Chi-squared distributions in variance components models with many parameters:
Update from Chris Amos
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It was pointed out to me at about 5:10 pm on Monday, that in fact when
one has a complicated mixture of chi-squared variates and observes a
test statistic that the p-value is actually just the sum of the
p-values from each of the mixtures times the weights.  In my talk, I
was somewhat confused by the fact that in my own research I had been
trying to find critical values for a certain significance level and
that does require integrating over the mixtures, which is hard to do
analytically. However, if you are given a certain test statistic, one
can just sum up the weighted p-values.  Thus, if you have, say 1/4
chisquare0, 1/2 chisquare1 and 1/4 chisquare3, one just takes the
p-value to be 1/4*0 + 1/2(p-value from chisquare 1) + 1/4(p-value from
chisquare 3). That makes life very simple! The person who pointed this
out to me is Dale Nyholt.  It might be worth mentioning.  Also, Dale
has a nice editorial in Am J Hum Genetics on assessing significance
from linkage tests of sib pairs (and DZ twins) that is worth
referencing.

Nyholt, D:  (2000). All LODs are not created equal.  Am J Hum Genet,
67:282-288.