Package {bfbin2arm}

Type: Package
Title: Bayes Factor Design for Two-Arm Binomial Trials
Version: 0.1.2
Date: 2026-05-06
Description: Design and analysis of one- and two-stage two-arm binomial clinical phase II trials using Bayes factors. Implements Bayes factors for point-null and directional hypotheses, predictive densities under different hypotheses, and power and sample size calibration.
Depends: R (≥ 4.0.0)
Imports: stats, VGAM, dplyr, parallel, utils, ggplot2, patchwork
Suggests: knitr, rmarkdown, testthat (≥ 3.0.0)
License: GPL-3
Encoding: UTF-8
RoxygenNote: 7.3.3
VignetteBuilder: knitr
URL: https://arxiv.org/abs/2511.23144, https://arxiv.org/abs/2603.01715, https://arxiv.org/abs/2502.02914
NeedsCompilation: no
Packaged: 2026-05-07 13:55:45 UTC; riko
Author: Riko Kelter [aut, cre]
Maintainer: Riko Kelter <rkelter@uni-koeln.de>
Repository: CRAN
Date/Publication: 2026-05-07 15:31:48 UTC

bfbin2arm: Bayesian Power for Two-Arm Binomial Bayes Factors

Description

Functions for Bayesian power and sample size calculation in two-arm binomial trials using Bayes factors for point-null and directional hypotheses.

Details

The package provides simulation-free and simulation-based methods to calibrate Bayes factor designs in two-arm phase II trials with binary endpoints, including power and type-I-error calculations and visualization tools.

Author(s)

Maintainer: Riko Kelter rkelter@uni-koeln.de

See Also

Useful links:

Bayes factor BF-0: H- vs H0

Description

Bayes factor BF-0: H- vs H0

Usage

BFminus0(BFminus1, BF01)

Arguments

BFminus1

Value of BF_{-1}.

BF01

Value of BF_{01}.

Value

Numeric scalar, BF_{-0}.

Bayes factor BF-1: H- vs H1

Description

Bayes factor BF-1: H- vs H1

Usage

BFminus1(y1, y2, n1, n2, a_1_a = 1, b_1_a = 1, a_2_a = 1, b_2_a = 1)

Arguments

y1

Number of successes in arm 1 (control).

y2

Number of successes in arm 2 (treatment).

n1

Sample size in arm 1.

n2

Sample size in arm 2.

a_1_a, b_1_a

Shape parameters of the Beta prior for p_1 under the alternative (analysis prior).

a_2_a, b_2_a

Shape parameters of the Beta prior for p_2 under the alternative (analysis prior).

Value

Numeric scalar, BF_{-1}.

Bayes factor BF+0: H+ vs H0

Description

Bayes factor BF+0: H+ vs H0

Usage

BFplus0(BFplus1, BF01)

Arguments

BFplus1

Value of BF_{+1}.

BF01

Value of BF_{01}.

Value

Numeric scalar, BF_{+0}.

Bayes factor BF+1: H+ vs H1

Description

Bayes factor BF+1: H+ vs H1

Usage

BFplus1(y1, y2, n1, n2, a_1_a = 1, b_1_a = 1, a_2_a = 1, b_2_a = 1)

Arguments

y1

Number of successes in arm 1 (control).

y2

Number of successes in arm 2 (treatment).

n1

Sample size in arm 1.

n2

Sample size in arm 2.

a_1_a, b_1_a

Shape parameters of the Beta prior for p_1 under the alternative (analysis prior).

a_2_a, b_2_a

Shape parameters of the Beta prior for p_2 under the alternative (analysis prior).

Value

Numeric scalar, BF_{+1} = posterior odds / prior odds for H+ vs H1.

Bayes factor BF+-: H+ vs H-

Description

Bayes factor BF+-: H+ vs H-

Usage

BFplusMinus(BFplus1, BFminus1)

Arguments

BFplus1

Value of BF_{+1}.

BFminus1

Value of BF_{-1}.

