Version:	0.1.5
Date:	2025-05-15
Author:	Samuel Pawel [aut, cre]
Maintainer:	Samuel Pawel <samuel.pawel@uzh.ch>
Title:	Power and Sample Size Calculations for Bayes Factor Analysis
Description:	Implements z-test, t-test, and normal moment prior Bayes factors based on summary statistics, along with functionality to perform corresponding power and sample size calculations as described in Pawel and Held (2025) <doi:10.1080/00031305.2025.2467919>.
License:	GPL-3
Encoding:	UTF-8
Imports:	lamW
Suggests:	roxygen2, tinytest, knitr
VignetteBuilder:	knitr
NeedsCompilation:	no
RoxygenNote:	7.3.1
URL:	https://github.com/SamCH93/bfpwr
BugReports:	https://github.com/SamCH93/bfpwr/issues
Packaged:	2025-05-15 07:37:20 UTC; sam
Repository:	CRAN
Date/Publication:	2025-05-15 08:20:06 UTC

z-test Bayes factor

Description

This function computes the Bayes factor that quantifies the evidence that the data (in the form of an asymptotically normally distributed parameter estimate with standard error) provide for a point null hypothesis with a normal prior assigned to the parameter under the alternative. The standard error is assumed to be known.

Usage

bf01(estimate, se, null = 0, pm, psd, log = FALSE)

Arguments

Value

Bayes factor in favor of the null hypothesis over the alternative (\text{BF}_{01} > 1 indicates evidence for the null hypothesis, whereas \text{BF}_{01} < 1 indicates evidence for the alternative)

Author(s)

Samuel Pawel

Examples

bf01(estimate = 0.2, se = 0.05, null = 0, pm = 0, psd = 2)

Binomial Bayes factor

Description

This function computes the Bayes factor for testing a binomial proportion p based on x observed successes out of n trials. Two types of tests are available:

• Test of a point null hypothesis: The Bayes factor quantifies the evidence for H_0 \colon p = p_0 against H_1 \colon p \neq p_0. A beta prior is assigned to the proportion p under the alternative hypothesis H_1.

• Test of a directional null hypothesis: The Bayes factor quantifies the evidence for H_0 \colon p \leq p_0 against H_1 \colon p > p_0. A beta prior that is truncated to the range [0, p_0] under the null H_0 and to (p_0, 1] under the alternative H_1 is assigned to the proportion p under the corresponding hypothesis.

Usage

binbf01(
  x,
  n,
  p0 = 0.5,
  type = c("point", "direction"),
  a = 1,
  b = 1,
  log = FALSE
)

Arguments

Value

Bayes factor in favor of the null hypothesis over the alternative (\text{BF}_{01} > 1 indicates evidence for the null hypothesis, whereas \text{BF}_{01} < 1 indicates evidence for the alternative)

Author(s)

Samuel Pawel

See Also

pbinbf01, nbinbf01

Examples

## example on Mendelian inheritance from ?stats::binom.test
binbf01(x = 682, n = 925, p0 = 3/4, a = 1, b = 1, type = "point")
## 18.6 => strong evidence for the hypothesized p = 3/4 compared to other p

## with directional hypothesis
binbf01(x = 682, n = 925, p0 = 3/4, a = 1, b = 1, type = "direction")
## 1.5 => only anecdotal evidence for p <= 3/4 over p > 3/4

## Particle-counting experiment from Stone (1997) with point null
binbf01(x = 106298, n = 527135, p0 = 0.2, a = 1, b = 1, type = "point")
## 8.1 => moderate evidence for the alternative over the null

## Coin flip experiment from Bartos et al. (2023) with point null
binbf01(x = 178079, n = 350757 , p0 = 0.5, a = 5100, b = 4900, type = "point")
## => 1/1.72e+17 extreme evidence in favor of the alternative over the null

Sample size determination for z-test Bayes factor

Description

This function computes the required sample size to obtain a Bayes factor (bf01) more extreme than a threshold k with a specified target power.

