The GENMOD Procedure

## Example 29.6: Log Odds Ratios and the ALR Algorithm

Since the respiratory data in Example 29.5 are binary, you can use the ALR algorithm to model the log odds ratios instead of using working correlations to model associations. Here, a "fully parameterized cluster" model for the log odds ratio is fit. That is, there is a log odds ratio parameter for each unique pair of responses within clusters, and all clusters are parameterized identically. The following statements fit the same regression model for the mean as in Example 29.5 but use a regression model for the log odds ratios instead of a working correlation. The LOGOR=FULLCLUST option specifies a fully parameterized log odds ratio model.

```   proc genmod data=resp;
class id center;
model outcome=center2 active female age baseline / dist=bin;
repeated  subject=id(center) / logor=fullclust;
run;
```

The results of fitting the model are displayed in Output 29.6.1 along with a table that shows the correspondence between the log odds ratio parameters and the within cluster pairs.

Output 29.6.1: Results of Model Fitting

 The GENMOD Procedure

 Log Odds Ratio Parameter Information Parameter Group Alpha1 (1, 2) Alpha2 (1, 3) Alpha3 (1, 4) Alpha4 (2, 3) Alpha5 (2, 4) Alpha6 (3, 4)

 Analysis Of GEE Parameter Estimates Empirical Standard Error Estimates Parameter Estimate Standard Error 95% Confidence Limits Z Pr > |Z| Intercept -0.9266 0.4513 -1.8111 -0.0421 -2.05 0.0400 center2 0.6287 0.3486 -0.0545 1.3119 1.80 0.0713 active 1.2611 0.3406 0.5934 1.9287 3.70 0.0002 female 0.1024 0.4362 -0.7526 0.9575 0.23 0.8144 age -0.0162 0.0125 -0.0407 0.0084 -1.29 0.1977 baseline 1.8980 0.3404 1.2308 2.5652 5.58 <.0001 Alpha1 1.6109 0.4892 0.6522 2.5696 3.29 0.0010 Alpha2 1.0771 0.4834 0.1297 2.0246 2.23 0.0259 Alpha3 1.5875 0.4735 0.6594 2.5155 3.35 0.0008 Alpha4 2.1224 0.5022 1.1381 3.1068 4.23 <.0001 Alpha5 1.8818 0.4686 0.9634 2.8001 4.02 <.0001 Alpha6 2.1046 0.4949 1.1347 3.0745 4.25 <.0001

You can fit the same model by fully specifying the z-matrix. The following statements create a data set containing the full z-matrix.

```   data zin;
keep id center z1-z6 y1 y2;
array zin(6) z1-z6;
set resp ;
by center id;
if first.id
then do;
t = 0;
do m = 1 to 4;
do n = m+1 to 4;
do j = 1 to 6;
zin(j) = 0;
end;
y1 = m;
y2 = n;
t + 1;
zin(t) = 1;
output;
end;
end;
end;
run;
proc print data=zin (obs=12);
run;
```

Output 29.6.2 displays the full z-matrix for the first two clusters. The z-matrix is identical for all clusters in this example.

Output 29.6.2: Full z-Matrix Data Set

 Obs z1 z2 z3 z4 z5 z6 center id y1 y2 1 1 0 0 0 0 0 1 1 1 2 2 0 1 0 0 0 0 1 1 1 3 3 0 0 1 0 0 0 1 1 1 4 4 0 0 0 1 0 0 1 1 2 3 5 0 0 0 0 1 0 1 1 2 4 6 0 0 0 0 0 1 1 1 3 4 7 1 0 0 0 0 0 1 2 1 2 8 0 1 0 0 0 0 1 2 1 3 9 0 0 1 0 0 0 1 2 1 4 10 0 0 0 1 0 0 1 2 2 3 11 0 0 0 0 1 0 1 2 2 4 12 0 0 0 0 0 1 1 2 3 4

The following statements fit the model for fully parameterized clusters by fully specifying the z-matrix. The results are identical to those shown previously.

```   proc genmod data=resp;
class id center;
model outcome=center2 active female age baseline / dist=bin;
repeated  subject=id(center) / logor=zfull
zdata=zin
zrow =(z1-z6)
ypair=(y1 y2) ;
run;
```