Offset vs fixed factor in a mixed poisson model

Message du 18/01/13 ? 21h29
De : "Highland Statistics Ltd" 
A : v_coudrain at voila.fr
Copie ? : r-sig-mixed-models at r-project.org
Objet : Re: Offset vs fixed factor in a mixed poisson model

On 18/01/2013 16:09, v_coudrain at voila.fr wrote:

Dear Alain,

Thank you for your reply. I tried to understand what you said, but have some difficulties:

If you use a covariate as an offset then you essentially saying: double
the value of the variable used for the offset, double the numbers
(strictly speaking: the expected value).

What do you mean wirh "double the value"? Does it mean that if the value of the offset double, then the expected value of my response variable should 

And if I have offset(logx), then doubling the log of my variable will double the estimate of the response variable?

Valerie,
Yes...indeed that is what the offset is doing. Double the value of the 
x....you assume that the expected value of your response also doubles. 
Just write out the equation for a Poisson and you will see:

Y_i ~ Poisson(mu_i)
E(Y_i) = mu_i
mu_i = exp(alpha + beta * z + 1 * log(x))
= x* exp(alpha + beta * z)

Double x....double mu


Keep in mind that when you analyse a ratio you implicitly do the same; 
1/2 = 100/ 200 = 0.5

Quite often sampling effort is used as an offset as it is not really interesting to model a
cause-effect relationship between sampling effort and your response.

Indeed I don't directly have different sampling effort, but I am testing species richness in 3 years in a growing population, such that the abundance of 

strongly increased between the year. The situation is quite similar as if we had increased the sampling effort over the years.

If you have a model with:
glm(y ~ x, family = poisson)
glm(y ~ x + offset(z), family = poisson)
and x is significant in the first model...but not in the second, then
either the offset explains most variation, or x and the offset are
highly correlated? Plot x versus z...and plot x versus log(z)...

x and z are indeed quite correlated, but it would be "nice" to see if x still explains some variation in my data independently of z.

'would be nice' and collinearity don't go together very well.

Ben Bolker suggested that the parameter estimate for using a variable as an offset should be about one. What is your opinion on this?

Ben is a clever cookie....and he is right.

Alain

Best,
Val?rie

-- 

Dr. Alain F. Zuur
First author of:

1. Analysing Ecological Data (2007).
Zuur, AF, Ieno, EN and Smith, GM. Springer. 680 p.
URL: www.springer.com/0-387-45967-7


2. Mixed effects models and extensions in ecology with R. (2009).
Zuur, AF, Ieno, EN, Walker, N, Saveliev, AA, and Smith, GM. Springer.
http://www.springer.com/life+sci/ecology/book/978-0-387-87457-9


3. A Beginner's Guide to R (2009).
Zuur, AF, Ieno, EN, Meesters, EHWG. Springer
http://www.springer.com/statistics/computational/book/978-0-387-93836-3


4. Zero Inflated Models and Generalized Linear Mixed Models with R. (2012) Zuur, Saveliev, Ieno.
http://www.highstat.com/book4.htm

Other books: http://www.highstat.com/books.htm


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