elicit_prior {abtest}

Elicit Prior

Description

Function for eliciting a prior distribution.

Usage

elicit_prior(
  q,
  prob,
  what = "logor",
  hypothesis = "H1",
  mu_beta = 0,
  sigma_beta = 1
)

Arguments

q

vector with quantiles for the quantity of interest.

prob

vector with probabilities corresponding to the quantiles (e.g., for the median the corresponding element of prob would need to be .5).

what

character specifying for which quantity a prior should be elicited. Either "logor" (i.e., log odds ratio) , "or" (i.e., odds ratio), "rrisk" (i.e., relative risk, the ratio of the "success" probability in the experimental and the control condition), or "arisk" (i.e., absolute risk, the difference of the "success" probability in the experimental and control condition).

hypothesis

character specifying whether the provided quantiles correspond to a two-sided prior (i.e., "H1"), a one-sided prior with lower truncation point (i.e., "H+"), or a one-sided prior with upper truncation point (i.e., "H-").

mu_beta

prior mean of the nuisance parameter \beta (i.e., the grand mean of the log odds). The default is 0.

sigma_beta

prior standard deviation of the nuisance parameter \beta (i.e., the grand mean of the log odds). The default is 1.

Details

It is assumed that the prior on the grand mean of the log odds (i.e., \beta) is not the primary target of prior elicitation and is fixed (e.g., to a standard normal prior). The reason is that the grand mean nuisance parameter \beta is not the primary target of inference and changes in the prior on this nuisance parameter do not affect the results much in most cases (see Kass & Vaidyanathan, 1992). Nevertheless, it should be emphasized that the implemented approach allows users to set the prior parameters mu_beta and sigma_beta flexibly; the only constraint is that this takes place before the prior on the test-relevant log odds ratio parameter \psi is elicited. The elicit_prior function allows the user to elicit a prior not only in terms of the log odds ratio parameter \psi, but also in terms of the odds ratio, the relative risk (i.e., the ratio of the "success" probability in the experimental and the control condition), or the absolute risk (i.e., the difference of the "success" probability in the experimental and control condition). In case the prior is not elicited for the log odds ratio directly, the elicited prior is always translated to the closest corresponding normal prior on the log odds ratio. The prior parameters mu_psi and sigma_psi are obtained using least squares minimization.

Value

list with the elicited prior parameters. Specifically, this list consists of:

• mu_psi (prior mean for the normal prior on the test-relevant log odds ratio).

• sigma_psi (prior standard deviation for the normal prior on the test-relevant log odds ratio),

• mu_beta (prior mean for the normal prior on the grand mean of the log odds),

• sigma_beta (prior standard deviation for the normal prior on the grand mean of the log odds).

Note that the prior on the grand mean of the log odds is not part of the elicitation and is assumed to be fixed by the user (using the arguments mu_beta and sigma_beta). Consequently, the returned values for mu_beta and sigma_beta simply correspond to the input values.

Author(s)

Quentin F. Gronau

References

Kass, R. E., & Vaidyanathan, S. K. (1992). Approximate Bayes factors and orthogonal parameters, with application to testing equality of two binomial proportions. Journal of the Royal Statistical Society, Series B, 54, 129-144. doi: 10.1111/j.2517-6161.1992.tb01868.x

Gronau, Q. F., Raj K. N., A., & Wagenmakers, E.-J. (2021). Informed Bayesian Inference for the A/B Test. Journal of Statistical Software, 100. doi: 10.18637/jss.v100.i17

See Also

The plot_prior function allows the user to visualize the elicited prior distribution.

Examples

# elicit prior
prior_par <- elicit_prior(q = c(0.1, 0.3, 0.5),
                          prob = c(.025, .5, .975),
                          what = "arisk")
print(prior_par)

# plot elicited prior (absolute risk)
plot_prior(prior_par = prior_par, what = "arisk")

# plot corresponding normal prior on log odds ratio
plot_prior(prior_par = prior_par, what = "logor")