Labs 6 7 CaseControl Analysis Logistic Regression Henian

Labs 6 & 7 Case-Control Analysis ----Logistic Regression Henian Chen, M. D. , Ph. D. Applied Epidemiologic Analysis P 8400 Fall 2002

Logistic Regression for Intercept only SAS Program proc logistic data=case_control 978 descending; model status=; run; * Descending: to get the probability and OR for dependent variable=1 SAS Output The LOGISTIC Procedure Model Information Data Set Response Variable Number of Response Levels Number of Observations Model Optimization Technique WORK. CASE_CONTROL 978 status 2 978 binary logit Fisher's scoring Applied Epidemiologic Analysis P 8400 Fall 2002

Logistic Regression for Intercept only SAS Output Response Profile Ordered Value 1 2 Total Frequency 200 778 status 1 0 Probability modeled is status=1. Model Convergence Status Convergence criterion (GCONV=1 E-8) satisfied. -2 Log L = 990. 8635 Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error Intercept 1 -1. 3584 0. 0793 Wald Chi-Square Pr > Chi. Sq 293. 5837 <. 0001 Applied Epidemiologic Analysis P 8400 Fall 2002

Logistic Regression for Intercept only Log [Y/(1 -Y)] = α Y = eα / (1+ eα) = exp(α) / [1 + exp(α)] In our model, α = -1. 3584, -1. 3584 is the log odds of cancer for total sample. The odds (eα) is 0. 2571. Y = exp(-1. 3584) / [1 + exp(-1. 3584)] =0. 2045 =200/(200+778) Y is related to α in Logistic Model Applied Epidemiologic Analysis P 8400 Fall 2002

Logistic Regression for Dichotomous Predictor Alcohol Consumption (alcgrp): 0=0 -39 gm/day; 1=40+ gm/day SAS Program proc logistic data=case_control 978 descending; model status=alcgrp; run; SAS Output Model Fit Statistics Criterion -2 Log L Intercept Only 990. 863 Intercept and Covariates 901. 036 Likelihood Ratio Test G = 990. 863 – 901. 036 = 89. 827 df = 1 The model with variable ‘alcgrp’ is significantly. Applied Epidemiologic Analysis P 8400 Fall 2002

Logistic Regression for Dichotomous Predictor SAS Output Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error Intercept alcgrp 1 1 -2. 5911 1. 7641 0. 1925 0. 2132 Wald Chi-Square Pr > Chi. Sq 181. 1314 68. 4372 <. 0001 Odds Ratio Estimates Effect Point Estimate alcgrp 5. 836 95% Wald Confidence Limits 3. 843 8. 864 -2. 5911 is the log odds of cancer for light drinkers (alcgrp=0). Log odds of cancer for heavy drinkers (alcgrp=1) is – 0. 827 (-2. 5911 + 1. 7641). Y = 0. 0697 for light drinkers, and 0. 3043 for heavy drinkers. OR = exp(β) = exp(1. 7641) = 5. 836 Heavy drinkers (alcgrp=1) are about 6 times more likely to get cancer than light drinkers (alcgrp=0). OR is not related to α in Logistic Model Applied Epidemiologic Analysis P 8400 Fall 2002

Logistic Regression for Ordinal Predictor Alcohol Consumption (alcgrp 4): 0=0 -39 gm/day; 1=40 -79 gm/day 2=80 -119 gm/day; 3=120+ gm/day SAS Program proc logistic data=case_control 978 descending; model status=alcgrp 4; run; SAS Output Model Fit Statistics Criterion -2 Log L Intercept Only Intercept and Covariates 990. 863 846. 467 Likelihood Ratio Test G = 990. 863 – 846. 467 = 144. 396 df = 1 The model with variable ‘alcgrp 4’ is significantly. Applied Epidemiologic Analysis P 8400 Fall 2002

