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MANOVA

MANOVA. All statistical methods we have learned so far have only one continuous DV and one or more IVs which may be continuous or categorical There are cases with multiple DVs.

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MANOVA

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  1. MANOVA • All statistical methods we have learned so far have only one continuous DV and one or more IVs which may be continuous or categorical • There are cases with multiple DVs. • All statistical methods that deal with multiple DVs or multiple variables without DV/IV specification are grouped into multivariate statistics • Multivariate analysis of variance (MANOVA) as an example • Why MANOVA instead of multiple ANOVA for each DV? • Experimentwise error rate • See what univariate analysis cannot see • The power of MANOVA test generally decreases with the number of variables that do not differ among groups. So be cautious in including variables in MANOVA (or any other multivariate statistical methods.

  2. Advantage of MANOVA Data Shape; Input Sex $ Height Width @@; datalines; Male 69 70 Male 68 74 Male 75 80 Male 78 85 Male 68 68 Male 63 68 Male 72 74 Male 63 66 Male 71 76 Male 72 78 Male 71 73 Male 70 73 Male 56 59 Male 77 83 Female 72 79 Female 64 65 Female 74 74 Female 72 75 Female 82 84 Female 69 68 Female 76 76 Female 68 65 Female 78 79 Female 70 71 Female 60 61 ; proc glm ; class Sex; model Height Width=Sex / solution; manova h=Sex; lsmeans Sex; title "MANOVA: Differences in Height and Width between sexes"; run; The SAS program will run both individual ANOVAs and a MANOVA. Run and explain output.

  3. Fisher Iris Data • Collected by Dr. Edgar Anderson, published in Fisher (1936) • Sepal length and width, petal length and width (in cm) of fifty plants for each of three types of iris • Iris setosa, diploid with 38 chromosomes • Iris versicolor, hexaploid (108 chromosomes) • Iris virginica, tetroploid • Fisher, R.A. (1936). "The Use of Multiple Measurements in Taxonomic Problems". Annals of Eugenics 7: 179–188.

  4. SAS Program Data Iris; input SepalLen SepalWid PetalLen PetalWid Species $ @@; cards; 5.1 3.5 1.4 0.2 Is 7.0 3.2 4.7 1.4 Ive 6.3 3.3 6.0 2.5 Ivi 4.9 3.0 1.4 0.2 Is 6.4 3.2 4.5 1.5 Ive 5.8 2.7 5.1 1.9 Ivi 4.7 3.2 1.3 0.2 Is 6.9 3.1 4.9 1.5 Ive 7.1 3 5.9 2.1 Ivi 4.6 3.1 1.5 0.2 Is 5.5 2.3 4 1.3 Ive 6.3 2.9 5.6 1.8 Ivi 5 3.6 1.4 0.2 Is 6.5 2.8 4.6 1.5 Ive 6.5 3 5.8 2.2 Ivi 5.4 3.9 1.7 0.4 Is 5.7 2.8 4.5 1.3 Ive 7.6 3 6.6 2.1 Ivi 4.6 3.4 1.4 0.3 Is 6.3 3.3 4.7 1.6 Ive 4.9 2.5 4.5 1.7 Ivi 5 3.4 1.5 0.2 Is 4.9 2.4 3.3 1 Ive 7.3 2.9 6.3 1.8 Ivi 4.4 2.9 1.4 0.2 Is 6.6 2.9 4.6 1.3 Ive 6.7 2.5 5.8 1.8 Ivi 4.9 3.1 1.5 0.1 Is 5.2 2.7 3.9 1.4 Ive 7.2 3.6 6.1 2.5 Ivi 5.4 3.7 1.5 0.2 Is 5 2 3.5 1 Ive 6.5 3.2 5.1 2 Ivi 4.8 3.4 1.6 0.2 Is 5.9 3 4.2 1.5 Ive 6.4 2.7 5.3 1.9 Ivi 4.8 3 1.4 0.1 Is 6 2.2 4 1 Ive 6.8 3 5.5 2.1 Ivi 4.3 3 1.1 0.1 Is 6.1 2.9 4.7 1.4 Ive 5.7 2.5 5 2 Ivi 5.8 4 1.2 0.2 Is 5.6 2.9 3.6 1.3 Ive 5.8 2.8 5.1 2.4 Ivi 5.7 4.4 1.5 0.4 Is 6.7 3.1 4.4 1.4 Ive 6.4 3.2 5.3 2.3 Ivi 5.4 3.9 1.3 0.4 Is 5.6 3 4.5 1.5 Ive 6.5 3 5.5 1.8 Ivi 5.1 3.5 1.4 0.3 Is 5.8 2.7 4.1 1 Ive 7.7 3.8 6.7 2.2 Ivi 5.7 3.8 1.7 0.3 Is 6.2 2.2 4.5 1.5 Ive 7.7 2.6 6.9 2.3 Ivi 5.1 3.8 1.5 0.3 Is 5.6 2.5 3.9 1.1 Ive 6 2.2 5 1.5 Ivi 5.4 3.4 1.7 0.2 Is 5.9 3.2 4.8 1.8 Ive 6.9 3.2 5.7 2.3 Ivi 5.1 3.7 1.5 0.4 Is 6.1 2.8 4 1.3 Ive 5.6 2.8 4.9 2 Ivi 4.6 3.6 1 0.2 Is 6.3 2.5 4.9 1.5 Ive 7.7 2.8 6.7 2 Ivi 5.1 3.3 1.7 0.5 Is 6.1 2.8 4.7 1.2 Ive 6.3 2.7 4.9 1.8 Ivi 4.8 3.4 1.9 0.2 Is 6.4 2.9 4.3 1.3 Ive 6.7 3.3 5.7 2.1 Ivi 5 3 1.6 0.2 Is 6.6 3 4.4 1.4 Ive 7.2 3.2 6 1.8 Ivi 5 3.4 1.6 0.4 Is 6.8 2.8 4.8 1.4 Ive 6.2 2.8 4.8 1.8 Ivi 5.2 3.5 1.5 0.2 Is 6.7 3 5 1.7 Ive 6.1 3 4.9 1.8 Ivi

