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Unformatted text preview: 201 MULTIPLE COMPARISONS When the hypothesis H : μ 1 = μ 2 = ··· = μ t is rejected, the inference is that at least one of the t population means differs from the rest. The next question is which means are different from others? Is μ 1 6 = μ 2 ? is μ 6 6 = μ 7 ? is the average ( μ 1 + μ 2 + μ 3 ) / 3 different from ( μ 4 + μ 5 + μ 6 ) / 3? etc. Many times our question will not result in a simple com parison of whether a difference like μ 2 μ 3 = 0 or not. It may be a more complicated question that requires a comparison like μ 1 ( μ 2 + μ 3 ) / 2 = 0 to be made. Not all questions can be formulated as comparisons. To enable us to understand what kinds of questions can be formulated as comparisons, we define a tt linear form to be a linear combination of the means μ 1 , μ 2 , ··· μ t of the form: ‘ = a 1 μ 1 + a 2 μ 2 + ··· + a t μ t for given numbers a 1 , a 2 , . . . , a t which satisfy ∑ t i =1 a i = 0. Let us look at some specific examples. Examples: Suppose t = 5 i.e., we consider the means μ 1 , μ 2 , μ 3 , μ 4 , and μ 5 . • The linear form ‘ = μ 2 μ 3 has a i values a 1 = 0 , a 2 = 1 , a 3 = 1 , a 4 = 0 , a 5 = 0 . Note that ∑ a i = 0 as required. 202 • The linear form ‘ = ( μ 1 + μ 2 ) / 2 ( μ 3 + μ 4 ) / 2 has a i values a 1 = 1 / 2 , a 2 = 1 / 2 , a 3 = 1 / 2 , a 4 = 1 / 2 , a 5 = 0 . Again, ∑ a i = 0. A point estimate of a linear form is called a linear contrast , and is given by ˆ ‘ = a 1 ¯ y 1 . + a 2 ¯ y 2 . + a 3 ¯ y 3 . + ··· + a t ¯ y t . with ∑ a i = 0. The estimated variance of ˆ ‘ is ˆ V ( ˆ ‘ ) = s 2 W t X i =1 a 2 i n i where n i is the number of observations taken from the ith population. To test the hypothesis H : ‘ = 0 we can use the t statistic t = ˆ ‘ s ˆ V ( ˆ ‘ ) with degrees of freedom same as that of the within mean square s 2 W . Two contrasts ˆ ‘ 1 = ∑ i a i ¯ y i. and ˆ ‘ 2 = ∑ i b i ¯ y i. are orthogonal whenever ∑ i a i b i = 0. This is only defined when n 1 = n 2 = ··· = n t = n . If all linear contrasts in a set ˆ ‘ 1 , ˆ ‘ 2 , . . . , ˆ ‘ t 1 203 are pairwise orthogonal – i.e., every possible pair is orthogonal – then the set is said to be mutually orthogonal set of linear contrasts. Given t means μ 1 , μ 2 , . . . , μ t , and sample means ¯ y 1 . , ¯ y 2 . , . . . , ¯ y t. (all based on the same number n of observations) it is the case that the maximum number of mutually orthogonal contrasts that exist is ( t 1). There are many ( t 1) sets of con trasts that are mutually orthogonal. Given such a maximum mutually orthogonal set ˆ ‘ 1 , ˆ ‘ 2 , . . . , ˆ ‘ t 1 , these linear contrasts, before the samples are taken, tt are random variables which are independent....
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This note was uploaded on 01/23/2010 for the course STAT 213 taught by Professor Hao during the Spring '10 term at Internet2.
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