GEE for Binary Data with Logit Link Function
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The GENMOD Procedure

Example 42.5 GEE for Binary Data with Logit Link Function

Output 42.5.1 displays a partial listing of a SAS data set of clinical trial data comparing two treatments for a respiratory disorder. See “Gee Model for Binary Data” in the SAS/STAT Sample Program Library for the complete data set. These data are from Stokes, Davis, and Koch (2000).

Patients in each of two centers are randomly assigned to groups receiving the active treatment or a placebo. During treatment, respiratory status, represented by the variable outcome (coded here as 0=poor, 1=good), is determined for each of four visits. The variables center, treatment, sex, and baseline (baseline respiratory status) are classification variables with two levels. The variable age (age at time of entry into the study) is a continuous variable.

Explanatory variables in the model are Intercept ($x_{ij1}$), treatment ($x_{ij2}$), center ($x_{ij3}$), sex ($x_{ij4}$), age ($x_{ij5}$), and baseline ($x_{ij6}$), so that $x^\prime =[x_{ij1},x_{ij2},\ldots ,x_{ij6}]$ is the vector of explanatory variables. Indicator variables for the classification explanatory variables can be automatically generated by listing them in the CLASS statement in PROC GENMOD. To be consistent with the analysis in Stokes, Davis, and Koch (2000), the four classification explanatory variables are coded as follows via options in the CLASS statement:

\[ \begin{array}{llllll} x_{ij2}& =& \left\{ \begin{array}{l} \mbox{0 placebo} \\ \mbox{1 active} \end{array}\right.& x_{ij3}& =& \left\{ \begin{array}{l} \mbox{0 center 1} \\ \mbox{1 center 2} \end{array}\right. \\ \end{array} \]
\[ \begin{array}{llllll} x_{ij4}& =& \left\{ \begin{array}{l} \mbox{0 male} \\ \mbox{1 female} \end{array}\right. & x_{ij6}& =& \left\{ \begin{array}{l} \mbox{0 0} \\ \mbox{1 1} \end{array}\right. \\ \end{array} \]

Suppose $y_{ij}$ represents the respiratory status of patient i at the jth visit, $j=1,\ldots ,4$, and $\mu _{ij}=\mr {E}(y_{ij})$ represents the mean of the respiratory status. Since the response data are binary, you can use the variance function for the binomial distribution $v(\mu _{ij})=\mu _{ij}(1-\mu _{ij})$ and the logit link function $g(\mu _{ij}) = \log (\mu _{ij}/(1-\mu _{ij}))$. The model for the mean is $g(\mu _{ij})={\mb {x}_{ij}}^\prime \bbeta $, where $\bbeta $ is a vector of regression parameters to be estimated.

Output 42.5.1: Respiratory Disorder Data

Obs center id treatment sex age baseline visit1 visit2 visit3 visit4 visit outcome
1 1 1 P M 46 0 0 0 0 0 1 0
2 1 1 P M 46 0 0 0 0 0 2 0
3 1 1 P M 46 0 0 0 0 0 3 0
4 1 1 P M 46 0 0 0 0 0 4 0
5 1 2 P M 28 0 0 0 0 0 1 0
6 1 2 P M 28 0 0 0 0 0 2 0
7 1 2 P M 28 0 0 0 0 0 3 0
8 1 2 P M 28 0 0 0 0 0 4 0
9 1 3 A M 23 1 1 1 1 1 1 1
10 1 3 A M 23 1 1 1 1 1 2 1
11 1 3 A M 23 1 1 1 1 1 3 1
12 1 3 A M 23 1 1 1 1 1 4 1
13 1 4 P M 44 1 1 1 1 0 1 1
14 1 4 P M 44 1 1 1 1 0 2 1
15 1 4 P M 44 1 1 1 1 0 3 1
16 1 4 P M 44 1 1 1 1 0 4 0
17 1 5 P F 13 1 1 1 1 1 1 1
18 1 5 P F 13 1 1 1 1 1 2 1
19 1 5 P F 13 1 1 1 1 1 3 1
20 1 5 P F 13 1 1 1 1 1 4 1


The GEE solution is requested with the REPEATED statement in the GENMOD procedure. The option SUBJECT=ID(CENTER) specifies that the observations in any single cluster are uniquely identified by both center and id. An equivalent specification is SUBJECT=ID*CENTER. Since the same id values are used in each center, one of these specifications is needed. If id values were unique across all centers, SUBJECT=ID would be specified.

The option TYPE=UNSTR specifies the unstructured working correlation structure. The MODEL statement specifies the regression model for the mean with the binomial distribution variance function. The following SAS statements perform the GEE model fit: 

proc genmod data=resp descend;
   class id treatment(ref="P") center(ref="1") sex(ref="M")
      baseline(ref="0") / param=ref;
   model outcome=treatment center sex age baseline / dist=bin;
   repeated subject=id(center) / corr=unstr corrw;
run;

These statements first fit the generalized linear (GLM) model specified in the MODEL statement. The parameter estimates from the generalized linear model fit are not shown in the output, but they are used as initial values for the GEE solution. The DESCEND option in the PROC GENMOD statement specifies that the probability that outcome = 1 be modeled. If the DESCEND option had not been specified, the probability that outcome = 0 would be modeled by default.

Information about the GEE model is displayed in Output 42.5.2. The results of GEE model fitting are displayed in Output 42.5.3. Model goodness-of-fit criteria are displayed in Output 42.5.4. If you specify no other options, the standard errors, confidence intervals, Z scores, and p-values are based on empirical standard error estimates. You can specify the MODELSE option in the REPEATED statement to create a table based on model-based standard error estimates.

Output 42.5.2: Model Fitting Information

The GENMOD Procedure

GEE Model Information
Correlation Structure Unstructured
Subject Effect id(center) (111 levels)
Number of Clusters 111
Correlation Matrix Dimension 4
Maximum Cluster Size 4
Minimum Cluster Size 4


Output 42.5.3: Results of Model Fitting

Working Correlation Matrix
  Col1 Col2 Col3 Col4
Row1 1.0000 0.3351 0.2140 0.2953
Row2 0.3351 1.0000 0.4429 0.3581
Row3 0.2140 0.4429 1.0000 0.3964
Row4 0.2953 0.3581 0.3964 1.0000

Analysis Of GEE Parameter Estimates
Empirical Standard Error Estimates
Parameter   Estimate Standard Error 95% Confidence Limits Z Pr > |Z|
Intercept   -0.8882 0.4568 -1.7835 0.0071 -1.94 0.0519
treatment A 1.2442 0.3455 0.5669 1.9214 3.60 0.0003
center 2 0.6558 0.3512 -0.0326 1.3442 1.87 0.0619
sex F 0.1128 0.4408 -0.7512 0.9768 0.26 0.7981
age   -0.0175 0.0129 -0.0427 0.0077 -1.36 0.1728
baseline 1 1.8981 0.3441 1.2237 2.5725 5.52 <.0001


Output 42.5.4: Model Fit Criteria

GEE Fit Criteria
QIC 512.3416
QICu 499.6081


The nonsignificance of age and sex make them candidates for omission from the model.

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