Question

Why is the Mean Square due to Error a better estimate of the population variance than the Mean Square due to Treatment? When is the Mean Square due to Treatment also a good estimate for the population variance? Why?

Answer 1

Here we will approach the question by assuming a one way fixed effects model.

Here we will show that the MSE is always the unbiased estimator for population vatiance whereas the MS due to treatment is unbiased under special condition.

Suppose there are n observations classified into k classes, the number of observations in the ith class being ni. Let yij be the jth observation in the ith level.

Here our linear model is : yij=i+eij, i=1,2,...,k;j=1,2,...,ni,

where i :fixed effect due to the ith level of the treatment,i=1,2,....,k

eij:errors which are assumed to be normally distributed with means zero and common variance 2.

Let =(nii )/ni :general effect;

and i=i-:additional effect of the ith level over the general mean effect.

Then the model can be written as yij=+i+eij

nii=nii - ni =0

The least square estimators of i=yi0-y00, i=1,2,...k; =y00;

yi0:mean of the ith class=+i +ei0;

y00:grand mean of all obseravations.=+ e00

Total SS=SST+SSE

where Total SS= (yij-y00)2

SST= ni(yi0-y00)2

SSE= (yij-yi0)2

= (eij-eio)2 = eij2-   niei02

E(SSE)= E(eij2) - E( niei02)

=n 2 -ni( 2/ni) =n2  - k 2 =(n-k) 2

E(MSE)= 2  

Now SST= ni(yi0-y00)2= ni(i + ei0 - e00)2

E(SST)= nii2 + E[ ni(ei0-e00)2 ]

= nii2 + E[niei02-ne002]

= nii2  + [ ni 2/ni-n 2/n]

= nii2 +(k-1)2

E(MSA)= 2 +( nii2 )/(k-1)

So when the null hypothesis Ho=1=..........=k =0 is true only then MSA is the unbiased estimator of  2.. Otherwise it is greater than 2, unlike MSE which is always the unbiased estimator of 2.

Thus the Mean Square due to Error a better estimate of the population variance than the Mean Square due to Treatment. And the Mean square due to treatment is a good estimator of population variance only when the null hypothesis is true.