out." In principle, this calls for constructing "residualized" matrices corresponding to ST and SB after the parameter vector excluding the regression coefficients has been transformed as in Section 4. This means that wherever Y appears in the formulas for ST and SB (either explicitly or through Ψ), it has to be replaced by Y - Ŷ + ̂, the vector of residuals of Y from its regression on X. In practice, however, the process of residualizing the actual criterion scores is by-passed, and the adjustments are made directly on the sums of squares.
Equivalence of t tests for Significance of the Difference Between Two Means and for Significance of Regression Weight of Criterion on Dummy Variate Indicating Group Membership
We have seen, in Equation 4, that the regression weight in question is given by
b = Ȳ1 - Ȳ2, (1)
where Ȳ1 is the criterion mean of the group for which X = 1, and Ȳ2 is that of the group with X = 0.
The t statistic for testing the significance of b is easily derivable from the more
familiar t statistic for the significance of the correlation coefficient r, namely,
We need only express r in terms of b by the relation
r = bsx/sy
which, substituted in the expression for t, yields
The variance(in the total sample) for the dummy variace X is