Example 22.6: Repeated Measures, 2 Response Levels, 3 Populations
In this multi-population repeated measures example, from
Guthrie (1981), subjects from three groups have their
responses (0 or 1) recorded in each of four trials. The
analysis of the marginal probabilities is directed at
assessing the main effects of the repeated measurement
factor (Trial) and the independent variable
(Group), as well as their interaction. Although the
contingency table is incomplete (only thirteen of the
sixteen possible responses are observed), this poses no
problem in the computation of the marginal probabilities.
The following statements produce Output 22.6.1 through
Output 22.6.5:
title 'Multi-Population Repeated Measures';
data group;
input a b c d Group wt @@;
datalines;
1 1 1 1 2 2 0 0 0 0 2 2 0 0 1 0 1 2 0 0 1 0 2 2
0 0 0 1 1 4 0 0 0 1 2 1 0 0 0 1 3 3 1 0 0 1 2 1
0 0 1 1 1 1 0 0 1 1 2 2 0 0 1 1 3 5 0 1 0 0 1 4
0 1 0 0 2 1 0 1 0 1 2 1 0 1 0 1 3 2 0 1 1 0 3 1
1 0 0 0 1 3 1 0 0 0 2 1 0 1 1 1 2 1 0 1 1 1 3 2
1 0 1 0 1 1 1 0 1 1 2 1 1 0 1 1 3 2
;
proc catmod data=group;
weight wt;
response marginals;
model a*b*c*d=Group _response_ Group*_response_
/ freq nodesign;
repeated Trial 4;
title2 'Saturated Model';
run;
Output 22.6.1: Analysis of Multiple-Population Repeated Measures
| Multi-Population Repeated Measures |
| Saturated Model |
| Response |
a*b*c*d |
Response Levels |
13 |
| Weight Variable |
wt |
Populations |
3 |
| Data Set |
GROUP |
Total Frequency |
45 |
| Frequency Missing |
0 |
Observations |
23 |
| Population Profiles |
| Sample |
Group |
Sample Size |
| 1 |
1 |
15 |
| 2 |
2 |
15 |
| 3 |
3 |
15 |
|
Output 22.6.2: Response Profiles
| Multi-Population Repeated Measures |
| Saturated Model |
| Response Profiles |
| Response |
a |
b |
c |
d |
| 1 |
0 |
0 |
0 |
0 |
| 2 |
0 |
0 |
0 |
1 |
| 3 |
0 |
0 |
1 |
0 |
| 4 |
0 |
0 |
1 |
1 |
| 5 |
0 |
1 |
0 |
0 |
| 6 |
0 |
1 |
0 |
1 |
| 7 |
0 |
1 |
1 |
0 |
| 8 |
0 |
1 |
1 |
1 |
| 9 |
1 |
0 |
0 |
0 |
| 10 |
1 |
0 |
0 |
1 |
| 11 |
1 |
0 |
1 |
0 |
| 12 |
1 |
0 |
1 |
1 |
| 13 |
1 |
1 |
1 |
1 |
|
Output 22.6.3: Response Frequencies
| Multi-Population Repeated Measures |
| Saturated Model |
| Response Frequencies |
| Sample |
Response Number |
| 1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
| 1 |
0 |
4 |
2 |
1 |
4 |
0 |
0 |
0 |
3 |
0 |
1 |
0 |
0 |
| 2 |
2 |
1 |
2 |
2 |
1 |
1 |
0 |
1 |
1 |
1 |
0 |
1 |
2 |
| 3 |
0 |
3 |
0 |
5 |
0 |
2 |
1 |
2 |
0 |
0 |
0 |
2 |
0 |
|
Output 22.6.4: Analysis of Variance Table
| Multi-Population Repeated Measures |
| Saturated Model |
| Analysis of Variance |
| Source |
DF |
Chi-Square |
Pr > ChiSq |
| Intercept |
1 |
354.88 |
<.0001 |
| Group |
2 |
24.79 |
<.0001 |
| Trial |
3 |
21.45 |
<.0001 |
| Group*Trial |
6 |
18.71 |
0.0047 |
| Residual |
0 |
. |
. |
|
Output 22.6.5: Parameter Estimates
| Multi-Population Repeated Measures |
| Saturated Model |
