Variance under the Hurdle Beta-Binomial

Computes the unconditional variance via the law of total variance: $$ \text{Var}[Y] = q \cdot V_{\text{ZT}} + q (1 - q) \cdot (E_{\text{ZT}})^2, $$ where \(E_{\text{ZT}} = n\mu / (1 - p_0)\) is the zero-truncated mean and \(V_{\text{ZT}}\) is the zero-truncated variance.

Usage

hurdle_variance(n, q, mu, kappa)

Arguments

n

Integer vector of trial sizes (\(n \ge 1\)).

q

Numeric vector of participation probabilities, each in \([0, 1]\).

mu

Numeric vector of intensity means in \((\varepsilon, 1 - \varepsilon)\).

kappa

Numeric vector of concentrations (\(> 0\)).

Value

A numeric vector of variances (non-negative).

Details

The zero-truncated variance is derived from the untruncated moments: $$ V_{\text{ZT}} = \frac{E_{\text{BB}}[Y^2]}{1 - p_0} - \left(\frac{E_{\text{BB}}[Y]}{1 - p_0}\right)^2, $$ where $$ E_{\text{BB}}[Y^2] = \text{Var}_{\text{BB}} + (n\mu)^2 = n\mu(1 - \mu) \frac{n + \kappa}{1 + \kappa} + n^2 \mu^2. $$

The result is clamped to be non-negative via pmax(., 0).

References

Ghosal, S., Ghosh, S., and Moores, M. (2020). “Hierarchical beta-binomial models for batch effects in cytometry data.” Journal of the Royal Statistical Society: Series A, 183(4), 1579–1601.

See also

Other distributions: compute_p0(), compute_ztbb_mean(), dbetabinom(), dhurdle_betabinom(), dztbetabinom(), hurdle_mean(), pbetabinom(), rbetabinom(), rhurdle_betabinom(), rztbetabinom()

Examples

hurdle_variance(n = 10, q = 0.7, mu = 0.3, kappa = 5)
#> [1] 5.635486

# q = 0 always gives 0
hurdle_variance(n = 10, q = 0, mu = 0.3, kappa = 5)
#> [1] 0