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).
calc_post_norm(internal_data, response, prior, internal_sd = NULL)A tibble of the internal data.
Name of response variable
A distributional object corresponding to a normal distribution, a t distribution, or a mixture distribution of normal and/or t components
Standard deviation of internal response data if
assumed known. It can be left as NULL if assumed unknown
distributional object
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.
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)