Value

Numeric scalar, BF_{+-}.

Exact frequentist OCs for one two-stage two-arm BF design

Description

Exact frequentist OCs for one two-stage two-arm BF design

Usage

compute_freq_twostage_oc_2arm(
  n1_1,
  n1_2,
  n2_1,
  n2_2,
  k,
  k_f,
  test,
  p1_power,
  p2_power,
  p_null_grid = NULL,
  a_0_a,
  b_0_a,
  a_1_a,
  b_1_a,
  a_2_a,
  b_2_a,
  a_1_a_Hminus,
  b_1_a_Hminus,
  a_2_a_Hminus,
  b_2_a_Hminus
)

Exact frequentist operating characteristics for a fixed two-stage two-arm BF design

Description

Internal helper. Computes the exact frequentist rejection probability of the two-stage design with futility stopping at interim and efficacy decision at final.

Usage

freq_oc_twostage_twoarm_fixed(
  n1_1,
  n1_2,
  n2_1,
  n2_2,
  k,
  k_f,
  test,
  p1,
  p2,
  a_0_a,
  b_0_a,
  a_1_a,
  b_1_a,
  a_2_a,
  b_2_a,
  a_1_a_Hminus,
  b_1_a_Hminus,
  a_2_a_Hminus,
  b_2_a_Hminus
)

Fixed-sample frequentist type-I error supremum over a null grid

Description

Internal helper. For a given fixed-sample design (n1, n2), computes the supremum of the frequentist type-I error over a grid of null parameter configurations, using freq_oc_twoarm_fixed().

Usage

freq_t1e_sup_fixed(
  n1,
  n2,
  k,
  k_f,
  test,
  p_null_grid = NULL,
  a_0_a,
  b_0_a,
  a_1_a,
  b_1_a,
  a_2_a,
  b_2_a,
  a_1_a_Hminus,
  b_1_a_Hminus,
  a_2_a_Hminus,
  b_2_a_Hminus,
  alpha_target = NULL,
  tol_excess = 1e-04
)

Two-stage frequentist type-I error supremum over a null grid

Description

Internal helper. For a given two-stage design (n1_1, n1_2, n2_1, n2_2), computes the supremum of the two-stage frequentist type-I error over a grid of null parameter configurations, using freq_oc_twostage_twoarm_fixed().

Usage

freq_t1e_twostage_twoarm_sup(
  n1_1,
  n1_2,
  n2_1,
  n2_2,
  k,
  k_f,
  test,
  p_null_grid = NULL,
  a_0_a,
  b_0_a,
  a_1_a,
  b_1_a,
  a_2_a,
  b_2_a,
  a_1_a_Hminus,
  b_1_a_Hminus,
  a_2_a_Hminus,
  b_2_a_Hminus,
  alpha_target = NULL,
  tol_excess = 1e-04
)

Sample size calibration for two-arm binomial Bayes factor designs

Description

Searches over a grid of total sample sizes n to find the smallest n such that Bayesian power, Bayesian type-I error, and probability of compelling evidence under H0 meet specified design criteria. Optionally, frequentist type-I error and power constraints are also evaluated. Unequal fixed randomisation between the two arms is allowed via alloc1 and alloc2.

Usage

ntwoarmbinbf01(
  k = 1/3,
  k_f = 1/3,
  power = 0.8,
  alpha = 0.05,
  pce_H0 = 0.9,
  test = c("BF01", "BF+0", "BF-0", "BF+-"),
  nrange = c(10, 150),
  n_step = 1,
  progress = TRUE,
  compute_freq_t1e = FALSE,
  p1_grid = seq(0.01, 0.99, 0.02),
  p2_grid = seq(0.01, 0.99, 0.02),
  p1_power = NULL,
  p2_power = NULL,
  a_0_d = 1,
  b_0_d = 1,
  a_0_a = 1,
  b_0_a = 1,
  a_1_d = 1,
  b_1_d = 1,
  a_2_d = 1,
  b_2_d = 1,
  a_1_a = 1,
  b_1_a = 1,
  a_2_a = 1,
  b_2_a = 1,
  output = c("plot", "numeric"),
  a_1_d_Hminus = 1,
  b_1_d_Hminus = 1,
  a_2_d_Hminus = 1,
  b_2_d_Hminus = 1,
  a_1_a_Hminus = 1,
  b_1_a_Hminus = 1,
  a_2_a_Hminus = 1,
  b_2_a_Hminus = 1,
  alloc1 = 0.5,
  alloc2 = 0.5
)

