countreg

Dispersion Tests

Description

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

Usage

disptest(object, 
  type = c("lrtNB2", "scoreNB2", "scoreNB2adj", "scoreNB1", "scoreNB1adj", "scoreKatz"), 
  trafo = NULL, alternative = c("greater", "two.sided", "less"))

Arguments

`object`	a fitted Poisson GLM of class `“glm”` as fitted by `glm` with family `poisson`.
`type`	type of test, one of `lrtNB2`, `scoreNB2`, `scoreNB2adj`, `scoreNB1`, `scoreNB1adj`, `scoreKatz`. See details.
`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 \(\mathsf{E}[y] = \mu\) which is assumed to be equal to the variance \(\mathsf{VAR}[y] = \mu\). disptest 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.

Type “lrtNB2” is the LRT comparing the classical Poisson and negative binomial regression models. Note that this test has a non-standard null distribution here, since the negative binomial shape parameter (called theta in glm.nb) is on the boundary of the parameter space under the null hypothesis. Hence the asymptotic distribution of the LRT is that of the arithmetic mean of a point mass at zero and a \(\chi^2_1\) distribution, implying that the \(p\)-value is half that of the classical case.

Type “scoreNB2” corresponds to the statistic \(T_1\) in Dean and Lawless (1989), type “scoreNB2adj” is their \(T_a\). “scoreNB2” also appears in Lee (1986). Type “scoreNB1” corresponds to the statistic \(P_C\) in Dean (1992), type “scoreNB1adj” is her \(P'_C\). Type “scoreKatz” is the score test against Katz alternatives derived by Lee (1986), these distributions permit overdispersion as well as underdispersion. The score tests against NB1 and NB2 alternatives are also the score tests against Generalized Poisson type 1 and type 2 alternatives (Yang, Hardin, and Addy, 2009).

Value

An object of class “htest”.

References

Cameron AC, Trivedi PK (1990). “Regression-based Tests for Overdispersion in the Poisson Model”. Journal of Econometrics, 46, 347–364.

Cameron AC, Trivedi PK (2005). Microeconometrics: Methods and Applications. Cambridge: Cambridge University Press.

Cameron AC, Trivedi PK (2013). Regression Analysis of Count Data, 2nd ed. Cambridge: Cambridge University Press.

Dean CB (1992). “Testing for Overdispersion in Poisson and Binomial Regression Models”. Journal of the American Statistical Association, 87, 451–457.

Dean C, Lawless JF (1989). “Tests for Detecting Overdispersion in Poisson Regression Models”. Journal of the American Statistical Association, 84, 467–472.

Jaggia S, Thosar S (1993). “Multiple Bids as a Consequence of Target Management Resistance: A Count Data Approach”. Review of Quantitative Finance and Accounting, 3, 447–457.

Lee LF (1986). “Specification Test for Poisson Regression Models”. International Economic Review, 27, 689–706.

Yang Z, Hardin JW, Addy CL (2009). “A Note on Dean’s Overdispersion Test”. Journal of Statistical Planning and Inference, 139 (10), 3675–3678.

Examples

library("countreg")

## Data with overdispersion
data("RecreationDemand", package = "AER")
rd_p <- glm(trips ~ ., data = RecreationDemand, family = poisson)

## Cameron and Trivedi (2013), p. 248
disptest(rd_p, type = "lrtNB2", alternative = "greater")


    Overdispersion test

data:  rd_p
z = 1407.7, p-value < 2.2e-16
alternative hypothesis: greater

## Data with underdispersion
data("TakeoverBids", package = "countreg")
tb_p <- glm(bids ~ . + I(size^2), data = TakeoverBids, family = poisson)

## Jaggia and Thosar (1993), Table 3
## testing overdispersion
disptest(tb_p, type = "scoreNB2", alternative = "greater")


    Overdispersion test

data:  tb_p
z = -1.0173, p-value = 0.8455
alternative hypothesis: greater

disptest(tb_p, type = "scoreNB2adj", alternative = "greater")


    Overdispersion test

data:  tb_p
z = -0.29924, p-value = 0.6176
alternative hypothesis: greater

## testing underdispersion
disptest(tb_p, type = "scoreKatz", alternative = "two.sided")


    Dispersion test

data:  tb_p
z = -2.5206, p-value = 0.01172
alternative hypothesis: two.sided