Beyond ANOVA (Part-4)

[Neeraj Kaushik]

Now lets start working practically on ANCOVA.

Attached is ancova2.sav file
where dep var is Sales & indep var is Territory

We want to ascertain whether Sales in various territories is same or not? i.e. whether there's significant relationship between Sales & Territory.

While working for this, we consider that Advt may be another variable which may influence the Sales, hence this is taken as co-variate.

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So before starting with ANCOVA lets verify whether our data is fit for it. This is done by verifying the 2 assumptions:

1. Independence of covariate & treatment effect
2. Homogenity of regression slopes

The first one basically means that the covariate should not be different across the various categories of indep var. For this we can conduct an ANOVA or t­‐test using the groups as the independent variable and the covariate as the dep var, this analysis should be non­‐significant i.e. p-value or Sig value shd be > 0.05

For the 2nd assumption, we'll use Levene's test. But as of now lets start with SPSS.
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Step-1
Run ANOVA with dep var: Advt & Indep var: Territory
Sig- value of 0.152 indicates there's no sig relation between co-variate & indep var i.e. co-var doesnt vary among the various categories of indep var territory.

We'll check for the 2nd assumption later.
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Step-2
Go to Analyze-->General Linear Model (GLM)-->Univariate
{This is the menu for ANCOVA as well as n-way ANOVA}

//First lets work without co-variate.

Put Sales in dep var & territory in Fixed factors
{There're 6 other menu which we'll use}.
First click on Options
Click on
Descriptives (to get mean score of dep var for various categories of indep var.) &
Estimates of effect size (this will give Partial Eta squared which works like r-sq of regression. This explains how much variations of dep var are explained by any of the var)

Click Continue-->OK
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The Table-1 in output gives info about no.of items in various categories of indep var.

Table-2 gives Descriptive scores
So basically we're working on Null hypothesis that there is no sig difference between 3.22 (mean of A), 4.88 (mean of B) & 4.85 (mean of C).

Table-3 gives Tests of between subjects effects.
Lets understand it thoroughly.
The total variations in Sales accounts to 683 which are corrected to 110.967 {Plz see these'll never change in any of the further tables}.
Out of these 16.844 are explained by indep var territory (quite low!!).
Since there's only 1 indep var, the corrected model gives the same fig as given by indep var i.e. 16.844
Partial Eta squared gives value 0.152 (Same for corrected model & indep var). Plz note the same value is written below the table as r-sq=0.152
So 15.2.% variations of dep var are explained by territory.
Sig value of Territory is 0.108 indicating there's no sig relationship between territory & Sales.
{Remember the decision rule: If Sig value>0.05 Null hypothesis accepted}
Max variations are explained by Intercept i.e. sill there're many var which explains variations of dep var.

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Now its time for introducing co-variate. Run GLM again, this time put Advt in co-variates. Click OK..

See Table-3 and compare it with earlier table of Tests of between subjects effects.
First thing: Variations explained by Corrected model improved to 31.92 and r-sq also increased to 0.288
Second thing: Variations explained by Intercept go drastically low. But still Intercept explains 49% of variations of dep var (Partial eta squared). It indicates there're still var that explains variations of dep var.
Advt explains 16% variations (Partial eta squared) & its sig value indicates it is significantly related with sales.
& Territory's Partial eta squared increased to 0.242

Why & How??

Plz note this is result of ANCOVA.
When the effect of advt is moderated i.e. if we do away the effect of Advt then Territory better explains the Sales.
Plz note the Sig value of Territory is now 0.027 which indicates that now it is significantly related with Sales.

If we dnt undertake advt then we concluded that Sales is same in each territory but now in the presence of a co-variate, we modify our results & concluded Sales is related to territory if effect of advt is moderated.

Next question comes which territories are significantly different. That needs paired comparison {remember Post hoc in ANOVA}

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Rest of the options in next Post.

[Preeti Jain]

Thank u sir................

--
Preeti Jain

[Neeraj Kaushik]

Dear Preeti
Plz try to work on the file & report if anything is not clear or any mistakes.
Regards
Neeraj

[Dr. Anagha Shukre]

Many thanx....Sir,

[Radha garg]

Great sir...... But i just want to know that in previous table of Test of between subjects, Partial Eta squared value was same for corrected model and  territory (indep var) i.e. .152, whereas in last table of Test of between subjects, both  Partial Eta squared value (corrected model and  territory) are different i.e. .288 and .242. I think both have the same meaning. so, why it's different. 

[Neeraj Kaushik]

See Radha
In first table there's only 1 indep var hence Corrected model have its value but in 2nd analysis, we've a moderated var (co-var) hence the overall r-sq (denoted by partial eta squared here) is changed.
Plz note partial eta squared is not the total of all partial eta squared of various var.

Best wishes
Neeraj

[Radha garg]

I already understand this thing sir... But i want to know the reason of difference between the value of eta squared value in corrected model and territory i.e. .288 and .242 in last table only. As in previous analysis, both value are same.

Sorry...if i m asking the wrong question again. Actually, this is only thing which is not clear to me.

Regards

Radha

On Friday, December 21, 2012 10:40:55 AM UTC+5:30, Dr Neeraj Kaushik wrote:

[Dr Neeraj Kaushik]

Dear Radha
As I say again & again None of the question is a stupid questions. Every questions contains so much in itself.
However the same is not true for answers. There may be stupid answers.

Coming to ur question. Look again in first case we've not considered any extraneous variable. So whatever variance explained (0.152) is there, it is coz of indep var hence the same value or eta squared for the corrected model as well as indep var.
In 2nd attepmt, we've considered one co-variate which is supposedly influence the dep var. So now when we compute the table, its indirect influence is there. Technically speaking when we control the variations of co-variate then the variance explained by indep var will increase to 0.242 and overall Corrected model will also improve.

I cudnt find any more explanation in any of the books :-(

Best wishes
Neeraj

[Radha garg]

 Thanks for the explanation sir..... Now my doubt has been somewhat clear.....

Regards

Radha

[Radha garg]

Respected Sir,

Pls also explain the 2nd assumption of ANOCOVA here i.e. homogeneity of regression slopes.

How we can check it to proceeding further to use the ANOCOVA. 

Thanks and Regards

Radha 

On Friday, December 21, 2012 10:40:55 AM UTC+5:30, Dr Neeraj Kaushik wrote:

[Neeraj Kaushik]

Ooops!!! Forget to write about that.
Will write asap.