Logistic Regression for Ordinal Predictor SAS Output Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error Intercept alcgrp 4 1 1 -2. 4866 1. 0453 0. 1459 0. 0934 Wald Chi-Square Pr > Chi. Sq 290. 4172 125. 2007 <. 0001 Odds Ratio Estimates Effect alcgrp 4 Point Estimate 2. 844 95% Wald Confidence Limits 2. 368 3. 416 OR = exp(1. 0453) = 2. 844. Men with alcgrp 4=1 are about 3 times more likely to get cancer than men with alcgrp 4=0. This OR is also for alcgrp 4= 1 vs. alcgrp 4=2; or alcgrp 4=2 vs. alcgrp 4=3. OR = exp[(3 -1)*1. 0453] = exp(2. 0906) = 8. 090 for alcgrp 4=1 vs. alcgrp 4=3 OR = exp[(3 -0)*1. 0453] = exp(3. 1359) = 23. 009 for alcgrp 4=0 vs. alcgrp 4=3 Applied Epidemiologic Analysis P 8400 Fall 2002

Logistic Regression for Continuous Predictor Alcohol Consumption (alcohol): daily consumption in grams SAS Program proc logistic data=case_control 978 descending; model status=alcohol; run; SAS Output Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error Intercept alcohol 1 1 -2. 9741 0. 0261 0. 1807 0. 00232 Wald Chi-Square Pr > Chi. Sq 270. 9266 126. 4179 <. 0001 Odds Ratio Estimates Effect Point Estimate alcohol 1. 026 95% Wald Confidence Limits 1. 022 Applied Epidemiologic Analysis P 8400 Fall 2002 1. 031

Logistic Regression for Continuous Predictor OR = exp(0. 0261) = 1. 026. The odds of cancer increase by a factor of 1. 026 for each unit in alcohol consumption OR = exp[40*(0. 0261)] = exp(1. 044) = 2. 8406 for a 40 -grams increase in alcohol consumption per day OR = exp[120*(0. 0261)] = 22. 825 for a man who drinks 160 grams per day compare with a man who is similar in other respects but drinks 40 grams per day. Applied Epidemiologic Analysis P 8400 Fall 2002

Interaction in Logistic Regression model status = α + β 1 alcgrp + β 2 tobgrp β 1 : the effect of alcohol on cancer, controlling for tobacco (i. e. , the same OR across levels of tobacco) β 2 : the effect of tobacco on cancer, controlling for alcohol (i. e. , the same OR across levels of alcohol) model status = α + β 1 alcgrp + β 2 tobgrp + β 3 alcgrp*tobgrp β 1 : the effect of alcohol on cancer among non-smokers (tobgrp=0) β 2 : the effect of tobacco on cancer among non-drinkers (alcgrp=0) β 3 : interaction between smokers and drinkers Applied Epidemiologic Analysis P 8400 Fall 2002

Interaction in Logistic Regression model status = -3. 33 + 2. 28 (alcgrp) + 1. 38 (tobgrp) – 0. 98 (alcgrp*tobgrp) A: alcgrp=0 & tobgrp=0 B: alcgrp=1 & tobgrp=0 C: alcgrp=0 & tobgrp=1 D: alcgrp=1 & tobgrp=1 Log odds 2. 28*0 + 1. 38*0 – 0. 98*0*0 = 0. 00 2. 28*1 + 1. 38*0 – 0. 98*1*0 = 2. 28*0 + 1. 38*1 – 0. 98*0*1 = 1. 38 2. 28*1 + 1. 38*1 – 0. 98*1*1 = 2. 68 A vs. B A vs. C A vs. D B vs. D C vs. D Odds Ratio 9. 78 = 9. 78/1. 00 3. 97 = 3. 97/1. 00 14. 59 = 14. 59/1. 00 1. 49 = 14. 59/9. 78 3. 68 = 14. 59/3. 97 Applied Epidemiologic Analysis P 8400 Fall 2002 odds 1. 00 9. 78 3. 97 14. 59