Usage

nbf01(
  k,
  power,
  usd,
  null = 0,
  pm,
  psd,
  dpm = pm,
  dpsd = psd,
  nrange = c(1, 10^5),
  lower.tail = TRUE,
  integer = TRUE,
  analytical = TRUE,
  ...
)

Arguments

Details

It is assumed that the standard error of the future parameter estimate is of the form \code{se} =\code{usd}/\sqrt{\code{n}}. For example, for normally distributed data with known standard deviation sd and two equally sized groups of size n, the standard error of an estimated standardized mean difference is \code{se} = \code{sd}\sqrt{2/n}, so the corresponding unit standard deviation is \code{usd} = \code{sd}\sqrt{2}. See the vignette for more information.

Value

The required sample size to achieve the specified power

Note

A warning message will be displayed in case that the specified target power is not achievable under the specified analysis and design priors.

Author(s)

Samuel Pawel

See Also

pbf01, powerbf01, bf01

Examples

## point alternative (analytical and numerical solution available)
nbf01(k = 1/10, power = 0.9, usd = 1, null = 0, pm = 0.5, psd = 0,
      analytical = c(TRUE, FALSE), integer = FALSE)

## standardized mean difference (usd = sqrt(2), effective sample size = per group size)
nbf01(k = 1/10, power = 0.9, usd = sqrt(2), null = 0, pm = 0, psd = 1)
## this is the sample size per group (assuming equally sized groups)

## z-transformed correlation (usd = 1, effective sample size = n - 3)
nbf01(k = 1/10, power = 0.9, usd = 1, null = 0, pm = 0.2, psd = 0.5)
## have to add 3 to obtain the actual sample size

## log hazard/odds ratio (usd = 2, effective sample size = total number of events)
nbf01(k = 1/10, power = 0.9, usd = 2, null = 0, pm = 0, psd = sqrt(0.5))
## have to convert the number of events to a sample size

Sample size determination for binomial Bayes factor

Description

This function computes the required sample size to obtain a binomial Bayes factor (binbf01) more extreme than a threshold k with a specified target power.

Usage

nbinbf01(
  k,
  power,
  p0 = 0.5,
  type = c("point", "direction"),
  a = 1,
  b = 1,
  dp = NA,
  da = a,
  db = b,
  dl = 0,
  du = 1,
  lower.tail = TRUE,
  nrange = c(1, 10^4),
  ...
)

Arguments

Value

The required sample size to achieve the specified power

Author(s)

Samuel Pawel

See Also

pbinbf01, binbf01

Examples

## sample size parameters
pow <- 0.9
p0 <- 3/4
a <- 1
b <- 1
k <- 1/10

## Not run: 
## sample sizes for directional testing
(nH1 <- nbinbf01(k = k, power = pow, p0 = p0, type = "direction", a = a,
                 b = b, da = a, db = b, dl = p0, du = 1))
(nH0 <- nbinbf01(k = 1/k, power = pow, p0 = p0, type = "direction", a = a,
                 b = b, da = a, db = b, dl = 0, du = p0, lower.tail = FALSE))
nseq <- seq(1, 1.1*max(c(nH1, nH0)), length.out = 100)
powH1 <- pbinbf01(k = k, n = nseq, p0 = p0, type = "direction", a = a,
                  b = b, da = a, db = b, dl = p0, du = 1)
powH0 <- pbinbf01(k = 1/k, n = nseq, p0 = p0, type = "direction", a = a,
                  b = b, da = a, db = b, dl = 0, du = p0, lower.tail = FALSE)
matplot(nseq, cbind(powH1, powH0), type = "s", xlab = "n", ylab = "Power", lty = 1,
        ylim = c(0, 1), col = c(2, 4), las = 1)
abline(h = pow, lty = 2)
abline(v = c(nH1, nH0), col = c(2, 4), lty = 2)
legend("topleft", legend = c("H1", "H0"), lty = 1, col = c(2, 4))

