The REG Procedure

## Multivariate Tests

The MTEST statement described in the "MTEST Statement" section can test hypotheses involving several dependent variables in the form where L is a linear function on the regressor side, 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 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: 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 ofSquares MeanSquare 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 9.75 Adj R-Sq 0.7058 Coeff Var 23.5004

 Parameter Estimates Variable DF ParameterEstimate StandardError 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 ofSquares MeanSquare 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 29.0378

 Parameter Estimates Variable DF ParameterEstimate StandardError 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

 CanonicalCorrelation AdjustedCanonicalCorrelation ApproximateStandardError SquaredCanonicalCorrelation Eigenvalues of Inv(E)*H = CanRsq/(1-CanRsq) Test of H0: The canonical correlations inthecurrent row and all that follow are zero Eigenvalue Difference Proportion Cumulative LikelihoodRatio ApproximateF 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.