PROC GLIMMIX: AN OVERVIEW By William E. Jackman. PROC GLIMMIX: AN OVERVIEW A new SAS/STAT Product...

PROC GLIMMIX: AN OVERVIEW

By William E. Jackman


• 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)


• 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


• 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?


• 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.


• General form of the GENMOD procedure• PROC GENMOD options ;• CLASS variables ;• MODEL response=effects / dist= link=

options ;• REPEATED SUBJECT=subjects-effects /

options ;• RUN ;


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


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


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


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 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


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.


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 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 - 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 OVERVIEWThe 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.


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


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 ;


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.


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 ;


• 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.


Parameters Estimation Methods in PROC GLIMMIX

• The GLIMMIX procedure has two basic modes of parameter estimation: GLM-mode and GLMM-mode.

• 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.


Parameter Estimation for generalized linear models

• Normal distribution: restricted maximum likelihood

• All other known distributions: maximum likelihood

• Unknown distributions: quasi-likelihood


Parameter Estimation for generalized linear models with overdispersion

• Parameters are estimated using maximum likelihood

• An overdispersion parameter can be estimated from the Pearson statistic


Parameter Estimation for generalized linear mixed models

• Pseudo-likelihood


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 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 OVERVIEWPROC 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 versus PROC GENMODPROC 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 unit-specific.)

• 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 pseudo-likelihood method.


PROC GLIMMIX versus PROC GENMODPROC GENMOD

• cannot accommodate random effects• fits only population-average models• computes the parameter estimates by a

moment-based method.


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.


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)


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 4B 3 68 1 1 P 3 68 1 7 B 4 62 0 1 P 4 62 0 35B 5 76 0 0 P 5 76 0 9 B 6 76 1 1 P 6 76 1 7


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;


• 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.

PROC GLIMMIX: AN OVERVIEW By William E. Jackman. PROC GLIMMIX: AN OVERVIEW A new SAS/STAT Product...

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