Re: Probit analysis
- From: "Peter / Labo" <peterpc5j@xxxxxxxxxxx>
- Date: Wed, 17 Sep 2008 11:50:10 -0700 (PDT)
In a log-likelihood ratio (Wilks) test, you would fit the probit model
for each subsample, calculate the maximal log-likelihood, add them and
multiply by minus two to get the deviance. You would fit the model
with exactly the same structure but with parameters equal for both
samples, that is, join the samples and do the fit as if you did for
the individual samples. This will also give you a log-likelihood and
deviance, the latter of which should be larger than the deviance when
you had double this much parameters to estimate. Now calculate the
difference in deviances, which will be your test statistic, to be
compared with a chi-square distribution. The exact shape of the chi-sq
is determined by degrees of freedom, which is, in this case, the
number of parameters that you forced to be equal across samples in
your null model. For example, if you have a quadratic probit model,
you have z = b0+b1*X+b2*X^2 ; in the model where samples are treated
separately, you actually estimate 6 parameters: b0, b1 and b2 for each
sample. In the model where you have joined samples into one
supersample, there's only one set of b0, b1 and b2, 3 parameters in
total. You would therefore look at the quantiles of the chi-square
distribution with 6-3 = 3 degrees of freedom. The p-value of the
comparison would be the area of the chi-square distribution "above"
the test statistic (sometimes written as G²), in other words, 1 minus
the cumulative distribution function. There are tables and software to
determine this.
There are other ways to explain this (with design matrices) and
variations, such as looking at the possibility that, as an example,
b0's are equal across samples but the other b's are not. It is also
good to realize that there are some conditions to be met: the sample
size must be sufficiently large compared to the number of parameters,
high prevalence of zero- frequencies (i.e., all "successes" or all
"failures" in a certain condition) affects the test statistic
distribution, and, probably not your case but good to remember, the
test doesn't work for tests of null hypothesis in which parameter
values are constrained to the boundary of the parameter space. Other
techniques (e.g. bootstrapping) are available when an assumption feels
shaky.
.
- References:
- Re: Probit analysis
- From: ark561
- Re: Probit analysis
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