Residuals for uncounted models — residuals.uncounted • uncounted

Extracts residuals of the specified type from a fitted uncounted model.

Usage

# S3 method for class 'uncounted'
residuals(
  object,
  type = c("response", "pearson", "deviance", "anscombe", "working"),
  ...
)

Arguments

object: An object of class "uncounted".
type: Type of residual: "response" (default), "pearson", "deviance", or "anscombe".
...: Additional arguments (ignored).

Value

A numeric vector of residuals with the same length as the number of observations.

Details

Four residual types are supported:

Response residuals

$$r_i = m_i - \hat\mu_i$$

Pearson residuals

$$r_i = \frac{m_i - \hat\mu_i}{\sqrt{V(\hat\mu_i)}}$$ where the variance function $V(\mu)$ depends on the model:

Poisson: $V(\mu) = \mu$
Negative Binomial: $V(\mu) = \mu + \mu^2/\theta$
OLS/NLS: $V(\mu) = \mu^2 \hat\sigma^2$ (delta-method approximation on the log scale)

Deviance residuals

Signed square root of the individual deviance contributions: $$r_i = \mathrm{sign}(m_i - \hat\mu_i)\,\sqrt{d_i}$$ where $d_i$ is the $i$-th term in the deviance sum (see deviance.uncounted). For OLS/NLS models, standardised log-scale residuals are returned instead.

Anscombe residuals

Variance-stabilising residuals.

For Poisson (McCullagh & Nelder, 1989):
\deqn{r_i = \frac{3\bigl(m_i^{2/3} - \hat\mu_i^{2/3}\bigr)}
      {2\,\hat\mu_i^{1/6}}}

For Negative Binomial (Hilbe, 2011):
\deqn{r_i = \frac{3\bigl(m_i^{2/3} - \hat\mu_i^{2/3}\bigr)}
      {2\,\hat\mu_i^{1/6}\,(1 + \hat\mu_i/\theta)^{1/6}}}

For OLS/NLS: standardised log-scale residuals.

References

McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. Chapman & Hall.

Hilbe, J. M. (2011). Negative Binomial Regression, 2nd ed. Cambridge University Press.

Pierce, D. A. and Schafer, D. W. (1986). Residuals in generalized linear models. Journal of the American Statistical Association, 81(396), 977–986.

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 <- 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.

# Response residuals (default)
head(residuals(fit))
#> [1] -1.9426446 -1.2439567 -0.1250438 -0.5329768 -1.4553729 -1.0807200

# Pearson residuals
head(residuals(fit, type = "pearson"))
#> [1] -1.3937879 -0.6038371 -0.1178902 -0.3348827 -0.6231062 -0.6157254

# Deviance and Anscombe residuals
head(residuals(fit, type = "deviance"))
#> [1] -1.9711137 -0.6376606 -0.1201818 -0.3477951 -0.6544409 -0.6583125
head(residuals(fit, type = "anscombe"))
#> [1] -2.0906818 -0.6379992 -0.1201894 -0.3478822 -0.6547205 -0.6588482