Beyond ANOVA (Part 6.3) {1-way & 2-way MANOVA}

[Neeraj Kaushik]

Now that we are satisfied with the assumptions, we can proceed with MANOVA.
 
Dnt forget to remove split from file. 
Click on Data->Split file and click on Analyze all cases.
 
Obj 9: To check whether season is related with dep var Sales & Advt
 
File: amcova2.sav
Dep var: Sales & Advt
Indep var: Season
Technique used: One way MANOVA
 
Click on Analyze->General Linear Model->Multivariae
Put Sales & Advt in dep var & season in Fixed Factor
Click on OK
 
In output, Table-1 gives summary of indep var
Table-2 gives Multivariate test report
We'll see the output of indep var only (Ignore the Intercept details)
We're given 4 test statistics here. Lets discuss them first.
 
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Hotelling's trace is always larger than Pillai's trace, but when these two statistics are nearly equal, this indicates that the effect probably does not contribute much to the model. 
 
Roy's largest root is always less than or equal to Hotelling's trace. When these two statistics are equal, the effect is predominantly associated with just one of the dependent variables, there is a strong correlation between the dependent variables, or the effect does not contribute much to the model.
 
There is evidence that Pillai's trace is more robust than the other statistics to violations of model assumptions(Olson, 1974).
 
Each multivariate statistic is transformed into a test statistic with an approximate or exact F distribution.
 
The significance values of less than 0.05 indicating that the effects contribute to the model.
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So in Table-2 we found that season is not related to the dep var(s).
Table-2 just gives an idea about the overall relationship of indep var(s) with dep var(s) but to know in details, consult Table-3 Tests of between subjects effects.
 
Ignore the values of Corrected Model & Intercept. Look at the rows od season. It indiactes the separate effect of season of both dep var. Sig values > 0.05 indicates season has no effect on any of the dep var.
 
Since there's no effect of season on dep var there's no point in plotting charts of running Post hoc.
 
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Obj 9: To check whether territory is related with dep var Sales & Advt
 
Dep var: Sales & Advt
Indep var: Territory
Technique used: One way MANOVA 
 
Repeat the analsyis (Dnt use any of the other option) and see output.
 
Table-2 indicates a significant relationship between territory & dep var(s). All 4 statistics agree on it, In case there's any difference between the 4 statistics we'll consult Pillai's trace.
 
To know what about the relationship see Table-3
It indicates Territory has no relation with Sales & Advt. 
 
So now we get quite contrary results. Why it is so?
The answer is that Table-2 gives effect when both dep var(s) are taken together while Table-3 is equivalent to 1 way ANOVA.
 
The big advantage of MANOVA over ANOVA is that it takes correlations of 2 dep var into picture while 1 way ANOVA fails to do so.
 
Now recall the output of ANCOVA. Territory was found to be related with Sales when co-variate Advt was controlled. Likewise territory was related to Advt when sales was controlled. What is means is that there's some undercurrent going on between sales & advt (remember correlation of 0.7 between the two). Individually none seems to be related to territory but together they do.
 
Conclusion: Individually sales & advt doesn't relate to indep var territory but when both are taken together territory is found to related with them.
 
Next question: Which territory is diferent from the rest?
Obviosly we'll go with Post hoc analysis & Plots. But since there were no relationship was found in the Table: Tests of Between subjects Effects so the Post hoc analysis will not give any significant difference in any of the category.
 
So repeat the analysis. This time click on Plots and put territory on horizontal axis.
In the output charts show that while Advt was max in Territory A but Sale here was minimum.
 
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This concludes our discussion on One-way MANOVA.
 
Obj 10: To check whether territory & season together are related with dep var Sales & Advt
 
Dep var: Sales & Advt
Indep var: Territory & season
Technique used: Two way MANOVA 
 
Repeat the process with sales & advt as dep var and season, territory as Fixed factors.
The analysis will be done in the same way.
 
Table-2 repeats the results of obj 9 that territory has effect on dep var(s) but season has no effect.
Table-3 also repeats the results that Territory has no relation with Sales & Advt. 
 
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Happy Learning.
Neeraj

[Radha garg]

Well done sir.......... 

Thanks for explaining the concept of MNOVA beautifully. Now only MANOCOVA has been left..... :-)

Regards

Radha