Bayesian Discount Prior: Gaussian mean values

bdpnormal is used for estimating posterior samples from a Gaussian outcome where an informative prior is used. The prior weight is determined using a discount function. This code is modeled after the methodologies developed in Haddad et al. (2017).

Usage

bdpnormal(
  mu_t = NULL,
  sigma_t = NULL,
  N_t = NULL,
  mu0_t = NULL,
  sigma0_t = NULL,
  N0_t = NULL,
  mu_c = NULL,
  sigma_c = NULL,
  N_c = NULL,
  mu0_c = NULL,
  sigma0_c = NULL,
  N0_c = NULL,
  discount_function = "identity",
  alpha_max = 1,
  fix_alpha = FALSE,
  weibull_scale = 0.135,
  weibull_shape = 3,
  number_mcmc = 10000,
  method = "mc",
  compare = TRUE
)

Arguments

mu_t: scalar. Mean of the current treatment group.
sigma_t: scalar. Standard deviation of the current treatment group.
N_t: scalar. Number of observations of the current treatment group.
mu0_t: scalar. Mean of the historical treatment group.
sigma0_t: scalar. Standard deviation of the historical treatment group.
N0_t: scalar. Number of observations of the historical treatment group.
mu_c: scalar. Mean of the current control group.
sigma_c: scalar. Standard deviation of the current control group.
N_c: scalar. Number of observations of the current control group.
mu0_c: scalar. Mean of the historical control group.
sigma0_c: scalar. Standard deviation of the historical control group.
N0_c: scalar. Number of observations of the historical control group.
discount_function: character. Specify the discount function to use. Currently supports weibull, scaledweibull, and identity. The discount function scaledweibull scales the output of the Weibull CDF to have a max value of 1. The identity discount function uses the posterior probability directly as the discount weight. Default value is "identity".
alpha_max: scalar. Maximum weight the discount function can apply. Default is 1. For a two-arm trial, users may specify a vector of two values where the first value is used to weight the historical treatment group and the second value is used to weight the historical control group.
fix_alpha: logical. Fix alpha at alpha_max? Default value is FALSE.
weibull_scale: scalar. Scale parameter of the Weibull discount function used to compute alpha, the weight parameter of the historical data. Default value is 0.135. For a two-arm trial, users may specify a vector of two values where the first value is used to estimate the weight of the historical treatment group and the second value is used to estimate the weight of the historical control group. Not used when discount_function = "identity".
weibull_shape: scalar. Shape parameter of the Weibull discount function used to compute alpha, the weight parameter of the historical data. Default value is 3. For a two-arm trial, users may specify a vector of two values where the first value is used to estimate the weight of the historical treatment group and the second value is used to estimate the weight of the historical control group. Not used when discount_function = "identity".
number_mcmc: scalar. Number of Monte Carlo simulations. Default is 10000.
method: character. Analysis method with respect to estimation of the weight parameter alpha. Default method "mc" estimates alpha for each Monte Carlo iteration. Under "mc", the stochastic comparison probability is recomputed at each Monte Carlo draw, so the resulting alpha_discount is a random vector and may exhibit substantial Monte Carlo variability. Alternate value "fixed" estimates alpha once and holds it fixed throughout the analysis. See the bdpnormal vignette
vignette("bdpnormal-vignette", package="bayesDP") for more details.
compare: logical. Should a comparison object be included in the fit? For a one-arm analysis, the comparison object is simply the posterior chain of the treatment group parameter. For a two-arm analysis, the comparison object is the posterior chain of the treatment effect that compares treatment and control. If compare=TRUE, the comparison object is accessible in the final slot, else the final slot is NULL. Default is TRUE.

Value

bdpnormal returns an object of class "bdpnormal". The functions summary and print are used to obtain and print a summary of the results, including user inputs. The plot function displays visual outputs as well.

An object of class bdpnormal is a list containing at least the following components:

posterior_treatment

list. Entries contain values related to the treatment group:

alpha_discount numeric. Alpha value, the weighting parameter of the historical data.
p_hat numeric. The posterior probability of the stochastic comparison between the current and historical data.
posterior_mu vector. A vector of length number_mcmc containing the posterior mean of the treatment group. If historical treatment data is present, the posterior incorporates the weighted historical data.
posterior_sigma2 vector. A vector of length number_mcmc containing the posterior variance of the treatment group. If historical treatment data is present, the posterior incorporates the weighted historical data.
posterior_flat_mu vector. A vector of length number_mcmc containing Monte Carlo samples of the mean of the current treatment group under a flat/non-informative prior, i.e., no incorporation of the historical data.
posterior_flat_sigma2 vector. A vector of length number_mcmc containing Monte Carlo samples of the standard deviation of the current treatment group under a flat/non-informative prior, i.e., no incorporation of the historical data.
prior_mu vector. If historical treatment data is present, a vector of length number_mcmc containing Monte Carlo samples of the mean of the historical treatment group under a flat/non-informative prior.
prior_sigma2 vector. If historical treatment data is present, a vector of length number_mcmc containing Monte Carlo samples of the standard deviation of the historical treatment group under a flat/non-informative prior.

