Predictions from a Hurdle Beta-Binomial Model

Point predictions are computed at the posterior mean of the fixed effects (and state random effects for SVC models): $$ \hat{y}_i = \hat{q}(x_i) \cdot \hat\mu(x_i), $$ where $\hat{q}(x) = \mathrm{logistic}(x'\hat\alpha)$ and $\hat\mu(x) = \mathrm{logistic}(x'\hat\beta)$.

When interval = "credible", the full posterior distribution is propagated through the link function to produce empirical credible intervals. For each MCMC draw $m$: $$ \hat{y}_i^{(m)} = q^{(m)}(x_i) \cdot \mu^{(m)}(x_i), $$ and the $(\alpha/2,\; 1-\alpha/2)$ quantiles across draws give the interval bounds.

Usage

# S3 method for class 'hbb_fit'
predict(
  object,
  newdata = NULL,
  type = c("response", "extensive", "intensive"),
  interval = c("none", "credible"),
  level = 0.95,
  ndraws = NULL,
  state = NULL,
  ...
)

Arguments

object

An object of class "hbb_fit" returned by hbb.

newdata

An optional data frame containing the covariates for prediction. If NULL (default), in-sample predictions are returned using the training design matrix.

type

Character string: "response" (default), "extensive", or "intensive". See Prediction types above.

interval

Character string: "none" (default) for point predictions only, or "credible" for posterior credible intervals.

level

Numeric scalar in $(0, 1)$. Credible level when interval = "credible". Default is 0.95.

ndraws

Optional positive integer. If specified, the posterior draws are thinned to ndraws before computing intervals. Ignored when interval = "none". Default is NULL (use all draws).

state

Character string or character vector specifying states for SVC prediction.

Character of length 1:: If newdata is provided, interpreted as a column name in newdata. Otherwise, interpreted as a single state label to apply to all observations.
Character vector of length nrow(newdata):: Direct state labels for each observation.
NULL:: No state information. For SVC models, predictions use fixed effects only (with a warning).

...

Additional arguments (currently unused).

Value

A data frame with one row per observation. Columns:

fit: Numeric: point prediction (posterior mean).
lwr: Numeric: lower credible bound. Present only when interval = "credible".
upr: Numeric: upper credible bound. Present only when interval = "credible".

Details

Generates point predictions and optional posterior credible intervals from a fitted hurdle Beta-Binomial model. Supports both in-sample prediction (using the training data) and out-of-sample prediction (using newdata).

Prediction types

Three types of predictions are available:

"response": (Default.) The composite enrollment proportion $\hat{q}_i \cdot \hat\mu_i \in [0,1]$. This represents the unconditional expected proportion $E[Y_i/n_i]$.
"extensive": The participation probability $\hat{q}_i = \mathrm{logistic}(x_i'\hat\alpha) \in (0,1)$.
"intensive": The conditional enrollment share $\hat\mu_i = \mathrm{logistic}(x_i'\hat\beta) \in (0,1)$.

Newdata design matrix

When newdata is provided, the design matrix is constructed by:

Extracting the covariate columns named in object$formula$fixed from newdata.
Applying the same centering and scaling as the training data, using the stored hbb_data$x_center (column means) and hbb_data$x_scale (column SDs). This ensures that regression coefficients remain on the standardised scale.
Prepending an intercept column of ones.

All covariate columns must be present in newdata. Missing columns cause an informative error.

State-varying coefficients (SVC)

For SVC models, out-of-sample prediction requires specifying state (either a column name in newdata or a vector of state labels). The state labels must match hbb_data$group_levels from the training data. Unknown state labels cause an error.

If state = NULL for an SVC model, predictions use fixed effects only (i.e., the population-average prediction without state random effects), and a warning is issued.

Credible interval theory

The posterior credible interval for prediction $i$ at level $1-\alpha$ is $$ \bigl[Q_{\alpha/2}(\hat{y}_i^{(1:M)}),\; Q_{1-\alpha/2}(\hat{y}_i^{(1:M)})\bigr], $$ where $Q_p$ denotes the empirical $p$-quantile over $M$ posterior draws. This is a conditional credible interval for the expected proportion $E[Y_i/n_i | \theta]$, not a predictive interval for $Y_i$ itself (which would additionally account for sampling variability from the Beta-Binomial).

For computational efficiency, ndraws can be used to thin the draws before propagation. With $M = 4000$ total draws, setting ndraws = 500 reduces computation 8-fold while typically preserving interval accuracy.

References

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). Chapman and Hall/CRC.

Ghosal, R., Ghosh, S. K., and Maiti, T. (2020). Two-part regression models for longitudinal zero-inflated count data. Journal of the Royal Statistical Society: Series A, 183(4), 1603–1626.

Examples

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

# In-sample point predictions
pred <- predict(fit)

# Out-of-sample with credible intervals
new_df <- data.frame(poverty = c(0.1, 0.3, 0.5),
                     urban   = c(1, 0, 1))
pred_ci <- predict(fit, newdata = new_df,
                   interval = "credible", level = 0.95)

# Extensive-margin predictions only
pred_ext <- predict(fit, type = "extensive")

# SVC model with state assignment
pred_svc <- predict(fit_svc, newdata = new_df,
                    state = c("AL", "CA", "NY"),
                    interval = "credible")
} # }