Matched count regressions: Poisson, negative binomial, COM-Poisson, generalized Poisson, gamma-count, double Poisson

Fits a count regression with a log link from any of the package's count families, optionally zero-truncated and/or with unit fixed effects, returning an object that compare_models(), score(), dispersion_test(), dispersion_profile(), and the broom methods treat on the same footing as a cpb() or gec() fit. The families share one interface, so a gamma-count and a COM-Poisson can be compared to a CPB with identical degrees-of-freedom, log-likelihood, proper-score, weight, and robust-standard-error accounting rather than reconciled across packages.

Usage

count_reg(
  formula,
  data,
  family = c("poisson", "negbin", "compois", "mpcmp", "genpois", "gammacount",
    "doublepois"),
  truncated = FALSE,
  fe = NULL,
  offset = NULL,
  weights = NULL,
  se = c("analytic", "robust", "cluster", "none"),
  cluster = NULL
)

Arguments

formula: A model formula.
data: A data frame.
family: One of "poisson", "negbin", "compois", "mpcmp", "genpois", "gammacount", "doublepois" (see Families).
truncated: Logical; if TRUE, fit the zero-truncated form (requires all Y >= 1), the count analogue of the CPB's zero-truncated default.
fe: Optional column name(s) for fixed effects, entered as factor dummies (so the degrees of freedom count each absorbed intercept, matching cpb_fe()). Pass a vector of two columns for two-way (e.g. unit and time) fixed effects.
offset: Optional offset entered on the linear-predictor (log) scale – the log-mean for the mean-parameterized families, the log-rate ($\log\lambda$) for "compois". A numeric vector or the name of a column in data, e.g. $\log(\text{exposure})$ so the model becomes a rate model.
weights: Optional frequency weights: a numeric vector or the name of a column in data. A weight of $w_i$ is equivalent to $w_i$ copies of row $i$ in the log-likelihood, the information, and the score products, and nobs() returns the weight total. Survey (probability) weights call for se = "robust" or se = "cluster".
se: Standard errors: "analytic" (inverse information, default), "robust" (heteroskedasticity-consistent sandwich), "cluster" (cluster-robust; needs cluster), or "none".
cluster: Optional cluster identifier (a column name in data or a vector aligned to its rows) for se = "cluster"; supplying it selects se = "cluster" unless se is given explicitly.

Value

An object of class "count_reg". The component $theta holds the dispersion parameter on its natural scale (the negative-binomial size, the COM-Poisson $\nu$, the generalized-Poisson $\lambda$, the gamma-count $\alpha$, the double-Poisson $\theta$; NA for the Poisson); $fitted.values and $residuals are on the mean scale (the conditional mean E(Y | Y >= 1) for a zero-truncated fit), as fitted() and residuals() return them, and $mu is the family's natural parameter exp(offset + x'b).

Families

"poisson": the equidispersed baseline.
"negbin": negative binomial (overdispersion only); $theta is the size.
"compois": the classical Conway–Maxwell–Poisson in its rate parameterization, $\log\lambda = x'\beta$. The rate $\lambda$ is not the mean (though predict(type = "response") and fitted() return the mean); $theta is the dispersion $\nu$ ($\nu > 1$ underdispersed, $\nu = 1$ Poisson, $\nu < 1$ overdispersed).
"mpcmp": the COM-Poisson in Huang's (2017) mean parameterization, $\log\mathrm{E}(Y) = x'\beta$, with the same dispersion $\nu$; the coefficients are effects on the log mean and rate ratios are exact.
"genpois": the Consul–Jain generalized Poisson with constant dispersion $\lambda \in (-1, 1)$ and mean $\mu = \exp(x'\beta)$, so $\mathrm{Var}/\mathrm{Mean} = 1/(1-\lambda)^2$; $\lambda < 0$ is underdispersion, on a finite support that the pmf is renormalized over (see genpois-distribution).
"gammacount": Winkelmann's (1995) gamma-count renewal-process model, underdispersed when the waiting times between events are more regular than exponential ($\alpha > 1$); $\exp(x'\beta)$ is the long-run event rate and the exact mean is reported (see gammacount-distribution).
"doublepois": Efron's (1986) double Poisson with the exact normalizing constant, $\theta > 1$ underdispersed (see doublepois-distribution).

The Poisson is the equidispersed member of every family, so dispersion_test() tests each family's dispersion parameter against it, and dispersion_profile() compares the families' implied variance-to-mean curves against the data.

Runtime: the Poisson, negative binomial, gamma-count, and double Poisson and generalized Poisson families fit in under a second at a few hundred rows; the COM-Poisson families evaluate a normalizing sum per observation (and the mean parameterization solves a root per observation), so they take seconds at a few hundred rows and minutes at several thousand; se = "robust" and "cluster" add numerical scores at the same cost per parameter.

References

Huang, A. (2017). Mean-parametrized Conway–Maxwell–Poisson regression models for dispersed counts. Statistical Modelling, 17(6), 359-380. Winkelmann, R. (1995). Duration dependence and dispersion in count-data models. Journal of Business & Economic Statistics, 13(4), 467-474. Efron, B. (1986). Double exponential families and their use in generalized linear regression. Journal of the American Statistical Association, 81(395), 709-721. Consul, P. C. and Famoye, F. (1992). Generalized Poisson regression model. Communications in Statistics – Theory and Methods, 21(1), 89-109.

Examples

set.seed(1); n <- 400; x <- rnorm(n)
y <- rgammacount(n, exp(1 + 0.5 * x), alpha = 2)     # regular event timing
d <- data.frame(y = y, x = x)
m <- count_reg(y ~ x, data = d, family = "gammacount")
m
#> Gamma-count count regression  
#> Call:  count_reg(formula = y ~ x, data = d, family = "gammacount")
#> 
#> Coefficients:
#> (Intercept)           x 
#>      0.9847      0.4955 
#> 
#> alpha (gamma-count dispersion; > 1 underdispersed): 1.9406  [Var/Mean ~ 0.515]
compare_models(poisson = count_reg(y ~ x, d, family = "poisson"), gammacount = m)
#>            df    logLik      AIC      BIC logscore       rps
#> gammacount  3 -637.0878 1280.176 1292.150 1.592719 0.6721447
#> poisson     2 -661.4718 1326.944 1334.927 1.653679 0.6905665