Example 22.10: Direct Input of Response Functions and Covariance Matrix

The CATMOD Procedure

Example 22.10: 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 eighteen 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 via 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).

   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   0.25300
    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 22.10.1 through Output 22.10.3:

   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_ 
            / title='Main Effects for Sex and Age';
   run;

Output 22.10.1: Health Survey Data: Using Direct Input

Complex Sample Survey Analysis

Main Effects for Sex and Age

The CATMOD Procedure

Response Functions Directly Input from Data Set FBEING

Sample	Function Number	Response Function	Design Matrix
Sample	Function Number	Response Function	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

Output 22.10.2: ANOVA Table

Complex Sample Survey Analysis

Main Effects for Sex and Age

The CATMOD Procedure

Response Functions Directly Input from Data Set FBEING

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

Output 22.10.3: Parameter Estimates

Complex Sample Survey Analysis

Main Effects for Sex and Age

The CATMOD Procedure

Response Functions Directly Input from Data Set FBEING

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 (Output 22.10.2) shows that the additive model fits and that there is a significant effect of both sex and age. The following statements produce Output 22.10.4:

   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;

Output 22.10.4: Age<65 Contrast

Complex Sample Survey Analysis

Main Effects for Sex and Age

The CATMOD Procedure

Response Functions Directly Input from Data Set FBEING

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

The analysis of the contrast shows that there is no significant difference among the four age groups that are under age 65. Thus, the next model contains a binary age effect (less than 65 versus 65 and over). The following statements produce Output 22.10.5 through Output 22.10.7:

      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)')
            / title='Binary Age Effect (25-64 vs. 65-74)' ;
   quit;

Output 22.10.5: Design Matrix

Complex Sample Survey Analysis

Main Effects for Sex and Age

The CATMOD Procedure

Response Functions Directly Input from Data Set FBEING

Complex Sample Survey Analysis

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

The CATMOD Procedure

Response Functions Directly Input from Data Set FBEING

Sample	Function Number	Response Function	Design Matrix
Sample	Function Number	Response Function	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

Output 22.10.6: ANOVA Table

Complex Sample Survey Analysis

Main Effects for Sex and Age

The CATMOD Procedure

Response Functions Directly Input from Data Set FBEING

Complex Sample Survey Analysis

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

The CATMOD Procedure

Response Functions Directly Input from Data Set FBEING

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

Output 22.10.7: Parameter Estimates

Complex Sample Survey Analysis

Main Effects for Sex and Age

The CATMOD Procedure

Response Functions Directly Input from Data Set FBEING

Complex Sample Survey Analysis

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

The CATMOD Procedure

Response Functions Directly Input from Data Set FBEING

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 22.10.6 shows that the model fits (note that the goodness-of-fit statistic is the sum of the previous one (Output 22.10.2) plus the chi-square for the contrast matrix in Output 22.10.4). 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.

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