dispersiontest {AER} R Documentation

## 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.

### Details

The standard Poisson GLM models the (conditional) mean E[y] = mu which is assumed to be equal to the variance VAR[y] = mu. `dispersiontest` assesses the hypothesis that this assumption holds (equidispersion) against the alternative that the variance is of the form:

VAR[y] = mu + alpha * 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 trafo are trafo(mu) = mu^2 or 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.,

VAR[y] = (1 + alpha) * mu = dispersion * 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.

### Value

An object of class `"htest"`.

### 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

```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)
```

[Package AER version 1.1-9 Index]