Multivariate Analysis
Hypothesis Testing: Equality of Population Covariance Matrices

Univariate

Test of Equality of Population Variances

 Using two sample variances from independent samples test:

H0: σ12 = σ22 versus H1: σ12 <> σ22

Note: Larger variance goes in the numerator.

With degrees of freedom ν1 = n1 -1 and ν2 = n2 - 1.

Multivariate

Test of Equality of Population Covariance Matrices

 Hypotheses


Some Definitions

Let C be the pooled within covariance matrix.

Log-likelihood ratio Statistic

  • H0 is tested using a modified log-likelihood-ratio statistic:

    the product Mh is a chi-square with p(p+1)(k-1)/2 degrees of freedom.

    When all the ni are equal h reduces to:

    If the observed chi-square, Mh, is larger than the critical value then reject H0.

    Example

  • Suppose that two English tests -- a long one on paragraph comprehension and a short one on vocabulary -- were given to three classes of high school seniors, pursuing three different curricular programs. Test the assumption of equal covariance matrices in the three populations. The SSCP matrices for the three classes were as follows:

  • With n1=33, n2=37 and n3=24.

  • Test the hypothesis that Σ1 = Σ2 = Σ3.

    Stata Matrix Program

    
scalar k = 3    
scalar p = 2
scalar n1 = 33   
scalar n2 = 37    
scalar n3 = 24

matrix  s1 = (3352.71, 100.09 \ 100.09, 286.97)    
matrix  s2 = (2647.87, 149.22 \ 149.22, 251.57)
matrix  s3 = (2948.02, 61.00 \ 61.00, 181.83)

matrix  c1 = (1/(n1-1))*s1    
matrix  c2 = (1/(n2-1))*s2    
matrix  c3 = (1/(n3-1))*s3
matrix list c1    
matrix list c2    
matrix list c3

scalar n = n1+n2+n3
matrix c = (s1+s2+s3)/(n-k)    
matrix list c

scalar d = det(c)    
scalar d1 = det(c1)    
scalar d2 = det(c2)    
scalar d3 = det(c3)
scalar m = ((n-k)*log(d))-((n1-1)*log(d1) + (n2-1)*log(d2) + (n3-1)*log(d3))
display "m = " m

scalar h=1-((2*p*p+3*p-1)/(6*(p+1)*(k-1))*(1/(n1-1)+1/(n2-1)+1/(n3-1)-1/(n-k)))
display "h = " h

scalar chi = m*h  
display "chi = " chi
scalar df = p*(p+1)*(k-1)/2    
display "df = " df

display "prob = " chiprob(df, chi)

    Multivariate Course Page

    Phil Ender, 29Jan98