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# lect6 - IE 410 Lecture 6 More on Multiple Comparisons...

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IE 410 Lecture 6: More on Multiple Comparisons Pairwise Comparisons. So far we have learned: FSD LSD Scheffe's Method As ways to test all pairs of means

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Next are methods based on the "Studentized Range" First, what is the Studentized range? Suppose we have " a " observations Y i ~ NID( μ i , σ 2 ) Let W = Max[Y i ] - Min[Y i ]
That is, W is the range of the sample Suppose we have an estimate of σ 2 (call it S 2 ), with v degrees of freedom, which is independent of the Y i 's. Let S be the square root of this estimate.

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Finally let q = W/S. q is the "Studentized Range". We will denote it as q( a ,v) to remind ourselves of the degrees of freedom
q(a,v) is a random variable. The larger " a " is, the larger we expect q( a ,v) to be. (Why)? The distribution of q(a,v) has been derived and tabulated in table VIII in the appendix

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Now in the case of pairwise comparisons we have y i. ~ NID( μ i , σ 2 /n) And MS E /n as an estimate of σ 2 /n We can thus find a Confidence Interval on: W = Max[ y i. ] - Min[ y i. ]
Under H 0 : all means are equal The (one-sided upper) CI is : T( α ) = q( α ,a,N-a) n MS E That is: Pr{Max( y i. ) - Min( y i. ) T( α )} = 1- α

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TUKEY"S TEST Tukey's test says that any pair of treatment averages further apart than T( α ) are declared significantly different. Can think of it via the plot of treatment averages. Slide an interval of width T( α ) underneath to compare means.
y 3. y 4. y 1. y 2. * * * * T( α )

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NEUMAN-KEULS TEST Example: Suppose we have 5 treatment averages STEP 1. We use q( α ,5,v) n MS E to look at the whole range.
If all averages in range, stop: * * * * * In this example, all means not different

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Otherwise , if not all averages in range, * * * * * Then STEP 2. Use q( α , 4 ,v) n MS E to compare all (both in this case) ranges of 4 points
* * * * * If all points in range, stop .

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lect6 - IE 410 Lecture 6 More on Multiple Comparisons...

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