regression coefficients
Why is s assumed known and common to the k groups? I doubt if that is what was meant (although it was too imprecise to be at all sure). If `common' is a viable assumption, you can just fit a model with by-group regressions vs one with a common regression (which seems to be what you are testing) and use anova(). If not, the case k=2 encompasses the Welch t-test so exact distribution theory is not going to be possible, but by fitting a common model and three separate models and summing the -2log-lik for the latter you can easily get the LT test and refer it to its `standard' (asymptotic) Chi-squared distribution.
On Tue, 20 May 2003, Spencer Graves wrote:
I don't know of a simply function to do what you want, but I can give you part of the standard log(likelihood ratio) theory: Suppose b[i]|s ~ N.r(b, s^2*W[i]), i = 1, ..., k. Then the log(likelihood) is a sum of k terms of the following form: l[i] = (-0.5)*(r*log(2*pi*s^2)+log|W[i]| +(s^-2)*t(b[i]-b)%*%solve(W[i]%*%(b[i]-b) By differentiating with respect to b and setting to 0, we get the maximum likelihood estimate for b as follows: b.hat = solve(sum(solve(W[i]))%*%sum(solve(W[i])%*%b[i]) In words: b.hat = weighted average with weights inversely proportional to the variances. Then log(likelihood ratio) is as follows: log.LR = sum((s^-2)*t(b[i]-b.hat)%*%solve(W[i])%*%(b[i]-b.hat)) This problem should be in most good books on multivariate analysis. I would guess that log.LR probably has an F distribution with numerator degrees of freedom = r*(k-1) and with denominator degrees of freedom = degrees of freedom in the estimate of s. However, I don't remember for sure. It's vaguely possible that this is an "unsolved" problem. In the latter case, you should have all the pieces here to generate a Monte Carlo.
You have assumed s is known, in which case it is a Chi-squared distribution. If s is unknown, you need to maximize over it to get an LR test (separately under the null and the alternative).
lamack lamack wrote:
dear all, How can I compare regression coefficients across three (or more) groups?
Brian D. Ripley, ripley at stats.ox.ac.uk Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/ University of Oxford, Tel: +44 1865 272861 (self) 1 South Parks Road, +44 1865 272866 (PA) Oxford OX1 3TG, UK Fax: +44 1865 272595