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 |
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).