  5. 5.2 3.4 1.4 0.2 Is 6 2.9 4.5 1.5 Ive 6.4 2.8 5.6 2.1 Ivi 4.7 3.2 1.6 0.2 Is 5.7 2.6 3.5 1 Ive 7.2 3 5.8 1.6 Ivi 4.8 3.1 1.6 0.2 Is 5.5 2.4 3.8 1.1 Ive 7.4 2.8 6.1 1.9 Ivi 5.4 3.4 1.5 0.4 Is 5.5 2.4 3.7 1 Ive 7.9 3.8 6.4 2 Ivi 5.2 4.1 1.5 0.1 Is 5.8 2.7 3.9 1.2 Ive 6.4 2.8 5.6 2.2 Ivi 5.5 4.2 1.4 0.2 Is 6 2.7 5.1 1.6 Ive 6.3 2.8 5.1 1.5 Ivi 4.9 3.1 1.5 0.2 Is 5.4 3 4.5 1.5 Ive 6.1 2.6 5.6 1.4 Ivi 5 3.2 1.2 0.2 Is 6 3.4 4.5 1.6 Ive 7.7 3 6.1 2.3 Ivi 5.5 3.5 1.3 0.2 Is 6.7 3.1 4.7 1.5 Ive 6.3 3.4 5.6 2.4 Ivi 4.9 3.6 1.4 0.1 Is 6.3 2.3 4.4 1.3 Ive 6.4 3.1 5.5 1.8 Ivi 4.4 3 1.3 0.2 Is 5.6 3 4.1 1.3 Ive 6 3 4.8 1.8 Ivi 5.1 3.4 1.5 0.2 Is 5.5 2.5 4 1.3 Ive 6.9 3.1 5.4 2.1 Ivi 5 3.5 1.3 0.3 Is 5.5 2.6 4.4 1.2 Ive 6.7 3.1 5.6 2.4 Ivi 4.5 2.3 1.3 0.3 Is 6.1 3 4.6 1.4 Ive 6.9 3.1 5.1 2.3 Ivi 4.4 3.2 1.3 0.2 Is 5.8 2.6 4 1.2 Ive 5.8 2.7 5.1 1.9 Ivi 5 3.5 1.6 0.6 Is 5 2.3 3.3 1 Ive 6.8 3.2 5.9 2.3 Ivi 5.1 3.8 1.9 0.4 Is 5.6 2.7 4.2 1.3 Ive 6.7 3.3 5.7 2.5 Ivi 4.8 3 1.4 0.3 Is 5.7 3 4.2 1.2 Ive 6.7 3 5.2 2.3 Ivi 5.1 3.8 1.6 0.2 Is 5.7 2.9 4.2 1.3 Ive 6.3 2.5 5 1.9 Ivi 4.6 3.2 1.4 0.2 Is 6.2 2.9 4.3 1.3 Ive 6.5 3 5.2 2 Ivi 5.3 3.7 1.5 0.2 Is 5.1 2.5 3 1.1 Ive 6.2 3.4 5.4 2.3 Ivi 5 3.3 1.4 0.2 Is 5.7 2.8 4.1 1.3 Ive 5.9 3 5.1 1.8 Ivi ;

