Beyond ANOVA (Part -2)
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Dr Neeraj Kaushik
Dec 19, 2012, 12:05:24 AM12/19/12
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Lets understand what exactly ANOVA is & how it works, coz this knowledge & understanding is crucial for elder of ANOVA like ANCOVA, MANOVA etc:
When u use ANOVA, there're several groups to be compared for significant difference as regard their mean value of dep var is concerned e.g. compare the test results of Math subject in 3 schools A, B & C.
Now first we're assuming no extraneous var here (This is the differentiating factor in ANOVA & ANCOVA).
Lets start by calculating the mean score of math subject in the 3 schools i.e. Average math score in School A, school B & School C. Obviously the 3 are not identical. So now question comes what are the reasons for the variation in these average marks?
The variations in marks (dep var) are coz of 2 reasons:
(a) variations due to chance factor. It may also be understood as within group variations {we know everything in nature is different, so its natural that the students of same level & calibre when taught by the same teacher may not get the same marks. We say the variations are coz of inherent natural difference}
(b) variations due to the indep var {treatment or whatever we call it. e.g. here we call it subjects}. This is called as between group variations. Now schools will like to emphasize that the marks are different coz of their teaching pedagogy.
Hence total variation is the sum of "between group variations' + 'within group variations'
Now if we want to determine whether the variations due to chance are small as compared to variations due to treatment then first we need to find average of both. This is done by dividing them by their respective degree of freedom (NOT by traditional n). This converts them into mean scores. If we calculate the ratio of these 2 mean scores, it called as F-test.
F-test is a ratio of Mean Square (treatment) & Mean square (chance).
The tables values of F (which I call as max permissible difference or max permissible difference ratio here) denotes the max value of F-ratio which comes under natural circumstances.
Ex If table value of F=2 then it means if Mean Square (treatment) is twice that of Mean square (chance), it is ok. Results upto here are expected coz of chance factor. But when this calculated F-ratio is larger than table value then it means Mean Square (treatment) is much bigger than Mean square (chance), hence we're compelled to believe that the treatment is effective or in other words lets put it like this: At least one of the indep var is significantly different from the rest.
So decision rule:
When cal value of F < Table value of F ==> Accept Null hypothesis that there's no significant difference between the various indep var.
But in computer age
we use when Sig value > 0.05 ==>Accept Null hypothesis that there's no significant difference between the various indep var.
When u use ANOVA, there're several groups to be compared for significant difference as regard their mean value of dep var is concerned e.g. compare the test results of Math subject in 3 schools A, B & C.
Now first we're assuming no extraneous var here (This is the differentiating factor in ANOVA & ANCOVA).
Lets start by calculating the mean score of math subject in the 3 schools i.e. Average math score in School A, school B & School C. Obviously the 3 are not identical. So now question comes what are the reasons for the variation in these average marks?
The variations in marks (dep var) are coz of 2 reasons:
(a) variations due to chance factor. It may also be understood as within group variations {we know everything in nature is different, so its natural that the students of same level & calibre when taught by the same teacher may not get the same marks. We say the variations are coz of inherent natural difference}
(b) variations due to the indep var {treatment or whatever we call it. e.g. here we call it subjects}. This is called as between group variations. Now schools will like to emphasize that the marks are different coz of their teaching pedagogy.
Hence total variation is the sum of "between group variations' + 'within group variations'
Now if we want to determine whether the variations due to chance are small as compared to variations due to treatment then first we need to find average of both. This is done by dividing them by their respective degree of freedom (NOT by traditional n). This converts them into mean scores. If we calculate the ratio of these 2 mean scores, it called as F-test.
F-test is a ratio of Mean Square (treatment) & Mean square (chance).
The tables values of F (which I call as max permissible difference or max permissible difference ratio here) denotes the max value of F-ratio which comes under natural circumstances.
Ex If table value of F=2 then it means if Mean Square (treatment) is twice that of Mean square (chance), it is ok. Results upto here are expected coz of chance factor. But when this calculated F-ratio is larger than table value then it means Mean Square (treatment) is much bigger than Mean square (chance), hence we're compelled to believe that the treatment is effective or in other words lets put it like this: At least one of the indep var is significantly different from the rest.
So decision rule:
When cal value of F < Table value of F ==> Accept Null hypothesis that there's no significant difference between the various indep var.
But in computer age
we use when Sig value > 0.05 ==>Accept Null hypothesis that there's no significant difference between the various indep var.
Rekha Mishra
Dec 19, 2012, 3:06:21 AM12/19/12
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Thanks Sir.
Radha garg
Dec 19, 2012, 12:22:05 PM12/19/12
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Can you please explain the same example with Mathematical calculation like factor score example.
Though i understand the theoretical concept of ANOVA with this post, but i could't understand that how this three groups are divided into the two group of within group variation and between group variation. May be the practical example of it make me more clear.
Thanks and Regards
Radha
On Wednesday, December 19, 2012 10:35:24 AM UTC+5:30, Dr Neeraj Kaushik wrote:
Preeti Jain
Dec 19, 2012, 2:09:00 PM12/19/12
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Three groups are not divided in to 2 groups. They are by using following formulas
With in= Sigma(X1-mean of X1)2+sigma(X2 -mean of X2)2+sigma(X3-mean of X3)2
N-K
Between=
N1(mean of X1-grand mean)2+N2(mean of X2-grand mean)2+N3(mean of X3-grand mean)
K-1
Preeti Jain
Radha garg
Dec 20, 2012, 2:37:13 AM12/20/12
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Regards
Radha
Neeraj Kaushik
Dec 20, 2012, 4:09:09 AM12/20/12
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Hi Radha
Preeti has explained it well. I'll take examples in next posts.
Best wishes
Neeraj
On 12/20/12, Radha garg <gargr...@gmail.com> wrote:
>
> Thanks Preeti.........for clarification. Now, the things are clear to me.
>
> Regards
> Radha
>
>
> On Thursday, December 20, 2012 12:39:00 AM UTC+5:30, preeti jain wrote:
>>
>>
>> Three groups are not divided in to 2 groups. They are by using
>> following formulas
>>
>>
>> With in= *Sigma(X1-mean of X1)2+sigma(X2 -mean of
>> X2)2+sigma(X3-mean
>> of X3)2*
>>
>> N-K
>>
>> Between=
>>
>> * N**1(mean of X1-gr**and mean)2+N2(mean of
>> X2-grand
>> mean)2+N3(mean of X3-grand mean)*
>> <javascript:>>wrote:
Preeti has explained it well. I'll take examples in next posts.
Best wishes
Neeraj
On 12/20/12, Radha garg <gargr...@gmail.com> wrote:
>
> Thanks Preeti.........for clarification. Now, the things are clear to me.
>
> Regards
> Radha
>
>
> On Thursday, December 20, 2012 12:39:00 AM UTC+5:30, preeti jain wrote:
>>
>>
>> Three groups are not divided in to 2 groups. They are by using
>> following formulas
>>
>>
>> X2)2+sigma(X3-mean
>> of X3)2*
>>
>> N-K
>>
>> Between=
>>
>> * N**1(mean of X1-gr**and mean)2+N2(mean of
>> X2-grand
>> mean)2+N3(mean of X3-grand mean)*
>> <javascript:>>wrote:
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