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PROC GLIMMIX: AN OVERVIEW By William E. Jackman PROC GLIMMIX: AN OVERVIEW • • • • • A new SAS/STAT Product Experimental in SAS 9.1 Production in SAS 9.2. %GLIMMIX macro Combines and extends statistical features found in other SAS procedures • Part of a succession of SAS procedures which have extended the General Linear Model (GLM) PROC GLIMMIX: AN OVERVIEW • • • • • • • • Regression Analysis Basics Y = B0 + B1 X1 +B2 X2 ... + Bn Xn + e y = Xβ + ε (matrix notation) ε ~ N(0, α2 In) Estimation by ordinary least squares (OLS). Essence of the General Linear Model (GLM) Y's and the X's go by several names Covariates PROC GLIMMIX: AN OVERVIEW • The GLM underlies PROC REG and PROC GLM • Both procedures use OLS to fit the GLM to data with continuous response variable • Same assumptions about residuals • PROC REG has advantages for continuous effects (regressors). • PROC GLM has advantages for discrete effects (regressors). PROC GLIMMIX: AN OVERVIEW • Indicator (dummy) variables and interactions * PROC REG: must be created in data step * PROC GLM: use class & model statements • Which Procedure to use? * Interested primarily in effect of continuous variables (covariates)? * Interested primarily in effect of grouping variables? PROC GLIMMIX: AN OVERVIEW • The generalized linear model (GzLM) extends (or generalizes) the GLM. • Presented in 1972; expanded in 1989. • Non-normal data from exponential family • Linearity is achieved through the link function. • Implemented, for example, in PROC GENMOD • PROC GENMOD can also handle correlated residuals. PROC GLIMMIX: AN OVERVIEW • General form of the GENMOD procedure • PROC GENMOD options ; • CLASS variables ; • MODEL response=effects / dist= link= options ; • REPEATED SUBJECT=subjects-effects / options ; • RUN ; PROC GLIMMIX: AN OVERVIEW Example of the GENMOD procedure for Poisson regression proc genmod data=skin ; class city age ; model cases=city age / offset=log_pop dist=poi link=log ; run ; where log_pop = log of the population PROC GLIMMIX: AN OVERVIEW The generalized linear model (GzLM) • • • • Canonical link functions most common. Obtained from probability density function Default in PROC GENMOD For the Poisson distribution the default link function is the log of the response variable. • log(μ) = Xβ • Inverse link functions • μ = eη PROC GLIMMIX: AN OVERVIEW Logistic Regression: A special case of the generalized linear model (GzLM) • • • • • • Response variable from binomial distribution Part of the exponential family so GzLM applies Link function is the logit. logit(pi) = ln(pi / (1-pi)) Can be done with PROC GENMOD Input from David Schlotzhauer of SAS Institute PROC GLIMMIX: AN OVERVIEW FURTHER EXTENSIONS OF THE GLM • • • • GLM and GzLM cannot handle random effects. Fixed effects-interest only in levels specified Random effects-inference to other levels PROC GENMOD and PROC LOGISTIC cannot handle random effects. PROC GLIMMIX: AN OVERVIEW PROC MIXED: An extension of the GLM • Can handle random effects and correlated errors • fixed effects only model • y = Xβ + ε • mixed model • y = Xβ + Zγ + ε PROC GLIMMIX: AN OVERVIEW Mixed models distinguish between G-side random effects and R-side random effects. • G-side random effects correspond to covariates (regressors) in the model which are random. • R-side random effects correspond to the residuals in the model. PROC GLIMMIX: AN OVERVIEW Example of PROC MIXED syntax proc mixed ; class id time gender ; model z = gender age gender*age ; random intercept / subject=id ; *** G-side effects go here. ; repeated time /subject=id type=ar(1) ; *** R-side effects go here. ; run ; PROC GLIMMIX: AN OVERVIEW PROC MIXED: a linear mixed model (LMM) • PROC MIXED allows for random intercepts for each subject. • models the correlation in the repeated measures within each subject. • has rich variety of covariance matrices for dealing with correlated residuals. • Unlike GzLM’s, LMM’s require a normally distributed response variable. PROC GLIMMIX: AN OVERVIEW • PROC GLIMMIX - PUTTING IT ALL TOGETHER • A Generalized Linear Mixed Model (GzLMM) • Combines and extends features of GzLM’s and LMM’s • Enables modeling random effects and correlated errors for non-normal data PROC GLIMMIX: AN OVERVIEW The Generalized Linear Mixed Model (GzLMM) • A linear predictor can contain random effects: η = Xβ + Z γ • The random effects are normally distributed • The conditional mean, μ|γ, relates to the linear predictor through a link function: g(μ|γ) = η • The conditional distribution (given γ) of the data belongs to the exponential family of distributions. PROC GLIMMIX: AN OVERVIEW Other new features of PROC GLIMMIX include: • low-rank smoothing based on mixed models • new features for LS-means comparisons and display. • SAS programming statements allowed within the procedure • Fits models to multivariate data with different distributions or links PROC GLIMMIX: AN OVERVIEW General form of the GLIMMIX procedure: • PROC GLIMMIX options ; • programming