  6. SAS program (cont.) proc glm data=Iris; class Species; model SepalLen SepalWid PetalLen PetalWid=Species/solution; manova h=_all_; /* Print multivariate tests together with characteristic roots and vectors of: E-1 * H. */ means Species / tukey; title "MANOVA test"; run; proc discrim pool = test slpool = 0.05; class Species; var SepalLen SepalWid PetalLen PetalWid; priors proportional; title "Discriminant function analysis"; run; proc stepdisc ; class Species; var SepalLen SepalWid PetalLen PetalWid; title "Stepwise discriminant function analysis"; run; ‘pool=yes|no’: assuming equal|unequalcovariance matrices ‘pool=test’: test the equal covariance assumption, with slpool specifiying the significance level and with subsequent analysis depending on the outcome of the test. Run and explain

  7. Interpretation of MANOVA • If the multivariate test is • not significant, report no group differences among the mean vectors • significant, perform univariate ANOVA and relevant contrasts • Correlation among variables that may lead to significant MANOVA test but no significant ANOVA test. • Contrasts • Prior (planned): Certain theory predicts which treatments should be different • Post hoc (unplanned): Not sure which treatments should be different • Control of experimentwise error rate

  8. MANOVA Assumptions • Independence assumption: All observations are independent (residuals are uncorrelated) • Multivariate normality • Sphericity assumption in repeated measures • Homoscedasticity (equal variance and covariance) assumption: Each sample (group) has the same covariance matrix (compound symmetry) • Linearity assumption: Relationship among variables are linear.

  9. Discriminant function analysis Centroid N: number of genes (rows), c: number of clusters, b: number of time points or replicates (columns).

  10. Multi-group DFA

  11. GLM and repeated measures data dental; input person gender$ y1-y4; datalines; 1 F 21.0 20.0 21.5 23.0 2 F 21.0 21.5 24.0 25.5 3 F 20.5 24.0 24.5 26.0 4 F 23.5 24.5 25.0 26.5 5 F 21.5 23.0 22.5 23.5 6 F 20.0 21.0 21.0 22.5 7 F 21.5 22.5 23.0 25.0 8 F 23.0 23.0 23.5 24.0 9 F 20.0 21.0 22.0 21.5 10 F 16.5 19.0 19.0 19.5 11 F 24.5 25.0 28.0 28.0 12 M 26.0 25.0 29.0 31.0 13 M 21.5 22.5 23.0 26.5 14 M 23.0 22.5 24.0 27.5 15 M 25.5 27.5 26.5 27.0 16 M 20.0 23.5 22.5 26.0 17 M 24.5 25.5 27.0 28.5 18 M 22.0 22.0 24.5 26.5 19 M 24.0 21.5 24.5 25.5 20 M 23.0 20.5 31.0 26.0 21 M 27.5 28.0 31.0 31.5 22 M 23.0 23.0 23.5 25.0 23 M 21.5 23.5 24.0 28.0 24 M 17.0 24.5 26.0 29.5 25 M 22.5 25.5 25.5 26.0 26 M 23.0 24.5 26.0 30.0 27 M 22.0 21.5 23.5 25.0 ; procglm; class gender; model y1-y4=gender / nouni; repeated age 4 (8101214); means gender; lsmeans gender / pdiff; run;

  12. Adjusted F • Greenhouse-Geisser Epsilon measures by how much the sphericity assumption is violated. Epsilon is then used to adjust for the potential bias in the F statistic. • Epsilon = 1: the sphericity assumption is met perfectly. • minimum Epsilon = 1/(k - 1), where k is the number of levels in the repeated measure factor. For k = 3, the minimum Epsilon = ½. • The Huynh-Feldt epsilon its a correction of the Greenhouse-Geisser epsilon because the latter is too conservative.

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