# Calculating SAM priors This function is exactly from the SAMprior R package version 1.1.1. It is included to support the runSAM function in this package. ```r 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. ## Returns Displays the SAM prior as a mixture of an informative prior (constructed based on the historical data) and a non-informative prior. ## Details 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**). 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. ## 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. ## Examples ```r set.seed(123) ## Examples for binary endpoints ## Suppose that the informative prior constructed based on historical data is ## beta(40, 60) prior.historical <- RBesT::mixbeta(c(1, 40, 60)) ## Data of the control arm data.control <- stats::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 <- RBesT::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 ) graphics::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 <- RBesT::mixnorm(c(1, prior.mean, prior.se), sigma = sigma) ## Data of the control arm data.control <- stats::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 <- RBesT::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 ) graphics::plot(SAM.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`

SAM_prior function