modelling proportions, with aggregated data, and the new/old lme4

Joerg,

What is the point in the Poisson model in that way (assuming
log(total) as offset)? This would be a binomial model; if you observe
N events from a Poisson distribution of which X are successes (or
marked in some way) with probability p, then X will be binomial(p, N).

Best regards,
Fredrik Nilsson

PS. Here's a practical "proof", the real is sketched below.
invlogit<-function(x) exp(x)/(1+exp(x))
n<-10
nn<-ceiling(exp(rnorm(n)+3))
pp<-seq(0.25,.75,length=n)
ca<-rbinom(rep(1,n),nn,prob=pp)
co<-nn-ca
fa<-gl(n,1)

#binomial model
test.glm<-glm(cbind(ca,co)~fa-1, family=binomial)
#conditional poisson
ptest.glm<-glm(ca~fa-1+offset(log(nn)), family=poisson)
summary(ptest.glm)
exp(coef(ptest.glm))
invlogit(coef(test.glm))

all.equal(exp(coef(ptest.glm)), invlogit(coef(test.glm)))

# proof.
X~Poisson(a) and ?Y~Poisson(b), X & Y independent -> X+Y~Poisson(a+b)
(e.g. by generating functions).

Prob(X=x| X+Y=n) = Prob(X=x, Y=n-x)/Prob(X+Y=n) =
Prob(X=x)*Prob(Y=n-x)/Prob(X+Y=n) = a^x/x! exp(a) b^(n-x)/(n-x)!
exp(b) /((a+b)^n/n! exp(a+b)) =
n!/x!/(n-x)! (a/(a+b))^x (b/(a+b))^(n-x) = n!/x!/(n-x)! (a/(a+b))^x
(1-a/(a+b))^(n-x) , i.e. binomial(x,n,p) with p=a/(a+b).


2012/3/19 Ben Bolker <bbolker at gmail.com>:

Joerg Luedicke <joerg.luedicke at ...> writes:

I would certainly check out a Poisson model with the number of
successes as outcome and successes+failures as an offset.

?That seems odd to me; the Poisson+offset model
should be appropriate when p<<1 (i.e. for very small p,
the Poisson variance mu=n*p is approximately the same as
the binomial variance n*p*(1-p); in this case p is not small.

?Of course, I could be wrong.

On Sun, Mar 18, 2012 at 11:51 AM, Rolf Turner <r.turner at ...> wrote:

On 19/03/12 07:10, Ben Bolker wrote:

<SNIP>

head(dat)
? successes failures id sex subdist
1 ? ? ? 560 ? ? ?726 ?1 ? F ? ? ? 4
2 ? ? ? 844 ? ? ?510 ?1 ? M ? ? ? 4

?[snip]

In community #1, which is in subdistrict #4, there are 560 women

? you mean 510, right?

<SNIP>

Sure looks like 560 to me. ?Time for a trek to the optometrist, Ben?

?Looked at the wrong line (failures in men rather than successes in women).