Computes the deviance (twice the difference between the saturated and fitted
log-likelihoods) for a fitted uncounted model.
Usage
# S3 method for class 'uncounted'
deviance(object, ...)Details
The formula depends on the estimation method:
Poisson: $$D = 2\sum_{i}\bigl[m_i\log(m_i/\hat\mu_i) - (m_i - \hat\mu_i)\bigr]$$ with the convention \(0 \log 0 = 0\).
Negative Binomial: $$D = 2\sum_{i}\bigl[m_i\log(m_i/\hat\mu_i) - (m_i+\theta)\log\bigl((m_i+\theta)/(\hat\mu_i+\theta)\bigr)\bigr]$$
OLS / NLS: $$D = \sum_{i}(\log m_i - \log\hat\mu_i)^2$$ i.e., the residual sum of squares on the log scale.
References
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. Chapman & Hall.
Examples
set.seed(123)
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 > 1 (max = 3.271). Population size estimates may be unreliable. Consider using constrained = TRUE or simplifying cov_alpha.
deviance(fit_po)
#> [1] 17.61431