Average Marginal Effects Decomposition for Hurdle Beta-Binomial Models

The hurdle Beta-Binomial model implies two channels through which covariates affect the expected outcome. The extensive margin governs whether a center serves infant/toddler children at all ($q_i = \mathrm{logistic}(X_i' \alpha)$), while the intensive margin governs the enrollment share conditional on participation ($\mu_i = \mathrm{logistic}(X_i' \beta)$).

By the product rule, the marginal effect of covariate $k$ on $E[y_i / n_i]$ decomposes as: $$ \frac{\partial E[y_i/n_i]}{\partial x_{ik}} = \underbrace{\alpha_k \, q_i(1-q_i) \, \mu_i}_{\text{extensive}} + \underbrace{\beta_k \, \mu_i(1-\mu_i) \, q_i}_{\text{intensive}}. $$

The Average Marginal Effect averages this over all $N$ observations: $$ \mathrm{AME}_k = \frac{1}{N} \sum_{i=1}^{N} \bigl[\alpha_k \, q_i(1-q_i) \, \mu_i + \beta_k \, \mu_i(1-\mu_i) \, q_i\bigr] = \mathrm{AME}_k^{\mathrm{ext}} + \mathrm{AME}_k^{\mathrm{int}}. $$

Usage

ame(fit, cholesky = NULL, sandwich = NULL, level = 0.95, n_draws = NULL)

Arguments

fit: An object of class "hbb_fit" returned by hbb. Must contain a CmdStanR fit with parameters alpha[1:P], beta[1:P], log_kappa, and an hbb_data element with design matrix X.
cholesky: An optional object of class "hbb_cholesky" returned by cholesky_correct. If provided, uses the Cholesky-corrected draws (cholesky$theta_corrected) and posterior mean (cholesky$theta_hat) for AME computation. If NULL (default), uses raw MCMC draws from the fit object.
sandwich: An optional object of class "hbb_sandwich" returned by sandwich_variance. If provided, computes delta-method Wald confidence intervals for each AME component. If NULL (default), Wald results are omitted.
level: Numeric scalar in $(0, 1)$. Confidence level for posterior credible intervals and Wald confidence intervals. Default is 0.95.
n_draws: Integer or NULL. Number of MCMC draws to use. If NULL (default), uses all available draws. If an integer, subsamples by systematic thinning to approximately n_draws draws.

Value

An S3 object of class "hbb_ame" containing:

theta_draws_used: Numeric matrix of dimension M_use x D containing the (possibly subsampled) draws.
theta_hat: Numeric vector of length D: point estimate (posterior mean or Cholesky theta_hat).
ext_ame_draws: Numeric matrix M_use x P: extensive AME draws for each covariate.
int_ame_draws: Numeric matrix M_use x P: intensive AME draws for each covariate.
total_ame_draws: Numeric matrix M_use x P: total AME draws for each covariate.
ext_summary: Data frame with posterior summaries of extensive AME (7 columns: covariate, post_mean, post_median, ci_lo, ci_hi, post_sd, pr_positive).
int_summary: Data frame with posterior summaries of intensive AME.
total_summary: Data frame with posterior summaries of total AME.
decomp_table: Data frame for non-intercept covariates with columns: covariate, ext_ame, ext_ci_lo, ext_ci_hi, int_ame, int_ci_lo, int_ci_hi, total_ame, total_ci_lo, total_ci_hi, ext_share, int_share, sign_pattern.
reversal_probs: Named numeric vector: for each non-intercept covariate, the posterior probability of opposing signs between extensive and intensive components.
wald_summary: Data frame of Wald-based AME inference, or NULL if sandwich was not provided.
ame_ext_hat: Numeric vector of length P: extensive AME point estimates at theta_hat.
ame_int_hat: Numeric vector of length P: intensive AME point estimates at theta_hat.
ame_total_hat: Numeric vector of length P: total AME point estimates at theta_hat.
mean_q: Numeric vector of length M_use: mean participation probability across observations per draw.
mean_mu: Numeric vector of length M_use: mean conditional intensity across observations per draw.
N, P, D, M_total, M_use, level, cov_labels: Dimensional and metadata scalars.

Details

Computes the Average Marginal Effect (AME) of each covariate on the expected IT enrollment share $E[y_i / n_i] = q_i \mu_i$, decomposed into extensive-margin and intensive-margin contributions.

Poverty Reversal

A covariate exhibits a "reversal" when its extensive and intensive AME components have opposing signs. For the poverty variable in the NSECE application: $\alpha_{\mathrm{poverty}} < 0$ (higher poverty reduces participation) but $\beta_{\mathrm{poverty}} > 0$ (higher poverty increases IT share among servers). The posterior probability of reversal is $\Pr(\mathrm{AME}_k^{\mathrm{ext}} < 0 \;\text{AND}\; \mathrm{AME}_k^{\mathrm{int}} > 0)$.

Population-Average AME

This function computes the population-average AME (PA-AME) using global (population-level) coefficients $\alpha$ and $\beta$ only, without state random effects. This is consistent with the sandwich-corrected inference framework and answers the question: "what is the average marginal effect for a new (arbitrary) state?"

Wald Inference

When sandwich is provided, the function also computes delta-method Wald confidence intervals for each AME component. The numerical gradient of $\mathrm{AME}_k(\theta)$ with respect to $\theta$ is computed via central differences (step size $\epsilon = 10^{-5}$), and the Wald variance is: $$ \mathrm{Var}(\mathrm{AME}_k) \approx \nabla_\theta \mathrm{AME}_k(\hat\theta)^\top V_{\mathrm{sand}}\, \nabla_\theta \mathrm{AME}_k(\hat\theta). $$

Performance

The computation is vectorised over observations within each MCMC draw. The key insight is that the N-vectors $q'$, $\mu$, $\mu'$, $q$ are independent of the covariate index $k$, so: $$ \mathrm{AME}_k^{\mathrm{ext}} = \alpha_k \cdot \overline{q'(1-q) \mu}, \quad \mathrm{AME}_k^{\mathrm{int}} = \beta_k \cdot \overline{\mu'(1-\mu) q}, $$ reducing the per-draw cost from $O(NP)$ to $O(N)$.

References

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{
# After fitting:
fit <- hbb(y | trials(n_trial) ~ poverty + urban, data = my_data,
           weights = "weight")
sand <- sandwich_variance(fit)
chol <- cholesky_correct(fit, sand)

# Full AME decomposition with Wald comparison
ame_result <- ame(fit, cholesky = chol, sandwich = sand)
print(ame_result)

# Extract decomposition table
ame_decomposition(ame_result)
} # }