calc_post_norm {beastt}

Calculate Posterior Normal

Description

Calculate a posterior distribution that is normal (or a mixture of normal 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 mean).

Usage

calc_post_norm(internal_data, response, prior, internal_sd = NULL)

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 normal distribution, a t distribution, or a mixture distribution of normal and/or t components

internal_sd

Standard deviation of internal response data if assumed known. It can be left as NULL if assumed unknown

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 normally distributed such that y_i \sim N(\theta, \sigma_I^2), i=1,\ldots,N_I. If \sigma_I^2 is assumed known, the posterior distribution for \theta is written as

\pi( \theta \mid \boldsymbol{y}, \sigma_{I}^2 ) \propto \mathcal{L}(\theta \mid \boldsymbol{y}, \sigma_{I}^2) \; \pi(\theta),

where \mathcal{L}(\theta \mid \boldsymbol{y}, \sigma_{I}^2) is the likelihood of the response data from the internal arm and \pi(\theta) is a prior distribution on \theta (either a normal distribution, a t distribution, or a mixture distribution with an arbitrary number of normal and/or t components). Any t components of the prior for \theta are approximated with a mixture of two normal distributions.

If \sigma_I^2 is unknown, the marginal posterior distribution for \theta is instead written as

\pi( \theta \mid \boldsymbol{y} ) \propto \left\{ \int_0^\infty \mathcal{L}(\theta, \sigma_{I}^2 \mid \boldsymbol{y}) \; \pi(\sigma_{I}^2) \; d\sigma_{I}^2 \right\} \times \pi(\theta).

In this case, the prior for \sigma_I^2 is chosen to be \pi(\sigma_{I}^2) = (\sigma_I^2)^{-1} such that \left\{ \int_0^\infty \mathcal{L}(\theta, \sigma_{I}^2 \mid \boldsymbol{y}) \; \pi(\sigma_{I}^2) \; d\sigma_{I}^2 \right\} becomes a non-standardized t distribution. This integrated likelihood is then approximated with a mixture of two normal distributions.

If internal_sd is supplied a positive value and prior corresponds to a single normal distribution, then the posterior distribution for \theta is a normal distribution. If internal_sd = NULL or if other types of prior distributions are specified (e.g., mixture or t distribution), then the posterior distribution is a mixture of normal distributions.

Value

distributional object

Examples

library(distributional)
library(dplyr)
post_treated <- calc_post_norm(internal_data = filter(int_norm_df, trt == 1),
                               response = y,
                               prior = dist_normal(50, 10),
                               internal_sd = 0.15)