Summary Method for hbb_fit Objects

Produces a comprehensive summary of a fitted hurdle Beta-Binomial model, including fixed-effect estimates with standard errors and confidence intervals, dispersion parameter summary, MCMC diagnostics, and optionally design effect ratios from the sandwich variance.

Usage

# S3 method for class 'hbb_fit'
summary(object, sandwich = NULL, level = 0.95, ...)

Arguments

object: An object of class "hbb_fit" returned by hbb.
sandwich: An optional object of class "hbb_sandwich" returned by sandwich_variance. If provided, produces sandwich-based Wald inference. If NULL (default), produces posterior-based inference.
level: Numeric scalar in $(0, 1)$. Confidence/credible level. Default is 0.95.
...: Currently unused; included for S3 method consistency.

Value

An S3 object of class "summary.hbb_fit" containing:

fixed_effects: Data frame with columns: parameter, estimate, se, ci_lower, ci_upper, rhat, ess_bulk. One row per fixed-effect parameter ($D = 2P + 1$).
dispersion: Named list with kappa_hat (point estimate on the natural scale), kappa_se (delta-method SE or posterior SD), and kappa_ci (2-vector confidence limits on the $\kappa$ scale).
random_effects: NULL for base/weighted models. For SVC models, a list with element tau (posterior mean of the random-effect scale parameters).
diagnostics: Named list with n_divergent, n_max_treedepth, ebfmi, max_rhat, min_ess_bulk, min_ess_tail.
model_info: Named list with N, P, S (or NULL), zero_rate, model_type, formula.
sandwich_used: Logical: whether sandwich SEs were used.
DER: Named numeric vector of Design Effect Ratios if sandwich was provided, NULL otherwise.
level: The confidence level used.
call: The original model call.

Inference mode

When sandwich is supplied, standard errors and confidence intervals are computed from the sandwich variance $V_{\mathrm{sand}}$ (Wald-type intervals): $$ \mathrm{CI}_{1-\alpha}(\theta_p) = \hat\theta_p \pm z_{(1+\mathrm{level})/2}\, \sqrt{V_{\mathrm{sand},pp}}. $$ When sandwich is NULL, standard errors are posterior standard deviations and intervals are quantile-based credible intervals from the MCMC draws.

Dispersion

The dispersion parameter $\kappa$ controls overdispersion relative to the Binomial. It is estimated on the log scale (log_kappa) and back-transformed via the delta method: $$\mathrm{SE}(\hat\kappa) = \hat\kappa \cdot \mathrm{SE}(\widehat{\log\kappa}).$$ The confidence interval on the $\kappa$ scale is obtained by exponentiating the interval for $\log\kappa$.

References

Williams, M. R. and Savitsky, T. D. (2021). Uncertainty estimation for pseudo-Bayesian inference under complex sampling. International Statistical Review, 89(1), 72–107. doi:10.1111/insr.12376

Examples

if (FALSE) { # \dontrun{
fit  <- hbb(y | trials(n_trial) ~ poverty + urban, data = my_data)
sand <- sandwich_variance(fit)

# Summary with sandwich SEs (recommended for survey data)
s <- summary(fit, sandwich = sand, level = 0.95)
print(s)

# Summary with posterior SDs (for unweighted models)
s0 <- summary(fit)
print(s0)
} # }