Distribution function of the ratio of two hazards

Computes $\Pr(\lambda_t / \lambda_c \le q)$ (or the upper tail $\Pr(\lambda_t / \lambda_c > q)$) for hazards with independent Gamma distributions, $\lambda_t \sim \textrm{Gamma}(a_t, b_t)$ and $\lambda_c \sim \textrm{Gamma}(a_c, b_c)$ (shape and rate). This is the posterior probability used for a time-to-event endpoint evaluated by the hazard ratio with a concurrent, external or hypothetical control.

Usage

phazardratio(q, shape_t, rate_t, shape_c, rate_c, lower.tail = TRUE)

Arguments

q: Numeric vector of quantiles (hazard ratios).
shape_t, rate_t: Positive shape and rate of the treatment hazard.
shape_c, rate_c: Positive shape and rate of the control hazard.
lower.tail: Logical; if TRUE (default), probabilities are $\Pr(\lambda_t / \lambda_c \le q)$, otherwise $\Pr(\lambda_t / \lambda_c > q)$.

Value

A numeric vector of probabilities. All arguments except lower.tail are recycled to a common length; each must have length 1 or the common length.

Details

Because $2 b_k \lambda_k$ has the chi-squared distribution with $2 a_k$ degrees of freedom, $(b_t \lambda_t / a_t) / (b_c \lambda_c / a_c)$ has the F distribution with $2 a_t$ and $2 a_c$ degrees of freedom, and $$\Pr(\lambda_t / \lambda_c \le q) = F_{2 a_t, 2 a_c}\left(\frac{q \, a_c \, b_t}{a_t \, b_c}\right),$$ which is computed exactly by pf.

Examples

# Posterior probability that the hazard ratio is below 0.7 with 20 events
# in 300 months of follow-up versus 30 events in 250 months (Gamma(0.1, 1)
# priors)
phazardratio(0.7, shape_t = 20.1, rate_t = 301, shape_c = 30.1,
             rate_c = 251)
#> [1] 0.7923543