posterior_control

list. Similar entries as posterior_treament. Only present if a control group is specified.

final

list. Contains the final comparison object, dependent on the analysis type:

One-arm analysis: vector. Posterior chain of the mean.
Two-arm analysis: vector. Posterior chain of the mean difference comparing treatment and control groups.

args1

list. Entries contain user inputs. In addition, the following elements are output:

arm2 binary indicator. Used internally to indicate one-arm or two-arm analysis.
intent character. Denotes current/historical status of treatment and control groups.

Details

bdpnormal uses a two-stage approach for determining the strength of historical data in estimation of a mean outcome. In the first stage, a discount function is used that that defines the maximum strength of the historical data and discounts based on disagreement with the current data. Disagreement between current and historical data is determined by stochastically comparing the respective posterior distributions under noninformative priors. With Gaussian data, the comparison is the proability (p) that the current mean is less than the historical mean. The comparison metric p is then input into the discount function and the final strength of the historical data is returned (alpha).

In the second stage, posterior estimation is performed where the discount function parameter, alpha, is used incorporated in all posterior estimation procedures.

To carry out a single arm (OPC) analysis, data for the current treatment (mu_t, sigma_t, and N_t) and historical treatment (mu0_t, sigma0_t, and N0_t) must be input. The results are then based on the posterior distribution of the current data augmented by the historical data.

To carry our a two-arm (RCT) analysis, data for the current treatment and at least one of current or historical control data must be input. The results are then based on the posterior distribution of the difference between current treatment and control, augmented by available historical data.

For more details, see the bdpnormal vignette:
vignette("bdpnormal-vignette", package="bayesDP")

References

Haddad, T., Himes, A., Thompson, L., Irony, T., Nair, R. MDIC Computer Modeling and Simulation working group.(2017) Incorporation of stochastic engineering models as prior information in Bayesian medical device trials. Journal of Biopharmaceutical Statistics, 1-15.

Examples

# One-arm trial (OPC) example
fit <- bdpnormal(
  mu_t = 30, sigma_t = 10, N_t = 50,
  mu0_t = 32, sigma0_t = 10, N0_t = 50,
  method = "fixed"
)
summary(fit)
#> 
#>     One-armed bdp normal
#> 
#> data:
#>   Current treatment: mu_t = 30, sigma_t = 10, N_t = 50
#>   Historical treatment: mu0_t = 32, sigma0_t = 10, N0_t = 50
#> Stochastic comparison (p_hat) - treatment (current vs. historical data): 0.326
#> Discount function value (alpha) - treatment: 0.326
#> 95 percent CI: 
#>  27.9583  33.0088
#> posterior sample estimate:
#> mean of treatment group
#>  30.4735
if (FALSE) { # \dontrun{
plot(fit)
} # }

# Two-arm (RCT) example
fit2 <- bdpnormal(
  mu_t = 30, sigma_t = 10, N_t = 50,
  mu0_t = 32, sigma0_t = 10, N0_t = 50,
  mu_c = 25, sigma_c = 10, N_c = 50,
  mu0_c = 25, sigma0_c = 10, N0_c = 50,
  method = "fixed"
)
summary(fit2)
#> 
#>     Two-armed bdp normal
#> 
#> data:
#>   Current treatment: mu_t = 30, sigma_t = 10, N_t = 50
#>   Current control: mu_c = 25, sigma_c = 10, N_c = 50
#>   Historical treatment: mu0_t = 32, sigma0_t = 10, N0_t = 50
#>   Historical control: mu0_c = 25, sigma0_c = 10, N0_c = 50
#> Stochastic comparison (p_hat) - treatment (current vs. historical data): 0.33
#> Stochastic comparison (p_hat) - control (current vs. historical data): 0.993
#> Discount function value (alpha) - treatment: 0.33
#> Discount function value (alpha) - control: 0.993
#> alternative hypothesis: two.sided
#> 95 percent CI: 
#>  2.3734  8.6811
#> posterior sample estimates:
#> treatment group  control group
#>           30.50          25.00
if (FALSE) { # \dontrun{
plot(fit2)
} # }