Function for eliciting a prior distribution.
elicit_prior(
q,
prob,
what = "logor",
hypothesis = "H1",
mu_beta = 0,
sigma_beta = 1
)vector with quantiles for the quantity of interest.
vector with probabilities corresponding to the quantiles (e.g., for
the median the corresponding element of prob would need to be .5).
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).
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-").
prior mean of the nuisance parameter \(\beta\) (i.e., the grand mean of the log odds). The default is 0.
prior standard deviation of the nuisance parameter \(\beta\) (i.e., the grand mean of the log odds). The default is 1.
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.
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.
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. 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. 10.18637/jss.v100.i17
The plot_prior function allows the user to visualize
the elicited prior distribution.
# NOT RUN {
# 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")
# }
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