Binomial marginal likelihood of the KL calibration's hypothetical results

The KL criterion compares the posterior of the power parameter under two hypothetical target results. With binomial likelihoods that posterior is the Beta prior times the marginal likelihood m(gamma) = Z_T(gamma) / Z_S(gamma) of the binomial power prior (see binomial_power_prior_log_marginal()), which does not depend on the Beta prior. It is therefore tabulated once per hypothetical result, on a grid of power parameters dense near 0 where Beta priors with small shapes put their quadrature nodes, and interpolated during the optimisation.

The hypothetical results are the expected responder counts of the design: the control arm at the design's control rate, the treatment arm at that rate plus the hypothetical risk difference. The binomial likelihood is defined for fractional counts, as the normal criterion's hypothetical estimate is the expected one.

Usage

npp_kl_binomial_log_marginals(
  source_counts,
  n_control,
  n_treatment,
  control_rate,
  effects,
  n_lattice = 1000L,
  n_grid = 801L
)

Arguments

source_counts

List of the four source counts.

n_control, n_treatment

Target arm sizes.

control_rate

The design's target control rate.

effects

Named vector of hypothetical risk differences.

n_lattice

Number of lattice points.

n_grid

Number of power parameters in the table.

Value

A list with gamma and one vector of log marginal likelihoods per element of effects.