SAMprior (version 2.0.0)

SAM_prior: 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 betaMix
SAM_prior(if.prior, nf.prior, weight, ...)
# S3 method for gammaMix
SAM_prior(if.prior, nf.prior, weight, ...)
# S3 method for normMix
SAM_prior(if.prior, nf.prior, weight, ..., sigma)

Value

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

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.

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.

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.

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; 79(4), 2857-2868.

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)