Example 30.10 Direct Input of Response Functions and Covariance Matrix

This example illustrates the ability of PROC CATMOD to operate on an existing vector of functions and the corresponding covariance matrix. The estimates under investigation are composite indices summarizing the responses to 18 psychological questions pertaining to general well-being. These estimates are computed for domains corresponding to an age-by-sex cross-classification, and the covariance matrix is calculated using the method of balanced repeated replications. The analysis is directed at obtaining a description of the variation among these domain estimates. The data are from Koch and Stokes (1979).

In the following statements, the first row of the `fbeing` data set contains the response functions for the variables `b1``b10`, while the remaining rows contain the covariance matrix. From the PROC CATMOD statements, the READ option in the RESPONSE statement says that you are inputting the response functions and their covariance matrix, while the PROFILE= option in the FACTORS statement tells you that the variables `b1``b5` correspond to the effects for `sex`='male' at the five different age groupings, and `b6``b10` likewise correspond to the effects for `sex`='female'. See the section Inputting Response Functions and Covariances Directly for more information about using the READ option.

```data fbeing(type=est);
input   b1-b5   _type_ \$  _name_ \$  b6-b10 #2;
datalines;
7.93726   7.92509   7.82815   7.73696   8.16791  parms    .
7.24978   7.18991   7.35960   7.31937   7.55184
0.00739   0.00019   0.00146  -0.00082   0.00076  cov      b1
0.00189   0.00118   0.00140  -0.00140   0.00039
0.00019   0.01172   0.00183   0.00029   0.00083  cov      b2
-0.00123  -0.00629  -0.00088  -0.00232   0.00034
0.00146   0.00183   0.01050  -0.00173   0.00011  cov      b3
0.00434  -0.00059  -0.00055   0.00023  -0.00013
-0.00082   0.00029  -0.00173   0.01335   0.00140  cov      b4
0.00158   0.00212   0.00211   0.00066   0.00240
0.00076   0.00083   0.00011   0.00140   0.01430  cov      b5
-0.00050  -0.00098   0.00239  -0.00010   0.00213
0.00189  -0.00123   0.00434   0.00158  -0.00050  cov      b6
0.01110   0.00101   0.00177  -0.00018  -0.00082
0.00118  -0.00629  -0.00059   0.00212  -0.00098  cov      b7
0.00101   0.02342   0.00144   0.00369   0.00253
0.00140  -0.00088  -0.00055   0.00211   0.00239  cov      b8
0.00177   0.00144   0.01060   0.00157   0.00226
-0.00140  -0.00232   0.00023   0.00066  -0.00010  cov      b9
-0.00018   0.00369   0.00157   0.02298   0.00918
0.00039   0.00034  -0.00013   0.00240   0.00213  cov     b10
-0.00082   0.00253   0.00226   0.00918   0.01921
;
```

The following statements produce Output 30.10.1:

```proc catmod data=fbeing;
title 'Complex Sample Survey Analysis';
response read b1-b10;
factors sex \$ 2, age \$ 5 / _response_=sex age
profile=(male     '25-34',
male     '35-44',
male     '45-54',
male     '55-64',
male     '65-74',
female   '25-34',
female   '35-44',
female   '45-54',
female   '55-64',
female   '65-74');
model _f_=_response_
/ design title='Main Effects for Sex and Age';
run;
```

Output 30.10.1: Health Survey Data: Using Direct Input

 Complex Sample Survey Analysis

Main Effects for Sex and Age

The CATMOD Procedure

Response Functions and Design Matrix
Sample Function
Number
Response
Function
Design Matrix
1 2 3 4 5 6
1 1 7.93726 1 1 1 0 0 0
2 7.92509 1 1 0 1 0 0
3 7.82815 1 1 0 0 1 0
4 7.73696 1 1 0 0 0 1
5 8.16791 1 1 -1 -1 -1 -1
6 7.24978 1 -1 1 0 0 0
7 7.18991 1 -1 0 1 0 0
8 7.35960 1 -1 0 0 1 0
9 7.31937 1 -1 0 0 0 1
10 7.55184 1 -1 -1 -1 -1 -1