## sample sizes for point null testing
(nH1 <- nbinbf01(k = k, power = pow, p0 = p0, type = "point", a = a,
                 b = b, da = a, db = b))
(nH0 <- nbinbf01(k = 1/k, power = pow, p0 = p0, type = "point", a = a,
                 b = b, dp = p0, lower.tail = FALSE, nrange = c(1, 10^5)))
nseq <- seq(1, max(c(nH1, nH0)), length.out = 100)
powH1 <- pbinbf01(k = k, n = nseq, p0 = p0, type = "point", a = a,
                  b = b, da = a, db = b, dl = 0, du = 1)
powH0 <- pbinbf01(k = 1/k, n = nseq, p0 = p0, type = "point", a = a,
                  b = b, dp = p0, lower.tail = FALSE)
matplot(nseq, cbind(powH1, powH0), type = "s", xlab = "n", ylab = "Power", lty = 1,
        ylim = c(0, 1), col = c(2, 4), las = 1)
abline(h = pow, lty = 2)
abline(v = c(nH1, nH0), col = c(2, 4), lty = 2)
legend("topleft", legend = c("H1", "H0"), lty = 1, col = c(2, 4))

## End(Not run)

Normal moment prior Bayes factor

Description

This function computes the Bayes factor that quantifies the evidence that the data (in the form of an asymptotically normally distributed parameter estimate with standard error) provide for a point null hypothesis with a normal moment prior assigned to the parameter under the alternative.

Usage

nmbf01(estimate, se, null = 0, psd, log = FALSE)

Arguments

Details

A normal moment prior has density f(x \mid \code{null}, \code{psd}) = N(x \mid \code{null}, \code{psd}^2) \times (x - \code{null})/ \code{psd}^2 with N(x \mid m, v) the normal density with mean m and variance v evaluated at x.

Value

Bayes factor in favor of the null hypothesis over the alternative (\text{BF}_{01} > 1 indicates evidence for the null hypothesis, whereas \text{BF}_{01} < 1 indicates evidence for the alternative)

Author(s)

Samuel Pawel

References

Johnson, V. E. and Rossell, D. (2010). On the use of non-local prior densities in Bayesian hypothesis tests. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(2):143–170. doi:10.1111/j.1467-9868.2009.00730.x

Pramanik, S. and Johnson, V. E. (2024). Efficient alternatives for Bayesian hypothesis tests in psychology. Psychological Methods, 29(2):243–261. doi:10.1037/met0000482

See Also

nmbf01, pnmbf01, nnmbf01, powernmbf01

Examples

nmbf01(estimate = 0.25, se = 0.05, null = 0, psd = 0.5/sqrt(2)) # mode at 0.5

Sample size determination for normal moment prior Bayes factor

Description

This function computes the required sample size to obtain a normal moment prior Bayes factor (nbf01) more extreme than a threshold k with a specified target power.

Usage

nnmbf01(
  k,
  power,
  usd,
  null = 0,
  psd,
  dpm,
  dpsd,
  nrange = c(1, 10^5),
  lower.tail = TRUE,
  integer = TRUE,
  ...
)

Arguments

Details

It is assumed that the standard error of the future parameter estimate is of the form \code{se} =\code{usd}/\sqrt{\code{n}}. For example, for normally distributed data with known standard deviation sd and two equally sized groups of size n, the standard error of an estimated standardized mean difference is \code{se} = \code{sd}\sqrt{2/n}, so the corresponding unit standard deviation is \code{usd} = \code{sd}\sqrt{2}. See the vignette for more information.

Value

The required sample size to achieve the specified power

Author(s)

Samuel Pawel

See Also

nmbf01, pnmbf01, powernmbf01

Examples

nnmbf01(k = 1/10, power = 0.9, usd = 1, null = 0, psd = 0.5/sqrt(2), dpm = 0.5, dpsd = 0)

Sample size calculations for t-test Bayes factor

Description

This function computes the required sample size to obtain a t-test Bayes factor (tbf01) more extreme than a threshold k with a specified target power.

