Performs a likelihood ratio test between two nested uncounted models
fitted by maximum likelihood (Poisson or Negative Binomial). The test
statistic is
Value
An object of class "uncounted_lrtest" with components
statistic, df, p_value, boundary, model
descriptions, and log-likelihoods.
Details
$$LR = 2\bigl(\ell_2 - \ell_1\bigr)$$
where \(\ell_1\) and \(\ell_2\) are the maximised log-likelihoods of the simpler and more complex models, respectively. Under \(H_0\) the statistic follows a \(\chi^2\) distribution with degrees of freedom equal to the difference in the number of estimated parameters.
Boundary correction for Poisson vs NB. When comparing a Poisson model (H0) against a Negative Binomial model (H1), the dispersion parameter \(\theta\) lies on the boundary of its parameter space under \(H_0\) (\(\theta \to \infty\)). The standard \(\chi^2\) approximation is invalid in this setting. Following Self & Liang (1987), the p-value is corrected as
$$p = 0.5\,\Pr(\chi^2_1 > LR)$$
which corresponds to a 50:50 mixture of a point mass at zero and a \(\chi^2_1\).
This function is intended for comparing count models only (Poisson and NB). OLS/NLS models are not supported because they use a pseudo-log-likelihood. The models must be fitted to the same data (identical number of observations).
References
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(42)
df <- data.frame(
N = rep(1000, 30),
n = rpois(30, lambda = 50)
)
df$m <- rpois(30, lambda = df$N^0.5 * (df$n / df$N)^0.8)
fit_po <- estimate_hidden_pop(data = df, observed = ~m, auxiliary = ~n,
reference_pop = ~N, method = "poisson")
#> Warning: Some alpha values < 0 (min = -1.28). Consider using constrained = TRUE.
fit_nb <- estimate_hidden_pop(data = df, observed = ~m, auxiliary = ~n,
reference_pop = ~N, method = "nb")
#> Warning: Some alpha values < 0 (min = -1.244). Consider using constrained = TRUE.
# Test for overdispersion (Poisson vs NB)
lrtest(fit_po, fit_nb)
#> Likelihood ratio test
#> ----------------------------------------
#> Model 1: POISSON (logLik = -52.51 )
#> Model 2: NB (logLik = -52.52 )
#> LR statistic: 0 on 1 df
#> (Boundary-corrected: 0.5 * P(chi2 > LR), Self & Liang 1987)
#> p-value: 0.5