Analysis of Variance
Source DF Chi-Square Pr > ChiSq
Intercept 1 28089.07 <.0001
sex 1 65.84 <.0001
age 4 9.21 0.0561
Residual 4 2.92 0.5713

Analysis of Weighted Least Squares Estimates
Effect Parameter Estimate Standard
Error
Chi-
Square
Pr > ChiSq
Intercept 1 7.6319 0.0455 28089.07 <.0001
sex 2 0.2900 0.0357 65.84 <.0001
age 3 -0.00780 0.0645 0.01 0.9037
4 -0.0465 0.0636 0.54 0.4642
5 -0.0343 0.0557 0.38 0.5387
6 -0.1098 0.0764 2.07 0.1506

The analysis of variance table in Output 30.10.1 shows that the additive model fits and that there is a significant effect of both sex and age. The following statements produce Output 30.10.2:

```   contrast 'No Age Effect for Age<65' all_parms 0 0 1 0 0 -1,
all_parms 0 0 0 1 0 -1,
all_parms 0 0 0 0 1 -1;
run;
```

The analysis of the contrast shows that there is no significant difference among the four age groups that are under age 65.

Output 30.10.2: Health Survey Data: Age<65 Contrast

 Complex Sample Survey Analysis

Main Effects for Sex and Age

The CATMOD Procedure

Analysis of Contrasts
Contrast DF Chi-Square Pr > ChiSq
No Age Effect for Age<65 3 0.72 0.8678

The next model contains a binary age effect (under 65 versus 65 and over). The following statements produce Output 30.10.3:

```   model _f_=(1  1  1,
1  1  1,
1  1  1,
1  1  1,
1  1 -1,
1 -1  1,
1 -1  1,
1 -1  1,
1 -1  1,
1 -1 -1)
(1='Intercept' ,
2='Sex'       ,
3='Age (25-64 vs. 65-74)')
/ design title='Binary Age Effect (25-64 vs. 65-74)' ;
run;
quit;
```

Output 30.10.3: Health Survey Data: Age<65 Model

 Complex Sample Survey Analysis

Binary Age Effect (25-64 vs. 65-74)

The CATMOD Procedure

Response Functions and Design Matrix
Sample Function
Number
Response
Function
Design Matrix
1 2 3
1 1 7.93726 1 1 1
2 7.92509 1 1 1
3 7.82815 1 1 1
4 7.73696 1 1 1
5 8.16791 1 1 -1
6 7.24978 1 -1 1
7 7.18991 1 -1 1
8 7.35960 1 -1 1
9 7.31937 1 -1 1
10 7.55184 1 -1 -1

Analysis of Variance
Source DF Chi-Square Pr > ChiSq
Intercept 1 19087.16 <.0001
Sex 1 72.64 <.0001
Age (25-64 vs. 65-74) 1 8.49 0.0036
Residual 7 3.64 0.8198

Analysis of Weighted Least Squares Estimates
Effect Parameter Estimate Standard
Error
Chi-
Square
Pr > ChiSq
Model 1 7.7183 0.0559 19087.16 <.0001
2 0.2800 0.0329 72.64 <.0001
3 -0.1304 0.0448 8.49 0.0036

The analysis of variance table in Output 30.10.3 shows that the model fits (note that the goodness-of-fit statistic is the sum of the previous one (Output 30.10.1) plus the chi-square for the contrast matrix in Output 30.10.2). The age and sex effects are significant. Since the second parameter in the table of estimates is positive, males (the first level for the sex variable) have a higher predicted index of well-being than females. Since the third parameter estimate is negative, those younger than age 65 (the first level of age) have a lower predicted index of well-being than those 65 and older.