Example 30.8 Repeated Measures, Logistic Analysis of Growth Curve

The following data, from a longitudinal study reported in Koch et al. (1977), are from patients in four populations (2 diagnostic groups 2 treatments) who are measured at three times to assess their response (n=normal or a=abnormal) to treatment:

title 'Growth Curve Analysis';
data growth2;
input Diagnosis $Treatment$ week1 $week2$ week4 \$ count @@;
datalines;
mild std n n n 16    severe std n n n  2
mild std n n a 13    severe std n n a  2
mild std n a n  9    severe std n a n  8
mild std n a a  3    severe std n a a  9
mild std a n n 14    severe std a n n  9
mild std a n a  4    severe std a n a 15
mild std a a n 15    severe std a a n 27
mild std a a a  6    severe std a a a 28
mild new n n n 31    severe new n n n  7
mild new n n a  0    severe new n n a  2
mild new n a n  6    severe new n a n  5
mild new n a a  0    severe new n a a  2
mild new a n n 22    severe new a n n 31
mild new a n a  2    severe new a n a  5
mild new a a n  9    severe new a a n 32
mild new a a a  0    severe new a a a  6
;


The analysis is directed at assessing the effect of the repeated measurement factor, Time, as well as the independent variables, Diagnosis (mild or severe) and Treatment (std or new). The RESPONSE statement is used to compute the logits of the marginal probabilities. The times used in the design matrix (0, 1, 2) correspond to the logarithms (base 2) of the actual times (1, 2, 4). The following statements produce Output 30.8.1 through Output 30.8.4:

proc catmod data=growth2 order=data;
title2 'Reduced Logistic Model';
weight count;
population Diagnosis Treatment;
response logit;
model week1*week2*week4=(1 0 0 0,  /* mild, std */
1 0 1 0,
1 0 2 0,

1 0 0 0,  /* mild, new */
1 0 0 1,
1 0 0 2,

0 1 0 0,  /* severe, std */
0 1 1 0,
0 1 2 0,

0 1 0 0,  /* severe, new */
0 1 0 1,
0 1 0 2)
(1='Mild diagnosis, week 1',
2='Severe diagnosis, week 1',
3='Time effect for std trt',
4='Time effect for new trt')
/ freq design;
contrast 'Diagnosis effect, week 1' all_parms 1 -1 0 0;
contrast 'Equal time effects' all_parms 0 0 1 -1;
quit;


The samples and the response numbers are defined in Output 30.8.1, and the frequency distribution of the response numbers within the samples is displayed.

Output 30.8.1: Logistic Analysis of Growth Curve

 Growth Curve Analysis Reduced Logistic Model

The CATMOD Procedure

Data Summary
Response week1*week2*week4 Response Levels 8
Weight Variable count Populations 4
Data Set GROWTH2 Total Frequency 340
Frequency Missing 0 Observations 29

Population Profiles
Sample Diagnosis Treatment Sample Size
1 mild std 80
2 mild new 70
3 severe std 100
4 severe new 90

Response Profiles
Response week1 week2 week4
1 n n n
2 n n a
3 n a n
4 n a a
5 a n n
6 a n a
7 a a n
8 a a a

Output 30.8.2 displays the design matrix specified in the MODEL statement, and the observed logits of the marginal probabilities are displayed in the Response Function column.

Output 30.8.2: Response Frequencies

Response Frequencies
Sample Response Number
1 2 3 4 5 6 7 8
1 16 13 9 3 14 4 15 6
2 31 0 6 0 22 2 9 0
3 2 2 8 9 9 15 27 28
4 7 2 5 2 31 5 32 6

Response Functions and Design Matrix
Sample Function
Number
Response
Function
Design Matrix
1 2 3 4
1 1 0.05001 1 0 0 0
2 0.35364 1 0 1 0
3 0.73089 1 0 2 0
2 1 0.11441 1 0 0 0
2 1.29928 1 0 0 1
3 3.52636 1 0 0 2
3 1 -1.32493 0 1 0 0
2 -0.94446 0 1 1 0
3 -0.16034 0 1 2 0
4 1 -1.53148 0 1 0 0
2 0.00000 0 1 0 1
3 1.60944 0 1 0 2

The analysis of variance table in Output 30.8.3 shows that the data can be adequately modeled by two parameters that represent diagnosis effects at week 1 and two log-linear time effects (one for each treatment). Both of the time effects are significant.

Since the estimate of the logit for the severe diagnosis effect (parameter 2) is more negative than it is for the mild diagnosis effect (parameter 1), there is a smaller predicted probability of the first response (normal) for the severe diagnosis group.

Output 30.8.3: ANOVA and Parameter Estimates

Analysis of Variance
Source DF Chi-Square Pr > ChiSq
Mild diagnosis, week 1 1 0.28 0.5955
Severe diagnosis, week 1 1 100.48 <.0001
Time effect for std trt 1 26.35 <.0001
Time effect for new trt 1 125.09 <.0001
Residual 8 4.20 0.8387

Analysis of Weighted Least Squares Estimates
Effect Parameter Estimate Standard
Error
Chi-
Square
Pr > ChiSq
Model 1 -0.0716 0.1348 0.28 0.5955
2 -1.3529 0.1350 100.48 <.0001
3 0.4944 0.0963 26.35 <.0001
4 1.4552 0.1301 125.09 <.0001

The analysis of contrasts (Output 30.8.4) shows that the diagnosis effect at week 1 is highly significant. In other words, those subjects with a severe diagnosis have a significantly higher probability of abnormal response at week 1 than those subjects with a mild diagnosis.

Output 30.8.4: Contrasts

Analysis of Contrasts
Contrast DF Chi-Square Pr > ChiSq
Diagnosis effect, week 1 1 77.02 <.0001
Equal time effects 1 59.12 <.0001

The analysis of contrasts (Output 30.8.4) also shows that the time effect for the standard treatment is significantly different from the one for the new treatment. The table of parameter estimates (Output 30.8.3) shows that the time effect for the new treatment (parameter 4) is stronger than it is for the standard treatment (parameter 3).