### Example 32.4 Log-Linear Model, Three Dependent Variables

This analysis reproduces the predicted cell frequencies for Bartlett’s data by using a log-linear model of no three-variable interaction (Bishop, Fienberg, and Holland, 1975, p. 89). Cuttings of two different lengths (`Length`=short or long) are planted at one of two time points (`Time`=now or spring), and their survival status (`Status`=dead or alive) is recorded.

As in the text, the variable levels are simply labeled 1 and 2. The following statements produce Output 32.4.1 through Output 32.4.3:

```data bartlett;
input Length Time Status wt @@;
datalines;
1 1 1 156     1 1 2  84     1 2 1 84     1 2 2 156
2 1 1 107     2 1 2 133     2 2 1 31     2 2 2 209
;
```
```title 'Bartlett''s Data';
proc catmod data=bartlett;
weight wt;
model Length*Time*Status=_response_
/ noparm pred=freq;
loglin Length|Time|Status @ 2;
title2 'Model with No 3-Variable Interaction';
quit;
```

Output 32.4.1: Analysis of Bartlett's Data: Log-Linear Model

 Bartlett's Data Model with No 3-Variable Interaction

The CATMOD Procedure

Data Summary
Response Length*Time*Status Response Levels 8
Weight Variable wt Populations 1
Data Set BARTLETT Total Frequency 960
Frequency Missing 0 Observations 8

Population Profiles
Sample Sample Size
1 960

Response Profiles
Response Length Time Status
1 1 1 1
2 1 1 2
3 1 2 1
4 1 2 2
5 2 1 1
6 2 1 2
7 2 2 1
8 2 2 2

Maximum Likelihood Analysis
Maximum likelihood computations converged.

Maximum Likelihood Analysis of Variance
Source DF Chi-Square Pr > ChiSq
Length 1 2.64 0.1041
Time 1 5.25 0.0220
Length*Time 1 5.25 0.0220
Status 1 48.94 <.0001
Length*Status 1 48.94 <.0001
Time*Status 1 95.01 <.0001
Likelihood Ratio 1 2.29 0.1299

The analysis of variance table shows that the model fits since the likelihood ratio test for the three-variable interaction is nonsignificant. All of the two-variable interactions, however, are significant; this shows that there is mutual dependence among all three variables.

The predicted values table (Output 32.4.2) displays observed and predicted values for the generalized logits.

Output 32.4.2: Response Function Predicted Values

Maximum Likelihood Predicted Values for Response Functions
Function
Number
Observed Predicted Residual
Function Standard
Error
Function Standard
Error
1 -0.29248 0.105806 -0.23565 0.098486 -0.05683
2 -0.91152 0.129188 -0.94942 0.129948 0.037901
3 -0.91152 0.129188 -0.94942 0.129948 0.037901
4 -0.29248 0.105806 -0.23565 0.098486 -0.05683
5 -0.66951 0.118872 -0.69362 0.120172 0.024113
6 -0.45199 0.110921 -0.3897 0.102267 -0.06229
7 -1.90835 0.192465 -1.73146 0.142969 -0.17688

The predicted frequencies table (Output 32.4.3) displays observed and predicted cell frequencies, their standard errors, and residuals.

Output 32.4.3: Predicted Frequencies

Maximum Likelihood Predicted Values for Frequencies
Length Time Status Observed Predicted Residual
Frequency Standard
Error
Frequency Standard
Error
1 1 1 156 11.43022 161.0961 11.07379 -5.09614
1 1 2 84 8.754999 78.90386 7.808613 5.096139
1 2 1 84 8.754999 78.90386 7.808613 5.096139
1 2 2 156 11.43022 161.0961 11.07379 -5.09614
2 1 1 107 9.750588 101.9039 8.924304 5.096139
2 1 2 133 10.70392 138.0961 10.33434 -5.09614
2 2 1 31 5.47713 36.09614 4.826315 -5.09614
2 2 2 209 12.78667 203.9039 12.21285 5.09614