1. Log-binomial regression: relative risk instead of odds ratio. Review of log-binomial regression: Am J Epidemiol 2005; 162:

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1 Lecture Log-binomial regression: relative risk instead of odds ratio Review of log-binomial regression: Am J Epidemiol 2005; 162: Stokes, etal. (2000) Categorical Data Analysis Using the SAS System, 2nd Edition McCullagh, Nelder (1989) Generalized Linear Models, Second Edition Harrell (2001) Regression Modeling Strategies (Springer) 1 Odds and risk for rare events Logistic regression gives odds ratios from regression coefficients: exp( ˆØ X ) = odds ratio for 1-unit increase in predictor X If sample number of events is A, number of non-events is B, odds = number with event number without event = A B risk = number with event number with event + number without event = A B + A Risk is a rate, a percent, and a probability. 2

2 Rare events: the number of events A is very small relative to the number of non-events B, B º B + A, so A B º A B + A that is, odds º risk (rate). When event is rare in both groups, odds ratio º relative risk. But when events are not rare, odds ratios are not good estimates of relative risk. Odds ratio is always farther than relative risk from null value 1. Alternative: estimate relative risk directly. 3 Log-binomial regression: estimating relative risks directly Logistic regression: binomial response y, mean chance of event is º(x) and µ º(x) log = Ø 0 + Ø 1 x 1 º(x) Log-binomial regression: binomial response y, mean chance of event is º(x) and log º(x) = Ø 0 + Ø 1 x 4

3 With log-binomial model, we estimate relative risk (risks ratio), not odds ratio. Regression coefficient for predictor x is Ø 1 = Ø 0 + Ø 1 (x + 1) Ø 0 + Ø 1 (x) = log p(x + 1) log p(x) µ p(x + 1) = log p(x) = log relative risk for unit increase in x =) exp(ø 1 )=relative risk for 1-unit increase in X 5 Obesity in NHANES 2004 NHANES 2004 data for children and adults people age 10 to 50 (n = 6116) Event = obesity, defined as BMI 30, or 95th percentile for children Rate of obesity by age and gender: 6

4 Main-effects logistic regression model gives odds ratios for age and gender: Proc Logistic descending data=under50; class gender; model obese = age gender; Odds Ratio Estimates and Profile-Likelihood Confidence Intervals Effect Unit Estimate 95% Confidence Limits age gender F vs M Are these odds ratios good estimates of corresponding relative risks? 7 Log-binomial regression in Proc Genmod Log binomial model: log º(gender, age) = Ø 0 + Ø 1 age + Ø 2 gender Use Proc Genmod, specify binomial distribution and log link: Proc Genmod descending data=under50; class gender; model obese = age gender/ type3 dist=bin link=log; type3 requests Type 3 sums of squares 8

5 Standard Wald 95% Wald Parameter DF Estimate Error Confidence Limits Chi-Square Pr > ChiSq Intercept <.0001 age <.0001 gender F gender M Scale NOTE: The scale parameter was held fixed. LR Statistics For Type 3 Analysis Chi- Source DF Square Pr > ChiSq age <.0001 gender Regression coefficients for age and gender are log(relative risk). Proc Genmod does not back-transform automatically. 9 Back-transforming regression coefficients 1 Proc Genmod descending data=under50; class gender; model obese = age gender/ type3 dist=bin link=log; ODS output ParameterEstimates = reg_coef; proc print data=reg_coef; Data back_transf; set reg_coef; RR = exp(estimate); left_ci=exp(lowerwaldcl); right_ci = exp(upperwaldcl); proc print; var parameter RR left_ci right_ci; 10

6 Lower Upper Prob Obs Parameter Level1 DF Estimate StdErr WaldCL WaldCL ChiSq ChiSq 1 Intercept < age < gender F gender M Scale Obs Parameter RR left_ci right_ci 1 Intercept age gender gender Scale Back-transforming regression coefficients 2 Proc Genmod descending data=under50; class gender; model obese = age gender/ type3 dist=bin link=log; lsmeans gender / diff CL exp; estimate age age 1 ; label effect value Back-transform mean difference between genders Back-transform ˆØ 1 age for age = 1 12

7 gender Least Squares Means Exponentiated Exponentiated gender Exponentiated Lower Upper F obesity rates, CI M Differences of gender Least Squares Means Exponentiated Exponentiated gender _gender Lower Upper Exponentiated Lower Upper F M Contrast Estimate Results Mean Mean L Beta Standard Label Estimate Confidence Limits Estimate Error Alpha age Relative risks from log-binomial model Effect Estimate 95% Confidence Limits age gender F vs M Odds ratios from logistic regression Effect Estimate 95% Confidence Limits age gender F vs M Annual change for age: 2% vs 3%; for gender 18% vs 23% 14

8 Why doesn t everyone use log-binomial instead of logistic regression? 1. Logistic is numerically more stable: log-binomial does not always converge to produce an answer. 2. Logistic is conventional approach, software more developed. 15