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Unformatted text preview: Solutions to Assignment 5 Stat 512, Spring 2010 1. (a) Define the factors as A: plant ( i = 1 , 2 , 3 , 4) and B: leaf ( j = 1 , 2 , 3). Factor B is nested within A. The linear model is Y ijk = μ + α i + β j ( i ) + ² ijk , where α i ∼ N (0 ,σ 2 α ), β + j ( i ) ∼ N (0 ,σ 2 β ), and ² ijk ∼ N (0 ,σ 2 ). > # read in data, convert plant and leaf to factors > turnip < read.table("Turnip.txt", header=T) > plant < factor(turnip$plant) > contrasts(plant) < contr.sum > leaf < factor(turnip$leaf) > contrasts(leaf) < contr.sum > calcium < turnip$calcium > # fit nested model > result < lm( calcium ~ plant/leaf ) > anova(result) Analysis of Variance Table Response: calcium Df Sum Sq Mean Sq F value Pr(>F) plant 3 7.5603 2.5201 378.727 3.805e12 *** plant:leaf 8 2.6302 0.3288 49.409 5.090e08 *** Residuals 12 0.0799 0.0067 Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 The expected mean squares are E (MS A ) = σ 2 + nσ 2 β + nbσ 2 α E (MS B ( A ) ) = σ 2 + nσ 2 β E (MS Err ) = σ 2 so the MOM estimates are ˆ σ 2 = MS Err ˆ σ 2 β = MS B ( A ) MS Err n ˆ σ 2 α = MS A MS B ( A ) nb > # extract MS and etimate variance components > tmp < anova(result) > names(tmp) [1] "Df" "Sum Sq" "Mean Sq" "F value" "Pr(>F)" > MSA < tmp$"Mean Sq"[1] > MSB.A < tmp$"Mean Sq"[2] > MSE < tmp$"Mean Sq"[3] > a < 4 > b < 3 > n < 2 > # variance components: plants, leaves and error > sigma2.A< (MSA  MSB.A) / (n*b) > sigma2.B < (MSB.A  MSE) / n > sigma2 < MSE > sigma2.A [1] 0.3652234 > sigma2.B [1] 0.1610604 > sigma2 [1] 0.006654167 The test for no plant effect H : σ 2 α = 0 shown in the table is not correct, The correct test is based on F = MS A / MS B ( A ) . > # no plant effect > Fplant < MSA/MSB.A > Fplant [1] 7.665167 > 1  pf(Fplant, 3, 8) [1] 0.009725121 The test for no leaf effect H : σ 2 β = 0 shown in the table is correct. It is based on F = MS B ( A ) / MS Err and is highly significant, (b) i. Correlation between measurements from the same leaf: Corr( Y ijk ,Y ijk ) = σ 2 α + σ 2 β σ 2 α + σ 2 β + σ 2 > (sigma2.A + sigma2.B)/(sigma2.A + sigma2.B + sigma2) [1] 0.9875142 ii. Correlation between measurements from the same plant and different leaves: Corr( Y ijk ,Y ij k ) = σ 2 α σ 2 α + σ 2 β + σ 2 > sigma2.A /(sigma2.A + sigma2.B + sigma2) [1] 0.6853019 (c) To get an estimate and interval for the grand mean, note that the means for the plants...
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 Spring '11
 John

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