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Hello:
Could you possibly help me with the deviance formula for the binomial model.
Suppose y_i ~ Bi(n_i, p_i), i from 1 to N.
The deviance formula is well known, i.e. in Dobson on p 77:
D = 2 * sum(i from 1 to N) of two terms:
1) y_i * log(y_i / fitted y_i)
2) A similar term for non-occurences, i.e. (n_i - y_i).
I am interested in computing the deviance for the null model, when the fitted value of p_i does not depend on i. Denote it p_0. In that case, the first term under the sum becomes
y_i * log(p_i / p_0)
where p_i is the fitted probability for the saturated model, p_i = y_i / n_i;
The corresponding test is callled Likelihood Ratio in, this SAS output (statistic = 71.05):
http://www.ats.ucla.edu/stat/sas/output/sas_logit_output.htm Now here's the problem: I came across a white paper where D is computed based only on the first term (it doesn't derive the result). The second term that describes the non-occurrences is not there. I checked manually vs both SAS and R and the formula works. So I am wondering where the second term has gone. I tried to derive it myself, but the second term doesn't cancel out.
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Regards,
Nik
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