MANOVA Dig it! Anova vs. Manova Why not multiple Anovas? Anovas run separately cannot take into account the pattern of covariation among the dependent measures It may be possible that multiple Anovas may show no differences while the Manova brings them out MANOVA is sensitive not only to mean differences but also to the direction and size of correlations among the dependents Anova vs. Manova Consider the following 2 group and 3 group scenarios, regarding two DVs Y1 and Y2 If we just look at the marginal distributions of the groups on each separate DV, the overlap suggests a statistically

significant difference would be hard to come by for either DV However, considering the joint distributions of scores on Y1 and Y2 together Anova vs. Manova Now we can look for the greatest possible effect along some linear combination of Y1 and Y2 The linear combination of the DVs created makes the differences among group means on this new dimension look as Anova vs. Manova So, by measuring multiple DVs you increase your chances for finding a group difference In this sense, in many cases such a test has

more power than the univariate procedure, but this is not necessarily true as some seem to believe Also conducting multiple ANOVAs increases the chance for type 1 error and MANOVA can in some cases help control for the inflation Kinds of research questions Which DVs are contributing most to the difference seen on the linear combination of the DVs? Discriminant analysis As mentioned, the Manova regards the linear combination of DVs, the individual Anovas do not take into account DV interrelationships If you are really interested in group differences on the individual DVs, then Manova is not appropriate Different Multivariate test criteria Hotellings Trace Wilks Lambda, Pillais Trace

Roys Largest Root Whats going on here? Which to use? The Multivariate Test of Significance Thinking in terms of an F statistic, how is the typical F calculated in an Anova calculated? As a ratio of B/W (actually mean b/t sums of squares and within sums of squares) Doing so with matrices involves calculating* BW-1 We take the between subjects matrix and post multiply by the inverted error matrix Example Dataset example 1: Experimental 2: Counseling 3: Clinical Psy Program Silliness Pranksterism

1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 60 57 65 63 60 62 66 61

63 68 52 59 59 58 62 8 7 13 15 12 15 16 11 12 16 17 20 23 19 21 Example

Hypothesis Error Between-Subjects SSCP Matrix group dv1 dv2 dv1 dv2 dv1 210.000 -90.000 88.000 80.000 dv2 -90.000 90.000 80.000 126.000 B matrix W matrix

Based on Type III Sum of Squares To find the inverse of a matrix one must find the matrix such that A-1A = I where I is the identity matrix 1s on the diagonal, 0s on the off diagonal For a two by two matrix its not too bad Example We find the inverse by first finding the determinate of the original matrix and multiply its inverse by the adjoint of that matrix of interest* Our determinate here is 4688 and so our result for W-1 is a11 a12 D

a a 22 21 D a11a22 a12 a21 1 D D 1 a22 a 21 a12 a11 .0269 .0171 .0171 .0188 You might for practice verify that multiplying this matrix by W will result in a matrix of 1s on the diagonal and zeros off-diagonal

Example With this new matrix BW-1, we could find the eigenvalues and eigenvectors associated with it.* For more detail and a different understanding of what were doing, click the icon; for some the detail helps. For the more practically minded just see the R code below The eigenvalues of BW-1 are 210 90 .0269 .0171 7.18 5.27 90 90 .0171 .0188 3.95 3.23 Lets examine the computer output for that data

Multivariate Testsc Effect program Pillai's Trace Wilks' Lambda Hotelling's Trace Roy's Largest Root Value 1.095 .073 10.405 10.179 F Hypothesis df 7.261 4.000 14.864 4.000 26.013 4.000 b 61.074

2.000 Error df 24.000 22.000 20.000 12.000 Sig. .001 .000 .000 .000 b. The statistic is an upper bound on F that yields a lower bound on the significance level. c. Design: Intercept+program Partial Eta Squared .548 .730 .839 .911 Wilks and Roys

Well start with Wilks lamda It is calculated as we presented before |W|/|T| = .0729 It actually is the product of the inverse of the eignvalues+1 (1/11.179)*(1/1.226) =.073 Next, take a look at the value of Roys largest root It is the largest eigenvalue of the BW-1 matrix s 1 Wilks ' i 1 1 i Multivariate Testsc Effect program Pillai's Trace

