ABSTRACT
Up to now the problem at hand has been to determine whether the posttest means were equal among treatment groups after taking into account the dependency on pretest scores. For many of the statistical analyses presented, it was assumed that the variance of the pretest measurements was equal to the variance of the posttest measurements. Only with one of the simulations in the previous chapter did we test to see how the power of the statistical tests presented fared when the variances were unequal. Sometimes the assumption of equality of variance may be in doubt and it becomes necessary to test this assumption directly. The null and alternative hypotheses may be written as
H :
H :
σ = σ
σ ≠ σ
or equivalently
H : 0
H : 0
σ −σ =
σ −σ ≠
where X and Y refer to the pretest and posttest measurements, respectively. Pittman (1939) and Morgan (1939) first tackled this problem more than 50 years ago. They showed that if E(X) = µX, E(Y) = µY, Var(X) = 2Xσ , Var(Y)= 2Yσ , and the correlation between X and Y was equal to ρ, then the covariance between the difference scores,
i i iD Y X= − (10.1) and sum scores,
i i iS Y X= + (10.2) can be written as
( ) ( ) ( ) ( ) ( ) ( )
Cov D,S Cov Y X,X Y Var X Cov X,Y Cov X,Y Var Y
= − +
= + − −
= σ −σ
(10.3)
Notice that the right side of Eq. (10.3) is equal to the form of the null hypothesis to be tested. If we can find a test statistic which tests whether the
covariance of the difference and sum scores equals 0, this would be equivalent to a test of whether the variances of X and Y were equal. Recall that the correlation between two variables is the ratio of their covariance to the product of their individual standard deviations. Then the correlation between the difference and sum scores, ρsd, is
cov(D,S) σ −σρ = = σ σ σ σ
, (10.4)
where σd and σs are the standard deviations of the difference and sum scores, respectively. Eq. (10.4) is also similar to the null hypothesis to be tested with the exception of the presence of a denominator term. The denominator term maps the domain of the covariance from the interval ( ),−∞ ∞ to the interval (-1, 1). Thus a test of Ho: σX = σY is equivalent to a test for 0 correlation between the sum and difference scores.
