Multivariate Analysis of Variance (MANOVA)

ANOVA:

- Used to test for group differences in an IV(s) on 1 DV

MANOVA:

- Used to test for group differences in an IV(s) on multiple DVs

- Effectively an ANOVA looking for group differences on a single composite DV.

- Composite DV: a new composite DV is created which is weighted combination of the individual DVs

- Made up from a weighted combination of the old DVs

- If MANOVA is performed with 3 DVs:

           - new DV = b1 x DV1 + b2 x DV2 + b3 x DV3 + a

- Weights are chosen (through complex procedures) that produce the biggest group differences on the new DV.

- When yoy have multiple IVs, a different 'new' DV is used for each effect - as different weights may be required for each IV to maximise the group differences.

- Good protection against inflation of type I error rate 

        - carrying out lots of seperate ANOVAs could lead to spurious significances (multiple comparisons)

- Can reveal effects not detected by ANOVA 

         - rarely occurs 

- More complex

- Additonal assumptions 

- Outcome can be difficult to interpret 

- Can be less powerful than ANOVA under certain circumstances 

        - i.e., less sensitive to detecting a real effect

- Moderately correlated DVs work best 

        - indicvates that the concept under study (e.g. well-being) consists of different (but related) dimensions 

- MANOVA poor choice when very low or high correlations 

        - high correlations suggest that multiple dimesions do not exist, but probably measuiring the same thing several times 

        - in this case choose one DV and perform ANOVA 

        - very low correlations - MANOVA less powerful        

Outliers 

- Check using boxplots or histograms 

- Each DV mjust be checked for outliers in every cell

         - SPSS 'Split File' function is a convienent way of splitting the entire sample into seperate groups 

Multivariate Normality:

- Individual IVs should be normally distrubuted 

- Composite Dvs should also be normally distributed 

- Checking second assumption is unrealistic, checking first is normally sufficient 

- If you have equal cell sizes and 15 or more ppts. in each group MANOVA is relatively robust to violations of multivariate normality

Homogeneity of Variance - Box's M test

- Tests if variances and covariances of DVs are the same in each cell

- If Box's M signficant (p<o.o5) then they are significantly different & assumption is violated 

- If Box's M non-significant then variances - covariances are the same & assumption is met 

- However, MANOVA is robust to violations of this assumption if cell sizes are equal 

TIP: where possible, planning your research so you have equal(ish) cell sizes and reasonable sample size - avoids potential problems with assumptions 

- Other assumptions may be required if equivalent ANOVA requires them 

        - Repeated Measures MANOVA requires sphericity assumption to be met 

        - MANCOVA requires a linear relationship between the covariate and each DV.

1) Look at Multivariate Tests table for MANOVA results 

          - SPSS gives you 4 different statistics 

                 - Pillai's Trace, Wilk's Lambada, Hotelling's Trace & Roy's largest root

          - Each gives an F-value & p-value

                  - Use Wilk's Lambada; although others often give similar results 

Only is MANOVA is signficant then:

2) Do 'post-hoc' analyses using ANOVA on each individual DV to see where the IV(s) have had an effect. Use significance correction (Bonferroni) for multiple tests 

- Table of correlations shows if correlations are not very high or very low to suggest whether it is worth perfoming MANOVA.

- Table of means indicates the mean score for the 2 groups on each DV

- Box's M is non-signifcant - indicates homogeneity of variance-covariance assumption sucessfully met 

- 

- Multivariate Tests indicates the MANOVA results, use Wilk's Lambada to see if there is a significant effect 

- If MANOVA results are signficant, do seperate analysis on ANOVAs on each DV to see where differences are

- ANOVA requires assumption of homegeneity of variance to be met- variance for each group should be equal 

- Apply a Bonferreroni correction for the three tests 

"The __________  data were analysed using MANOVA with IV(s) entered as the IVs, and DVs entered as the DVs. The analysis revealed a significant/non-signficant multivariate effect of IV (Wilk's Lambada = ?(value on output), F(?,?) = ?(F value), p=?). ANOVA performed on each DV seperately showed that even after applying a Bonferroni correction for the three tests of ?(sig. value) a significant/non-signficant effect on IV was observed (F(df1,df2)= ? (F value), p=?), with _____ (mean) report significantly/non-signficantly higher/lower IV than _____ (mean). There were no significant group differences  on either DV (F(df1,df2) =? (Fvalue), p=?), or DV(F(df1,df2)=?(Fvalue), p=?)"