Edna Huelsenbeck wrote:
>> Calculate the maximum likelihood under each hypothesis and denote
> the likelihoods L0 and L1 for the null and alternative hypotheses,
> respectively. The null hypothesis can be tested using a likelihood
> ratio test. The test statistic is -2 (lnL0 - lnL1). The null hypothesis
> is a special case of the alternative, and it is tempting to think that
> -2 (lnL0 - lnL1) follows a chi-square distribution with three degrees
> of freedom. I suspect that the chi-square approximation would be
> quite good, although technically the null hypothesis is at the
> edge of the parameter space.
When the hypotheses differ by one parameter, with the null hypothesis at
the edge (boundary) of the parameter space, the "naive" chi-squared
distribution with 1 d.f. is reasonably close to the "correct"
(asymptotic) "chi-squared-bar" distribution. At least, things like the
useful %-points of the distribution are not far off. When there are 2
or more parameters (the example which initiated this thread has 2) which
are on the parameter space boundary in the null hypothesis, things get
considerably more complicated and I think the difference between the
naive chi-squared distribution and the correct distributions could be
more important.
Some of this is discussed, for analogous parameters, in papers by Ota,
Waddell, Hasegawa, Shimodaira and Kishino (2000: Mol. Biol. Evol.
17(5):798--803) and by Goldman and Whelan (2000: Mol. Biol. Evol.
17(6):sorry, don't have the page numbers to hand).
> If you are concerned about the
> quality of the chi-square approximation, you can simulate the
> null distribution of the test statistic (i.e., using parametric
> bootstrapping).
Absolutely.
Nick Goldman
-----------------------------------------------------------------------
Nick Goldman, Dept of Zoology, tel: +44-(0)1223-336649
Downing St, Cambridge CB2 3EJ, U.K. fax: +44-(0)1223-336679
N.Goldman at zoo.cam.ac.ukhttp://www.zoo.cam.ac.uk/zoostaff/goldman
---