Posterior of the treatment effect under the binomial power prior

The binomial conditional power prior: uniform priors on the source control rate u and the target control rate v, a uniform prior on the common treatment effect theta over (-min(u, v), 1 - max(u, v)), which has density 1 / (1 - |u - v|), and the source likelihood raised to the power gamma.

The rates are discretised on the lattice of binomial_npp_prior_kernels(): the midpoints of N equal cells of \([0, 1]\), with the treatment effect on the multiples of the cell width, so that every treatment rate is a lattice point. The lattice covers the whole unit square, so it follows the posterior wherever the target data move it, including far into the tails of the source likelihood when the two studies conflict. Nodes placed on the quantiles of the source and target control rates' own likelihoods, as an earlier version of this function used, miss that region: under conflict the posterior probability of benefit was off by 0.016 and the posterior mean by up to 0.07 against Stan.

Only the target control rates and treatment rates where the target likelihood exceeds 1e-20 of its maximum are visited, and every source control rate, so the cost is N times the number of target lattice points per arm times the number of treatment effects they reach.

With no target patients the target counts are zero and the result is the prior.

Usage

binomial_power_prior_posterior(
  power_parameter,
  n_control_source,
  n_successes_control_source,
  n_treatment_source,
  n_successes_treatment_source,
  n_control,
  n_successes_control,
  n_treatment,
  n_successes_treatment,
  n_lattice = 1000L,
  control_rate = NULL
)

Arguments

power_parameter

The power parameter gamma, in \([0, 1]\).

n_control_source, n_successes_control_source

Source control arm.

n_treatment_source, n_successes_treatment_source

Source treatment arm.

n_control, n_successes_control

Target control arm.

n_treatment, n_successes_treatment

Target treatment arm.

n_lattice

Number of lattice points N on the rates.

control_rate

Target control rate to condition on, or NULL to integrate it out. Conditioning puts the target control rate's whole mass on the lattice cell that contains it, which confines the treatment effect to the differences that keep the target treatment rate in \([0, 1]\).

Value

A grid_posterior() list.