Lemma 1: Independence of Location-only and Scale-only Test Statistics under the Null Hypothesis for Normally Distributed Traits
Let TLocation=βˆ1/(S/Sxx) be our location-only test statistic, testing the linear effect of x on Y in a sample of size n, where S2=(1/(n−2))∑(yi−βˆ0−βˆ1xi)2, βˆ0=y¯−βˆ1x¯, βˆ1=Sxy/Sxx, Sxy=∑(xi−x¯)(yi−y¯), and Sxx=∑(xi−x¯)2 (x = G in Equation 2), and let TScale be our scale-only test statistic, here defined as Levene’s test statistic for equality of variances.14
Lemma 1: For the conditional normal model Yi∼N(β0+β1xi,σxi2), where xi = 0,1 or 2, TLocation and TScale are independent if σ02=σ12=σ22.
Proof: For fixed x, Y is normally distributed with constant variance σ2 and mean E[Y|x]=β0+β1x. The density of Y is(2πσ2)−n/2exp[−12σ2∑(yi−β0−β1xi)2].
This is an exponential family with three parameters θ=(θ1,θ2,θ3)=(β1/σ2,−1/2σ2,β0/σ2) for which the sufficient statistics T=(T1,T2,T3)=(∑xiyi,∑yi2,∑yi) are complete. If σ02=σ12=σ22, TScale is approximately distributed as a F3−1,n−3 variable, and it does not depend on θ (i.e., TScale is ancillary for θ). Thus, TScale is independent of T (see page 152 in Lehmann and Romano40). Because TLocation is a function of T, TLocation and TScale are therefore independent under the null.
Note that the proof of independence holds regardless of the version of Levene’s test statistic chosen, provided that the approximation to the F distribution (or some other distribution not depending on θ) is justifiable. Similar statements of independence with analogous proofs can be obtained for other choices of location test statistics such as the analysis of variance (ANOVA) F-statistic.