## Example 39.7 Log-Linear Model for Count Data

In this example the data, from Thall and Vail (1990), concern the treatment of people suffering from epileptic seizure episodes. These data are also analyzed in Diggle, Liang, and Zeger (1994). The data consist of the number of epileptic seizures in an eight-week baseline period, before any treatment, and in each of four two-week treatment periods, in which patients received either a placebo or the drug Progabide in addition to other therapy. A portion of the data is displayed in Table 39.12. See "Gee Model for Count Data, Exchangeable Correlation" in the SAS/STAT Sample Program Library for the complete data set.

Table 39.12 Epileptic Seizure Data

Patient ID

Treatment

Baseline

Visit1

Visit2

Visit3

Visit4

104

Placebo

11

5

3

3

3

106

Placebo

11

3

5

3

3

107

Placebo

6

2

4

0

5

.

.

.

101

Progabide

76

11

14

9

8

102

Progabide

38

8

7

9

4

103

Progabide

19

0

4

3

0

.

.

.

Model the data as a log-linear model with (the Poisson variance function) and

where

number of epileptic seizures in interval

length of interval

The correlations between the counts are modeled as , (exchangeable correlations). For comparison, the correlations are also modeled as independent (identity correlation matrix). In this model, the regression parameters have the interpretation in terms of the log seizure rate displayed in Table 39.13.

Table 39.13 Interpretation of Regression Parameters

Treatment

Visit

Placebo

Baseline

1–4

Progabide

Baseline

1–4

The difference between the log seizure rates in the pretreatment (baseline) period and the treatment periods is for the placebo group and for the Progabide group. A value of indicates a reduction in the seizure rate.

Output 39.7.1 lists the first 14 observations of the data, which are arranged as one visit per observation:

Output 39.7.1 Partial Listing of the Seizure Data
Obs id y visit trt bline age
1 104 5 1 0 11 31
2 104 3 2 0 11 31
3 104 3 3 0 11 31
4 104 3 4 0 11 31
5 106 3 1 0 11 30
6 106 5 2 0 11 30
7 106 3 3 0 11 30
8 106 3 4 0 11 30
9 107 2 1 0 6 25
10 107 4 2 0 6 25
11 107 0 3 0 6 25
12 107 5 4 0 6 25
13 114 4 1 0 8 36
14 114 4 2 0 8 36

Some further data manipulations create an observation for the baseline measures, a log time interval variable for use as an offset, and an indicator variable for whether the observation is for a baseline measurement or a visit measurement. Patient 207 is deleted as an outlier, as in the Diggle, Liang, and Zeger (1994) analysis. The following statements prepare the data for analysis with PROC GENMOD:

```data new;
set thall;
output;
if visit=1 then do;
y=bline;
visit=0;
output;
end;
run;

data new;
set new;
if id ne 207;
if visit=0 then do;
x1=0;
ltime=log(8);
end;
else do;
x1=1;
ltime=log(2);
end;
run;
```

For comparison with the GEE results, an ordinary Poisson regression is first fit. The results are shown in Output 39.7.2.

Output 39.7.2 Maximum Likelihood Estimates
The GENMOD Procedure

Analysis Of Maximum Likelihood Parameter Estimates
Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald Chi-Square Pr > ChiSq
Intercept 1 1.3476 0.0341 1.2809 1.4144 1565.44 <.0001
x1 1 0.1108 0.0469 0.0189 0.2027 5.58 0.0181
trt 1 -0.1080 0.0486 -0.2034 -0.0127 4.93 0.0264
x1*trt 1 -0.3016 0.0697 -0.4383 -0.1649 18.70 <.0001
Scale 0 1.0000 0.0000 1.0000 1.0000

 Note: The scale parameter was held fixed.

The GEE solution is requested with the REPEATED statement in the GENMOD procedure. The SUBJECT=ID option indicates that the variable id describes the observations for a single cluster, and the CORRW option displays the working correlation matrix. The TYPE= option specifies the correlation structure; the value EXCH indicates the exchangeable structure.

