[R] inverse prediction and Poisson regression

[Vincent Philion]

Hello to all, I'm a biologist trying to tackle a "fish" (Poisson Regression) which is just too big for my modest understanding of stats!!!

Here goes...

I want to find good literature or proper mathematical procedure to calculate a confidence interval for an inverse prediction of a Poisson regression using R. 

I'm currently trying to analyse a "dose-response" experiment. 

I want to calculate the dose (X) for 50% inhibition of a biological response (Y). My "response" is a "count" data that fits a Poisson distribution perfectly. 

I could make my life easy and calculate: "dose response/control response" = % of total response... and then use logistic regression, but somehow, that doesn't sound right.
 
Should I just stick to logistic regression and go on with my life? Can I be cured of this paranoia?
;-)

I thought a Poisson regression would be more appropriate, but I don't know how to "properly" calculate the dose equivalent to 50% inhibition. i/e confidence intervals, etc on the "X" = dose. Basically an "inverse" prediction problem.

By the way, my data is "graphically" linear for Log(Y) = log(X) where Y is counts and X is dose.

I use a Poisson regression to fit my dose-response experiment by EXCLUDING the response for dose = 0, because of log(0)

Under "R" = 

> 	glm.dose <- glm(response[-1] ~ log(dose[-1]),family=poisson())

(that's why you see the "dose[-1]" term. The "first" dose in the dose vector is 0. 

This is really a nice fit. I can obtain a nice slope (B) and intercept (A):

log(Y) = B log(x) + A

I do have a biological value for dose = 0 from my "control". i/e Ymax = some number with a Poisson error again

So, what I want is EC50x :

Y/Ymax = 0.5 = exp(B log(EC50x) + A) / Ymax

exp((log(0.5) + Log(Ymax)) - A)/B) = EC50x

That's all fine, except I don't have a clue on how to calculate the confidence intervals of EC50x or even if I can model this inverse prediction with a Poisson regression. In OLS linear regression, fitting X based on Y is not a good idea because of the way OLS calculates the slope and intercept. Is the same problem found in GLM/Poisson regression? Moreover, I also have a Poisson error on Ymax that I would have to consider, right?

Help!!!!


-- 
Vincent Philion, M.Sc. agr.
Phytopathologiste
Institut de Recherche et de Développement en Agroenvironnement (IRDA)
3300 Sicotte, St-Hyacinthe
Québec
J2S 7B8

téléphone: 450-778-6522 poste 233
courriel: vincent.philion at irda.qc.ca
Site internet : www.irda.qc.ca