Biostatistics : One-Way Analysis of Variance : Decomposing the Variance and Its Meaning

**DECOMPOSING THE VARIANCE AND ITS MEANING**

Cochran’s theorem is the basis for the sums of
squares having independent chisquare distributions when Equation 13.1 holds
[see Rao (1997), page 4]. It can be deduced from Cochran’s theorem in the case
of the one-way ANOVA that Σ(*X _{ij}* – )

*X _{ij} *is normally distributed with mean

The variance is *σ*^{2}

*X _{ij} *is the

*X _{i}*. is the average of all
observations in the

*X *is the average over all the observations in all
groups

Let *Q*, *Q*_{1}, and *Q*_{2} refer to total sum of squares, within-groups sum of
squares, and between-groups sum of squares, respectively. We have that *Q* = *Q*_{1}
+ *Q*_{2} *Q* = S(*X _{ij}* – )

The mathematical relationship between this *F* statistic and the sample multiple
correlation coefficient *R*^{2}
(discussed in Chapter 12) is as follows: *R*^{2}
= (*n _{b}* – 1)

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