Calculate Posterior Beta — calc_post

Calculate a posterior distribution that is beta (or a mixture of beta components). Only the relevant treatment arms from the internal dataset should be read in (e.g., only the control arm if constructing a posterior distribution for the control response rate).

calc_post_beta(internal_data, response, prior)

Arguments

internal_data: This can either be a propensity score object or a tibble of the internal data.
response: Name of response variable
prior: A distributional object corresponding to a beta distribution or a mixture distribution of beta components

Value

distributional object

Details

For a given arm of an internal trial (e.g., the control arm or an active treatment arm) of size $N_I$, suppose the response data are binary such that $y_i \sim \mbox{Bernoulli}(\theta)$, $i=1,\ldots,N_I$. The posterior distribution for $\theta$ is written as

$$\pi( \theta \mid \boldsymbol{y} ) \propto \mathcal{L}(\theta \mid \boldsymbol{y}) \; \pi(\theta),$$

where $\mathcal{L}(\theta \mid \boldsymbol{y})$ is the likelihood of the response data from the internal arm and $\pi(\theta)$ is a prior distribution on $\theta$ (either a beta distribution or a mixture distribution with an arbitrary number of beta components). The posterior distribution for $\theta$ is either a beta distribution or a mixture of beta components depending on whether the prior is a single beta distribution or a mixture distribution.

Examples

library(dplyr)
#> 
#> Attaching package: ‘dplyr’
#> The following objects are masked from ‘package:stats’:
#> 
#>     filter, lag
#> The following objects are masked from ‘package:base’:
#> 
#>     intersect, setdiff, setequal, union
library(distributional)
calc_post_beta(internal_data = filter(int_binary_df, trt == 1),
                              response = y,
                              prior = dist_beta(0.5, 0.5))
#> <distribution[1]>
#> [1] Beta(56, 26)