BFs {BayesRep}

Sceptical Bayes factor

Description

Computes the sceptical Bayes factor

Usage

BFs(to, so, tr, sr, truncate = FALSE, zo = NULL, zr = NULL, c = NULL)

Arguments

to

Original effect estimate

so

Standard error of the original effect estimate

tr

Replication effect estimate

sr

Standard error of the replication effect estimate

truncate

Logical indicating whether advocacy prior should be truncated to direction of the original effect estimate (i.e., a one-sided test). Defaults to FALSE

zo

Original z-value zo = to/so (alternative parametrization for to and so)

zr

Replication z-value zr = tr/sr (alternative parametrization for tr and sr)

c

Relative variance c = so^2/sr^2 (alternative parametrization for so and sr)

Details

The sceptical Bayes factor is a summary measure of the following two-step reverse-Bayes procedure for assessing replication success:

1. Use the data from the original study to determine the standard deviation \tau_{\gamma} of a sceptical normal prior \theta \sim \mathrm{N}(0, \tau_{\gamma}^2) such that the Bayes factor contrasting the null hypothesis H_0: \theta = 0 to the sceptic's hypothesis H_{\mathrm{S}}: \theta \sim \mathrm{N}(0, \tau_{\gamma}^2) equals a specified level \gamma \in (0, 1]. This prior represents a sceptic who remains unconvinced about the presence of an effect at level \gamma.

2. Use the data from the replication study to compare the sceptic's hypothesis H_{\mathrm{S}}: \theta \sim \mathrm{N}(0, \tau_{\gamma}^2) to the advocate's hypothesis H_{\mathrm{A}}: \theta \sim f(\theta \, | \, \mathrm{original~study}). The prior of the effect size under H_{\mathrm{A}} is its posterior based on the original study and a uniform prior, thereby representing the position of an advocate of the original study. Replication success at level \gamma is achieved if the Bayes factor contrasting H_{\mathrm{S}} to H_{\mathrm{A}} is smaller than \gamma, which means that the replication data favour the advocate over the sceptic at a higher level than the sceptic's initial objection. The sceptical Bayes factor \mathrm{BF}_{\mathrm{S}} is the smallest level \gamma at which replication success can be established.

The function can be used with two input parametrizations, either on the absolute effect scale (to, so, tr, sr) or alternatively on the relative z-scale (zo, zr, c). If an argument on the effect scale is missing, the z-scale is automatically used and the other non-missing arguments on the effect scale ignored.

Value

The sceptical Bayes factor \mathrm{BF}_{\mathrm{S}}. \mathrm{BF}_{\mathrm{S}} < 1 indicates replication success, the smaller the value of \mathrm{BF}_{\mathrm{S}} the higher the degree of replication success. It is possible that the result of the replication is so inconclusive that replication success cannot be established at any level. In this case, the sceptical Bayes factor does not exist and the function returns NaN.

Author(s)

Samuel Pawel

References

Pawel, S. and Held, L. (2022). The sceptical Bayes factor for the assessment of replication success. Journal of the Royal Statistical Society Series B: Statistical Methodology, 84(3): 879-911. doi:10.1111/rssb.12491

See Also

BFsSMD, BFslogOR

Examples

to <- 2
tr <- 2.5
so <- 1
sr <- 1
BFs(to = to, so = so, tr = tr, sr = sr)
BFs(zo = to/so, zr = tr/sr, c = so^2/sr^2)