Usage

ntbf01(
  k,
  power,
  null = 0,
  plocation = 0,
  pscale = 1/sqrt(2),
  pdf = 1,
  type = c("two.sample", "one.sample", "paired"),
  alternative = c("two.sided", "less", "greater"),
  dpm = plocation,
  dpsd = pscale,
  lower.tail = TRUE,
  integer = TRUE,
  nrange = c(2, 10^4),
  ...
)

Arguments

Value

The required sample size to achieve the specified power

Author(s)

Samuel Pawel

See Also

ptbf01, powertbf01, tbf01

Examples

 ## example from Schönbrodt and Wagenmakers (2018, p.135)
 ntbf01(k = 1/6, power = 0.95, dpm = 0.5, dpsd = 0, alternative = "greater")
 ntbf01(k = 1/6, power = 0.95, dpm = 0.5, dpsd = 0.1, alternative = "greater")
 ntbf01(k = 6, power = 0.95, dpm = 0, dpsd = 0, alternative = "greater",
        lower.tail = FALSE, nrange = c(2, 10000))

Cumulative distribution function of the z-test Bayes factor

Description

This function computes the probability of obtaining a Bayes factor (bf01) more extreme than a threshold k with a specified sample size.

Usage

pbf01(k, n, usd, null = 0, pm, psd, dpm = pm, dpsd = psd, lower.tail = TRUE)

Arguments

Details

It is assumed that the standard error of the future parameter estimate is of the form \code{se} =\code{usd}/\sqrt{\code{n}}. For example, for normally distributed data with known standard deviation sd and two equally sized groups of size n, the standard error of an estimated standardized mean difference is \code{se} = \code{sd}\sqrt{2/n}, so the corresponding unit standard deviation is \code{usd} = \code{sd}\sqrt{2}. See the vignette for more information.

Value

The probability that the Bayes factor is less or greater (depending on the specified lower.tail) than the specified threshold k

Author(s)

Samuel Pawel

See Also

nbf01, powerbf01, bf01

Examples

## point alternative (psd = 0)
pbf01(k = 1/10, n = 200, usd = 2, null = 0, pm = 0.5, psd = 0)

## normal alternative (psd > 0)
pbf01(k = 1/10, n = 100, usd = 2, null = 0, pm = 0.5, psd = 2)

## design prior is the null hypothesis (dpm = 0, dpsd = 0)
pbf01(k = 10, n = 1000, usd = 2, null = 0, pm = 0.3, psd = 2, dpm = 0, dpsd = 0, lower.tail = FALSE)

## draw a power curve
nseq <- round(exp(seq(log(10), log(10000), length.out = 100)))
plot(nseq, pbf01(k = 1/10, n = nseq, usd = 2, null = 0, pm = 0.3, psd = 0), type = "l",
     xlab = "n", ylab = bquote("Pr(BF"["01"] <= 1/10 * ")"), ylim = c(0, 1),
     log = "x", las = 1)

## standardized mean difference (usd = sqrt(2), effective sample size = per group size)
n <- 30
pbf01(k = 1/10, n = n, usd = sqrt(2), null = 0, pm = 0, psd = 1)

## z-transformed correlation (usd = 1, effective sample size = n - 3)
n <- 100
pbf01(k = 1/10, n = n - 3, usd = 1, null = 0, pm = 0.2, psd = 0.5)

## log hazard/odds ratio (usd = 2, effective sample size = total number of events)
nevents <- 100
pbf01(k = 1/10, n = nevents, usd = 2, null = 0, pm = 0, psd = sqrt(0.5))

Cumulative distribution function of the binomial Bayes factor

Description

This function computes the probability of obtaining a binomial Bayes factor (binbf01) more extreme than a threshold k with a specified sample size.