The following statements perform the analysis:

```proc genmod data=new;
class id;
model y=x1 | trt / d=poisson offset=ltime;
repeated subject=id / corrw covb type=exch;
run;
```

These statements first fit a generalized linear model (GLM) to these data by maximum likelihood. The estimates are not shown in the output, but are used as initial values for the GEE solution.

Information about the GEE model is displayed in Output 39.7.3. The results of fitting the model are displayed in Output 39.7.4. Compare these with the model of independence displayed in Output 39.7.2. The parameter estimates are nearly identical, but the standard errors for the independence case are underestimated. The coefficient of the interaction term, , is highly significant under the independence model and marginally significant with the exchangeable correlations model.

Output 39.7.3 GEE Model Information
The GENMOD Procedure

GEE Model Information
Correlation Structure Exchangeable
Subject Effect id (58 levels)
Number of Clusters 58
Correlation Matrix Dimension 5
Maximum Cluster Size 5
Minimum Cluster Size 5

Output 39.7.4 GEE Parameter Estimates
Analysis Of GEE Parameter Estimates
Empirical Standard Error Estimates
Parameter Estimate Standard Error 95% Confidence Limits Z Pr > |Z|
Intercept 1.3476 0.1574 1.0392 1.6560 8.56 <.0001
x1 0.1108 0.1161 -0.1168 0.3383 0.95 0.3399
trt -0.1080 0.1937 -0.4876 0.2716 -0.56 0.5770
x1*trt -0.3016 0.1712 -0.6371 0.0339 -1.76 0.0781

Table 39.14 displays the regression coefficients, standard errors, and normalized coefficients that result from fitting the model with independent and exchangeable working correlation matrices.

Table 39.14 Results of Model Fitting

Variable

Correlation Structure

Coef.

Std. Error

Coef./S.E.

Intercept

Exchangeable

1.35

0.16

8.56

Independent

1.35

0.03

39.52

Visit

Exchangeable

0.11

0.12

0.95

Independent

0.11

0.05

2.36

Treat

Exchangeable

0.11

0.19

0.56

Independent

0.11

0.05

2.22

Exchangeable

0.30

0.17

1.76

Independent

0.30

0.07

4.32

The fitted exchangeable correlation matrix is specified with the CORRW option and is displayed in Output 39.7.5.

Output 39.7.5 Working Correlation Matrix
Working Correlation Matrix
Col1 Col2 Col3 Col4 Col5
Row1 1.0000 0.5941 0.5941 0.5941 0.5941
Row2 0.5941 1.0000 0.5941 0.5941 0.5941
Row3 0.5941 0.5941 1.0000 0.5941 0.5941
Row4 0.5941 0.5941 0.5941 1.0000 0.5941
Row5 0.5941 0.5941 0.5941 0.5941 1.0000

If you specify the COVB option, you produce both the model-based (naive) and the empirical (robust) covariance matrices. Output 39.7.6 contains these estimates.

Output 39.7.6 Covariance Matrices
Covariance Matrix (Model-Based)
Prm1 Prm2 Prm3 Prm4
Prm1 0.01223 0.001520 -0.01223 -0.001520
Prm2 0.001520 0.01519 -0.001520 -0.01519
Prm3 -0.01223 -0.001520 0.02495 0.005427
Prm4 -0.001520 -0.01519 0.005427 0.03748

Covariance Matrix (Empirical)
Prm1 Prm2 Prm3 Prm4
Prm1 0.02476 -0.001152 -0.02476 0.001152
Prm2 -0.001152 0.01348 0.001152 -0.01348
Prm3 -0.02476 0.001152 0.03751 -0.002999
Prm4 0.001152 -0.01348 -0.002999 0.02931

The two covariance estimates are similar, indicating an adequate correlation model.