N old active Obs N Mean Sum =========================================================================== 0 0 115 115 0.1565217 18.0000000 1 350 350 0.0714286 25.0000000 1 0 119 119 0.1680672 20.0000000 1 318 318 0.1540881 49.0000000 It appears the effect of treatment is mostly in younger patients

13. PROC LOGISTIC DATA=temp DESCENDING; MODEL CVD = active old active_old; CONTRAST 'A v P (Young)' active 1 /ESTIMATE=BOTH; CONTRAST 'A v P (Old)' active 1 active_old 1 /ESTIMATE=BOTH; * Will give us beta1 + beta 3; RUN;

14. SAS OUTPUT Response Profile Ordered Total Value cvd Frequency 1 1 112 2 0 790 Probability modeled is cvd=1. Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 15.7787 3 0.0013 Score 14.7851 3 0.0020 Wald 14.0735 3 0.0028

15. The LOGISTIC Procedure Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.6843 0.2566 43.0730 <.0001 active 1 -0.8806 0.3301 7.1180 0.0076 old 1 0.0850 0.3549 0.0573 0.8108 active_old 1 0.7771 0.4395 3.1261 0.0770 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits active 0.415 0.217 0.792 old 1.089 0.543 2.183 active_old 2.175 0.919 5.147 b1 b2 b3 Odds CVD (A v P) for younger patients = exp(b1) = 0.415 Odds CVD (A v P) for older patients = exp(b1 + b3) = exp(-0.11) = 0.90 2.175 = 0.90/.415 Ratio of Odds Ratios

16. CONTRAST 'A v P (Old)' active 1 active_old 1 /ESTIMATE=BOTH; Computes 1*beta1 + 0*beta2 + 1*beta3 = beta1 + beta3 Plus test and 95%CI Contrast Rows Estimation and Testing Results Standard Lower Upper Contrast Type Row Estimate Error Alpha Limit Limit A v P (Young) PARM 1 -0.8806 0.3301 0.05 -1.5275 -0.2337 A v P (Young) EXP 1 0.4145 0.1368 0.05 0.2171 0.7916 A v P (Old) PARM 1 -0.1035 0.2902 0.05 -0.6723 0.4653 A v P (Old) EXP 1 0.9017 0.2617 0.05 0.5105 1.5925     Exp(     

17. Patients in the active group were at 36% lower risk of CVD compared to the placebo group (OR: 0.64; 95% CI:0.42-0.98). Analyses by age showed that the benefit for active treatment was greatest in younger patients. In patients < age 55 the CVD risk was 58% lower in the active treatment (OR: 0.42) where for patients over 55 years of age the CVD risk was only 10% lower (OR:.90). The test for interaction between treatment and age approached significance (p=.07). Description of Findings

18. Logistic Regression for Case Control Studies Same analyses as prospective study Outcome: –Y = 1 is a case –Y = 0 is a control Model log (odds) of being a case Odds ratios have same meaning Estimating probability of being a case not appropriate

19. Example Colon Polyp Study Cases (N=574) –Patients diagnosed with colorectal polyps from colonoscopy Controls (N=707) –Patients clear of colorectal polyps from colonoscopy Risk Factors Under Study –FH of colon cancer –Smoking and alcohol –Reproductive history factors –Obesity and adiposity (weight to hip measures)

20. Example Colon Polyp Study Variables –CC Status (1=case, 2=control) –Age (years) –FH colon cancer (1=Y, 0=N) –Current Smoking (1=Y, 0=N) –Gender (1=Men, 0 = Women) –Waist to Hip Ratio Variables Names –CC, AGE, FHCC, SMOKERS, MEN, and WHIP

21. PROC LOGISTIC DATA=temp ; MODEL cc = age fhcc smokers men whip; UNITS whip = 0.1 ; Response Profile Ordered Total Value CC1 Frequency 1 1 561 2 2 690 Probability modeled is CC=1. Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 165.4379 5 <.0001 Score 155.7546 5 <.0001 Wald 139.8082 5 <.0001

22. Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -4.0683 0.5953 46.7054 <.0001 AGE 1 0.0497 0.00618 64.8156 <.0001 FHCC 1 -0.4434 0.1505 8.6798 0.0032 smokers 1 0.5272 0.1623 10.5537 0.0012 men 1 0.8379 0.1503 31.0610 <.0001 WHIP 1 0.7491 0.6287 1.4197 0.2335 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits AGE 1.051 1.038 1.064 FHCC 0.642 0.478 0.862 smokers 1.694 1.233 2.329 men 2.312 1.722 3.104 WHIP 2.115 0.617 7.253 UNITS whip = 0.1 ; Effect Unit Estimate 95% Confidence Limits WHIP 0.1000 1.078 0.953 1.219

23. Interaction Model Is relationship of waist to hip ratio different for men and women Define interaction term –Whip * men

24. PROC LOGISTIC DATA=temp DESCENDING; MODEL cc = age fhcc smokers men whip whip_men; Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -2.4771 0.8632 8.2349 0.0041 AGE 1 0.0511 0.00626 66.7467 <.0001 FHCC 1 -0.4528 0.1511 8.9866 0.0027 smokers 1 0.5487 0.1631 11.3203 0.0008 men 1 -2.5148 1.3103 3.6838 0.0549 WHIP 1 -1.2470 1.0235 1.4846 0.2231 whip_men 1 3.7225 1.4392 6.6897 0.0097 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits AGE 1.052 1.040 1.065 FHCC 0.636 0.473 0.855 smokers 1.731 1.257 2.383 men 0.081 0.006 1.055 WHIP 0.287 0.039 2.136 whip_men 41.367 2.464 694.576 P-value for women

25. Some Practical Aspects for Analyses Divide continuous variable of interest into 3-5 categories and compute relative odds for increasing categories. Summarize results using beta coefficient using factor as continuous variable.

26. Example Omega-3 Intake and CHD Omega-3 Intake CHD N Odds Ratio (95% CI) I 401.00 II 421.08 (0.80 – 1.45) III 370.92 (0.68 – 1.32) IV 350.89 (0.66 - 1.25) V 240.61 (0.34 – 0.98) Beta (SE) 0.30 (.15)

27. Advantages Can determine if risk increases linearly with increasing levels of factor No assumptions of pattern of risk when using categories Can determine if there is a threshold effect Eliminates possible effect of outliers.

28. Analysis Create indicator variables for quintiles of omega-3 and run logistic regression Run regression using omega-3 as continuous variable

29. In Class Exercise Investigate whether the odds of CVD increases linearly with age Divide age into 4-categories –< 50; 50-54; 55-59; 60+ Two CVD endpoints: –Clinical – major CVD –Second – major + minor CVD Compute percent with CVD with each age category Run logistic regression with 3 indicator variables using < 50 as reference group Run logistic regression using age as continuous variable