Lecture 14—Wednesday, October 8, 2008

What was covered?

Terminology Defined

Predictive simulation

Inferential uncertainty


where the distribution shown is the multivariate normal distribution, the multivariate analog of the ordinary univariate normal distribution.

For a given choice of β this equation can be used to predict the log mean richness of each of the 29 Galapagos islands in the data set. Repeating this process for the n different samples of β yields an n × 29 matrix of log means.

Predictive uncertainty: uncertainty in the individual outcomes

Predictive simulation from a Bayesian perspective

Negative binomial regression in WinBUGS

Here α and β are positive parameters (called the shape and scale parameters, respectively) and is the gamma function. If then the mean and variance of X are as follows.

where .

model{
for(i in 1:n) {
y[i]~dpois(mustar[i])
mustar[i]<-rho[i]*mu[i]
#log link
log(mu[i])<-b0+b1*x[i]
rho[i]~dgamma(alpha,alpha)
}
b0~dnorm(0,.00001)
b1~dnorm(0,.00001)
alpha<-exp(logalpha)
logalpha~dnorm(0,0.0001)
}

Generalized linear models (continued)

A family tree of statistical models

Exponential family of distributions

The Poisson distribution is a member of the exponential family

Details on the probability distributions and link functions used in GLIMs

Probability Distribution
Canonical Link g(μ) Other links supported in R Historic name for these models
Poisson
log:
identity, sqrt
Poisson regression, loglinear model
Normal (Gaussian)
identity:
log, inverse
ordinary linear regression
Binomial
logit:
probit, cloglog, log
logistic regression, probit analysis
Gamma
inverse:
identity, log
gamma regression
"Negative Binomial"
log, sqrt, identity
negative binomial regression

Link functions versus Transformations of the Response

Exponentiating this yields the nth root of the product of the individual values. This is called the geometric mean. There is a famous mathematical inequality that relates the geometric and arithmetic means that states that the geometric mean is always less than or equal to the arithmetic mean. Thus the log-Arrhenius model can perhaps be interpreted as estimating the geometric mean of the data as a function of log(area).

The offset in count data regression models

Thus by including an offset we end up fitting a model for the rate of occurrence, as was desired.

Cited References

Course Home Page


Jack Weiss
Phone: (919) 962-5930
E-Mail: jack_weiss@unc.edu
Address: Curriculum in Ecology, Box 3275, University of North Carolina, Chapel Hill, 27516
Copyright © 2008
Last Revised--October 15, 2008
URL: http://www.unc.edu/courses/2008fall/ecol/563/001/docs/lectures/lecture14.htm