Re: Chi-square for binomial samples

Michael McLaughlin <mmclaughlin1@xxxxxxx> wrote in
Xuofg.24929$ZW3.3160@dukeread04:">news:Xuofg.24929$ZW3.3160@dukeread04:

Scenario:

N experimenters take samples from a common population, effectively
infinite, BUT these samples are of various sizes. They each look for
a given (fixed) attribute of interest and record their frequency of
success.

Were their samples all of the same size, then every statistics book
ever written would discuss how to compute the associated chi-square
statistic.

Most chi-square statistics that I have seen (and there are many) assume
variable numbers in each stratum. Can you give us some examples?

Moreover, computing a maximum-likelihood value for p is
still straightforward since every term in the log-likelihood is
well-defined.

Question:

Is there an accepted method to compute chi-square under the conditions
stated -- where p is assumed known but sample size is variable? Is
this even a sensible question?

The chi-square is just the sum of squared deviations from the expected.
Usual basis for expected is row-sum x column-sum/total. Are you proposing
to substitute a known p (times the row total presumably) for that number?
If so, you may want to investigate GLMs with offsets.

The analogous question could be asked wrt the beta-binomial
distribution as well. There, the scenario described is, in fact, the
norm. The question could be asked again, a fortiori, wrt the
hypergeometric distribution (two parameters nominally fixed).


It sounds as though you are asking for a test of equal proportions among
k multiple samples. Results (successes, failures and associated
marginals) would conventionally be displayed in a 2 column x k row table
with row and column sums. The X-squared statistic would be compared to
chi-squared distribution with (k-1) degrees of freedom. The row totals do
not need to be equal. The usual conditon for validity is than the number
of cells with counts below 5 is fewer than 20%.

Your added information about "p being known" may be a ringer. What sort
of global hypothesis are you thinking of testing when the proportion of
successes is already given?

--
David Winsemius
.

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