MVDA - Multivariate analysis of variance and Descriptive Data Analysis

Week 5: MANOVA and Descriptive DA

The goal of MANOVA and of Descriptive DA is the optimal prediction of differences between group means on several interval variables.

Why? Often very natural to compare groups on more than one variable, for example:

• quality and quantity of task performance;

• different stress reaction (e.g. emotions, physiological measures).

Compared to ANOVA:

ANOVA: one dependent variable -->(univariate)

MANOVA: two or more dependent variables -->(multivariate)

Check the assumption of equality of the covariance matrices and discuss robustness ofthe MANOVA against violations of the assumption of multivariate normality.

To check for the equality of covariance matrices, in the table: Test of Equality of Covariance Matrices, look at Box’s M. If higher than 0.05 then not significant.

To check for robustness of non-normality: number of participants per group has to be ≥20:

The design is balanced when robust to unequal covariance matrices.

Consider the mean values in the descriptive statistics. Which groups differ a lot on characteristic x (e.g. physical complaints)? Which groups differ only a little?

Look at descriptive statistics table and the appropriate box, in this case: physical complaints. Compare the values under Mean.

Given your crude assessment in the previous question, is it plausible to expect a multivariate effect?

If there are differences under sample group means, then yes, a multivariate effect is expected.

Example: Is there a significant effect of occupation on physical complaints, experience of hostility, and/or dissatisfaction?

We look at the multivariate tests table, in the Occupation box. Then, we report F statistic, df, and p value. If the tests are higher than p≤.05, there is a significant effect.

For which variables is the univariate effect significant?

We look at the Tests of Between-Subjects Effects table. Then, you report the tests were the p<0.05.

What is the answer to the previous question be if we apply a Bonferroni correction for multiple testing?

For Bonferroni, we divide the Alpha by number of categories (dependent variables). For example, if there are three categories (e.g.: hostility, physical complaint, dissatisfaction). The alpha then becomes smaller and we have to check if the p values are still smaller. Often, they aren’t or only few remain significant. Example:

Before Bonferroni:

Physical complaints F(2,177) = 3.196,p=.043 -->significant

Hostility F(2,177) = 3.405,p=.035 ---> significant

Dissatisfaction F(2,177) = 4.511,p=.013 --> significant

After Bonferroni, Withα=.05/3 =.0167:

Only Dissatisfaction F(2,177) = 4.511,p=.013 -->significant.

Interpret the significant effect(s) (after Bonferroni correction) using the table with multiple comparisons. Why is the Tukey HSD correction applied?

The Tukey test is invoked when you need to determine if the interaction among three or more variables is mutually statistically significant. We look at the multiple comparisons table with Tukey HSD applied and look at the category with significant effect. Example:

Dissatisfaction: catering different from management, (Mdif f= 2.767,p=.009)

What is theoretically the maximum number of discriminant functions?

imax= min(k−1,p) = min(2,3) = 2

How many discriminant functions are significant?

We look at the Wilk’s Lamda box. Then, under chi square. We count how many p scores are below 0.05, and that’s the number of significant discriminant functions. Report wilk’s lamda and p value.

Use the Structure matrix to interpret the significant functions.

Example: All variables high correlations with F1→general (workplace stress) function

Which group has the highest mean value on the first discriminant function?

We look at the table Functions at Group Centroids. Then, under F1 look for the highest number to get the mean value score.

How good is the overall classification? For which group does the classification work best?

Under the classification table it says: x%( e.g.: 40.6 %) of original grouped cases correctly classified. Within the classification table, check which group whichever number is highest belongs to, and that’s the group where the classification works best.