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Suppose I have fitted a Poisson GLM to model rates as follows: 

> fit.1=glm(response~X1+X2+ offset(log(population)),family=poisson,data=...)

I can get the estimated rates by using two new values for X1 and X2 and population=1 as follows:

>new.data=data.frame(X1=new.X1,X2=new.X2,population=1)
>estimated.rates=predict(fit.1,newdata=new.data,type="response")

Following Subsection 13.4.5 in Introduction to Linear Regression Analysis, 5th Edition, I then double check the Deviance Residuals using qqnorm and found that they are not even approximately normally distributed. So to revise my model, I used a box and cox transformation (lambda) to improve the model and fitted a new quasi-Poisson model:

> fit.2=glm(I(response^lambda)~X1+X2+ offset(log(population)),family=quasipoisson,data=...)

I can again estimate the rates similarly as before: 

> estimated.rates.2=predict(fit.2,newdata=new.data,type="response")

The estimated.rates.2 is on the scale of the transformed response (using the box and cox transformation).

Is it possible to back transform the rates estimated.rates.2 into the original scale (i.e. similar to estimated.rates)?

Stat
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    Why do you need the residuals to be normal? – Glen_b Apr 06 '14 at 10:15
  • We need to check the it and correct it if they are not. See e.g. the reference I gave [here](http://stats.stackexchange.com/questions/92394/checking-residuals-for-normality-in-generalised-linear-models/92413#92413). – Stat Apr 06 '14 at 14:27
  • The quote you give is not a reason to transform. It's an observation that is sometimes but not always true. – Glen_b Apr 06 '14 at 16:45
  • I don't understand it. Can you explain more? – Stat Apr 06 '14 at 16:49
  • I checked the residuals and found that they are not normal. So I used Box cox transformation. BTW, this is not my question, my question is something else. – Stat Apr 06 '14 at 16:52
  • Your question is based on a misapprehension. There's no need for them to be normal. – Glen_b Apr 06 '14 at 17:18
  • That's not correct Glen. This is why I gave a reference. In that book, they have fitted a poisson GLM (together with other glm models) and performed a qqplot to double check the normality. Like any other lm, we need to check it as well. – Stat Apr 06 '14 at 17:23
  • A Poisson GLM assumes that data is conditionally Poisson, not normal. Why would anyone be surprised to discover non-normality for something assumed non-normal? You said (of checking normality) "We need to check the it and correct it if they are not." -- I'm sorry, but as a general statement that's simply false. Residuals (any of the kinds in common use) in GLMs can be plainly non-normal when nothing whatever is wrong with the model. I repeat what I said: there's no need for them to be normal. It can be useful to look at the plot, but non-normality is not *of itself* an indication of a problem. – Glen_b Oct 01 '14 at 04:14

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