Source: Manly, Bryan F.J. Multivariate Statistical Methods: A Primer, Third Edition, CRC Press,07/2004.Page 1 of 5Tutorial Solutions– Week 4 (HT)Question 1:When you have several dependent variables and several samples/groups the four statisticsthat may be used to identify differences between group means are Pillai’s trace, Wilk’sLambda, Roy’s largest root, and Lawes-Hotelling trace. Briefly, describe and compare.Solution:All have F equivalents. All compare some form of variation either within, between or totalSS.Wilk’s: compares variation within groups to variation both within and between groups(total) based on SS. A small Wilk’s indicates that the variation within is relatively small anda significant difference between groups.Roy’s looks at linear combination of variables that maximises ratio of between sample SSand within sample SS. This ratio is lambda and the largest is the maximum latent root(eigenvalue). A large lambda indicates a significant difference between groups.Pillai’s trace also considers lambda and large lambda indicates a significant differencebetween groups.Lawes also considers lambda and large lambda indicates a significant difference betweengroups.Question 2:How important are the assumption of MVN and equal covariances to Hotelling’s T2 and thefour statistics commonly used in MANOVA analysis?Solution:Chapter 4.2 Hotelling’s T2 is multivariate t-test. Some deviation from MVN is not tooimportant and moderate differences between population covariance matrices is fine.All 4 MANOVA statistics assume MVN and equal covariance matrices, with Pillai’s trace themost robust to deviations. All 4 tests are fairly robust to unequal sample sizes(unbalanced).Question 3:When testing for normality is it possible to have non-significant univariate results andsignificant multivariate results? Why?Solution:Chapter 4.4: Yes, because of accumulation of small-ish deviation from normality acrossmany variables can lead to significant deviation from MVN.Question 4:How does multiple testing affect Type I error rates? Explain how the Bonferroni correctionfor multiple testing works.Solution:Chapter 4.4: multiple testing increases the chance of Type I error – reject Ho when samplesare not really from different populations. Specific multivariate tests like Hotelling’s T2 are anadvantage over series of univariate tests.Source: Manly, Bryan F.J. Multivariate Statistical Methods: A Primer, Third Edition, CRC Press,07/2004.Page 2 of 5Question 5:Complete the exercise at the end of Chapter 4 of Manly. The data file ‘mandiblefull.dat’ isavailable on the Study Desk. Some R code to get you started is provided in ‘mandibleMANOVA.R’The variable names and codes are as follows:X1 – length of mandibleX2 – breadth of mandible below 1st molarX3 – breadth of articular condyleX4 – height of mandible below 1st molarX5 – length of 1st molarX6 – breadth of 1st molarX7 – length of 1st to 3rd molar inclusive (1st to 2nd for cuon)X8 – length from 1st to 4th premolar inclusiveX9 – breadth of lower canineSex – Male (1), Female (2), Unknown (0)Group – Thai (modern) dogs (1), golden jackals (2), cuons (3), Indian wolves (4),Thai (prehistoric) dogs (5).Solution:> Y (cory mf.manova1 summary(mf.manova1) #default test is Pillai’sDf Pillai approx F num Df den Df Pr(>F)as.factor(Group) 4 2.5892 13.662 36 268 summary(mf.manova1, test=”Wilks”)Df Wilks approx F num Df den Df Pr(>F)as.factor(Group) 4 0.0021936 27.666 36 241.57 summary(mf.manova1, test=”Roy”)Df Roy approx F num Df den Df Pr(>F)as.factor(Group) 4 16.348 121.7 9 67 summary(mf.manova1, test=”Hotelling-Lawley”)Df Hotelling-Lawley approx F num Df den Df Pr(>F)as.factor(Group) 4 25.129 43.627 36 250 0.9). Somecorrelation is needed for MANOVA however, it is possible that some of these highlycorrelated variables should be removed from the analysis.There is a significant difference (p #to subset and run individual comparisons in MANOVA:> mf.manova2 summary(mf.manova2)Df Pillai approx F num Df den Df Pr(>F)as.factor(Group) 1 0.83288 8.8603 9 16 0.0001013 ***Residuals 24—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1> ##To run Hotellings on only 2 groups> library(DescTools)> G51 (HotellingsT2Test(cbind(X1, X2, X3, X4, X5, X6, X7, X8,X9) ~ Group, data=G51))Hotelling’s two sample T2-testdata: cbind(X1, X2, X3, X4, X5, X6, X7, X8, X9) by GroupT.2 = 8.8603, df1 = 9, df2 = 16, p-value = 0.0001013alternative hypothesis: true location difference is not equal to c(0,0,0,0,0,0,0,0,0)When comparing just 5-prehistoric dogs and 1-modern dogs using either MANOVA orHotelling’s T2 the two species are significantly different in ‘size’ (p mf.manova3 summary(mf.manova3)Df Pillai approx F num Df den Df Pr(>F)as.factor(Group) 1 0.9137 23.527 9 20 9.423e-09 ***Residuals 28—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1> mf.manova4 summary(mf.manova4)Df Pillai approx F num Df den Df Pr(>F)as.factor(Group) 1 0.97745 81.894 9 17 3.222e-12 ***Residuals 25—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1> mf.manova5 summary(mf.manova5)Source: Manly, Bryan F.J. Multivariate Statistical Methods: A Primer, Third Edition, CRC Press,07/2004.Page 4 of 5Df Pillai approx F num Df den Df Pr(>F)as.factor(Group) 1 0.91706 17.198 9 14 4.213e-06 ***Residuals 22—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1The mean ‘size’ described by nine variables differs significantly (p mf_sexF)as.factor(Group) 3 2.32644 20.3400 27 159 boxplot(G5, xlab=”size variables for prehistoric dogs (2)”, ylab=”size (mm)”)Source: Manly, Bryan F.J. Multivariate Statistical Methods: A Primer, Third Edition, CRC Press,07/2004.Page 5 of 5

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