Convert CV(alpha) to Variance

Converts a coefficient of variation specification for \(\alpha\) to the implied variance of \(K_J\) under the A1 approximation.

Usage

cv_alpha_to_variance(mu_K, cv_alpha)

Arguments

mu_K

Numeric; target prior mean of \(K_J\).

cv_alpha

Numeric; target coefficient of variation for \(\alpha\).

Value

Numeric; implied variance of \(K_J\).

Details

Under the A1 approximation: $$\text{CV}(\alpha) = 1/\sqrt{a} = \frac{\sqrt{\sigma^2_K - m}}{m}$$

where \(m = \mu_K - 1\). Inverting: $$\sigma^2_K = m + (\text{CV}(\alpha) \cdot m)^2 = m(1 + \text{CV}(\alpha)^2 \cdot m)$$

See also

vif_to_variance for VIF-based specification

Examples

# CV(alpha) = 0.5 means moderate prior concentration
var_K <- cv_alpha_to_variance(mu_K = 5, cv_alpha = 0.5)

# Verify round-trip
fit <- DPprior_a1(J = 50, mu_K = 5, var_K = var_K)
1 / sqrt(fit$a)  # Should be approximately 0.5
#> [1] 0.5