Basic Concepts of ANOVA – Advanced

Property 1: If a sample is made as described in Definition 1, with the xij independently and normally distributed and with all σj2 equal, then

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Proof: The first two of these results from Theorem 2 of Chi-square Distribution. Also by Theorem 2 of Chi-square Distribution,

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where MSj = SSj / dfj and dfj = nj – 1. Thus,

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If all the σj2 are equal, the third assertion follows by Property 2 of Chi-square Distribution.

Property 2:

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Proof: The assertion about the degrees of freedom is clear. The proof of the assertion about the SS is as follows:

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Property 3:

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Proof: Since ē  = 0 and eij = xij – j, by Definition 1, we have

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