## Microsoft word - model paper.doc

**From GLM to GLIMMIX-Which Model to Choose? **
Patricia B. Cerrito, University of Louisville, Louisville, KY

**ABSTRACT **

The purpose of this paper is to investigate several SAS procedures that are used in linear predictive models in

SAS/Stat. The primary focus will be on the correct choice of model given the designated outcome variable, and the

combination of input variables. Procedures to be discussed include GLM, LOGISTIC, GENMOD, MIXED, and

GLIMMIX. PROC GLIMMIX is a relatively new SAS procedure, although it has been available as a macro for some

time.

There are three main types of variables used in linear models: nominal, ordinal, and interval. Nominal is defined as

categorical (such as gender); ordinal is defined as categorical that can be ordered from least to most (such as

employee evaluation rank); interval data can define ratios. While all of the models discussed can include all three

types of input variables, the model choice is different if the outcome variable is interval or nominal.

Another consideration for model choice is whether the input variables are fixed effects or random effects. Fixed

effects are definitive, and will not change regardless of the sample data collection. Random effects can change when

the experiment is replicated. Examples of random effects include subjects in a drug study, choice of items to

compare between retail stores for market basket price differences, and classrooms in an education study. Examples

will be discussed.

**INTRODUCTION **

An inappropriate model will provide inappropriate results. For those users of SAS who know SAS/Stat and PROC

GLM, there are other models that are more appropriate to the collected data. It is necessary to fit the model to the

data-not the data to the model (to a man with a hammer….) If regression is not appropriate because the

assumptions are violated, change the model. There are several models readily available in SAS/Stat (Figure 1).

**Figure 1. Linear Models Available in SAS/Stat **
Generalized Linear Mixed Model PROC GLIMMIX
Each model serves a different purpose, and should be used with different types of data. The purpose of this paper is to focus on model choice; it is not intended to provide all details concerning the use of each model. Should the investigator choose one of the models, details are available in on-line docs. Items that must be considered in model choice are
1. Type of outcome variable-whether nominal, ordinal, or interval 2. Type of input variable-whether nominal ordinal, or interval
3. Type of input variable-whether fixed or random effect 4. Choice of covariance matrix format for random effects 5. Choice of link function for non-normal residuals

As the complexity of the data increases, so, too, does the complexity of the model. Choices must be made, choices

that impact model outcomes. Consider Table 1, which gives some indication as to how the models should be used.

**Table 1. Outline of Model Choice **

Model
**LOGISTIC **Binary

Categorical, Interval, Fixed Effects only Log-Normal
Categorical, Interval, Fixed Effects only Normality

This paper will discuss the different models, and how to define outcomes and inputs, along with a consideration of

the assumptions as listed in Table 1.

**PROC ANOVA and PROC REG **

ANOVA should only be used for a balanced design in which every categorical choice is divided equally. If there are

three treatments, then each treatment should have exactly the same number of observations. This procedure

requires less computing time compared to PROC GLM. However, since a completely balanced design almost never

happens with large samples, there is really no need to use ANOVA instead of GLM.

PROC REG can only use interval or ordinal variables as inputs. In order to include nominal data, dummy variables

need to be created. Too many nominal inputs requires considerable programming effort. Essentially, for each level of

a nominal variable, PROC REG creates a new regression line that is parallel to the regression lines for all other

levels of the same variable. While PROC REG has diagnostics that are of value, the same diagnostics have now

been incorporated into PROC GLM. For this reason, it is better to use PROC GLM for all standard analyses.

**PROC GLM **

In the past, PROC GLM was the most sophisticated procedure for performing a linear models analysis. It can use

both interval and categorical variables as inputs; it now contains all of the diagnostic elements provided by PROC

REG, and it does not require a balanced design. In addition, PROC GLM uses the Type III Sum of Squares to

examine multiple types of treatments simultaneously.

The one problem with PROC GLM is that is was never intended to be used with random effects. Special cases of

random effects, such as nested designs and split plot designs have been developed for use with PROC GLM.

Repeated measures, also, can be examined using PROC GLM provided that there are few subjects dropping out in

the later time measurements. However, PROC GLM has become the model of choice that is used, and very little

consideration is usually given to whether the inputs are fixed or random effects.

Repeated measures represent a random effect since the choice of time points to collect measurements is somewhat

arbitrary on the part of the investigator. Inputs such as age that are divided into blocks are also random effects since

the blocks are arbitrary. For the same reason, Likert scales are random effects since it is somewhat arbitrary

whether a 4-point or a 5-point scale is used. However, in many cases, these inputs are entered into PROC GLM as if

they were fixed effects.

