Test Statistics for MANOVA

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# Test Statistics for MANOVA - PowerPoint PPT Presentation

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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### 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 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 ANOVA
• ANOVA does this by decomposing the variance of the sample into treatment and error components:
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
• 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 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 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
• All can be determined using the eigenvalues of A = HE-1
• 3 of the 4 can all be also calculated from H and E