#### Transcript Panel on Collaboration Discussion Questions

Introduction to logistic regression and Generalized Linear Models Karen Bandeen-Roche, PhD Department of Biostatistics Johns Hopkins University July 14, 2011 Introduction to Statistical Measurement and Modeling Data motivation Osteoporosis data Scientific question: Can we detect osteoporosis earlier and more safely? Some related statistical questions: How does the risk of osteoporosis vary as a function of measures commonly used to screen for osteoporosis? Does age confound the relationship of screening measures with osteoporosis risk? Do ultrasound and DPA measurements discriminate osteoporosis risk independently of each other? Outline Why we need to generalize linear models Generalized Linear Model specification Systematic, random model components Maximum likelihood estimation Logistic regression as a special case of GLM Systematic model / interpretation Inference Example Regression for categorical outcomes Why not just apply linear regression to categorical Y’s? Linear model (A1) will often be unreasonable. Assumption of equal variances (A3) will nearly always be unreasonable. Assumption of normality will never be reasonable Introduction: Regression for binary outcomes Yi = 1{event occurs for sampling unit i} = 1 if the event occurs = 0 otherwise. pi = probability that the event occurs for sampling unit i := Pr{Yi = 1} Begin by generalizing random model (A5): Probability mass function: Bernoulli Pr{Yi = 1} = pi; Pr{Yi = 0} = 1-pi all other yi occur with 0 probability fYi(y) : Pr{Yiy} piy(1 pi)1y. Binary regression By assuming Bernoulli: (A3) is definitely not reasonable Var(Yi ) = pi(1-pi) Variance is not constant: rather a function of the mean Systematic model Goal remains to describe E[Yi|xi] Expectation of Bernoulli Yi = pi To achieve a reasonable linear model (A1): describe some function of E[Yi|xi] as a linear function of covariates g(E[Yi|xi]) = xi’β Some common g: log, log{p/(1-p)}, probit General framework: Generalized Linear Models Random model Y~a density or mass function, fY, not necessarily normal Technical aside: fY within the “exponential family” Systematic model g(E[Yi|xi]) = xi’β = ηi “g” = “link function”; “xi’β” = “linear predictor” Reference: Nelder JA, Wedderburn RWM, Generalized linear models, JRSSA 1972; 135:370-384. Types of Generalized Linear Models Model (link function) Linear Logistic Log-linear Proportional hazards Response Distribution Regression Coef Interp Continuous Gaussian Change in ave(Y) per unit change in X Binary Binomial Log odds ratio Times to events/counts Poisson Log relative rate Times to events Semiparametric Log hazard Estimation Estimation: maximizes L(β,a;y,X) = n f (y ;μ(x ,β),a) i 1 Yi i i General method: Maximum likelihood (Fisher) Given {Y1,...,Yn} distributed with joint density or mass function fY(y;θ), a likelihood function L(θ;y) is any function (of θ) that is proportional to fY(y;θ). If sampling is random, {Y1,...,Yn} are statistically independent, and L(θ;y) α product of individual f. Maximum likelihood The maximum likelihood estimate (MLE), , maximizes L(θ;y): L(θ) Θ Θ Under broad assumptions MLEs are asymptotically Unbiased (consistent) Efficient (most precise / lowest variance) Logistic regression Yi binary with pi = Pr{Yi = 1} Example: Yi = 1{person i diagnosed with heart disease} Simple logistic regression (1 covariate) Random Model: Bernoulli / Binomial Systematic Model: log{pi/(1- pi)}= β0 + β1xi log odds; logit(pi) Parameter interpretation β0 = log(heart disease odds) in subpopulation with x=0 β1 = log{px+1/(1-px+1)}- log{px/(1-px)} Logistic regression Interpretation notes β1 = log{px+1/(1-px+1)}- log{px/(1-px)} = log exp(β1)log = px1/(1px1) px/(1px) px1/(1px1) px/(1px) = odds ratio for association of prevalent heart disease with each (say) one year increment in age = factor by which odds of heart disease increases / decreases with each 1-year cohort of age Multiple logistic regression Systematic Model: log{pi/(1- pi)}= β0 + β1xi1 + … + βpxip Parameter interpretation β0 = log(heart disease odds) in subpopulation with all x=0 βj = difference in log outcome odds comparing subpopulations who differ by 1 on xj, and whose values on all other covariates are the same “Adjusting for,” “Controlling for” the other covariates One can define variables contrasting outcome odds differences between groups, nonlinear relationships, interactions, etc., just as in linear regression Logistic regression - prediction Translation from ηi to pi log{pi/(1- pi)}= β0 + β1xi1 + … + βpxip Then pi e β0β1xi1...βpxip 1e β0β1xi1...βpxip e ηi 1e ηi = logistic function of ηi Graph of pi versus ηi has a sigmoid shape GLMs - Inference The negative inverse Hessian matrix of the log likelihood function characterizes Var( ) (adjunct) SE( ) obtained as square root of the jth diagonal entry Typically, substituting for β “Wald” inference applies the paradigm from Lecture 2 0 j is asympotically ~ N(0,1) under H0: βj= β0j Z= j SE ( j ) Z provides a test statistic for H0: βj= β0j versus HA: βj≠ β0j ± z(1-α/2) SE{ } =(L,U) is a (1-α)x100% CI for βj {exp(L),exp(U)} is a (1-α)x100% CI for exp(βj) GLMs: “Global” Inference Analog: F-testing in linear regression The only difference: log likelihoods replace SS Hypothesis to be tested is H0: βj1=...=βjk = 0 Fit model excluding xj1,...,xjpj: Save -2 log likelihood = Ls Fit “full” (or larger) model adding xj1,...,xjpj to smaller model. Save -2 log likelihood = LL Test statistic S = Ls - LL Distribution under null hypothesis: χ2pj Define rejection region based on this distribution Compute S Reject or not as S is in rejection region or not GLMs: “Global” Inference Many programs refer to “deviance” rather than -2 log likelihood This quantity equals the difference in -2 log likelihoods between ones fitted model and a “saturated model” Deviance measures “fit” Differences in deviances can be substituted for differences in -2 log likelihood in the method given on the previous page Likelihood ratio tests have appealing optimality properties Outline: A few more topics Model checking: Residuals, influence points ML can be written as an iteratively reweighted least squares algorithm Predictive accuracy Framework generalizes easily Main Points Generalized linear modeling provides a flexible regression framework for a variety of response types Continuous, categorical measurement scales Probability distributions tailored to the outcome Systematic model to accommodate Measurement range, interpretation Logistic regression Binary responses (yes, no) Bernoulli / binomial distribution Regression coefficients as log odds ratios for association between predictors and outcomes Main Points Generalized linear modeling accommodates description, inference, adjustment with the same flexibility as linear modeling Inference “Wald”- statistical tests and confidence intervals via parameter estimator standardization “Likelihood ratio” / “global” – via comparison of log likelihoods from nested models