Usage

pbinbf01(
  k,
  n,
  p0 = 0.5,
  type = c("point", "direction"),
  a = 1,
  b = 1,
  dp = NA,
  da = a,
  db = b,
  dl = 0,
  du = 1,
  lower.tail = TRUE
)

Arguments

Value

The probability that the Bayes factor is less or greater (depending on the specified lower.tail) than the specified threshold k

Author(s)

Samuel Pawel

See Also

binbf01, nbinbf01

Examples

## compute probability that BF > 10 under the point null
a <- 1
b <- 1
p0 <- 3/4
k <- 10
nseq <- seq(1, 1000, length.out = 100)
powH0 <- pbinbf01(k = k, n = nseq, p0 = p0, type = "point", a = a, b = b,
                  dp = p0, lower.tail = FALSE)
plot(nseq, powH0, type = "s", xlab = "n", ylab = "Power")

## compare to normal approximation
pm <- a/(a + b) # prior mean under H1
psd <- sqrt(a*b/(a + b)^2/(a + b + 1)) # prior standard deviation under H1
pownormH0 <- pbf01(k = k, n = nseq, usd = sqrt(p0*(1 - p0)), null = p0,
                   pm = pm, psd = psd, dpm = p0, dpsd = 0, lower.tail = FALSE)
lines(nseq, pownormH0, type = "s", col = 2)
legend("right", legend = c("Exact", "Normal approximation"), lty = 1,
       col = c(1, 2))

## compute probability that BF < 1/10 under the p|H1 ~ Beta(a, b) alternative
a <- 10
b <- 5
p0 <- 3/4
k <- 1/10
powH1 <- pbinbf01(k = k, n = nseq, p0 = p0, type = "point", a = a, b = b,
                  da = a, db = b, dl = 0, du = 1)
plot(nseq, powH1, type = "s", xlab = "n", ylab = "Power")

## compare to normal approximation
pm <- a/(a + b) # prior mean under H1
psd <- sqrt(a*b/(a + b)^2/(a + b + 1)) # prior standard deviation under H1
pownormH1 <- pbf01(k = k, n = nseq, usd = sqrt(pm*(1 - pm)), null = p0,
                   pm = pm, psd = psd, dpm = pm, dpsd = psd)
lines(nseq, pownormH1, type = "s", col = 2)
legend("right", legend = c("Exact", "Normal approximation"), lty = 1,
       col = c(1, 2))

## probability that directional BF <= 1/10 under uniform [3/4, 1] design prior
pow <- pbinbf01(k = 1/10, n = nseq, p0 = 3/4, type = "direction", a = 1, b = 1,
                da = 1, db = 1, dl = 3/4, du = 1)
plot(nseq, pow, type = "s", xlab = "n", ylab = "Power")

Plot method for class "power.bftest"

Description

Plot method for class "power.bftest"

Usage

## S3 method for class 'power.bftest'
plot(
  x,
  nlim = c(2, 500),
  ngrid = 100,
  type = "l",
  plot = TRUE,
  nullplot = TRUE,
  ...
)

Arguments

Value

Plots power curves (if specified) and invisibly returns a list of data frames containing the data underlying the power curves

Author(s)

Samuel Pawel

See Also

powerbf01, powertbf01, powernmbf01

Examples

ssd1 <- powerbf01(k = 1/6, power = 0.95, pm = 0, psd = 1/sqrt(2), dpm = 0.5, dpsd = 0)
plot(ssd1, nlim = c(1, 8000))

power1 <- powerbf01(k = 1/2, n = 120, pm = 0, psd = 1/sqrt(2), dpm = 0.5, dpsd = 0)
plot(power1, nlim = c(1, 1000))

Cumulative distribution function of the normal moment prior Bayes factor

Description

This function computes the probability of obtaining a normal moment prior Bayes factor (nmbf01) more extreme than a threshold k with a specified sample size.

