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Test Statistics for MANOVA. BMTRY 726 2/28/14. Univariate ANOVA. ANOVA valuates differences in mean response among multiple treatment groups Given a random sample We rewrite m l as the overall mean plus the treatment effect of treatment l

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univariate anova
Univariate ANOVA

ANOVA valuates differences in mean response among multiple treatment groups

Given a random sample

We rewrite ml as the overall mean plus the treatment effect of treatment l

We test our hypothesis using the model

Grand

mean

Treatment

effect

Error

Sample

mean

Est. treatment

effect

Residual

univariate anova1
Univariate ANOVA
  • ANOVA does this by decomposing the variance of the sample into treatment and error components:
f test anova
F-Test ANOVA
  • Based on this decomposition, the resulting F test examines the ratio of MStrt to MSe:
  • We can rewrite this ratio as
extension to manova
Extension to MANOVA
  • MANOVA is very similar but we are comparing treatment response of a vector of outcomes
  • We know then that out variance decomposition can be written in vector notation as:
extension to manova1
Extension to MANOVA
  • We can think about adapting our modified F-statistic from ANOVA for the MANOVA case (how would matrices make a difference?)
  • This then is the matrix analog to the statistic Awe discussed for ANOVA
extension to manova2
Extension to MANOVA
  • However, A is a matrix when we need a single statistic…
  • Early statisticians came up with several (4 in this case) statistics that all are based on
    • Wilk’s lambda
    • Pillai’s trace
    • Lawely-Hotellingtrace
    • Roy’s greatest root
    • All follow an exact or approximate F distribution
    • All are based on the eigenvalues of A = HE-1
the 4 statistics
The 4 Statistics
  • All can be determined using the eigenvalues of A = HE-1
  • 3 of the 4 can all be also calculated from H and E