MANOVA Lecture 12Multivariate Analysis of CovarianceSlide 3Slide 4Slide 5Slide 6Different Multivariate test criteriaDifferent Multivariate test criteriaSlide 9Slide 10Slide 11Slide 12Slide 13Slide 14Which do you choose?Assessing DVsSlide 17Slide 18Slide 19Slide 20Slide 21Specific Comparisons and Trend AnalysisSlide 23Unequal samples42510011 0010 1010 1101 0001 0100 1011 MANOVA Lecture 12Nuance stuffPsy 524Andrew Ainsworth42510011 0010 1010 1101 0001 0100 1011 Multivariate Analysis of Covariance•The linear combination of DVs is adjusted for one or more Covariates.•The adjusted linear combinations of the DVs is the combination that would have been had all of the subjects scored the same on the CVs.* ( ) ( ) ( ) 1 ( )( )Y YZ Z ZYS S S S S-= -42510011 0010 1010 1101 0001 0100 1011 Multivariate Analysis of Covariance•Each subjects score is made up of the DVs and the CVs111110115108IQY w rat rw rat a� �� �� �� �= -� �� �� �� �-� �� �42510011 0010 1010 1101 0001 0100 1011 Multivariate Analysis of Covariance•So that each S is a combination of the original S plus the SSCP for the CVs and the covariances between the DVs and the CVs.[ ] [ ]1 212 2.00 64.67 54.6764.67 2090.89 1767.8954.67 1767.56 1494.22Z Y YZYY� �� �� �� �� �� �� �� �42510011 0010 1010 1101 0001 0100 1011 Multivariate Analysis of Covariance[ ] [ ]* ( ) ( ) ( ) 1 ( )1*( )2090.89 1767.56 64.672 64.67 54.671767.56 1494.22 54.67Y YZ Z ZYS S S S SS--= -� � � �= -� � � �� � � �42510011 0010 1010 1101 0001 0100 1011 Multivariate Analysis of Covariance•Calculating Wilk’s Lambda is the same and for the most part the F-test is the same except calculating s and DF2:( )2 22 22( ) ( ) 4( ) ( ) 5( ) 1 ( )( ) 22 2effecteffecterror errorerrorp q dfsp q dfp q df p q dfdf s df+ -=+ + -+ - + + -� � � �= - -� � � �� � � �42510011 0010 1010 1101 0001 0100 1011 Different Multivariate test criteria •Hotelling’s Trace•Wilk’s Lambda,•Pillai’s Trace•Roy’s Largest Root42510011 0010 1010 1101 0001 0100 1011 Different Multivariate test criteria•When there are only two levels for an effect s=1 and all of the tests should be identical•When there are more than two levels the tests should be nearly identical but this is not always the case42510011 0010 1010 1101 0001 0100 1011 Different Multivariate test criteria•When there are more than two levels there are multiple ways in which the data can be combined to separate the groups –(e.g. one dimension separates group 1 from groups 2 and 3, a second dimension separates group 2 from group 3, etc.)42510011 0010 1010 1101 0001 0100 1011 Different Multivariate test criteria•Wilk’s Lambda, Hotelling’s Trace and Pillai’s trace all pool the variance from all the dimensions to create the test statistic.•Roy’s largest root only uses the variance from the dimension that separates the groups most (the largest “root” or difference).42510011 0010 1010 1101 0001 0100 1011 Different Multivariate test criteria•The various formulas are (E is error and H is hypothesized effect):–Wilk’s Lambda - |E| |H + E| - It’s the ratio of error to effect plus error. Analogous to 1 – R2. Middle of the road in terms of how conservative a test it is.42510011 0010 1010 1101 0001 0100 1011 Different Multivariate test criteria•The various formulas are (E is error and H is hypothesized effect):–Hotelling’s trace – Trace(H/E)=C and you look up C in a table to get the F value. It is analogous to an F-test. Very liberal test.42510011 0010 1010 1101 0001 0100 1011 Different Multivariate test criteria•The various formulas are (E is error and H is hypothesized effect):–Pillai’s trace – Trace(H/(H + E)). Analogous to R2. Very conservative42510011 0010 1010 1101 0001 0100 1011 Different Multivariate test criteria•The various formulas are (E is error and H is hypothesized effect):–Roy’s Largest Root - (H/(H + E)) and it looks for the biggest difference. It is variable in terms of how conservative it is.42510011 0010 1010 1101 0001 0100 1011 Which do you choose?•For the most part stick with Wilk’s lambda. It’s the most widely used•Use Hotelling’s Trace if –Manipulated (experimental) variables–Very clean design with no internal validity problems•Pillai’s trace is the most conservative, but if your design has many problems (e.g. unbalanced, assumption violation, etc) pillai’s is supposed to be robust to these problems42510011 0010 1010 1101 0001 0100 1011 Assessing DVs•If multivariate test is significant •Run multiple univariate F-tests (one per DV) in order to see on which DVs there are group differences, this assumes uncorrelated DVs.42510011 0010 1010 1101 0001 0100 1011 Assessing DVs•The overall alpha level should be controlled for considering the multiple tests•The alpha levels can be divided equally or they can be set up to give more important tests a more liberal alpha level.1 21 (1 )(1 ) (1 )overall pa a a a= - - - -K42510011 0010 1010 1101 0001 0100 1011 Assessing DVs•If DVs are correlated than individual F-tests are problematic but usually this is ignored and univariate Fs interpreted anyway42510011 0010 1010 1101 0001 0100 1011 Assessing DVs•Roy-Bargman step down procedure–Can be used as follow-up to MANOVA or MANCOVA with correlated DVs or as alternative to multivariate analysis all together.42510011 0010 1010 1101 0001 0100 1011 Assessing DVs•Roy-Bargman step down procedure–The theoretically most important DV is analyzed as an individual univariate test (DV1).–The next DV (DV2), in terms of theoretical importance, is then analyzed using DV1 as a covariate. This controls for the relationship between the two DVs.–DV3 (in terms of importance) is assessed with DV1 and DV2 as covariates, etc.42510011 0010 1010 1101 0001 0100 1011 Assessing DVs•Discriminant Function analysis – –We will discuss this more later but…–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)42510011 0010 1010 1101 0001 0100 1011 Specific Comparisons and Trend Analysis•With a significant multivariate (and univariate) test and more than two groups, this needs to be followed with