In response to my own query (see below), The estimate theta from glm.nb is actually 1/phi or 1/alpha in some texts, where phi is the dispersion parameter for the negative binomial distribution. However, the dispersion estimate from glm.poisson.disp does not equal 1/theta because 1/theta is a maximum likelihood (ML) estimate whereas the dispersion estimate from glm.poisson.disp is based on the method of moments (MM). Correct me if I'm wrong! The two estimates for dispersion are fairly different. Paul and Banerjee (1998) observe that the test statistic discussed below (TNBI) is more conservative when one uses the moment generated estimates for phi, and that phi is underestimated by ML and overestimated by MM. However, the overestimation by MM is slight compared to the underestimation by ML. Does that then mean MM estimates of phi are to be preferred?
Alex Dear list, This is a question about overdispersion and the ML estimates of the parameters returned by the glm.poisson.disp (L. Scrucca) and glm.nb (Venables and Ripley) functions. Both appear to assume a negative binomial distribution for the response variable. Paul and Banerjee (1998) developed C(alpha) tests for "interaction and main effects, in an unbalanced two-way layout of counts involving two fixed factors, when data are Poisson distributed, and when data are extra dispersed". In R I coded their C(alpha) test statistic (called TNBI) for interaction for the case where the counts are extra-dispersed, as well as their test for extra-dispersion (called T_a). Using the Quine data set (Quine, 1975) the authors collapse the orginal 4x2x2x2 study design into a 2x4 table and obtained estimates of TNBI=10.36 and T_a=90.81. Using the dispersion estimate from glm.poisson.disp and the estimates for mu I got exactly the same results for TNBI and T_a. This made me happy. Now I thought to try the ML estimates from glm.nb to see if the results would be the same but I am having difficulty relating the dispersion phi from glm.poisson.disp to theta estimated by glm.nb. According to the R help for glm.poisson.disp " Var(y_i) = mu_i(1+mu_i*phi) ". The help for glm.nb lead me to a book by V&R (1994) which indicates that Var(y)=mu+mu^2/theta. From this I gathered that phi=1/theta but the estimates do not relate to each other in this way unless one is in error. In a document by L.P. Ammann he says a "negative binomial model can be specified with mean mu and dispersion phi by taking theta=mu/(phi-1)". I had a problem implementing this because in my mind mu is a vector whereas phi and theta are scalars. Alex Hanke Department of Fisheries and Oceans St. Andrews Biological Station 531 Brandy Cove Road St. Andrews, NB Canada E5B 2L9 ______________________________________________ [EMAIL PROTECTED] mailing list https://www.stat.math.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