Wilks' Lambda Hotelling's Trace Roy's Largest Root Value 1.095 .073 10.405 10.179 F Hypothesis df 7.261 4.000 14.864 4.000 26.013 4.000 61.074b 2.000 Error df 24.000 22.000 20.000

12.000 Sig. .001 .000 .000 .000 b. The statistic is an upper bound on F that yields a lower bound on the significance level. c. Design: Intercept+program 0 10.179 0 .226 Partial Eta Squared .548 .730 .839 .911

Pillais and Hotellings s i V i 1 1 i Pillais trace is actually the total of our eigenvalues for the B(B+W)-1 matrix* Essentially the sum of the variance accounted in the variates Here we see it is the sum of the eigenvalue/1+eigenvalue ratios 10.179/11.179 + .226/1.226 = 1.095 Now look at Hotellings Trace It is simply the sum of the

eigenvalues of our Multivariate Testsc Effect program Pillai's Trace Wilks' Lambda Hotelling's Trace Roy's Largest Root Value 1.095 .073 10.405 10.179 F Hypothesis df 7.261 4.000 14.864 4.000 26.013 4.000

61.074b 2.000 Error df 24.000 22.000 20.000 12.000 Sig. .001 .000 .000 .000 b. The statistic is an upper bound on F that yields a lower bound on the significance level. c. Design: Intercept+program 0 10.179 0 .226

Partial Eta Squared .548 .730 .839 .911 Different Multivariate test criteria When there are only two levels for an effect that s = 1 and all of the tests will be identical When there are more than two levels the tests should be close but may not all be similarly sig or not sig Different Multivariate test criteria

As we saw, when there are more than two levels there are multiple ways in which the data can be combined to separate the groups Wilks Lambda, Hotellings Trace and Pillais trace all pool the variance from all the dimensions to create the test statistic. Roys largest root only uses the variance from the dimension that separates the groups most (the largest root or Which do you choose?

Wilks lambda is the traditional choice, and most widely used Wilks, Hotellings, and Pillais have shown to be robust (type I sense) to problems with assumptions (e.g. violation of homogeneity of covariances), Pillais more so, but it is also the most conservative usually. Roys is the more liberal test usually (though none are always most powerful), but it loses its strength when the differences lie along more than one dimension Some packages will even not provide statistics associated with it However in practice differences are often seen mostly along one dimension, and Roys is Guidelines Generally Wilks The others: Roys Greatest Characteristic Root: Uses only largest eigenvalue (of 1st linear combination)

Perhaps best with strongly correlated DVs Hotelling-Lawley Trace Perhaps best with not so correlated DVs Pillais Trace: Most robust to violations of assumption Post-hoc analysis Many run and report multiple univariate F-tests (one per DV) in order to see on which DVs there are group differences; this essentially assumes uncorrelated DVs. Furthemore if the DVs are correlated (as would be the reason for doing a Manova) then individual F-tests do not pick up on this, hence their utility of considering the set of DVs as a whole is problematic Multiple pairwise contrasts In a one-way setting one might instead consider performing the pairwise multivariate contrasts, i.e. 2 group

MANOVAs Hotellings T2 Doing so allows for the detail of individual comparisons that we usually want However type I error is a concern with multiple comparisons, so some correction would still be needed E.g. Bonferroni, False Discovery Rate Assessing DV importance Our previous discussion focused on group differences We might instead or also be interest in individual DV contribution to the group differences While in some cases univariate analyses may reflect DV importance in the multivariate analysis, better methods/approaches are available Discriminant Function Analysis It uses group membership as the DV and the Manova DVs as predictors of group membership*

Using this as a follow up to MANOVA will give you the relative importance of each DV predicting group membership (in a multiple regression sense) DFA Some suggest that interpreting the correlations of the p variables and the discriminant function (i.e. their loadings as we called them for cancorr) as studies suggest they are more stable from sample to sample So while the weights give an assessment of unique contribution, the loadings can give a sense of how much correlation a variable has with the underlying