Usage

pnmbf01(k, n, usd, null = 0, psd, dpm, dpsd, lower.tail = TRUE)

Arguments

Details

It is assumed that the standard error of the future parameter estimate is of the form \code{se} =\code{usd}/\sqrt{\code{n}}. For example, for normally distributed data with known standard deviation sd and two equally sized groups of size n, the standard error of an estimated standardized mean difference is \code{se} = \code{sd}\sqrt{2/n}, so the corresponding unit standard deviation is \code{usd} = \code{sd}\sqrt{2}. See the vignette for more information.

Value

The probability that the Bayes factor is less or greater (depending on the specified lower.tail) than the specified threshold k

Author(s)

Samuel Pawel

See Also

nmbf01, nnmbf01, powernmbf01

Examples

## point desing prior (psd = 0)
pnmbf01(k = 1/10, n = 200, usd = 2, null = 0, psd = 0.5/sqrt(2), dpm = 0.5, dpsd = 0)

## normal design prior to incorporate parameter uncertainty (psd > 0)
pnmbf01(k = 1/10, n = 200, usd = 2, null = 0, psd = 0.5/sqrt(2), dpm = 0.5, dpsd = 0.25)

## design prior is the null hypothesis (dpm = 0, dpsd = 0)
pnmbf01(k = 10, n = 200, usd = 2, null = 0, psd = 0.5/sqrt(2), dpm = 0, dpsd = 0,
        lower.tail = FALSE)

Power and sample size calculations for z-test Bayes factor

Description

Compute probability that z-test Bayes factor is smaller than a specified threshold (the power), or determine sample size to obtain a target power.

Usage

powerbf01(
  n = NULL,
  power = NULL,
  k = 1/10,
  sd = 1,
  null = 0,
  pm,
  psd,
  type = c("two.sample", "one.sample", "paired"),
  dpm = pm,
  dpsd = psd,
  nrange = c(1, 10^5)
)

Arguments

Details

This function provides a similar interface as stats::power.t.test. It also assumes that the data are continuous and that the parameter of interest is either a mean or a (standardized) mean difference. For some users, the low-level functions nbf01 (to directly compute the sample size for a fixed power) and pbf01 (to directly compute the power for a fixed sample size) may also be useful because they can be used for other data and parameter types.

Value

Object of class "power.bftest", a list of the arguments (including the computed one) augmented with method and note elements

Note

A warning message will be displayed in case that the specified target power is not achievable under the specified analysis and design priors.

Author(s)

Samuel Pawel

See Also

plot.power.bftest, nbf01, pbf01, bf01

Examples

## determine power
powerbf01(n = 100, pm = 0, psd = 1, dpm = 0.5, dpsd = 0)

## determine sample size
powerbf01(power = 0.99, pm = 0, psd = 1, dpm = 0.5, dpsd = 0)

Power and sample size calculations for binomial Bayes factor

Description

Compute probability that binomial Bayes factor (binbf01) is smaller than a specified threshold (the power), or determine sample size to obtain a target power.

Usage

powerbinbf01(
  n = NULL,
  power = NULL,
  k = 1/10,
  p0 = 0.5,
  type = c("point", "direction"),
  a = 1,
  b = 1,
  dp = NA,
  da = a,
  db = b,
  dl = 0,
  du = 1,
  nrange = c(1, 10^4)
)

Arguments

Details

This function provides a similar interface as stats::power.prop.test. For some users, the low-level functions nbinbf01 (to directly compute the sample size for a fixed power) and pbinbf01 (to directly compute the power for a fixed sample size) may also be useful.

Value

Object of class "power.bftest", a list of the arguments (including the computed one) augmented with method and note elements

Author(s)

Samuel Pawel

See Also

plot.power.bftest, pbinbf01, nbinbf01, binbf01

Examples

## determine sample size
(nres <- powerbinbf01(power = 0.8, p0 = 0.2, type = "direction", dl = 0.2))
## Not run: 
plot(nres, nlim = c(1, 250), ngrid = 250, type = "s")

## End(Not run)

## determine power
(powres <- powerbinbf01(n = 100, type = "point"))
## Not run: 
plot(powres)

## End(Not run)

Power and sample size calculations for normal moment prior Bayes factor

Description

Compute probability that normal moment prior Bayes factor is smaller than a specified threshold (the power), or determine sample size to obtain a target power.

