Probability Distribution 
Canonical Link g(μ)  Other links supported in R  Variance function  Historic name for these models 

Poisson 
log: 
identity, sqrt 
Poisson regression, loglinear model 

Normal (Gaussian) 
identity: 
log, inverse 
1 
ordinary linear regression 
Binomial 
logit: 
probit, cloglog, log 
logistic regression, probit analysis 

Gamma 
inverse: 
identity, log 
gamma regression 

Inverse Gaussian 
identity, inverse, log 
— 

"Negative Binomial" 
log, sqrt, identity 
negative binomial regression 
Since the last expression is a probability, the desired bounds on p are attained.
Recall that LR has an asymptotic chisquared distribution with degrees of freedom equal to the difference in the number of parameters estimated by the two models. Alternatively, the degrees of freedom is the number of parameters that are fixed to have a specific value in one model but are freely estimated in the other model.
Thus the deviance statistic can be used in a goodness of fit test. But just like the Pearson chisquared test, the G^{2} test, and the likelihood ratio test on which it is based, there are sample size and cell size issues that may make the asymptotic chisquared distribution suspect in specific applications.
Written this way L_{1} is the larger of the two models (more estimated parameters) and L_{0} is the simpler model in which some of those parameters have been assigned specific values (such as zero).
where D_{1} and D_{2} are the deviances of models 1 and 2, respectively.
Jack Weiss Phone: (919) 9625930 EMail: jack_weiss@unc.edu Address: Curriculum in Ecology, Box 3275, University of North Carolina, Chapel Hill, 27516 Copyright © 2006 Last RevisedFeb 26, 2006 URL: http://www.unc.edu/courses/2006spring/ecol/145/001/docs/lectures/lecture22.htm 