Output 40.5.1 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
(), treatment
(), center
(), sex
(), age
(), and baseline
(), so that 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:
Suppose represents the respiratory status of patient i at the jth visit, , and represents the mean of the respiratory status. Since the response data are binary, you can use the variance function for the binomial distribution and the logit link function . The model for the mean is , where is a vector of regression parameters to be estimated.
Output 40.5.1: 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 a single cluster be uniquely identified by center
and id
within center
. 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 40.5.2. The results of GEE model fitting are displayed in Output 40.5.3. Model goodness-of-fit criteria are displayed in Output 40.5.4. 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 40.5.2: Model Fitting Information
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 40.5.3: 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 |
The nonsignificance of age
and sex
make them candidates for omission from the model.