Usage

powernmbf01(
  n = NULL,
  power = NULL,
  k = 1/10,
  sd = 1,
  null = 0,
  psd,
  type = c("two.sample", "one.sample", "paired"),
  dpm,
  dpsd,
  nrange = c(1, 10^5)
)

Arguments

Details

This function provides a similar interface as stats::power.t.test. It also assumes that the data are continuous and that the parameter of interest is either a mean or a (standardized) mean difference. For some users, the low-level functions nnmbf01 (to directly compute the sample size for a fixed power) and pnmbf01 (to directly compute the power for a fixed sample size) may also be useful because they can be used for other data and parameter types.

Value

Object of class "power.bftest", a list of the arguments (including the computed one) augmented with method and note elements

Author(s)

Samuel Pawel

See Also

plot.power.bftest, nnmbf01, pnmbf01, nmbf01

Examples

## determine power
powernmbf01(n = 100, psd = 1, dpm = 0.5, dpsd = 0)

## determine sample size
powernmbf01(power = 0.99, psd = 1, dpm = 0.5, dpsd = 0)

Power and sample size calculations for t-test Bayes factor

Description

Compute probability that t-test Bayes factor is smaller than a specified threshold (the power), or determine sample size to obtain a target power.

Usage

powertbf01(
  n = NULL,
  power = NULL,
  k = 1/10,
  null = 0,
  plocation = 0,
  pscale = 1/sqrt(2),
  pdf = 1,
  type = c("two.sample", "one.sample", "paired"),
  alternative = c("two.sided", "less", "greater"),
  dpm = plocation,
  dpsd = pscale,
  nrange = c(2, 10^4)
)

Arguments

Details

This function provides a similar interface as stats::power.t.test. For some users, the low-level functions ntbf01 (to directly compute the sample size for a fixed power) and ptbf01 (to directly compute the power for a fixed sample size) may also be useful.

Value

Object of class "power.bftest", a list of the arguments (including the computed one) augmented with method and note elements

Author(s)

Samuel Pawel

See Also

plot.power.bftest, ptbf01, ntbf01, tbf01

Examples

## determine power
powertbf01(n = 146, k = 1/6, dpm = 0.5, dps = 0, alternative = "greater")

## determine sample size
powertbf01(power = 0.95, k = 1/6, dpm = 0.5, dps = 0, alternative = "greater")

Print method for class "power.bftest"

Description

Print method for class "power.bftest"

Usage

## S3 method for class 'power.bftest'
print(x, digits = getOption("digits"), ...)

Arguments

Value

Prints text summary in the console and invisibly returns the "power.bftest" object

Note

Function adapted from stats:::print.power.htest written by Peter Dalgaard

Author(s)

Samuel Pawel

See Also

powerbf01

Examples

powerbf01(power = 0.95, pm = 0, psd = 1, dpm = 0.5, dpsd = 0)
powerbf01(power = 0.95, pm = 0, psd = 1, dpm = 0.5, dpsd = 0, type = "one.sample")
powerbf01(power = 0.95, pm = 0, psd = 1, dpm = 0.5, dpsd = 0, type = "paired")
powerbf01(power = 0.95, pm = 1, psd = 0, dpm = 0.8, dpsd = 0, type = "paired")

Cumulative distribution function of the t-test Bayes factor

Description

This function computes the probability of obtaining a t-test Bayes factor (tbf01) more extreme than a threshold k with a specified sample size.

