Biology, images, analysis, design...
|"It has long been an axiom of mine that the little things are infinitely the most important" |
Poisson regression is used to model counts, whether it be counts of a rare mammal in quadrats of forest, or counts of disease or mortality expressed as rates (numbers per unit person-year).
Poisson regression assumes that the mean of the Poisson random variable is a function of explanatory variables:
E(Y ) = μ = λ = exp(α + β1X1 + ... + βkXk)
ln Y = Σ(ΒjXj) + ε,
where ε (the random variation component of Y) is Poisson distributed about E(Y) with a variance equal to E(Y).
Because the log of this function produces a linear combination of the predictors, this model is said to have a log link function. The exponent of a Poisson regression coefficient is a rate ratio corresponding to a one unit difference in the explanatory variable.
If (for example) we are considering cases of disease, we may find that each individual is observed for a different period of time. In this case one wants to model rates (counts per unit of time). We therefore write:
ln Y = ln (time) + Σ(ΒjXj) + ε,
log(Y/time)= + Σ(ΒjXj) + ε
The term ln(time) is known as the offset which is an explanatory variable with a known regression coefficient, in this case 1. It is the amount which must be added to estimate Y for any given X.
Assumptions of Poisson regressionThese include:
Parameter estimation and significance testing
Parameter estimation is carried out using a generalized linear model with a log link and Poisson errors. The log link ensures that all the fitted values are positive, while the Poisson errors take account of the fact that the data are integer and have variances equal to their means.
Assessing significance explained variation ('goodness of fit')
Procedures here are similar to those for logistic regression. Global goodness of fit tests can be made using the Pearson chi-squared and deviance test statistics. Large values of these statistics, and small P-values suggest that the model does not fit the observed data.
A simple Poisson regression model only allows for simple random variation and, as we pointed out in
As a word of caution, Gerard E.