Multivariate Analysis of Variance

As the results of coded multiple regression and analyses of variance are equivalent, so are the results of canonical analysis and multivariate analysis of variance. To illustrate this conceptual continuity within the general linear model, let us consider an example, discussed in Kerlinger and Pedhazur's Multiple Regression in Behavioral Research.

A Study of Attitude Change

This study described the impact of three kinds of appeal: democratic, fair play, and religious on attitudes toward two minorities, Jews and Blacks. In the table below are data from the study of attitudes toward minority Y₁ (Jews) and minority Y₂ (Blacks). A nine step rating scale for three types of appeals measured effectiveness of democratic (A₁), fair play (A₂), and religious (A₃) appeals. Higher scores indicate greater acceptance.

 

 

 

The obtained scores are plotted in the figure below.

 

 

The scores corresponding to democratic appeal were plotted as a square, the fair play appeal scores were marked as a circle and a triangle marks scores for the religious appeal.

The Religious Affinity Hypothesis

There is an affinity between Protestant religions and Judaism, both stressing the Old Testament. Let us hypothesize that this affinity makes the religious appeal the most efficient with respect to attitudes toward Jews. We may obtain orthogonal coding vectors by using Helmert contrasts [2 -1 -1] and [0 1 -1]. To test our hypothesis let us rearrange the [0 1 -1] vector as [1 0 -1] and use it as the first coding vector X₁. We also may to rearrange the [2 -1 -1] vector as [-1 2 -1] and use it as the second coding vector X₂, to remove the effect of the fair play appeal from the contrast between the religious-democratic appeals. The study was coded as shown in the table below.

 

 

 

The above data were analyzed by a computer program for canonical analysis tailored as to enhance features specific to multivariate analysis of variance.

Canonical Analysis

Using the CSSP program package, the obtained output can be interpreted as follows. First, notice that the correlation matrix, obtained as a part of the output, reflects the orthogonality of Helmert's coding vectors.

 

 

 

Two eigenvalues were extracted

 

 

and rotated

 

 

The Varimax rotated canonical loadings are

 

 

 

and reflect, within the first component, the tested hypothesis about the effectiveness of the religious appeal for enhancement of positive attitudes toward Jews. The effectiveness of this approach can be also confirmed by inspection of table, listing group means for the rotated solution

 

 

 

In the above table you may observe that the mean with the highest absolute value for the rotated canonical components is that for the first contrast - religious appeal.

Visualization of the Multivariate Analysis of Variance Solution

To visualize the rotated first canonical component,

 

 

 

 use Microsoft Excel to plot the values within the above table, as

 

Compare this rotated solution with the plot of the obtained data at the beginning of this chapter. The democratic (squares), fair play (circles) and religious (triangles) appeals are clearly separated and the hypothesis that the religious appeal is most efficient for enhancing positive attitudes toward Jews is clearly visible.

Multivariate Tests of Statistical Significance

The tests of significance discussed so far were the t-test for two groups where t² = F and degrees of freedom equal n - 2,

 

 

and the F-test for three or more groups where

 

 

 

where k equals the number of coded vectors and degrees of freedom associated with error equal n - k - 1. Within the multivariate analysis of variance context, the F ratio can be defined as

 

 

where the capital Greek letter lambda stands for Wilk's lambda, a multivariate analogue of the coefficient of alienation. In previous discussion we used symbol k to signify the number of coding vectors. Let us modify our notation and designate the number of coding vectors as p (predictor variables) and the number of criterion variables as q. The value s in the above formula is then defined as

 

The number of degrees of freedom for the coefficient of determination is defined as

 

The number of degrees of freedom for the coefficient of alienation  is defined as

 

 

 

The Wilk's lambda can be computed as a continuous product of  

 

For the example, the first eigenvalue equals .96 and the second eigenvalue is equal to .14, thus lambda in the above equation equals (1 - .96)(1 - .14) which is .034. The F ratio equals

 

 

 

since, for p = 2 and q = 2, s equals 2. The number of degrees of freedom for the coefficient of determination is 4. The number of degrees of freedom for the coefficient of alienation equals 22. The probability value associated with the obtained F_(4,22) is smaller than .01. The hypothesis that the religious appeal is the most efficient approach to enhance positive attitudes toward Jews is statistically significant.

There are other tests of significance within the context of the multivariate analysis of variance as Hotelling-Lawley trace, Pillai's trace and Roy's maximum root criterion. As the number of degrees of freedom increases, the Hotelling-Lawley trace, Pillai's trace, and Wilk's likelihood ratio converge, so the choice among these three statistics is not an important issue when the n is large. For maximum protection against finding a statistical significance when there is none, with small samples, Pillai's trace appears to be a preferred statistics.