P-values are the most commonly used tool to measure the evidence provided by the data against a model or hypothesis. Unfortunately, p-values are often incorrectly interpreted as the probability that the null hypothesis is true or as type I error probabilities. The `pcal`

package uses the calibrations developed by Sellke, Bayarri, and Berger

(2001) to calibrate p-values under a robust perspective and obtain measures of the evidence provided by the data in favor of point null hypotheses which are safer and more straightforward interpret:

`pcal()`

calibrates p-values so that they can be directly interpreted as either lower bounds on the posterior probabilities of point null hypotheses or as lower bounds on type I error probabilities. With this calibration one need not fear the misinterpretation of a type I error probability as the probability that the null hypothesis is true because they coincide. Note that the output of this calibration has both Bayesian and Frequentist interpretations.`bcal()`

calibrates p-values so that they can be interpreted as lower bounds on Bayes factors in favor of point null hypotheses.

Some utility functions are also included:

`bfactor_to_prob()`

turns Bayes factors into posterior probabilities using a formula from Berger and Delampady (1987).`bfactor_interpret()`

classifies the strength of the evidence implied by a Bayes factor according to the scales suggested by Jeffreys (1961) and Kass and Raftery (1995).`bfactor_log_interpret()`

is similar to`bfactor_interpret()`

but takes logarithms of Bayes factors as input.

The released version of `pcal`

can be installed from CRAN with:

install.packages("pcal")

The development version can be installed from GitHub using the `devtools`

package:

# install.packages("devtools") devtools::install_github("pedro-teles-fonseca/pcal")

First we need a p-value from any statistical test of a point null hypothesis:

x <- matrix(c(22, 13, 13, 23), ncol = 2) pv <- chisq.test(x)[["p.value"]] pv #> [1] 0.04377308

In classical hypothesis testing, if the typical 0.05 significance threshold is used then this p-value slightly below 0.05 would result in the rejection of the null hypothesis.

With `bcal()`

we can turn `pv`

into a lower bound for the Bayes factor in favor of the null hypothesis:

bcal(pv) #> [1] 0.3722807

We can also turn `pv`

into a lower bound for the posterior probability of the null hypothesis using `pcal()`

:

pcal(pv) #> [1] 0.2712861

This is an approximation to the minimum posterior probability of the null hypothesis that we would find by changing the prior distribution of the parameter of interest (under the alternative hypothesis) over wide classes of distributions. The output of `bcal()`

has an analogous interpretation for Bayes factors (instead of posterior probabilities).

Note that according to `pcal()`

the posterior probability that the null hypothesis is true is at least 0.27 (approximately), which implies that a p-value below 0.05 is not necessarily indicative of strong evidence against the null hypothesis.

One can avoid the specification of prior probabilities for the hypotheses by focusing solely on Bayes factors. To compute posterior probabilities for the hypotheses, however, prior probabilities must by specified. By default, `pcal()`

assigns a prior probability of 0.5 to the null hypothesis. We can specify different prior probabilities, for example:

pcal(pv, prior_prob = .95) #> [1] 0.8761354

In this case we obtain a higher lower bound because the null hypothesis has a higher prior probability.

Sellke, Bayarri, and Berger (2001) noted that a scenario in which they definitely recommend the aforementioned calibrations is when investigating fit to the null model with no explicit alternative in mind. Pericchi and Torres (2011) warned that despite the usefulness and appropriateness of these p-value calibrations they do not depend on sample size, and hence the lower bounds obtained with large samples may be conservative.

Since the output of `bcal(pv)`

is a Bayes factor, we can use `bfactor_to_prob()`

to turn it into a posterior probability:

bfactor_to_prob(bcal(pv)) # same as pcal(pv) #> [1] 0.2712861

Like `pcal()`

, `bfactor_to_prob()`

assumes a prior probability of 0.5 to the null hypothesis. We can change this default:

bfactor_to_prob(bcal(pv), prior_prob = .95) #> [1] 0.8761354

To classify the strength of the evidence in favor of the null hypothesis implied by a Bayes factor according to the scale suggested by Jeffreys

(1961) we can use `bfactor_interpret()`

:

bfs <- c(0.1, 2, 5, 20, 50, 150) bfactor_interpret(bfs) #> [1] "Negative" "Weak" "Substantial" "Strong" "Very Strong" #> [6] "Decisive"

Alternatively, we can use the interpretation scale suggested by Kass and Raftery (1995):

bfactor_interpret(bfs, scale = "kass-raftery") #> [1] "Negative" "Weak" "Positive" "Strong" "Strong" #> [6] "Very Strong"

Because Bayes factors are often reported on a logarithmic scale, there is also a `bfactor_log_interpret()`

function that interprets the logarithms of Bayes factors:

log_bfs <- log10(bfs) bfactor_log_interpret(log_bfs, base = 10) #> [1] "Negative" "Weak" "Substantial" "Strong" "Very Strong" #> [6] "Decisive" bfactor_log_interpret(log_bfs, scale = "kass-raftery", base = 10) #> [1] "Negative" "Weak" "Positive" "Strong" "Strong" #> [6] "Very Strong"

To compare Bayes factors with results from standard likelihood ratio tests it can be useful to obtain the strength of the evidence against the null hypothesis. If `bf`

is a Bayes factor in favor of the null hypothesis, one can use `1/bf`

as input to obtain the strength of the evidence against the null hypothesis:

# Evidence in favor of the null hypothesis bfactor_interpret(c(0.1, 2, 5, 20, 50, 150)) #> [1] "Negative" "Weak" "Substantial" "Strong" "Very Strong" #> [6] "Decisive" # Evidence against the null hypothesis bfactor_interpret(1/c(0.1, 2, 5, 20, 50, 150)) #> [1] "Strong" "Negative" "Negative" "Negative" "Negative" "Negative"

If you find a bug, please file an issue with a minimal reproducible example on GitHub. Feature requests are also welcome. You can contact me at pedro.teles.fonseca@phd.iseg.ulisboa.pt.

Berger, James O., and Mohan Delampady. 1987. “Testing Precise Hypotheses.” *Statistical Science* 2 (3): 317–35.

Jeffreys, Harold. 1961. *Theory of Probability*. 3rd ed. Oxford Classic Texts in the Physical Sciences. Oxford University Press.

Kass, Robert E., and Adrian E. Raftery. 1995. “Bayes Factors.” *Journal of the American Statistical Association* 90 (430): 773–95.

Pericchi, Luis, and David Torres. 2011. “Quick Anomaly Detection by the Newcomb—Benford Law, with Applications to Electoral Processes Data from the USA, Puerto Rico and Venezuela.” *Statistical Science* 26 (4): 502–16.

Sellke, Thomas, M. J. Bayarri, and James O. Berger. 2001. “Calibration of P Values for Testing Precise Null Hypotheses.” *The American Statistician* 55 (1): 62–71.