However, as is true in the special cases of split plots and nested effects, assuming the effects are fixed when they

are random will increase the size of the random error. That will decrease the overall size of the F-statistics. As a

result, the model will have non-significant F-statistics that should be significant.

Consider the following question, Should ordinal variables be defined as quantitative, or as classification variables in

PROC GLM? Since ANOVA assumes class levels (ie nominal data), and regression assumes interval data, there is

no real provision for ordinal variables. If defined as a class variable, many degrees of freedom will be used, but post-

hoc tests can be made. If defined as interval, only one degree of freedom is used in the model but post-hoc tests are

unavailable. Depending on the choice, model results can differ.

Sample GLM code is listed below:

**PROC** **GLM** DATA=WORK.SORT7659

;

MODEL hours= CourseLevel expectknownever
SINGULAR=

**1E-07** ;

LSMEANS CourseLevel expectknownever / PDIFF=ALL ;

**PROC LOGISTIC **

PROC LOGISTIC is very similar to PROC GLM, although it has a binary outcome variable rather than an interval

outcome. If the outcome is ordinal, PROC LOGISTIC can also be used, but with a complementary log-log link

function instead of the more standard log function.

Both PROC LOGISTIC and PROC GLM can place ordinal inputs either as class or as quantitative variables. Again,

consideration of the degrees of freedom and the necessity of post-hoc tests should be made before deciding where

to place the ordinal inputs.

Frequently, logistic regression is used to divide a population into high risk/low risk. However, this dichotomous

outcome is contrived. There could just as easily be 5 or 10 categories of risk. It is not necessary to reduce the

number of outcomes to 2 just to fit the results into a logistic model.

Logistic regression also defines odds ratios for the input variables. However, the default does not provide confidence

limits for them. Therefore, the user should always use the option to print confidence limits. In addition, the user

should examine the c-statistics. It is comparable to the r2 for the general linear model. If the outcome variable only

has two levels, logistic regression can also print a classification table and a receiver operating curve. They can be

used to define a cut-point to divide the population into the high/low categories. Standard code is given below:

**PROC** **LOGISTIC** DATA=WORK.SORT7975

;

SELECTION=NONE

LINK=PROBIT

CLPARM=WALD

CLODDS=WALD

ALPHA=

**0.05** ;

OUTPUT

OUT=SASUSER.PRED3492(LABEL="Logistic
regression predictions and statistics for
SASUSER.QURY0181") PREDPROBS=INDIVIDUAL;
For ordinal (or nominal outcomes with more than 2 levels), the code used is

**PROC** **LOGISTIC** DATA=WORK.SORT1118

;

SELECTION=NONE

LINK=CLOGLOG

CLPARM=WALD

CLODDS=WALD

ALPHA=

**0.05** ;

OUTPUT

OUT=SASUSER.PRED1881(LABEL="Logistic
regression predictions and statistics for
SASUSER.QURY0181") PREDPROBS=INDIVIDUAL;
There are some cautions in order concerning logistic regression. Logistic regression will ALWAYS inflate results, especially if the group sizes are very different and one of the groups represents a rare event, For example, if one group size is 95% and one is 5%, then one classification rule (put all subjects in class A) will be 95% accurate.
Poisson regression should be used for rare events instead. If possible, fresh data should be used to examine the

inflation rate of results.

**PROC MIXED **

PROC MIXED has two components, y=αX+γZ+ε. If γ=0, then the mixed model is identical to the general linear

model. If γ≠0, then there is some randomness in the model and some covariance between inputs. Special cases of

the mixed model are repeated measures, nested designs, and split plot designs. Before the introduction of PROC

MIXED, these three special cases were considered using PROC GLM, but with some changes to the error terms.

PROC MIXED is a superior method for these cases.

In order to use PROC MIXED, the covariance must be estimated in some way. If the investigator has no knowledge

of how the input random effects correlate, the default unstructured matrix is the optimal choice. PROC MIXED has a

number of possible covariance matrix designs that can be used-but only if the user has a good idea of the structure

of the matrix. Standard code is

**PROC** **MIXED** DATA = WORK.SORT5396

METHOD=REML

;

MODEL hours_modified= Applied CourseLevel Statistics
DDFM=CONTAIN OUTPM=WORK._PRE6476(LABEL="Predicted
; RANDOM CourseLevel / G TYPE=VC; LSMEANS Applied CourseLevel Statistics / PDIFF=ALL ;

**PROC GENMOD **

PROC GENMOD generalizes PROC LOGISTIC by allowing for more than binary outcomes. For the general linear

model (GLM), the model equation takes the form Y=α+βX+ε so that the estimate is

*y*ˆ = β

is assumed normally distributed with mean zero and constant variance. For the generalized linear model, the
estimate changes to

*g*(

*y*ˆ) =

*X*β where g is called a link function. If

*g*(

*y*ˆ) =
binary, then the model is the special case of logistic regression and PROC LOGISTIC can be used. If the outcome variable consists of count data then the link function

*g*( ˆ

*y*) = log( ˆ

*y*) can be used. The assumption here is that the residuals have a Poisson distribution. However, this same link function can be used under the assumption that the residuals are interval data. In this case, the residuals are assumed to form a gamma distribution, which also includes the special case of the exponential distribution. There are a number of other distributions that can be used as well. The problem is that the residual distribution of

*g*(

*y*ˆ) =

*X*β depends upon the model, and that model depends upon the choice of the link function. Possible link
functions are given in Table 2.

**Table 2. Examples of Link Functions in PROC GENMOD **

**occurrence) **
**Continuous **Normal

If the investigator has some domain knowledge that allows him to choose a link function, that function should be used. However, if the investigator cannot estimate the function, another way is to estimate Y=α+βX first using PROC GLM while saving the residuals in a dataset. The data can be used in PROC KDE to estimate the form of the distribution. The investigator can then choose the link function that comes closest to the kernel distribution. The kernel can be examined using the following code listed below. Figure 2 gives an example kernel density estimator.

**proc** **kde** data=sasuser.qury0181;

univar hours/gridl=

**0** gridu=

**25** out=sasuser.kdehours;

**run**;

**PROC** **GPLOT** DATA = sasuser.kdehours;

PLOT density * value /

**Figure 2. Results of PROC KDE **

** **

Standard code for PROC GENMOD is given below:

MODEL hours= Applied Statistics workhabits
LINK=LOG DIST=GAMMA TYPE3 CORRB LRCI CL ALPHA=0.05;
LSMEANS Applied Statistics workhabits / ALPHA=0.05;
PREDICTED=_predicted1 RESDEV=_resdev1 RESCHI=_reschi1; RUN; QUIT;

** **

PROC GLIMMIX

This procedure generalizes the GENMOD procedure to include error terms that are not normally distributed. It also

generalizes the MIXED procedure to allow for random effects in the model. However, the random effects must be

normal. The general format for GLIMMIX is
Class block a b; Model y=a b a*b / ddf=#; Random block a*block; Lsmeans a b a*b / diff;

Unlike PROC MIXED, PROC GLIMMIX does not have a repeated statement, and repeated measures are in the

RANDOM statement. Possible link functions are given in Table 3.

**Table 3. Link Functions for PROC GLIMMIX **

**Outcome **
**Binomial **
**Exponential **
**Gaussian **
**Geometric **
**Lognormal **
**Multinomial **
**Negbinomial **
** Tcentral **
** **

Sample code is given below:

MODEL hours_modified= Applied CourseLevel Statistics
DDFM=CONTAIN dist=gamma ; RANDOM CourseLevel / G TYPE=VC; LSMEANS Applied CourseLevel Statistics / PDIFF=ALL ; RUN; QUIT;

**EXAMPLES **

Consider the following examples:

• A test to compare the effectiveness of CT scans to x-ray in the detection of lung cancer. Each patient is
randomized to receive x-ray only or CT only. 10,000 patients are in the sample, limited to high-risk patients. The outcome variable is the occurrence of lung cancer.
• A randomized clinical trial to compare treatment of osteomyelitis (MRSA) with vancomycin and Zyvox.
Patients are treated according to protocol, with follow up at 1, 2, 6, 12 months after end of treatment. What if the study is observational rather than randomized?
In the first example, the occurrence of lung cancer is rare. Therefore, a Poisson distribution would better fit the study than a logistic regression. In the second, the measure of recurrence is a repeated measure. While it can also be
examined using survival analysis, the fact that measurements are at fixed intervals rather than continuous will also allow for a mixed models design.

**CONCLUSION **

While it is possible to use PROC GLIMMIX as the most complex of the models, it is not advisable. Even so, choices

as to random versus fixed effects, link function, and covariance matrix still have to be made. Therefore, the

investigator should use the simplest procedure that will accommodate the variable choices.

Patricia Cerrito University of Louisville Department of Mathematics Louisville, KY 40292 502-852-6826 502-852-7132 (fax)

[email protected]
SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. ® indicates USA registration. Other brand and product names are trademarks of their respective companies.

Source: http://www.technion.ac.il/usg/stat/SAS/SAS-Glimmix/From%20GLM%20to%20GLIMMIX-Which%20Model%20to%20Choose.pdf

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