Decomposing the Variance and Its Meaning

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Chapter: Biostatistics for the Health Sciences: One-Way Analysis of Variance

Biostatistics : One-Way Analysis of Variance : 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(Xij – )2 = Σ(Xij Xi)2 + Σ(Xi  )2, where the following holds:

Xij is normally distributed with mean μi

The variance is σ2

Xij is the jth observation from the ith group

Xi. is the average of all observations in the ith group

X is the average over all the observations in all groups

Let Q, Q1, and Q2 refer to total sum of squares, within-groups sum of squares, and between-groups sum of squares, respectively. We have that Q = Q1 + Q2 Q = S(Xij  )2 normalized has a chi-square distribution with nw + nb – 1 degrees of freedom; Q1 = Σ(XijXi.)2 has a chi-square distribution with nw degrees of free-dom; and Q2 = Σ(Xi. – )2 has a chi-square distribution with nb – 1 degrees of freedom. Q2 is independent of Σ(XijXi.)2. The symbol nb is the number of groups and nw is the number of degrees of freedom for error. The total sample size equals nnb. For Q2 to have a chi-square distribution when appropriately nor-malized, we need the null hypothesis that all μi are equal to be true. The F distri bution is obtained by taking [Q2/(nb – 1)]/[Q1/nw]. When the alternative holds, the normalized Q2 has what is called a noncentral chi-square distribution, and the ra-tio tends to be centered above 1. The distribution of [Q2/(nb – 1)]/[Q1/nw] is then called a noncentral F distribution.

The mathematical relationship between this F statistic and the sample multiple correlation coefficient R2 (discussed in Chapter 12) is as follows: R2 = (nb – 1)F/ {(nb – 1)F + nw} or F = {R2/(nb – 1)}/{(1 – R2)/nw}

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