Regression-adjusted test of equidispersion

Tests whether a fitted count model's dispersion departs from the Poisson, conditional on the covariates. For a fit from a family with a dispersion parameter (method = "lr", the default), the test is the likelihood-ratio test of that parameter against its Poisson value, with the Poisson refit on the same design (fixed effects included); its p-value comes from the asymptotic distribution or from a parametric bootstrap under the fitted Poisson, by the rule in Details. For a Poisson fit, method = "auxiliary" runs the Cameron–Trivedi (1990) auxiliary regression, which needs no alternative family.

Usage

dispersion_test(
  object,
  alternative = c("two.sided", "under", "over"),
  method = c("lr", "auxiliary"),
  B = NULL,
  cores = 1L
)

Arguments

object: A fitted cpb, cpb_fe, gec, gec_fe, or count_reg model (for method = "auxiliary", a count_reg fit with family = "poisson").
alternative: "two.sided" (default where the null is interior), "under" (underdispersion), or "over" (overdispersion).
method: "lr" (likelihood ratio against the Poisson refit; default) or "auxiliary" (the Cameron–Trivedi auxiliary regression for a Poisson fit).
B: Parametric-bootstrap replicates for method = "lr": NULL (the default) applies the calibration rule in Details (199 replicates when the bootstrap is needed), 0 requests the asymptotic distribution, and a positive whole number forces the bootstrap.
cores: Worker processes for the bootstrap replicates.

Value

An object of class c("dispersion_test", "htest") with the statistic, the p-value, the estimated dispersion parameter and its Poisson value, the two log-likelihoods, and the alternative; for method = "lr" also calibration ("asymptotic" or "parametric bootstrap"), B and B_ok (replicates requested and completed), boot (the simulated statistics), first_order_size (the asymptotic 5\ size), asymptotic_ok (whether the calibration rule admits the asymptotic distribution), and note where a statistic or p-value needs one.

Details

The Poisson is nested in every family at one value of the dispersion parameter: alpha = 1 (CPB), delta = 1 (GEC), nu = 1 (both COM-Poisson parameterizations), lambda = 0 (generalized Poisson), alpha = 1 (gamma-count), theta = 1 (double Poisson), and 1/theta = 0 (negative binomial). When that value is interior to the parameter space the likelihood-ratio statistic is asymptotically \(\chi^2_1\) under the null; a directional alternative ("under" or "over") uses the signed root \(r = \pm\sqrt{LR}\), positive when the estimate lies on the underdispersed side of the Poisson value, which is asymptotically standard normal. When the Poisson value is on the boundary (the CPB, whose alpha lives in (0, 1); the negative binomial, whose overdispersion parameter is non-negative) the asymptotic null distribution is the \(\tfrac12\chi^2_0 + \tfrac12\chi^2_1\) mixture (Self and Liang 1987; Andrews 2001) and the test is one-sided by construction ("under" for the CPB, "over" for the negative binomial). The CPB qualifies although its support moves with its parameters: near alpha = 1 its ceiling lambda / (1 - alpha) diverges, the log-likelihood is regular on that side, and its score there is the classical dispersion score \(-[(y - \lambda)^2 - y] / (2\lambda)\); the zero-truncated model reaches the same mixture through its efficient score. Because the Poisson is the CPB's limit as alpha approaches 1, the statistic cannot be negative in principle; when the max.support guard stops alpha short of that limit and the fitted log-likelihood lands just below the Poisson's, the statistic takes its boundary value 0. Any other negative statistic means the optimizer fell short, and no p-value is reported.

Calibration. Where the Poisson value is on the boundary of the family's parameter space (the CPB, the negative binomial), or the fit has unit fixed effects, the p-value is by default a parametric bootstrap with 199 replicates. The one-sided boundary statistic is skewed toward rejection in finite samples, and unit intercepts bias the dispersion estimate toward underdispersion by an amount of order 1/T; in both cases the asymptotic distribution over-rejects. Where the Poisson value is interior, the asymptotic distribution is used unless the estimated mean parameters shift it too far: to first order they move the signed root toward underdispersion by \(p/\sqrt{2n}\) standard deviations for \(p\) mean parameters on \(n\) observations (the weight total), the bias Dean and Lawless (1989) remove from the score test, and the bootstrap takes over when that shift pushes the first-order size of a 5\ simulates responses from the Poisson (zero-truncated Poisson) fitted to the same design, offset, weights, and unit effects, refits both models to each, and reports the share of simulated statistics at least as extreme as the observed one, counting the observed one, among the replicates whose two fits both succeeded. Simulating at the Poisson value itself keeps the bootstrap valid at the boundary. B = 0 requests the asymptotic distribution (with unit fixed effects it returns the statistic without a p-value), and a positive B sets the number of bootstrap replicates. Every replicate refits both models, so the bootstrap takes about B times as long as the original fit; cores spreads the replicates over worker processes.

References

Andrews, D. W. K. (2001). Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica, 69(3), 683-734. Cameron, A. C. and Trivedi, P. K. (1990). Regression-based tests for overdispersion in the Poisson model. Journal of Econometrics, 46(3), 347-364. Dean, C. and Lawless, J. F. (1989). Tests for detecting overdispersion in Poisson regression models. Journal of the American Statistical Association, 84(406), 467-472. Self, S. G. and Liang, K.-Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Association, 82(398), 605-610.

Examples

set.seed(2); n <- 400; x <- rnorm(n)
d <- data.frame(y = rgammacount(n, exp(1 + 0.4 * x), alpha = 2), x = x)
dispersion_test(count_reg(y ~ x, d, family = "gammacount"), alternative = "under")
#> 
#> Likelihood-ratio test of equidispersion (Gamma-count vs Poisson; asymptotic chi-square)
#> 
#> data: y ~ x
#> LR = 52.1022  (logLik: fitted family = -647.24, Poisson = -673.30),  p-value = 2.634e-13
#> alternative hypothesis: underdispersion
#> estimate: alpha = 1.9711  (Poisson value 1)
dispersion_test(count_reg(y ~ x, d, family = "poisson"), method = "auxiliary")
#> 
#> Cameron-Trivedi auxiliary regression test of equidispersion (Var = mu + a mu^2)
#> 
#> data: y ~ x
#> t = -9.4533,  p-value = < 2.2e-16
#> alternative hypothesis: two.sided
#> estimate: a = -0.1291  (Poisson value 0)