Usage

ptbf01(
  k,
  n,
  n1 = n,
  n2 = n,
  null = 0,
  plocation = 0,
  pscale = 1/sqrt(2),
  pdf = 1,
  dpm = plocation,
  dpsd = pscale,
  type = c("two.sample", "one.sample", "paired"),
  alternative = c("two.sided", "less", "greater"),
  lower.tail = TRUE,
  drange = "adaptive",
  ...
)

Arguments

Value

The probability that the Bayes factor is less or greater (depending on the specified lower.tail) than the specified threshold k

Author(s)

Samuel Pawel

See Also

tbf01, ntbf01, powertbf01

Examples

## example from Schönbrodt and Wagenmakers (2018, p. 135)
ptbf01(k = 1/6, n = 146, dpm = 0.5, dpsd = 0, alternative = "greater")
ptbf01(k = 6, n = 146, dpm = 0, dpsd = 0, alternative = "greater",
       lower.tail = FALSE)

## two-sided
ptbf01(k = 1/6, n = 146, dpm = 0.5, dpsd = 0)
ptbf01(k = 6, n = 146, dpm = 0, dpsd = 0, lower.tail = FALSE)

## one-sample test
ptbf01(k = 1/6, n = 146, dpm = 0.5, dpsd = 0, alternative = "greater", type = "one.sample")

t-test Bayes factor

Description

This function computes the Bayes factor that forms the basis of the informed Bayesian t-test from Gronau et al. (2020). The Bayes factor quantifies the evidence that the data provide for the null hypothesis that the standardized mean difference (SMD) is zero against the alternative that the SMD is non-zero. A location-scale t-distribution is assumed for the SMD under the alternative hypothesis. The Jeffreys-Zellner-Siow (JZS) Bayes factor (Rouder et al., 2009) is obtained as a special case by setting the location of the prior to zero and the prior degrees of freedom to one, which is the default.

The data are summarized by t-statistics and sample sizes. The following types of t-statistics are accepted:

• Two-sample t-test where the SMD represents the standardized mean difference between two group means (assuming equal variances in both groups)

• One-sample t-test where the SMD represents the standardized mean difference to the null value

• Paired t-test where the SMD represents the standardized mean change score

Usage

tbf01(
  t,
  n,
  n1 = n,
  n2 = n,
  plocation = 0,
  pscale = 1/sqrt(2),
  pdf = 1,
  type = c("two.sample", "one.sample", "paired"),
  alternative = c("two.sided", "less", "greater"),
  log = FALSE,
  ...
)

Arguments

Details

The Bayes factor is implemented as in equation (5) in Gronau et al. (2020), and using suitable truncation in case of one-sided alternatives. Integration is performed numerically with stats::integrate.

Value

Bayes factor in favor of the null hypothesis over the alternative (\text{BF}_{01} > 1 indicates evidence for the null hypothesis, whereas \text{BF}_{01} < 1 indicates evidence for the alternative)

Author(s)

Samuel Pawel

References

Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., Iverson, G. (2009). Bayesian t tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin & Review, 16(2):225-237. doi:10.3758/PBR.16.2.225

Gronau, Q. F., Ly., A., Wagenmakers, E.J. (2020). Informed Bayesian t-Tests. The American Statistician, 74(2):137-143. doi:10.1080/00031305.2018.1562983

See Also

powertbf01, ptbf01, ntbf01

Examples

## analyses from Rouder et al. (2009):
## values from Table 1
tbf01(t = c(0.69, 3.20), n = 100, pscale = 1, type = "one.sample")
## examples from p. 232
tbf01(t = c(2.24, 2.03), n = 80, pscale = 1, type = "one.sample")

## analyses from Gronau et al. (2020) section 3.2:
## informed prior
tbf01(t = -0.90, n1 = 53, n2 = 57, plocation = 0.350, pscale = 0.102, pdf = 3,
      alternative = "greater", type = "two.sample")
## default (one-sided) prior
tbf01(t = -0.90, n1 = 53, n2 = 57, plocation = 0, pscale = 1/sqrt(2), pdf = 1,
      alternative = "greater", type = "two.sample")