| Analysis of Weighted Least Squares Estimates |
| Effect |
Parameter |
Estimate |
Standard
Error |
Chi-
Square |
Pr > ChiSq |
| Intercept |
1 |
0.5833 |
0.0310 |
354.88 |
<.0001 |
| Group |
2 |
0.1333 |
0.0335 |
15.88 |
<.0001 |
| |
3 |
-0.0333 |
0.0551 |
0.37 |
0.5450 |
| Trial |
4 |
0.1722 |
0.0557 |
9.57 |
0.0020 |
| |
5 |
0.1056 |
0.0647 |
2.66 |
0.1028 |
| |
6 |
-0.0722 |
0.0577 |
1.57 |
0.2107 |
| Group*Trial |
7 |
-0.1556 |
0.0852 |
3.33 |
0.0679 |
| |
8 |
-0.0556 |
0.0800 |
0.48 |
0.4877 |
| |
9 |
-0.0889 |
0.0953 |
0.87 |
0.3511 |
| |
10 |
0.0111 |
0.0866 |
0.02 |
0.8979 |
| |
11 |
0.0889 |
0.0822 |
1.17 |
0.2793 |
| |
12 |
-0.0111 |
0.0824 |
0.02 |
0.8927 |
|
The analysis of variance table in Output 22.6.4 shows that
there is a significant interaction between the independent
variable Group and the repeated measurement factor
Trial. Thus, an intermediate model (not shown) is
fit in which the effects Trial and Group*
Trial are replaced by Trial(Group=1),
Trial(Group=2), and Trial(Group=3).
Of these three effects, only the last is significant, so it
is retained in the final model. The following statements
produce Output 22.6.6 and Output 22.6.7:
model a*b*c*d=Group _response_(Group=3)
/ noprofile noparm;
title2 'Trial Nested within Group 3';
quit;
Output 22.6.6: Final Model: Design Matrix
| Multi-Population Repeated Measures |
| Trial Nested within Group 3 |
| Response |
a*b*c*d |
Response Levels |
13 |
| Weight Variable |
wt |
Populations |
3 |
| Data Set |
GROUP |
Total Frequency |
45 |
| Frequency Missing |
0 |
Observations |
23 |
| Sample |
Function
Number |
Response
Function |
Design Matrix |
| 1 |
2 |
3 |
4 |
5 |
6 |
| 1 |
1 |
0.73333 |
1 |
1 |
0 |
0 |
0 |
0 |
| |
2 |
0.73333 |
1 |
1 |
0 |
0 |
0 |
0 |
| |
3 |
0.73333 |
1 |
1 |
0 |
0 |
0 |
0 |
| |
4 |
0.66667 |
1 |
1 |
0 |
0 |
0 |
0 |
| 2 |
1 |
0.66667 |
1 |
0 |
1 |
0 |
0 |
0 |
| |
2 |
0.66667 |
1 |
0 |
1 |
0 |
0 |
0 |
| |
3 |
0.46667 |
1 |
0 |
1 |
0 |
0 |
0 |
| |
4 |
0.40000 |
1 |
0 |
1 |
0 |
0 |
0 |
| 3 |
1 |
0.86667 |
1 |
-1 |
-1 |
1 |
0 |
0 |
| |
2 |
0.66667 |
1 |
-1 |
-1 |
0 |
1 |
0 |
| |
3 |
0.33333 |
1 |
-1 |
-1 |
0 |
0 |
1 |
| |
4 |
0.06667 |
1 |
-1 |
-1 |
-1 |
-1 |
-1 |
|
Output 22.6.6 displays the design matrix resulting from
retaining the nested effect.
Output 22.6.7: ANOVA Table
| Multi-Population Repeated Measures |
| Trial Nested within Group 3 |
| Analysis of Variance |
| Source |
DF |
Chi-Square |
Pr > ChiSq |
| Intercept |
1 |
386.94 |
<.0001 |
| Group |
2 |
25.42 |
<.0001 |
| Trial(Group=3) |
3 |
75.07 |
<.0001 |
| Residual |
6 |
5.09 |
0.5319 |
|
The residual goodness-of-fit statistic tests the joint
effect of Trial(Group=1) and Trial(
Group=2). The analysis of variance table in Output 22.6.7
shows that the final model fits, that there is a significant
Group effect, and that there is a significant
Trial effect in Group 3.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.
