Calculate an approximate (potentially inverse probability weighted) multivariate normal power prior for the log-shape and log-inverse-scale parameters of a Weibull likelihood for external time-to-event control data.
calc_power_prior_weibull(
external_data,
response,
event,
intercept,
shape,
approximation = c("Laplace", "MCMC"),
...
)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 Weibull shape and
intercept parameters)
Name of response variable
Name of event indicator variable (1: event; 0: censored)
Normal distributional object that is the initial prior for the intercept (i.e., log-inverse-scale) parameter
Integer value that is the scale of the half-normal prior for the shape parameter
Type of approximation to be used. Either Laplace or
MCMC. Laplace is used by default because it is faster than MCMC.
Arguments passed to rstan::sampling (e.g. iter, chains).
Multivariate Normal Distributional Object
Weighted participant-level response data from the control arm of an external study are incorporated into an approximated inverse probability weighted (IPW) power prior for the parameter vector \(\boldsymbol{\theta}_C = \{\log(\alpha), \beta\}\), where \(\beta = -\log(\sigma)\) is the intercept parameter of a Weibull proportional hazards regression model and \(\alpha\) and \(\sigma\) are the Weibull shape and scale parameters, respectively. When borrowing information from an external dataset of size \(N_{E}\), the IPW likelihood of the external response data \(\boldsymbol{y}_E\) with event indicators \(\boldsymbol{\nu}_E\) and weights \(\hat{\boldsymbol{a}}_0\) is defined as
$$\mathcal{L}_E(\alpha, \sigma \mid \boldsymbol{y}_E, \boldsymbol{\nu}_E, \hat{\boldsymbol{a}}_0) \propto \prod_{i=1}^{N_E} \left\{ \left( \frac{\alpha}{\sigma} \right) \left( \frac{y_i}{\sigma} \right)^{\alpha - 1} \exp \left( -\left( \frac{y_i}{\sigma} \right)^\alpha \right) \right\}^{\hat{a}_{0i} \nu_i} \left\{ \exp \left( -\left( \frac{y_i}{\sigma} \right)^\alpha \right) \right\}^{\hat{a}_{0i}(1 - \nu_i)}.$$
The initial priors for the intercept parameter \(\beta\) and the shape parameter
\(\alpha\) are assumed to be normal and half-normal, respectively, which can
be defined using the intercept and shape arguments.
The power prior for \(\boldsymbol{\theta}_C\) does not have a closed form, and
thus we approximate it via a bivariate normal distribution; i.e.,
$$\boldsymbol{\theta}_C \mid \boldsymbol{y}_E, \boldsymbol{\nu}_E, \hat{\boldsymbol{a}}_0
\; \dot\sim \; \mbox{MVN} \left( \tilde{\boldsymbol{\mu}}_0, \tilde{\boldsymbol{\Sigma}}_0 \right)$$.
If approximation = Laplace, then \(\tilde{\boldsymbol{\mu}}_0\) is the mode vector
of the IPW power prior and \(\tilde{\boldsymbol{\Sigma}}_0\) is the negative
inverse of the Hessian of the log IPW power prior evaluated at the mode. If
approximation = MCMC, then MCMC samples are obtained from the IPW power prior,
and \(\tilde{\boldsymbol{\mu}}_0\) and \(\tilde{\boldsymbol{\Sigma}}_0\)
are the estimated mean vector and covariance matrix of these MCMC samples.
Note that the Laplace approximation method is faster due to its use of
optimization instead of MCMC sampling.
The first element of the mean vector and the first row/column of covariance matrix correspond to the log-shape parameter (\(\log(\alpha)\)), and the second element corresponds to the intercept (\(\beta\), the log-inverse-scale) parameter.
Other power prior:
calc_power_prior_beta(),
calc_power_prior_norm()
library(distributional)
library(dplyr)
# This function can be used directly on the data
calc_power_prior_weibull(ex_tte_df,
response = y,
event = event,
intercept = dist_normal(0, 10),
shape = 50,
approximation = "Laplace")
#> <distribution[1]>
#> [1] MVN[2]
# Or this function can be used with a propensity score object
ps_obj <- calc_prop_scr(internal_df = filter(int_tte_df, trt == 0),
external_df = ex_tte_df,
id_col = subjid,
model = ~ cov1 + cov2 + cov3 + cov4)
calc_power_prior_weibull(ps_obj,
response = y,
event = event,
intercept = dist_normal(0, 10),
shape = 50,
approximation = "Laplace")
#> <distribution[1]>
#> [1] MVN[2]