Arguments

k

Evidence threshold for rejecting the null (inverted BF).

k_f

Evidence threshold for "compelling evidence" in favour of the null.

power

Target Bayesian power.

alpha

Target Bayesian type-I error.

pce_H0

Target probability of compelling evidence under H_0.

test

Character string, one of "BF01", "BF+0", "BF-0", "BF+-".

nrange

Integer vector of length 2 giving the search range for total n.

n_step

Step size for n.

progress

Logical; if TRUE, print progress to the console.

compute_freq_t1e

Logical; if TRUE, compute frequentist type-I error over a grid.

p1_grid, p2_grid

Grids of true proportions for frequentist T1E.

p1_power, p2_power

Optional true proportions for frequentist power.

a_0_d, b_0_d, a_0_a, b_0_a

Shape parameters for design and analysis priors under H_0.

a_1_d, b_1_d, a_2_d, b_2_d

Shape parameters for design priors under H_1 or H_+.

a_1_a, b_1_a, a_2_a, b_2_a

Shape parameters for analysis priors under H_1 or H_+.

output

"plot" or "numeric".

a_1_d_Hminus, b_1_d_Hminus, a_2_d_Hminus, b_2_d_Hminus

Optional design priors under H_- for directional tests.

a_1_a_Hminus, b_1_a_Hminus, a_2_a_Hminus, b_2_a_Hminus

Shape parameters for analysis priors under H-.

alloc1, alloc2

Fixed randomisation probabilities for arm 1 and arm 2; must be positive and sum to 1. Defaults are 0.5 and 0.5.

Value

If output = "plot", returns invisibly a list with recommended sample sizes and a ggplot object printed to the device. If output = "numeric", returns a list with recommended n and summary.

Examples

# Standard calibration with equal allocation: power 80%, type-I 5%, CE(H0) 80%

ntwoarmbinbf01(power = 0.8, alpha = 0.05, pce_H0 = 0.8, output = "numeric")


# 1:2 allocation (control:treatment) via alloc1 = 1/3, alloc2 = 2/3

ntwoarmbinbf01(power = 0.8, alpha = 0.05, pce_H0 = 0.8,
               alloc1 = 1/3, alloc2 = 2/3, output = "numeric")


# BF+0 directional test with plot

ntwoarmbinbf01(power = 0.8, alpha = 0.05, pce_H0 = 0.9,
               test = "BF+0", output = "plot")

Optimal two-stage two-arm Bayes-factor design for binary endpoints

Description

Computes an optimal two-stage two-arm Bayes-factor design for binary endpoints, minimizing the expected sample size under the null hypothesis while correcting the operating characteristics for the possibility of early stopping for futility.

Usage

optimal_twostage_2arm_bf(
  alpha = 0.05,
  beta = 0.2,
  k = 1/3,
  k_f = 3,
  n1_min = c(5, 5),
  n2_max = c(50, 50),
  alloc1 = 0.5,
  alloc2 = 0.5,
  power_cushion = 0,
  pceH0 = NULL,
  interim_fraction = c(0, 1),
  grid_step = 1L,
  coarse_step = 10L,
  progress = TRUE,
  max_iter = 10000L,
  ncores = getOption("bfbin2arm.ncores", 1L),
  compute_freq_oc = NULL,
  calibration_mode = c("Bayesian", "frequentist", "hybrid"),
  calibration_EN = NULL,
  p1_EN_H0 = NULL,
  p2_EN_H0 = NULL,
  alpha_freq = alpha,
  beta_freq = beta,
  p1_power = NULL,
  p2_power = NULL,
  p_null_grid = NULL,
  test = "BF01",
  a_0_d = 1,
  b_0_d = 1,
  a_0_a = 1,
  b_0_a = 1,
  a_1_d = 1,
  b_1_d = 1,
  a_2_d = 1,
  b_2_d = 1,
  a_1_a = 1,
  b_1_a = 1,
  a_2_a = 1,
  b_2_a = 1,
  a_1_d_Hminus = 1,
  b_1_d_Hminus = 1,
  a_2_d_Hminus = 1,
  b_2_d_Hminus = 1,
  a_1_a_Hminus = 1,
  b_1_a_Hminus = 1,
  a_2_a_Hminus = 1,
  b_2_a_Hminus = 1
)

Arguments

alpha

Numeric scalar, Bayesian type-I-error target.

beta

Numeric scalar, 1 minus the minimal Bayesian power target.

k

Numeric scalar, efficacy threshold; evidence against the null hypothesis is declared when the corresponding Bayes factor is smaller than k.

k_f

Numeric scalar, futility threshold; compelling evidence for the null hypothesis is declared when the corresponding Bayes factor is at least k_f.

n1_min

Numeric vector of length 2, minimum interim sample sizes for arms 1 and 2.

n2_max

Numeric vector of length 2, maximum final sample sizes for arms 1 and 2.

alloc1, alloc2

Positive numbers, allocation probabilities to arms 1 and 2.

power_cushion

Numeric scalar, optional extra power cushion used in the fixed-sample search of step 1.

pceH0

Optional numeric scalar in [0,1]. If specified, candidate two-stage designs must satisfy corrected CE_H0 >= pceH0.

interim_fraction

Numeric vector of length 2 giving lower and upper bounds for the interim sample size in each arm as a fraction of the fixed sample size.

grid_step

Positive integer giving the spacing of the interim design grid.

coarse_step

Positive integer giving the spacing of the coarse fixed-sample search grid in step 1.

progress

Logical; if TRUE, prints progress information.

max_iter

Integer, maximum number of total fixed-sample sizes searched in step 1.

ncores

Integer; number of parallel worker processes to use in the calibration. Defaults to getOption("bfbin2arm.ncores", 1L). In vignettes and examples, a conservative value (e.g. 1 or 2) is recommended for CRAN checks, whereas users can increase this to exploit all available cores on their own machines.

compute_freq_oc

Logical or NULL. Controls whether frequentist operating characteristics are computed for candidate two-stage designs during the search.

calibration_mode

Character string specifying the calibration mode. Must be one of "Bayesian", "frequentist", or "hybrid".

calibration_EN

Character string or NULL specifying whether the design is ranked by Bayesian or frequentist expected sample size under the null hypothesis.

p1_EN_H0, p2_EN_H0

Numeric scalars specifying the null response probabilities in control and treatment arm used when calibration_EN = "frequentist".

alpha_freq

Numeric scalar, frequentist type-I error target.

beta_freq

Numeric scalar, 1 minus the frequentist power target.

p1_power, p2_power

Numeric scalars specifying the success probabilities in control and treatment arm used for the frequentist power calculation.

p_null_grid

Optional numeric vector giving the grid of null response probabilities used for frequentist type-I-error maximization. If NULL, a default grid is used.

test

Character string, one of "BF01", "BF+0", "BF-0", "BF+-".

a_0_d, b_0_d, a_0_a, b_0_a

Shape parameters for design and analysis priors under H_0.

a_1_d, b_1_d, a_2_d, b_2_d

Shape parameters for design priors under H_1 or H_+.

a_1_a, b_1_a, a_2_a, b_2_a

Shape parameters for analysis priors under H_1 or H_+.

a_1_d_Hminus, b_1_d_Hminus, a_2_d_Hminus, b_2_d_Hminus

Optional design priors under H_- for directional tests.

a_1_a_Hminus, b_1_a_Hminus

Shape parameters of the analysis prior under the directional null hypothesis H0- for arm 1.

a_2_a_Hminus, b_2_a_Hminus

Shape parameters of the analysis prior under the directional null hypothesis H0- for arm 2.

Value

A list with the following components:

design

Four-element integer vector containing the selected two-stage design: interim sample sizes in arms 1 and 2 followed by final sample sizes in arms 1 and 2.

naive_oc

Named list of uncorrected fixed-sample operating characteristics and fixed-sample sizes found in step 1.

occ

Named numeric vector of corrected Bayesian operating characteristics for the selected two-stage design.

priors

List storing design hyperparameters and search settings.

freq_occ

Named numeric vector with fixed-sample and two-stage frequentist operating characteristics for the final design when frequentist calibration or reporting is active; otherwise NULL.

conv

Character string describing the search outcome. Typical values include "converged", "no_feasible_fixed", "no_interim_grid", and "no_feasible_design". In frequentist or hybrid calibration modes, additional informative status values may be returned when the best available design is returned although all requested constraints were not fully satisfied.

Examples

## Fast Bayesian example with small search space
res <- optimal_twostage_2arm_bf(
  alpha = 0.10,
  beta = 0.20,
  k = 1 / 3,
  k_f = 3,
  n1_min = c(3, 3),
  n2_max = c(12, 12),
  alloc1 = 0.5,
  alloc2 = 0.5,
  power_cushion = 0,
  pceH0 = NULL,
  interim_fraction = c(0.25, 0.75),
  grid_step = 2L,
  coarse_step = 4L,
  progress = FALSE,
  max_iter = 24L,
  calibration_mode = "Bayesian",
  test = "BF01",
  a_0_d = 1, b_0_d = 1,
  a_0_a = 1, b_0_a = 1,
  a_1_d = 1, b_1_d = 1,
  a_2_d = 1, b_2_d = 1,
  a_1_a = 1, b_1_a = 1,
  a_2_a = 1, b_2_a = 1
)
res$design
res$occ


res2 <- optimal_twostage_2arm_bf(
  alpha = 0.05,
  beta = 0.20,
  k = 1 / 3,
  k_f = 3,
  n1_min = c(5, 5),
  n2_max = c(20, 20),
  alloc1 = 0.5,
  alloc2 = 0.5,
  power_cushion = 0.02,
  pceH0 = 0.50,
  interim_fraction = c(0.25, 0.75),
  grid_step = 1L,
  coarse_step = 4L,
  progress = FALSE,
  max_iter = 40L,
  calibration_mode = "Bayesian",
  test = "BF+0",
  a_0_d = 1, b_0_d = 1,
  a_0_a = 1, b_0_a = 1,
  a_1_d = 1, b_1_d = 2,
  a_2_d = 2, b_2_d = 1,
  a_1_a = 1, b_1_a = 1,
  a_2_a = 1, b_2_a = 1,
  a_1_d_Hminus = 1, b_1_d_Hminus = 1,
  a_2_d_Hminus = 1, b_2_d_Hminus = 1,
  a_1_a_Hminus = 1, b_1_a_Hminus = 1,
  a_2_a_Hminus = 1, b_2_a_Hminus = 1
)
res2$design
res2$occ

Plot an optimal two-stage two-arm Bayes factor design

Description

Given the result from optimal_twostage_2arm_bf(), this function produces a six-panel base R plot showing the design schematic, operating characteristics, and the design and analysis priors under H_0 and H_1.

Usage

plot_twostage_2arm_bf(
  res,
  main = "Optimal two-stage two-arm Bayes factor design"
)

Arguments

res

A list returned by optimal_twostage_2arm_bf(), containing components $design, $naive_oc, $occ and $priors.

main

Character string with the main title of the plot.

Value

Invisibly returns NULL; called for its side effect of producing a plot.

