Output 39.5.1 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 at the th 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.
Obs | center | id | treatment | sex | age | baseline | visit1 | visit2 | visit3 | visit4 | visit | outcome |
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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 be modeled. If the DESCEND option had not been specified, the probability that would be modeled by default.
Information about the GEE model is displayed in Output 39.5.2. The results of GEE model fitting are displayed in Output 39.5.3. Model goodness-of-fit criteria are displayed in Output 39.5.4. If you specify no other options, the standard errors, confidence intervals, scores, and -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.
GEE Model Information | |
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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 |
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 |
GEE Fit Criteria | |
---|---|
QIC | 512.3416 |
QICu | 499.6081 |
The nonsignificance of age and sex make them candidates for omission from the model.