Beyond ANOVA (Part-6.1)

[Neeraj Kaushik]

Gud mrng Friends

I've just come out of class where I explained all intricacies of ANOVA, MANOVA, ANCONA to Pre PhD students of textile students in TIT&S.
So let me pen down what I discussed in class.

Plz see the attached file ancova2.sav
There are 4 variables in the file:
Territory (3 categories present here: A, B & C)
Season (3 categories present here: Summer, Winter & Rainy)
Sales & Advt which have metric data
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Before proceeding further, I've checked for Data Cleaning
Analyse->Descriptive Statistics->Explore
Go to Statistics-> Click Outlier
In output it'll give Boxplots, 5% trimmed means and extreme values.
All indicator no outlier is present

For checking the normality
Analyse->Non parametric test->Legacy dialog->1 sample KS test
Both sales & Advt are normnally distributed.

If we check Normality from explore menu Sales is given as Not-normally distributed. I dnt know why the same var comes differently. But we'll sort it out later.
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Now depending on our research obj (which emerges from the Review of literature) we can perform various type of analysis here. First I'm writing all possible research obj. In bracket is mentioned the Statistical technique to be applied).

Obj 1. To see the effect of season on Sales
Obj 2. To see the effect of territory on Sales
Obj 3. To see the effect of season on Advt
Obj 4. To see the effect of territory on Advt

In all 4 above obj we'll apply One-way ANOVA. Note important point here. Non metric indep var are termed as Factor in ANOVA.
We shd not forget to check for Homogenity of variance assumption (Levene's test) & use of Welch test in case assumption is violated.
This all we've done earlier in great details.

Obj 5. To see the effect of season & territory on Sales
Obj 6. To see the effect of season & territory on Advt

For these obj we shd work on Two-way ANOVA
Click on Analyze->General Linear Model->Univariate
Put Sales as dep var & territory & season as indep var. Click ok (No other menu to be used here as of now)
In output table see Table: Test of Between Subjects effects
Look at the last col of Sig avlues
All Sig values (Territory p-value=0.159, Season p-value=0.680, Interaction p-value=0.836)  are < 0.05 indicating that season & territory has no relation with sales.
Had any Sig value been < 0.05 then we wud have applied other options like Post hoc, Plots, Contrasts etc.

Repeat the procedure with dep var as Advt
Almost the same results.
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Obj 7. To see the effect of season & territory on Sales with Advt as co-variate
Here we'll use Two way ANCOVA. Now we're stepping on another dimension.

We feel Advt is somehow related with Sales so it might effect the effect of other indep var such as season & territory. So now we'll take Advt as co-variate
There're 2 criterion for th co-variate:
a) It must be on metric measurement (If its non-metric then we'll include it as just another factor
b) It must be somewhat correlated with dep var else how it'll effect other indep var

So before proceeding first check whether sales & advt are correlated. Pearson correlation coefficient is 0.7 (it is sgnificant too) So it is eligible for taking as co-variate.

Repeat the GLM analysis but this time put Advt as co-variate
Click ok

In output we see something different from previous table.
First Advt is significant (Sig value=0.000) but this we know earlier too (by pearson correlation coefficient)
But now territory is also significant (Sig value=0.002)

Conclusion: If we can control the effect of Advt then Territory is related with sales or Sales in various territories differ significantly.

Next questions comes: Which territories differ significantly?

Repeat the analysis and this time click on
Plots->Send territory to Horizontal axis & click on Add then Continue (This will give chart with Territory on x-axis & sales on y-axis. Remember here chart is made always for dep var)
Options->Send territory to Display Means for block
Click on compare mean effects. LSD is the default method. Choose Bonferroni from the drop menu.
{We're using this option for getting Post hoc as the usual Post hoc option is disables as soon as we introduce any co-variate)
Also click on Descriptive Statistics. Click Continue & OK.
 
Output:
The 2nd table descriptive statistics gives mean scores for every category of indep var(s).
We've already discussed Table-3 which indicates Territory is related to sales in the presence of advt.
Next Table-4 shows estimated mean scores for territory.
Compare it with total mean scores of territories. The value in table-4 are somewhat reduced.
In Table-2 the actual mean score of different territories are given while in table-4 the are modified nullifying the presence of co-variate advt.
Table-5 gives paired comparison of estimated mean scores. This table indicates that territoy 2 differs significantly from rest of territories.
Table-6 reiterates the things already explained in Table-5 i.e. the various territories differ in their sales.
Lastly the plots show the same thing that territory A differs significantly from the rest.
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Obj 8. To see the effect of season & territory on Sales with Advt as co-variate
Repeat the same analysis agin with Advt as dep var & sales as co-variate
We'll get almost same sort of results.

This concludes discussion on One way ANCOVA.

Rest obj in next post.

Happy Learning.
Neeraj

[Radha garg]

Respected Sir,

Thanks for more exercise of ANOCOVA. 

But, i think we should also check the assumption of Independence of co variate and Indep variable as mentioned earlier which means that co-variate should not have the significant relationship with Indep variable. But that's not a point here. Actually, i want to know that in above example if advt is different across with the various territory or have the significant relationship with territory means Advt. affect the dep and indep variable, then how we can control or neutralized its effect on both the variable.

Secondly, how we can use the post hoc test with the option of contrast here. As earlier we learned in various option of ANOCOVA that the contrast is same as post hoc. So, what is the utility of it here

Thanks and Regards

Radha 

[Neeraj Kaushik]

Dear Radha

Though I've not found any mention for checking the assumption of homogenity of var-covar between the indep var(s) & covariate but i'll for check it. Also m not sure of the very interpretation of "If we control the effect of co-variate then ......." Here how to control the co-variate? I've no idea as of now.

As regard Contrast, again i'll mention the same old story :-)
Use contrast when you know in advance which groups to compare, use Post hoc when you're not sure which one to compare.
Plz find attached the details of all contrast given in the Menu of Contrast (of ANCOVA, MANCOVA etc.)

Best wishes
Neeraj

[Radha garg]

Respected Sir,

Thanks for the reply. But i already understand this thing about contrast through our previous discussion. 

Actually, here i was looking for the application part of contrast here as it's application (various option for comparison between categories) is the different as compared to one way anova.

But thanks for the this file. I will read it and revert you back in case of any query.

Thanks and Regards

Radha 

[Neeraj Kaushik]

Dear Radha

Plz see the nice demos of Contrast in ANOVA on youtube. This video contain 4 parts, do watch all.
http://www.youtube.com/watch?v=F19BTj9NvZ8

Best wishes
Neeraj