The TRANSREG Procedure

Redundancy Analysis

Redundancy analysis (Stewart and Love, 1968) is a principal component analysis of multivariate regression predicted values. These first steps show the redundancy analysis results produced by PROC TRANSREG. The specification TSTANDARD= Z sets all variables to mean zero and variance one. METHOD= REDUNDANCY specifies redundancy analysis and outputs the redundancy variables to the OUT= data set. The MREDUNDANCY o-option outputs two sets of redundancy analysis coefficients to the OUT= data set.

The following statements produce Figure 104.61:

title 'Redundancy Analysis';

data x;
   input y1-y3 x1-x4;
   datalines;
 6  8  8 15 18 26 27
 1 12 16 18  9 20  8
 5  6 15 20 17 29 31
 6  9 15 14 10 16 22
 7  5 12 14  6 13  9
 3  6  7  2 14 26 22
 3  5  9 13 18 10 22
 6  3 11  3 15 22 29
 6  3  7 10 20 21 27
 7  5  9  8 10 12 18
;

proc transreg data=x tstandard=z method=redundancy;
   model identity(y1-y3) = identity(x1-x4);
   output out=red mredundancy replace;
run;

proc print data=red(drop=Intercept);
   format _numeric_ 4.1;
run;

Figure 104.61: Redundancy Analysis Example

Redundancy Analysis

Obs	_TYPE_	_NAME_	y1	y2	y3	x1	x2	x3	x4	Red1	Red2	Red3
1	SCORE	ROW1	0.5	0.6	-0.8	0.6	0.9	1.0	0.7	0.2	-0.5	-0.9
2	SCORE	ROW2	-2.0	2.1	1.5	1.1	-1.0	0.1	-1.7	1.6	-1.5	0.4
3	SCORE	ROW3	0.0	-0.1	1.2	1.4	0.7	1.5	1.2	1.0	0.8	-1.3
4	SCORE	ROW4	0.5	1.0	1.2	0.4	-0.8	-0.5	0.1	0.5	1.7	0.1
5	SCORE	ROW5	1.0	-0.4	0.3	0.4	-1.6	-1.0	-1.6	1.0	0.1	0.9
6	SCORE	ROW6	-1.0	-0.1	-1.1	-1.6	0.1	1.0	0.1	-0.8	-0.9	1.4
7	SCORE	ROW7	-1.0	-0.4	-0.6	0.2	0.9	-1.5	0.1	-1.0	-0.4	-1.3
8	SCORE	ROW8	0.5	-1.2	0.0	-1.5	0.3	0.4	1.0	-1.2	0.8	0.7
9	SCORE	ROW9	0.5	-1.2	-1.1	-0.3	1.3	0.2	0.7	-1.0	-0.9	-0.8
10	SCORE	ROW10	1.0	-0.4	-0.6	-0.6	-0.8	-1.1	-0.4	-0.4	0.8	0.7
11	M REDUND	Red1	.	.	.	0.7	-0.6	0.4	-0.1	.	.	.
12	M REDUND	Red2	.	.	.	0.3	-1.5	-0.6	1.9	.	.	.
13	M REDUND	Red3	.	.	.	-0.7	-0.7	0.3	-0.3	.	.	.
14	R REDUND	x1	.	.	.	.	.	.	.	0.8	-0.0	-0.6
15	R REDUND	x2	.	.	.	.	.	.	.	-0.6	-0.2	-0.7
16	R REDUND	x3	.	.	.	.	.	.	.	0.1	-0.2	-0.1
17	R REDUND	x4	.	.	.	.	.	.	.	-0.5	0.3	-0.5

The _TYPE_=’SCORE’ observations of the Red1–Red3 variables contain the redundancy variables. The nonmissing "M REDUND" values are coefficients for predicting the redundancy variables from the independent variables. The nonmissing "R REDUND" values are coefficients for predicting the independent variables from the redundancy variables.

The next steps show how to generate the same results manually. The data set is standardized, predicted values are computed, and principal components of the predicted values are computed. The following statements produce the redundancy variables, shown in Figure 104.62:

proc standard data=x out=std m=0 s=1;
   title2 'Manually Generate Redundancy Variables';
run;

proc reg noprint data=std;
   model y1-y3 = x1-x4;
   output out=p p=ay1-ay3;
run; quit;

proc princomp data=p cov noprint std out=p;
   var ay1-ay3;
run;

proc print data=p(keep=Prin:);
   format _numeric_ 4.1;
run;

Figure 104.62: Redundancy Analysis Example

Redundancy Analysis

Manually Generate Redundancy Variables

Obs	Prin1	Prin2	Prin3
1	0.2	-0.5	-0.9
2	1.6	-1.5	0.4
3	1.0	0.8	-1.3
4	0.5	1.7	0.1
5	1.0	0.1	0.9
6	-0.8	-0.9	1.4
7	-1.0	-0.4	-1.3
8	-1.2	0.8	0.7
9	-1.0	-0.9	-0.8
10	-0.4	0.8	0.7

The following statements produce the coefficients for predicting the redundancy variables from the independent variables, shown in Figure 104.63:

proc reg data=p outest=redcoef noprint;
   title2 'Manually Create Redundancy Coefficients';
   model Prin1-Prin3 = x1-x4;
run; quit;

proc print data=redcoef(keep=x1-x4);
   format _numeric_ 4.1;
run;

Figure 104.63: Redundancy Analysis Example

Redundancy Analysis

Manually Create Redundancy Coefficients

Obs	x1	x2	x3	x4
1	0.7	-0.6	0.4	-0.1
2	0.3	-1.5	-0.6	1.9
3	-0.7	-0.7	0.3	-0.3

The following statements produce the coefficients for predicting the independent variables from the redundancy variables, shown in Figure 104.64:

proc reg data=p outest=redcoef2 noprint;
   title2 'Manually Create Other Coefficients';
   model x1-x4 = prin1-prin3;
run; quit;

proc print data=redcoef2(keep=Prin1-Prin3);
   format _numeric_ 4.1;
run;

Figure 104.64: Redundancy Analysis Example

Redundancy Analysis

Manually Create Other Coefficients

Obs	Prin1	Prin2	Prin3
1	0.8	-0.0	-0.6
2	-0.6	-0.2	-0.7
3	0.1	-0.2	-0.1
4	-0.5	0.3	-0.5