Beyond ANOVA (Part 6.2) {MANOVA Assumptions}

[Neeraj Kaushik]

Continuing from the previous post:

Obj 8: To check whether Sales & Advt together depend on season.

Now we've 2 metric dep var and 1 indep Non metric var so we'll use One way MANOVA

Assumptions of MANOVA:
It follows all assumptions of 1 way ANOVA

1. Dep var shd be metric & normally distributed individually (Checked by 1 sample KS test. Sig value > 0.05 indicates data 
normally distributed)

2. Homogenity of variance i.e. Dep var shd have the same variance in all categories of indep var (Checked by Levene's test. 
if Sig value < 0.05 indiactes assumption is violated. However we've seen that when no. of observations in both categories are 
almost same we can ignore the Levene's test values. If no. of observations are not same, then we must accept Levene's test 
values and if significant then we shd attempt for the transformation of data)

Then some new assumptions are there

3. Multivariate Normality of dep var (So far we check whether individually dep var are normally distributed but now we're 
concerned whether together they're normally distributed or not?) But SPSS doesnt have the provision for checking this hence 
we've to rely on the individual normality. Though individual normality does not guarantee the multi variate normality. I 
explain it by quoting the example of a couple. A man & woman together makes a couple but its quite possible that how a man & 
woman behave indivudally may be quite different from how they behave as couple. Likewise individual normality doesn't 
guarantee multivariate normality.

4. Equality of co-variance matrices (For individual dep var, we were looking whether this var gor same variance or not among 
various categories of indep var but now since 2 or more dep var are present so we're interested in knowing about the equality 
of their co-variances (between the dep var). Now this is complex and is performed by matrix algebra)

Box's test is used for checking this assumption. But to make life tough, this test is very sensitive and often gives 
significant value thus indicating violation of assumption. It may be a case that Box's test is significant now coz of 
equality of co-variance assumption violation rather coz of violation of multivariate normality. So we'll use our common sense 
& will test for the covariance ourself to get the correct picture.
Again is no. of observations in each category of indep var are almost same, we can ignore Box's test results.

So lets first settle down with assumptions then we'll proceed with MANOVA.
In the file ancova2.sav (attached in previous post) Sales & Advt were found to be normally distributed. Both dep var had the 
same variance among various categories of season as well as territory (Levene's test confirmed it). We can cross check it in 
MANOVA.

Go to Analyze->General Linear Model->Multivariate
Put Sales & Advt in Dep var and season in Fixed Factor(s)
Click on Options->Homogenity tests. Click on Continue->OK
Table-1 gives info about the categories of indep var. There're 3 categories of season and all are equal in size.
Table-2 gives Box's test Sig value as 0.278 i.e. assumption of equality of covariance is not violated.
Table-4 gives Levene's test results which indicates assumptions of homogenity of variance is not violated.

Repeat this process with indep var territory
Table-1 indicates that ther're 3 categories of territory and they are not equal in number (though there's minor differences)
Box's Sig value is 0.035 indicating violation of assumption.
So lets check for our own how much differences are there in covarinces.

First lets understand what is covariance. Variations in one var is called Variance. When 2 var are there, it become 
covariance.

Now you may be surprised but SPSS has given the option for finding co-variance in Reliability analysis.

But since we need co-variances between the different categories of territory so first lest split the file on the basis of 
territory.
Go to Data->Split file->Click on Compare groups and now send territory to RHS block of "Groups based on"
Click OK.

Now go to Analyse->Scale->Reliability analysis
Send Sales & Advt on RHS block
Click on Statistics-> click on Covariance (given on top RHS block og Inter item)
Click Continue->OK

In output see Table-3 Inter-item covariance matrix (Box's test is checking whether is matrix is same in different categories 
or not?)
Compare the value of Sales & Advt for A (3.264), B(2.446) & C(2.276)
There doesn't seem to be enormous difference. Though statistically Box's test is right but general glance doesn’t tell big 
differences between these values.
So we can ignore Box's test value (given the info that there's much difference in the no. of observations for the 3 
territories).

So we can apply MANOVA now (in next post)......

Happy Learning
Neeraj