Monday, 14 December 2009

Box Test

The log-ANOVA test was proposed by Box (1953). In this procedure, clip_image002 sub-samples are created out of the clip_image004observations in the ith sample (where i=1,...,K). Each of the clip_image002[1] sub-samples hence contains clip_image006separate observations where,

clip_image008

And

clip_image010 (F 2.18)

Where,

clip_image012

And clip_image014is defined as

clip_image016

In the event that all values of clip_image018are equal (or extremely close to being equal), the Box statistic (clip_image020) can be defined as:

clip_image022 (F 2.19)

The clip_image020[1]statistic approximates an F-distribution with (K-1) and (clip_image024) degrees of freedom. However, where the clip_image018[1]values diverge (thus they are not equal), a generalisation developed by Scheffe (1959) can be used to obtain the clip_image026statistic:

clip_image028 (F 2.20)

In this equation, clip_image030 is defined as

clip_image032

Hence it can also be demonstrated that

clip_image034

that

clip_image036

The clip_image026[1]statistic also approximates an F-distribution with (K-1) and (clip_image024[1]) degrees of freedom. Other modifications of the Box method such as the Bargmann modification (Gartside, 1972) of the Box test have been proposed remove bias and to improve the approximation to the homogeneity condition of ANOVA.

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