Self and Liang (1987) and Stram and Lee (1994; see 1995 for erratum)
discuss many cases.
The general case of Q versus Q+1 random effects is a 50:50 mixture of
Chi-squared with Q+1 df and a Chi-squared with Q df. The multiplication
by 0.5 is the case when Q=0 i.e. 50:50 mixture of Chi-squared with 0
df (point mass zero) and Chi-squared with 1 df.
For Q versus Q+k, apart from special cases, you probably will not know the
correct null distribution.
Hope this helps,
On Fri, 8 Dec 2000, N.W. Galwey wrote:
> 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.
> Comments, please.
> Nick Galwey
> N.W. Galwey,
> Faculty of Agriculture,
> University of Western Australia,
> 35 Stirling Highway, Crawley.
> Western Australia 6009
> Tel.: +61 8 9380 1959 (direct line)
> +61 8 9380 2554 (switchboard)
> Fax: +61 8 9380 1108
> Asreml mailinglist archive: http://www.chiswick.anprod.csiro.au/lists/asreml
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