SAM_prior: Calculating SAM priors

In SAMprior: Self-Adapting Mixture (SAM) Priors

Calculating SAM priors

Description

The SAM_prior function is designed to display the SAM prior, given the informative prior (constructed from historical data), non-informative prior, and the mixture weight calculated using SAM_weight function (Yang, et al., 2023).

Usage

SAM_prior(if.prior, nf.prior, weight, ...)

## S3 method for class 'betaMix'
SAM_prior(if.prior, nf.prior, weight, ...)

## S3 method for class 'gammaMix'
SAM_prior(if.prior, nf.prior, weight, ...)

## S3 method for class 'normMix'
SAM_prior(if.prior, nf.prior, weight, ..., sigma)

Arguments

if.prior	Informative prior constructed from historical data, represented (approximately) as a mixture of conjugate distributions.
nf.prior	Non-informative prior used for the mixture.
weight	Weight assigned to the informative prior component (0 \leq weight \leq 1), which should be determined by SAM_weight function.
...	Additional parameters required for different endpoints.
sigma	Variance used for constructing the non-informative prior for continuous endpoints.

Details

SAM prior is constructed by mixing an informative prior \pi_1(\theta), constructed based on historical data, with a non-informative prior \pi_0(\theta) using the mixture weight w determined by SAM_weight function to achieve the degree of prior-data conflict (Schmidli et al., 2015, Yang et al., 2023).

Let \theta and \theta_h denote the treatment effects associated with the current arm data D and historical data D_h, respectively. Let \delta denote the clinically significant difference such that if |\theta_h - \theta| \ge \delta, then \theta_h is regarded as clinically distinct from \theta, and it is therefore inappropriate to borrow any information from D_h. Consider two hypotheses:

H_0: \theta = \theta_h, ~ H_1: \theta = \theta_h + \delta ~ or ~ \theta = \theta_h - \delta.

H_0 represents that D_h and D are consistent (i.e., no prior-data conflict) and thus information borrowing is desirable, whereas H_1 represents that the treatment effect of D differs from D_h to such a degree that no information should be borrowed.

The SAM prior uses the likelihood ratio test (LRT) statistics R to quantify the degree of prior-data conflict and determine the extent of information borrowing.

R = P(D | H_0, \theta_h) / P(D | H_1, \theta_h) = P(D | \theta = \theta_h) / \max(P(D | \theta = \theta_h + \delta), P(D | \theta = \theta_h - \delta)) ,

where P(D | \cdot) denotes the likelihood function. An alternative Bayesian choice is the posterior probability ratio (PPR):

R = P(D | H_0, \theta_h) / P(D | H_1, \theta_h) = P(H_0) / P( H_1) \times BF,

where P(H_0) and P(H_1) is the prior probabilities of H_0 and H_1 being true. BF is the Bayes Factor that in this case is the same as the LRT.

The SAM prior, denoted as \pi_{sam}(\theta), is then defined as a mixture of an informative prior \pi_1(\theta), constructed based on D_h and a non-informative prior \pi_0(\theta):

\pi_{sam}(\theta) = w\pi_1(\theta) + (1-w)\pi_0(\theta),

where the mixture weight w is calculated as:

w = R / (1 + R).

As the level of prior-data conflict increases, the likelihood ratio R decreases, resulting in a decrease in the weight w assigned to the informative prior and thus a decrease in information borrowing. As a result, \pi_{sam}(\theta) is data-driven and has the ability to self-adapt the information borrowing based on the degree of prior-data conflict.

Value

Displays the SAM prior as a mixture of an informative prior (constructed based on the historical data) and a non-informative prior.

Methods (by class)

• SAM_prior(betaMix): The function calculates the SAM prior for beta mixture distribution. The default nf.prior is set to be mixbeta(c(1,1,1)) which represents a uniform prior Beta(1,1).

• SAM_prior(gammaMix): The function calculates the SAM prior for gamma mixture distribution. The default nf.prior is set to be mixgamma(c(1,0.001,0.001)) which represents a vague gamma prior Gamma(0.001,0.001).

• SAM_prior(normMix): The function calculates the SAM prior for normal mixture distribution. The default nf.prior is set to be mixnorm(c(1,summary(if.prior)['mean'], sigma)) which represents a unit-information prior.

References

Yang P, Zhao Y, Nie L, Vallejo J, Yuan Y. SAM: Self-adapting mixture prior to dynamically borrow information from historical data in clinical trials. Biometrics 2023; 00, 1–12. https://doi.org/10.1111/biom.13927

Schmidli H, Gsteiger S, Roychoudhury S, O'Hagan A, Spiegelhalter D, Neuenschwander B. Robust meta-analytic-predictive priors in clinical trials with historical control information. Biometrics 2014; 70(4):1023-1032.

See Also

SAM_weight

Examples

set.seed(123)
## Examples for binary endpoints
## Suppose that the informative prior constructed based on historical data is
## beta(40, 60)
prior.historical <- mixbeta(c(1, 40, 60))
## Data of the control arm
data.control     <- rbinom(60, size = 1, prob = 0.42)
## Calculate the mixture weight of the SAM prior
wSAM <- SAM_weight(if.prior = prior.historical,
                   delta = 0.15,        ## Clinically significant difference
                   data = data.control  ## Control arm data
                   )
## Assume beta(1,1) as the non-informative prior used for mixture
nf.prior  <- mixbeta(nf.prior = c(1,1,1))
## Generate the SAM prior
SAM.prior <- SAM_prior(if.prior = prior.historical, ## Informative prior
                       nf.prior = nf.prior,         ## Non-informative prior
                       weight = wSAM                ## Mixture weight of the SAM prior
                       )
plot(SAM.prior)

## Examples for continuous endpoints
## Suppose that the informative prior constructed based on historical data is
## N(0, 3)
sigma      <- 3
prior.mean <- 0
prior.se   <- sigma/sqrt(100)
prior.historical <- mixnorm(c(1, prior.mean, prior.se), sigma = sigma)
## Data of the control arm
data.control <- rnorm(80, mean = 0, sd = sigma)
## Calculate the mixture weight of the SAM prior
wSAM <- SAM_weight(if.prior = prior.historical,
                   delta = 0.2 * sigma,    ## Clinically significant difference
                   data = data.control     ## Control arm data
                   )
## Assume unit-information prior N(0,3) as the non-informative prior used
## for the mixture
nf.prior         <- mixnorm(nf.prior = c(1,prior.mean, sigma),
                            sigma = sigma)
## Generate the SAM prior
SAM.prior <- SAM_prior(if.prior = prior.historical, ## Informative prior
                       nf.prior = nf.prior,         ## Non-informative prior
                       weight = wSAM                ## Mixture weight of the SAM prior
                       )
plot(SAM.prior)

SAMprior documentation built on Sept. 28, 2023, 1:07 a.m.