Examples

res <- optimal_twostage_2arm_bf(
  alpha = 0.10, beta = 0.20, k = 1/3, k_f = 3,
  n1_min = c(3, 3), n2_max = c(8, 8),
  alloc1 = 0.5, alloc2 = 0.5,
  power_cushion = 0,
  interim_fraction = c(0.5, 0.5),
  grid_step = 1,
  progress = FALSE,
  max_iter = 16,
  test = "BF01",
  a_0_d = 1, b_0_d = 1,
  a_0_a = 1, b_0_a = 1,
  a_1_d = 1, b_1_d = 1,
  a_2_d = 1, b_2_d = 1,
  a_1_a = 1, b_1_a = 1,
  a_2_a = 1, b_2_a = 1
)
if (is.numeric(res$design) && length(res$design) == 4 && !anyNA(res$design)) {
  plot_twostage_2arm_bf(res)
}

Posterior probability P(p2 <= p1 | data)

Description

Posterior probability P(p2 <= p1 | data)

Usage

postProbHminus(y1, y2, n1, n2, a_1_a = 1, b_1_a = 1, a_2_a = 1, b_2_a = 1)

Arguments

y1

Number of successes in arm 1 (control).

y2

Number of successes in arm 2 (treatment).

n1

Sample size in arm 1.

n2

Sample size in arm 2.

a_1_a, b_1_a

Shape parameters of the Beta prior for p_1 under the alternative (analysis prior).

a_2_a, b_2_a

Shape parameters of the Beta prior for p_2 under the alternative (analysis prior).

Value

Numeric scalar, posterior probability P(p_2 <= p_1 | y_1, y_2).

Posterior probability P(p2 > p1 | data) under independent Beta priors

Description

Uses Beta posteriors induced by the analysis priors to compute P(p_2 > p_1 \mid y_1, y_2).

Usage

postProbHplus(y1, y2, n1, n2, a_1_a = 1, b_1_a = 1, a_2_a = 1, b_2_a = 1)

Arguments

y1

Number of successes in arm 1 (control).

y2

Number of successes in arm 2 (treatment).

n1

Sample size in arm 1.

n2

Sample size in arm 2.

a_1_a, b_1_a

Shape parameters of the Beta prior for p_1 under the alternative (analysis prior).

a_2_a, b_2_a

Shape parameters of the Beta prior for p_2 under the alternative (analysis prior).

Value

Numeric scalar, posterior probability P(p_2 > p_1 | y_1, y_2).

Bayesian power, type-I error, and PCE(H0) for two-arm binomial Bayes factors

Description

Computes Bayesian power, Bayesian type-I error, and the probability of compelling evidence under H_0 (or H_- for BF+-), for a given sample size and Bayes factor test. Optionally, frequentist type-I error and frequentist power are computed by summing over the rejection region.

Usage

powertwoarmbinbf01(
  n1,
  n2,
  k = 1/3,
  k_f = 1/3,
  test = c("BF01", "BF+0", "BF-0", "BF+-"),
  a_0_d = 1,
  b_0_d = 1,
  a_0_a = 1,
  b_0_a = 1,
  a_1_d = 1,
  b_1_d = 1,
  a_2_d = 1,
  b_2_d = 1,
  a_1_a = 1,
  b_1_a = 1,
  a_2_a = 1,
  b_2_a = 1,
  output = c("numeric", "predDensmatrix", "t1ematrix", "ceH0matrix", "frequentist_t1e"),
  a_1_d_Hminus = 1,
  b_1_d_Hminus = 1,
  a_2_d_Hminus = 1,
  b_2_d_Hminus = 1,
  compute_freq_t1e = FALSE,
  p1_grid = seq(0.01, 0.99, 0.02),
  p2_grid = seq(0.01, 0.99, 0.02),
  p1_power = NULL,
  p2_power = NULL,
  a_1_a_Hminus = 1,
  b_1_a_Hminus = 1,
  a_2_a_Hminus = 1,
  b_2_a_Hminus = 1
)

