Sensitivity analysis for the binomial Bayes factor

Reports tipping points for the binomial Bayes factor: the smallest observation bias omega > 1 and the smallest rival-tilted prior Beta(1, M + 1) at which the Bayes factor first drops below threshold.

Usage

sens_binomial(y_W, y_R, threshold = 20, theta_cut = 0.5, M_max = 200L)

Arguments

y_W: Non-negative integer. Observed count favorable to the working theory.
y_R: Non-negative integer. Observed count favorable to the rival.
threshold: Positive numeric. Decision threshold the Bayes factor must remain at or above. Default 20.
theta_cut: Numeric in (0, 1). Cutpoint. Default 0.5.
M_max: Positive integer. Largest M searched in the prior sweep. Default 200.

Value

A list with elements:

bf: Bayes factor at the baseline (omega = 1, uniform prior).
omega_star: Bias tipping point. 0 if bf < threshold at baseline; NA_real_ if bf does not cross threshold for any reachable omega.
M_star: Smallest integer M >= 0 with Beta(1, M + 1) prior at which the Bayes factor drops below threshold. 0 if bf < threshold at baseline; NA_integer_ if the BF does not drop below threshold within [0, M_max].

Details

omega > 1 makes pro-\(H_1\) items more likely to be observed than they are in the universe, so the apparent dominance of pro- \(H_1\) evidence is partly an artifact of observation, and the Bayes factor falls. omega_star is the value at which this fall first crosses threshold. Beta(1, M + 1) priors place all density on \(\theta < 1\) and posit M pseudo-observations all favoring the rival. M_star is the smallest integer M at which the Bayes factor first drops below threshold.

If bf_binomial(y_W, y_R) < threshold at baseline, both tipping points are 0 (the conclusion fails before any perturbation).

Examples

s <- sens_binomial(7, 3)
s$bf
#> [1] 7.827586
s$omega_star
#> [1] 0
s$M_star
#> [1] 0