Example 29.4 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 29.4.1 through Output 29.4.3:

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 29.4.1 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 29.4.2) displays observed and predicted values for the generalized logits.

Output 29.4.2 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 29.4.3) displays observed and predicted cell frequencies, their standard errors, and residuals.

Output 29.4.3 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