In the ASREML manual, a method is given of testing the significance of a
single variance component by calculating the difference between the residual
log likelihoods from models with and without the component, then treating
this as a chi-square statistic with 1 d.f. but multiplying the P-value by
0.5 (approximation of Stram and Lee, 1994).
Can this method be extended to compare models that differ by >1 variance
Here is an attempt to do so.
The philosophy is that the chi-square statistic with 1 d.f. is the square of
an underlying variable, on which we are conducting a 1-tailed test (just as
F(1,nu2) is the square of t(nu2)), because only values at the right-hand end
of the underlying variable would cause us to reject H0. Now, chi-square
with 2 d.f. is similarly an integral of a bivariate distribution, in which
only values in the upper right-hand quadrant would cause us to reject H0.
And so on for higher numbers of d.f. Thus when p variance components are
being added to the model, the difference in residual log likelihoods should
be treated as a chi-square variable with p d.f., but the probability value
obtained should be multiplied by 0.5 ** p.
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