4 For a one way ANOVA parameter estimates for the cell means

4)         For a one way ANOVA, parameter estimates for the cell means model are Y1=15.42, Y2=18.52, Y3=15.00, Y4=9.74, Y5=13.34.   Estimate the factor effects model parameters using the sample grand mean assuming same number of observations for each level. Very zero-sum constraint.

Solution

let A be the factor with k levels A1,A2,A3,.....,Ak.

as in the question there are only 5 parameter estimates for the cell means model are given. so k=5

that is the factor A is divided into 5 levels.

let the ANOVA model be

Yij=u+ai +eij i=1,2,...,5       j=1,2,......,n

where Yij denote the jth value of the ith cell of the observations Y

u denote the mean effect due to factor A

ai denote the additional effect due to the ith level of factor A.

eij denote the error associated with (i,j)th value of Y

n=total number of observations in each level.

we have the the estimates of cell means as Y1,Y2,Y3,Y4,Y5

so the estimate of grand mean is Ybar=(Y1+Y2+Y3+Y4+Y5)/5   [since same number of observations for each level]

                                                       =(15.42+18.52+15+9.74+13.34)/5=14.404

we have the factor effects a1,a2,a,3,a4,a5.

so the estimates are

a1hat=additonal effect due to level A1=Y1-Ybar=15.42-14.404=1.016

a2hat=additonal effect due to level A2=Y2-Ybar=18.52-14.404=4.116

a3hat=additonal effect due to level A3=Y3-Ybar=15-14.404=0.596

a4hat=additonal effect due to level A4=Y4-Ybar=9.74-14.404=-4.664

a5hat=additonal effect due to level A5=Y5-Ybar=13.34-14.404=-1.064