Arguments

n1, n2

Sample sizes in arms 1 and 2.

k

Evidence threshold for rejecting the null (inverted BF).

k_f

Evidence threshold for "compelling evidence" in favour of the null.

test

Character string, one of "BF01", "BF+0", "BF-0", "BF+-".

a_0_d, b_0_d, a_0_a, b_0_a

Shape parameters for design and analysis priors under H_0.

a_1_d, b_1_d, a_2_d, b_2_d

Shape parameters for design priors under H_1 or H_+.

a_1_a, b_1_a, a_2_a, b_2_a

Shape parameters for analysis priors under H_1 or H_+.

output

One of "numeric", "predDensmatrix", "t1ematrix", "ceH0matrix", "frequentist_t1e".

a_1_d_Hminus, b_1_d_Hminus, a_2_d_Hminus, b_2_d_Hminus

Optional design priors under H_- for directional tests.

compute_freq_t1e

Logical; if TRUE, compute frequentist type-I error over a grid.

p1_grid, p2_grid

Grids of true proportions for frequentist T1E.

p1_power, p2_power

Optional true proportions for frequentist power.

a_1_a_Hminus, b_1_a_Hminus, a_2_a_Hminus, b_2_a_Hminus

Shape parameters for analysis priors under H_- (directional tests).

Value

Depending on output, either a named numeric vector with components Power, Type1_Error, CE_H0 (and optionally frequentist metrics) or matrices of predictive densities.

Examples

# Basic Bayesian power for BF01 test
powertwoarmbinbf01(n1 = 30, n2 = 30, k = 1/3, test = "BF01")

# Directional test BF+0 with frequentist type-I error
powertwoarmbinbf01(n1 = 40, n2 = 40, k = 1/3, k_f = 3,
                   test = "BF+0", compute_freq_t1e = TRUE)

# Predictive density matrices (advanced)
powertwoarmbinbf01(n1 = 25, n2 = 25, output = "predDensmatrix")

Predictive density under H0: p1 = p2 = p

Description

Beta-binomial predictive density for data (y1,y2) under H0.

Usage

predictiveDensityH0(y1, y2, n1, n2, a_0_d = 1, b_0_d = 1)

Arguments

y1

Number of successes in arm 1 (control).

y2

Number of successes in arm 2 (treatment).

n1

Sample size in arm 1.

n2

Sample size in arm 2.

a_0_d, b_0_d

Design-prior parameters for common p under H0.

Value

Numeric scalar, predictive density.

Predictive density under H1: p1 != p2

Description

Product of two independent Beta-binomial predictive densities.

Usage

predictiveDensityH1(y1, y2, n1, n2, a_1_d = 1, b_1_d = 1, a_2_d = 1, b_2_d = 1)

Arguments

y1

Number of successes in arm 1 (control).

y2

Number of successes in arm 2 (treatment).

n1

Sample size in arm 1.

n2

Sample size in arm 2.

a_1_d, b_1_d

Design-prior parameters for p1.

a_2_d, b_2_d

Design-prior parameters for p2.

Value

Numeric scalar, predictive density.

Predictive density under H-: p2 <= p1 (truncated prior)

Description

Predictive density under H-: p2 <= p1 (truncated prior)

Usage

predictiveDensityHminus_trunc(
  y1,
  y2,
  n1,
  n2,
  a_1_d = 1,
  b_1_d = 1,
  a_2_d = 1,
  b_2_d = 1
)

Arguments

y1

Number of successes in arm 1 (control).

y2

Number of successes in arm 2 (treatment).

n1

Sample size in arm 1.

n2

Sample size in arm 2.

a_1_d, b_1_d

Design-prior parameters for p1.

a_2_d, b_2_d

Design-prior parameters for p2.

Value

Numeric scalar, predictive density under H-.

