Multivariate Tests

The REG Procedure

Multivariate Tests

The MTEST statement described in the "MTEST Statement" section can test hypotheses involving several dependent variables in the form

$({L \beta} - {cj}) M = 0$

where L is a linear function on the regressor side, ${\beta}$ is a matrix of parameters, c is a column vector of constants, j is a row vector of ones, and M is a linear function on the dependent side. The special case where the constants are zero is

${L \beta M} = 0$

To test this hypothesis, PROC REG constructs two matrices called H and E that correspond to the numerator and denominator of a univariate F test:

$H & = & M^' ({LB} - {cj})^' (L(X'X)^{-}L^')^{-1} ({LB} - {cj})M \ E & = & M^' (Y'Y - B^' (X'X)B)M \$

These matrices are displayed for each MTEST statement if the PRINT option is specified.

Four test statistics based on the eigenvalues of E^-1 H or (E+H)^-1H are formed. These are Wilks' Lambda, Pillai's Trace, the Hotelling-Lawley Trace, and Roy's maximum root. These test statistics are discussed in Chapter 3, "Introduction to Regression Procedures."

The following statements perform a multivariate analysis of variance and produce Figures 55.50 through 55.54:

   * Manova Data from Morrison (1976, 190);
     data a;
        input sex $ drug $ @;
        do rep=1 to 4;
           input y1 y2 @;
           sexcode=(sex='m')-(sex='f');
           drug1=(drug='a')-(drug='c');
           drug2=(drug='b')-(drug='c');
           sexdrug1=sexcode*drug1;
           sexdrug2=sexcode*drug2;
           output;
        end;
        datalines;
     m a  5  6  5  4  9  9  7  6
     m b  7  6  7  7  9 12  6  8
     m c 21 15 14 11 17 12 12 10
     f a  7 10  6  6  9  7  8 10
     f b 10 13  8  7  7  6  6  9
     f c 16 12 14  9 14  8 10  5
     ;
     proc reg;
        model y1 y2=sexcode drug1 drug2 sexdrug1 sexdrug2;
        y1y2drug: mtest y1=y2, drug1,drug2;
        drugshow: mtest drug1, drug2 / print canprint;
     run;

The REG Procedure

Model: MODEL1

Dependent Variable: y1

Analysis of Variance
Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	5	316.00000	63.20000	12.04	<.0001
Error	18	94.50000	5.25000
Corrected Total	23	410.50000

Root MSE	2.29129	R-Square	0.7698
Dependent Mean	9.75000	Adj R-Sq	0.7058
Coeff Var	23.50039

Parameter Estimates
Variable	DF	Parameter Estimate	Standard Error	t Value	Pr > \|t\|
Intercept	1	9.75000	0.46771	20.85	<.0001
sexcode	1	0.16667	0.46771	0.36	0.7257
drug1	1	-2.75000	0.66144	-4.16	0.0006
drug2	1	-2.25000	0.66144	-3.40	0.0032
sexdrug1	1	-0.66667	0.66144	-1.01	0.3269
sexdrug2	1	-0.41667	0.66144	-0.63	0.5366

Figure 55.51: Multivariate Analysis of Variance: REG Procedure

The REG Procedure

Model: MODEL1

Dependent Variable: y2

Analysis of Variance
Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	5	69.33333	13.86667	2.19	0.1008
Error	18	114.00000	6.33333
Corrected Total	23	183.33333

Root MSE	2.51661	R-Square	0.3782
Dependent Mean	8.66667	Adj R-Sq	0.2055
Coeff Var	29.03782

Parameter Estimates
Variable	DF	Parameter Estimate	Standard Error	t Value	Pr > \|t\|
Intercept	1	8.66667	0.51370	16.87	<.0001
sexcode	1	0.16667	0.51370	0.32	0.7493
drug1	1	-1.41667	0.72648	-1.95	0.0669
drug2	1	-0.16667	0.72648	-0.23	0.8211
sexdrug1	1	-1.16667	0.72648	-1.61	0.1257
sexdrug2	1	-0.41667	0.72648	-0.57	0.5734

Figure 55.52: Multivariate Analysis of Variance: REG Procedure

The REG Procedure

Model: MODEL1

Multivariate Test: Y1Y2DRUG

Multivariate Statistics and Exact F Statistics
S=1 M=0 N=8
Statistic	Value	F Value	Num DF	Den DF	Pr > F
Wilks' Lambda	0.28053917	23.08	2	18	<.0001
Pillai's Trace	0.71946083	23.08	2	18	<.0001
Hotelling-Lawley Trace	2.56456456	23.08	2	18	<.0001
Roy's Greatest Root	2.56456456	23.08	2	18	<.0001

Figure 55.53: Multivariate Analysis of Variance: First Test

The four multivariate test statistics are all highly significant, giving strong evidence that the coefficients of drug1 and drug2 are not the same across dependent variables y1 and y2.

The REG Procedure

Model: MODEL1

Multivariate Test: DRUGSHOW

Error Matrix (E)
94.5	76.5
76.5	114

Hypothesis Matrix (H)
301	97.5
97.5	36.333333333

	Canonical Correlation	Adjusted Canonical Correlation	Approximate Standard Error	Squared Canonical Correlation	Eigenvalues of Inv(E)*H = CanRsq/(1-CanRsq)				Test of H0: The canonical correlations in the current row and all that follow are zero
	Canonical Correlation	Adjusted Canonical Correlation	Approximate Standard Error	Squared Canonical Correlation	Eigenvalue	Difference	Proportion	Cumulative	Likelihood Ratio	Approximate F Value	Num DF	Den DF	Pr > F
1	0.905903	0.899927	0.040101	0.820661	4.5760	4.5125	0.9863	0.9863	0.16862952	12.20	4	34	<.0001
2	0.244371	.	0.210254	0.059717	0.0635		0.0137	1.0000	0.94028273	1.14	1	18	0.2991

Figure 55.54: Multivariate Analysis of Variance: Second Test

The REG Procedure

Model: MODEL1

Multivariate Test: DRUGSHOW

Multivariate Statistics and F Approximations
S=2 M=-0.5 N=7.5
Statistic	Value	F Value	Num DF	Den DF	Pr > F
Wilks' Lambda	0.16862952	12.20	4	34	<.0001
Pillai's Trace	0.88037810	7.08	4	36	0.0003
Hotelling-Lawley Trace	4.63953666	19.40	4	19.407	<.0001
Roy's Greatest Root	4.57602675	41.18	2	18	<.0001

NOTE:

F Statistic for Roy's Greatest Root is an upper bound.

NOTE:

F Statistic for Wilks' Lambda is exact.

Figure 55.55: Multivariate Analysis of Variance: Second Test (continued)

The four multivariate test statistics are all highly significant, giving strong evidence that the coefficients of drug1 and drug2 are not zero for both dependent variables.

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