Relationship between two views of anova this section

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Unformatted text preview: freedom disappears for each group mean. In the end, we have Σni – k degrees of freedom. Another way to see this is to notice that SSresidual comes from calculating the sum of squares for each group separately, Σ(Xi – Mi)2, and then adding up over all the groups. The sum of squares for each group is closely related to the variance of the scores in that group; all we’d have to do to get the variance is to divide by the degrees of freedom, ni – 1. Therefore the sum of squares for each group has ni – 1 degrees of freedom. When we sum over all the groups, the nis get added up, and we subtract 1 k times. df residual = " n i # k If you compare the formulas for the three dfs, you’ll see that they add up: dftotal = dftreatment + dfresidual. This fits with the fact that the sums of squares add up (Eq. 1). SStotal is the sum of Σni – ! squares; k – 1 of these squares are in SStreatment and the remaining Σni – k squares are 1 in SSresidual. Relationship between MSresidual and MS for t ­tests. It might help to understand how MSresidual extends the idea of the mean square used in independent ­samples t ­tests. In both cases, we’re using the variability within the groups to estimate the population variance. In fact, when there are only two groups, ANOVA is the same as an independent ­samples t ­test, and MSresidual and MS are exactly the same. Here’s a reminder of the formula for MS for an independent ­samples t ­test: 2 MS = #( X A " M A) + #( X B " M B) df 2 Compare the numerator of this formula to Equation 3 for SSresidual (focus on the top line of Eq. 3). If there are 2 groups, meaning k = 2, then SSresidual and the numerator for MS are ! exactly the same (we call the groups 1 & 2 in one case and A & B in the other, but that’s just notation). The degrees of freedom in the two cases are also the same. For ANOVA, dfresidual equals Σni – k, which is n1 + n2 – 2 when there are two groups. This is the same as df for the independent ­samples t ­test (again, replacing 1 & 2 with A & B). Since MSresidual is defined as S...
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