Predictive density under H+: p2 > p1 (truncated prior)

Description

Predictive density under H+: p2 > p1 (truncated prior)

Usage

predictiveDensityHplus_trunc(
  y1,
  y2,
  n1,
  n2,
  a_1_d = 1,
  b_1_d = 1,
  a_2_d = 1,
  b_2_d = 1
)

Arguments

y1

Number of successes in arm 1 (control).

y2

Number of successes in arm 2 (treatment).

n1

Sample size in arm 1.

n2

Sample size in arm 2.

a_1_d, b_1_d

Design-prior parameters for p1.

a_2_d, b_2_d

Design-prior parameters for p2.

Value

Numeric scalar, predictive density under H+.

Prior probability P(p2 <= p1) under independent Beta priors

Description

Prior probability P(p2 <= p1) under independent Beta priors

Usage

priorProbHminus(a_1_a, b_1_a, a_2_a, b_2_a)

Arguments

a_1_a, b_1_a

Shape parameters of the Beta prior for p_1.

a_2_a, b_2_a

Shape parameters of the Beta prior for p_2.

Value

Numeric scalar, prior probability P(p2 <= p1).

Prior probability P(p2 > p1) under independent Beta priors

Description

Prior probability P(p2 > p1) under independent Beta priors

Usage

priorProbHplus(a_1_a, b_1_a, a_2_a, b_2_a)

Arguments

a_1_a, b_1_a

Shape parameters of the Beta prior for p_1.

a_2_a, b_2_a

Shape parameters of the Beta prior for p_2.

Value

Numeric scalar, prior probability P(p2 > p1).

Two-arm binomial Bayes factor BF01

Description

Computes the Bayes factor BF_{01} comparing the point-null H_0: p_1 = p_2 to the alternative H_1: p_1 \neq p_2 in a two-arm binomial setting with Beta priors.

Usage

twoarmbinbf01(
  y1,
  y2,
  n1,
  n2,
  a_0_a = 1,
  b_0_a = 1,
  a_1_a = 1,
  b_1_a = 1,
  a_2_a = 1,
  b_2_a = 1
)

Arguments

y1

Number of successes in arm 1 (control).

y2

Number of successes in arm 2 (treatment).

n1

Sample size in arm 1.

n2

Sample size in arm 2.

a_0_a, b_0_a

Shape parameters of the Beta prior for the common-p under the null model (analysis prior).

a_1_a, b_1_a

Shape parameters of the Beta prior for p_1 under the alternative (analysis prior).

a_2_a, b_2_a

Shape parameters of the Beta prior for p_2 under the alternative (analysis prior).

Value

Numeric scalar, the Bayes factor BF_{01}.

Examples

twoarmbinbf01(10, 20, 30, 30)

Bayes factor BF+0 for the directional alternative vs point-null

Description

Computes the Bayes factor BF_{+0} comparing the directional alternative hypothesis H_+ (p_2 > p_1) against the point-null H_0 (p_1 = p_2).

Usage

twoarmbinbf_plus0_direct(
  y1,
  y2,
  n1,
  n2,
  a_0_a = 1,
  b_0_a = 1,
  a_1_a = 1,
  b_1_a = 1,
  a_2_a = 1,
  b_2_a = 1
)

Arguments

y1, y2

Integer counts of successes in arms 1 and 2.

n1, n2

Integer sample sizes in arms 1 and 2.

a_0_a, b_0_a

Shape parameters of the analysis prior for the common response probability under H_0.

a_1_a, b_1_a

Shape parameters of the analysis prior for the response probability in arm 1 under H_+.

a_2_a, b_2_a

Shape parameters of the analysis prior for the response probability in arm 2 under H_+.

Details

Both marginal likelihoods are formed using the analysis priors, which represent inferential beliefs at the time the data are evaluated. The design priors are used only for computing Bayesian operating characteristics (predictive power / type-I error) and play no role here.

Value

Numeric scalar; the Bayes factor BF_{+0} = m_+(y_1, y_2) / m_0(y_1, y_2).