Calculate a (potentially inverse probability weighted) normal power prior using external data.
calc_power_prior_norm(
external_data,
response,
prior = NULL,
external_sd = NULL
)This can either be a prop_scr_obj created by calling
create_prop_scr() or a tibble of the external data. If it is just a
tibble the weights will be assumed to be 1. Only the external data for the
arm(s) of interest should be included in this object (e.g., external
control data if creating a power prior for the control mean)
Name of response variable
Either NULL or a normal distributional object that is the
initial prior for the parameter of interest (e.g., control mean) before the
external data are observed
Standard deviation of external response data if assumed
known. It can be left as NULL if assumed unknown
Normal power prior object
Weighted participant-level response data from an external study are incorporated into an inverse probability weighted (IPW) power prior for the parameter of interest \(\theta\) (e.g., the control mean if borrowing from an external control arm). When borrowing information from an external dataset of size \(N_{E}\), the IPW likelihood of the external response data \(y_E\) with weights \(\hat{\boldsymbol{a}}_0\) is defined as
$$\mathcal{L}_E(\theta \mid \boldsymbol{y}_E, \hat{\boldsymbol{a}}_0, \sigma_{E}^2) \propto \exp \left( -\frac{1}{2 \sigma_{E}^2} \sum_{i=1}^{N_{E}} \hat{a}_{0i} (y_i - \theta)^2 \right).$$
The prior argument should be either a distributional object with a family
type of normal or NULL, corresponding to the use of a normal initial
prior or an improper uniform initial prior (i.e., \(\pi(\theta) \propto
1\)), respectively.
The external_sd argument can be a positive value if the external standard
deviation is assumed known or left as NULL otherwise. If external_sd = NULL, then prior must be NULL to indicate the use of an improper
uniform initial prior for \(\theta\), and an improper prior is defined
for the unknown external standard deviation such that \(\pi(\sigma_E^2)
\propto (\sigma_E^2)^{-1}\). The details of the IPW power prior for each
case are as follows:
external_sd = positive value (\(\sigma_E^2\) known):With either a proper normal or an improper uniform initial prior, the IPW weighted power prior for \(\theta\) is a normal distribution.
external_sd = NULL (\(\sigma_E^2\) unknown):With improper priors for both \(\theta\) and \(\sigma_E^2\), the marginal IPW weighted power prior for \(\theta\) after integrating over \(\sigma_E^2\) is a non-standardized \(t\) distribution.
Defining the weights \(\hat{\boldsymbol{a}}_0\) to equal 1 results in a conventional normal (or \(t\)) power prior if the external standard deviation is known (unknown).
Other power prior:
calc_power_prior_beta()
library(distributional)
library(dplyr)
# This function can be used directly on the data
calc_power_prior_norm(ex_norm_df,
response = y,
prior = dist_normal(50, 10),
external_sd = 0.15)
#> <distribution[1]>
#> [1] N(82, 0.022)
# Or this function can be used with a propensity score object
ps_obj <- calc_prop_scr(internal_df = filter(int_norm_df, trt == 0),
external_df = ex_norm_df,
id_col = subjid,
model = ~ cov1 + cov2 + cov3 + cov4)
calc_power_prior_norm(ps_obj,
response = y,
prior = dist_normal(50, 10),
external_sd = 0.15)
#> <distribution[1]>
#> [1] N(0.71, 0.00035)