Intro to alphaN
In alphaN: Set Alpha Based on Sample Size Using Bayes Factors

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(alphaN)

We wish to determine which alpha level is equivalent to a Bayes factor of 1. I.e. only reject the null if the data is at least at likely under the null and under the alternative. To do this, we need a way to connect the $p$-value to the Bayes factor. The alphaN package does this for tests of coefficients in regression models.

Installation

You can install the development version of alphaN from GitHub with:

# install.packages("devtools")
devtools::install_github("jespernwulff/alphaN")

Basic functionality

This vignette provides an introduction to the basic functionality of alphaN. For full details on methodology, please refer to Wulff & Taylor (2023).

Setting the alpha level

Using the alphaN function, we can get the alpha level we need to use to obtain a desired level of evidence when testing a regression coefficient in regression model.

Here is an example: We are planning to run a linear regression model with 1000 observations. We thus set n = 1000. The default BF is 1 meaning that we want to avoid Lindley's paradox, i.e. we just want the null and the alternative to be at least equally likely when we reject the null.

alpha <- alphaN(n = 1000, BF = 1)
alpha

Therefore, to obtain evidence of at least 1, we should set our alpha to r round(alpha,4).

Plotting the relationship between the Bayes factor and p-value

The alphaN function works by mapping the p-value to the Bayes factor. This relationship can be shown using the JAB_plot. For instance:

JAB_plot(n = 1000, BF = 1)

The alpha level needed to achieve a Bayes factor of 1 is shown with a red triangle in the plot. Lines for achieving Bayes factors of 3 (moderate evidence) and 10 (strong evidence) are also shown by default. As it is evident a lower alpha level is needed to achieve higher evidence.

Alpha as a decreasing function of N

An important point of the procedure is that alpha will be set as a function of sample size. The larger the sample size, the lower the alpha needed such that a significant result can be interpreted as evidence for the alternative.

The graph below illustrates this relationship for previous example:

seqN <- seq(50, 1000, 1)
plot(seqN, alphaN(seqN), type = "l",
     xlab = "n", ylab = "Alpha")

Setting the prior

To set the alpha level as a function of sample size, we need to choose the prior carefully. alphaN allows the user to choose from four sensible prior options based on suggestions from the previous literature: Jeffreys' approximate BF (method = "JAB"), the minimal training sample (method = "min"), the robust minimal training sample (method = "robust"), and balanced Type-I and Type-II errors (method = "balanced"). method = "JAB" is a good choice for users who want to be conservative against small effects, method = "min" is for when the MLE is misspecified, method = "robust" is for when the MLE is misspecified and the sample size is small, and method = "balanced" is for when Type-II errors are costly.

For instance, to achieve evidence of 3 for 1,000 observations while we ensure balanced error rates, we run

alphaN(1000, BF = 3, method = "balanced")

The package contains the convenience function alphaN_plot that allows a quick comparison of alpha as a function of sample size for the four different methods: