hw3sol - Stat 511 Homework 3 Solution (pts:20) Spring 2011...

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Stat 511 Homework 3 Solution (pts:20) Spring 2011 1. Consider a factor eFects model for a 2-way ANOVA with 2 levels (a and b) of temperature and 2 levels of pressure (A and B). The table of cell means is: Pressure Temp. A B a μ aA μ aB b μ bA μ bB Please answer EACH of the following in two ways: in terms of cell means, and in words, e.g. the mean for pressure A, averaged over temperatures. OR, tell me the quantity is non-estimable. (a) If no constraints are put on the parameter vector, the β 0 vector and X matrix are: μα 1 α 2 β 1 β 2 αβ 11 αβ 12 αβ 21 αβ 22 1 1 0 1 0 1000 1 1 0 1 0 1 1 0 0 1 0100 1 1 0 0 1 1 0 1 1 0 0010 1 0 1 1 0 1 0 1 0 1 0001 1 0 1 0 1 What quantity is estimated by β 1 ? The quantity is non-estimable. What quantity is estimated by αβ 11 ? The quantity is non-estimable. (b) If ’set last to zero’ constraints are used, the β 0 vector and X matrix are: 1 β 1 αβ 11 11 1 1 11 0 0 10 1 0 10 0 0 What quantity is estimated by β 1 ? μ bA μ bB , the diFerence in mean response for Pressure A and B at Temperature b. 1
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Stat 511 Homework 3 Solution (pts:20) Spring 2011 What quantity is estimated by αβ 11 ? μ aA μ aB μ bA + μ bB , namely, an interaction contrast between Temperature and Pressure. (c) If ’sum to zero’ constraints are used, the β 0 vector and X matrix are: μα 1 β 1 αβ 11 111 1 1 11 1 1 1 1 1 1 1
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hw3sol - Stat 511 Homework 3 Solution (pts:20) Spring 2011...

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