pcal: Lower Bounds on Posterior Probabilities for Point Null...

In pedro-teles-fonseca/pcal: Calibration of P-Values for Point Null Hypotheses

Lower Bounds on Posterior Probabilities for Point Null Hypotheses

Description

Calibrate p-values under a robust perspective so that they can be directly interpreted as either lower bounds on the posterior probabilities of point null hypotheses or lower bounds on the probabilities of type I errors.

Usage

pcal(p, prior_prob = 0.5)

Arguments

p	A numeric vector with values in the [0,1] interval.
prior_prob	A numeric vector with values in the [0,1] interval. If length(p) == 1 then prior_prob can be of any positive length, but if length(p) > 1 then the length of prior_prob can only be 1 or equal to the length of p.

Details

pcal is a vectorized implementation of the calibration of p-values into lower bounds for the posterior probabilities of point null hypotheses (or lower bounds for the probabilities of type I errors) developed by \insertCitesellke2001;textualpcal. The calibration is:

α(p) = 1 / ( 1 + ( - e p log(p) ) ^ ( -1 ) )

where p is a p-value on a classical test statistic. This calibration assumes that both the null and the alternative hypotheses have 0.5 prior probability. We generalized the aforementioned calibration for prior probabilities other than 0.5:

α(p) = 1 / ( 1 + ( ( 1 - π ) / π ) ( - e p log(p) ) ^ ( -1 ) )

where π is the prior probability of the null hypothesis and 1 - π is the prior probability of the alternative hypothesis.

For each element of p, pcal returns an approximation of the smallest posterior probability of the null hypothesis that is found by changing the prior distribution of the parameter of interest (under the alternative hypothesis) over wide classes of distributions. Alternatively, the output of pcal can also be interpreted as lower bounds on the probabilities of type I errors, which means that this calibration has both Bayesian and Frequentist interpretations. \insertCitesellke2001;textualpcal noted that a scenario in which they definitely recommend this calibration is when investigating fit to the null model with no explicit alternative in mind. \insertCitepericchiTorres2011;textualpcal warn that despite the usefulness and appropriateness of this p-value calibration it does not depend on sample size, and hence the lower bounds obtained with large samples may be conservative.

The prior_prob argument is optional and is set to 0.5 by default, implying prior equiprobability of hypotheses. prior_prob can only be of length equal to the length of p, in which case each prior probability in prior_prob is used in the calibration of the corresponding element of p, or of length 1, in which case it will be recycled (if length(p) > 1) and the same prior_prob value is used in the calibration of all the elements of p.

Note that pcal(p, prior_prob) is equivalent to bfactor_to_prob(bcal(p), prior_prob).

Value

If length(p) > 1 then pcal returns a numeric vector with the same length as p, otherwise it returns a numeric vector with the same length as prior_prob. Warning messages are thrown if there are NA or NaN values in p or in prior_prob.

References

\insertAllCited

See Also

• bcal for a p-value calibration that returns lower bounds on Bayes factors in favor of point null hypotheses.

Examples

# Calibration of a typical "threshold" p-value:
# ----------------------------------------------------------------
pcal(.05)

# Calibration of typical "threshold" p-values:
# ----------------------------------------------------------------
pcal(c(.1, .05, .01, .005, .001))

# Application: chi-squared goodness-of-fit test,
# lower bound on the posterior probability of the null hypothesis:
# ----------------------------------------------------------------
data <- matrix(c(18, 12, 10, 12, 10, 23), ncol = 2)
pcal(chisq.test(data)[["p.value"]])

pedro-teles-fonseca/pcal documentation built on Nov. 4, 2022, 3:01 p.m.