By Daniel A. Murray

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**Example text**

Thus, we assume that under Model I, the eij's are completely independent and are normally distributed with mean zero and variance Furthermore, under Model II, the ai's are also completely independent of each other and of the eij's, and are normally distributed with mean zero and variance Now, first of all, we note that for a normal parent population with variance a 2 a;. a;. 1) where (j2 is an unbiased estimator of a 2 based on v degrees of freedom (see Appendix C). 10) that E(MS w) = a;. 3) that is, the ratio ofMS w to CJe2 is a x2[a(n - 1)] variable divided by a(n - 1).

6) i=] hence it has a-I degrees of freedom. Finally, since the within sum of squares (SSw) involves the deviations of the an observations from the a sample means, it has an - a = a(n - 1) degrees 2 For a geometric interpretation of degrees of freedom, see Walker (1940). 17 One-Way Classification of freedom. Alternatively, the degrees of freedom corresponding to the within groups can be argued as follows. Consider the component of the within sum of squares corresponding to the i-th factor level, namely, n L(Yij - yd.

For example, suppose there are Q such factors. Then the total sum of squares (SST) is partitioned as where SSA, SSB, ... , and SSQ represent the sums of squares associated with factors A, B, ... , and Q, respectively and which account in some sense for the variation that can be attributed to these factors or sources of variation. The experiment is so designed that each sum of squares reflects the effect of just one factor or that of the random error attributed to chance. Furthermore, all such sums of squares are made comparable by dividing each by an appropriate number known as the degrees of freedom.!