statements ; • CLASS variables ; • MODEL response=fixed-effects / DIST= LINK = options ; • RANDOM random-effects / options ; • RANDOM _RESIDUAL_ / options ; • RUN ; PROC GLIMMIX: AN OVERVIEW Like other mixed models, PROC GLIMMIX distinguishes between G-side random effects and R-side random effects. • G-side random effects correspond to covariates in the model which are random. • R-side random effects correspond to the residuals in the model. PROC GLIMMIX: AN OVERVIEW Example of a GzLMM using PROC GLIMMIX for Logistic Regression with Random Effects • proc glimmix data=example ; • class trt clinic ; • model y=trt / dist=binomial link=logit ; • random clinic trt*clinic ; • *** random intercept trt / subject=clinic ; • run ; PROC GLIMMIX: AN OVERVIEW • This example cannot be handled by PROC LOGISTIC since clinic is a random effect. • For logistic regression with fixed effect only, PROC GLIMMIX or PROC LOGISTIC can be used. Which should you use? • More input from David Schlotzhauer of the SAS Institute. PROC GLIMMIX: AN OVERVIEW Parameters Estimation Methods in PROC GLIMMIX • The GLIMMIX procedure has two basic modes of parameter estimation: GLM-mode and GLMMmode. • In GLM-mode, the data is never correlated and there can be no G-side random effect. • In the GLMM-mode, there might be random effects and/or correlated data. PROC GLIMMIX: AN OVERVIEW Parameter Estimation for generalized linear models • Normal distribution: restricted maximum likelihood • All other known distributions: maximum likelihood • Unknown distributions: quasi-likelihood PROC GLIMMIX: AN OVERVIEW Parameter Estimation for generalized linear models with overdispersion • Parameters are estimated using maximum likelihood • An overdispersion parameter can be estimated from the Pearson statistic PROC GLIMMIX: AN OVERVIEW Parameter Estimation for generalized linear mixed models • Pseudo-likelihood PROC GLIMMIX: AN OVERVIEW Using PROC GLIMMIX for Linear Mixed Models • In this example, the response variable is normally-distributed. • Proc glimmix data= grass ; • Class method variety ; • Model yield = method / dist=normal ; • Random variety method*variety ; • run ; • PROC GLIMMIX uses the residual/restricted maximum likelihood as does PROC MIXED. PROC GLIMMIX: AN OVERVIEW • PROC GLIMMIX can do much of what PROC LOGISTIC, PROC MIXED, PROC REG, and PROC GLM can do. • Could be viewed as a “super PROC” • Input from Jill Tao of the SAS Institute PROC GLIMMIX: AN OVERVIEW PROC GLIMMIX versus PROC MIXED Closely related but important differences • PROC GLIMMIX is not PROC MIXED with a LINK= and a DIST= option. • PROC GLIMMIX models non-normal data. PROC MIXED does not. • PROC GLIMMIX allows programming statements. PROC MIXED does not. • PROC GLIMMIX uses the RANDOM statement to model R-side random effects. PROC MIXED uses the REPEATED statement to model R-side random effects. • PROC GLIMMIX does not support the Kronecker and heterogeneous covariance structures as supported by PROC MIXED. PROC GLIMMIX: AN OVERVIEW • • • • PROC GLIMMIX versus PROC GENMOD PROC GLIMMIX fits unit-specific models with the G-side random effects fits population-average models without the G-side effects. (Without the G-side effects, there is no way to condition the response and make the estimates unitspecific.) provides sandwich estimators of covariance of fixed effects through the EMPIRICAL option when the model is processed by subjects. computes the parameter estimates by a pseudolikelihood method. PROC GLIMMIX: AN OVERVIEW PROC GLIMMIX versus PROC GENMOD PROC GENMOD • cannot accommodate random effects • fits only population-average models • computes the parameter estimates by a moment-based method. PROC GLIMMIX: AN OVERVIEW Applications Using the GLIMMIX Procedure (from "Statistical Analysis with the GLIMMIX Procedure") • Poisson Regression with Random Effects • An example of Beta Regression • Repeated Measures Data with Discrete Response • Introduction to Radial Smoothing Applications are explained in detail in the SAS course. PROC GLIMMIX: AN OVERVIEW Fitting Models To Multivariate Data In Which Observations Do Not All Have The Same Distribution Or Link • EXAMPLE: JOINT MODELS FOR BINARY AND POISSON DATA (from a paper by Oliver Schabenberger of the SAS Institute) PROC GLIMMIX: AN OVERVIEW data joint; length dist $7; input d$ patient age OKstatus response @@; if d = ’B’ then dist=’Binary’; else dist=’Poisson’; datalines; (only 3 lines shown) B 1 78 1 0 P 1 78 1 9 B 2 60 1 0 P 2 60 1 4 B 3 68 1 1 P 3 68 1 7 B 4 62 0 1 P 4 62 0 35 B 5 76 0 0 P 5 76 0 9 B 6 76 1 1 P 6 76 1 7 PROC GLIMMIX: AN OVERVIEW proc glimmix data=joint; class patient dist; model response(event=’1’) = dist dist*age dist*OKstatus / noint s dist=byobs(dist); random int / subject=patient; run; PROC GLIMMIX: AN OVERVIEW • The previous slide showed modeling correlations through G-side random effects. It could also be done through R-side random effects. This is presented in the SAS course “Statistical Analysis with the GLIMMIX Procedure” which expands upon this example.