AER (version 1.2-9)

# dispersiontest: Dispersion Test

## Description

Tests the null hypothesis of equidispersion in Poisson GLMs against the alternative of overdispersion and/or underdispersion.

## Usage

dispersiontest(object, trafo = NULL, alternative = c("greater", "two.sided", "less"))

## Arguments

object

a fitted Poisson GLM of class "glm" as fitted by glm with family poisson.

trafo

a specification of the alternative (see also details), can be numeric or a (positive) function or NULL (the default).

alternative

a character string specifying the alternative hypothesis: "greater" corresponds to overdispersion, "less" to underdispersion and "two.sided" to either one.

## Value

An object of class "htest".

## Details

The standard Poisson GLM models the (conditional) mean $$\mathsf{E}[y] = \mu$$ which is assumed to be equal to the variance $$\mathsf{VAR}[y] = \mu$$. dispersiontest assesses the hypothesis that this assumption holds (equidispersion) against the alternative that the variance is of the form: $$\mathsf{VAR}[y] \quad = \quad \mu \; + \; \alpha \cdot \mathrm{trafo}(\mu).$$ Overdispersion corresponds to $$\alpha > 0$$ and underdispersion to $$\alpha < 0$$. The coefficient $$\alpha$$ can be estimated by an auxiliary OLS regression and tested with the corresponding t (or z) statistic which is asymptotically standard normal under the null hypothesis.

Common specifications of the transformation function $$\mathrm{trafo}$$ are $$\mathrm{trafo}(\mu) = \mu^2$$ or $$\mathrm{trafo}(\mu) = \mu$$. The former corresponds to a negative binomial (NB) model with quadratic variance function (called NB2 by Cameron and Trivedi, 2005), the latter to a NB model with linear variance function (called NB1 by Cameron and Trivedi, 2005) or quasi-Poisson model with dispersion parameter, i.e., $$\mathsf{VAR}[y] \quad = \quad (1 + \alpha) \cdot \mu = \mathrm{dispersion} \cdot \mu.$$ By default, for trafo = NULL, the latter dispersion formulation is used in dispersiontest. Otherwise, if trafo is specified, the test is formulated in terms of the parameter $$\alpha$$. The transformation trafo can either be specified as a function or an integer corresponding to the function function(x) x^trafo, such that trafo = 1 and trafo = 2 yield the linear and quadratic formulations respectively.

## References

Cameron, A.C. and Trivedi, P.K. (1990). Regression-based Tests for Overdispersion in the Poisson Model. Journal of Econometrics, 46, 347--364.

Cameron, A.C. and Trivedi, P.K. (1998). Regression Analysis of Count Data. Cambridge: Cambridge University Press.

Cameron, A.C. and Trivedi, P.K. (2005). Microeconometrics: Methods and Applications. Cambridge: Cambridge University Press.

glm, poisson, glm.nb

## Examples

# NOT RUN {
data("RecreationDemand")
rd <- glm(trips ~ ., data = RecreationDemand, family = poisson)

## linear specification (in terms of dispersion)
dispersiontest(rd)
## linear specification (in terms of alpha)
dispersiontest(rd, trafo = 1)
## quadratic specification (in terms of alpha)
dispersiontest(rd, trafo = 2)
dispersiontest(rd, trafo = function(x) x^2)

## further examples
data("DoctorVisits")
dv <- glm(visits ~ . + I(age^2), data = DoctorVisits, family = poisson)
dispersiontest(dv)

data("NMES1988")
nmes <- glm(visits ~ health + age + gender + married + income + insurance,
data = NMES1988, family = poisson)
dispersiontest(nmes)
# }