Package 'SAMprior'

Generating Operating Characteristics of SAM Priors

Description

The get_OC function is designed to generate the operating characteristics of SAM priors (Yang, et al., 2023), including the relative bias, relative mean squared error, and type I error and power under a two-arm comparative trial design. As an option, the operating characteristic of robust MAP priors (Schmidli, et al., 2014) can also be generated for comparison.

Usage

get_OC(
  if.prior,
  theta.h,
  method.w,
  prior.odds,
  nf.prior,
  delta,
  n,
  n.t,
  decision,
  ntrial,
  if.MAP,
  weight,
  theta,
  theta.t,
  ...
)

## S3 method for class 'betaMix'
get_OC(
  if.prior,
  theta.h,
  method.w,
  prior.odds,
  nf.prior,
  delta,
  n,
  n.t,
  decision,
  ntrial,
  if.MAP,
  weight,
  theta,
  theta.t,
  ...
)

## S3 method for class 'normMix'
get_OC(
  if.prior,
  theta.h,
  method.w,
  prior.odds,
  nf.prior,
  delta,
  n,
  n.t,
  decision,
  ntrial,
  if.MAP,
  weight,
  theta,
  theta.t,
  ...,
  sigma
)

Arguments

if.prior	Informative prior constructed from historical data, represented (approximately) as a mixture of conjugate distributions.
theta.h	Estimate of the treatment effect based on historical data. If missing, the default value is set to be the posterior mean estimate from if.prior.
method.w	Methods used to determine the mixture weight for SAM priors. The default method is LRT (Likelihood Ratio Test), the alternative option can be PPR (Posterior Probability Ratio). See SAM_weight for more details.
prior.odds	The prior probability of $H_0$ being true compared to the prior probability of $H_1$ being true using PPR method. The default value is 1. See SAM_weight for more details.
nf.prior	Non-informative prior used for constructing the SAM prior and robust MAP prior.
delta	Clinically significant difference used for the SAM prior.
n	Sample size for the control arm.
n.t	Sample size for the treatment arm.
decision	Decision rule to compare the treatment with the control; see decision2S.
ntrial	Number of trials simulated.
if.MAP	Whether to simulate the operating characteristics of the robust MAP prior for comparison, the default value is FALSE.
weight	Weight assigned to the informative prior component ($0 \leq$ weight $\leq 1$) for the robust MAP prior, the default value is 0.5.
theta	A vector of the response rate (binary endpoints) or mean (continuous endpoints) for the control arm.
theta.t	A vector of the response rate (binary endpoints) or mean (continuous endpoints) for the treatment arm.
...	Additional parameters for continuous endpoints.
sigma	Variance to simulate the continuous endpoint under normality assumption.

Details

The get_OC function is designed to generate the operating characteristics of SAM priors, including the relative bias, relative mean squared error, and type I error, and power under a two-arm comparative trial design. As an option, the operating characteristics of robust MAP priors (Schmidli, et al., 2014) can also be generated for comparison.

The relative bias is defined as the difference between the bias of a method and the bias of using a non-informative prior. The relative mean squared error is the difference between the mean squared error (MSE) of a method and the MES of using a non-informative prior.

To evaluate type I error and power, the determination of whether the treatment is superior to the control is calculated based on function decision2S.

Value

Returns dataframe that contains the relative bias, relative MSE, type I error, and power for both SAM priors, as well as robust MAP priors. Additionally, the mixture weight of the SAM prior is also displayed.

Methods (by class)

• get_OC(betaMix): The function is designed to generate the operating characteristics of SAM priors for binary endpoints.

• get_OC(normMix): The function is designed to generate the operating characteristics of SAM priors for continuous endpoints.

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.

Examples

set.seed(123)
## Example of a binary endpoint
## Consider a randomized comparative trial designed to borrow information
## from historical data on the control. We assumed a non-informative prior
## beta(1, 1) and an informative prior beta(30, 50) after incorporating
## the historical data. The treatment is regarded as superior to the control
## if Pr(RR.t > RR.c | data) > 0.95, where RR.t and RR.c are response rates
##  of the treatment and control, respectively. The operating characteristics
##  were assessed under the scenarios of (RR.c, RR.t) = (0.3, 0.36) and (0.3, 0.56).
## OC <- get_OC(## Informative prior constructed based on historical data
##              if.prior = mixbeta(c(1, 30, 50)),
##              ## Non-informative prior used for constructing the SAM prior
##              nf.prior = mixbeta(c(1,1,1)),
##              delta    = 0.2,  ## Clinically significant difference
##              n = 35,          ## Sample size for the control arm
##              n.t = 70,        ## Sample size for the treatment arm
##              ## Decision rule to compare the whether treatment is superior
##              ## than the control
##              decision = decision2S(0.95, 0, lower.tail=FALSE),
##              ntrial   = 1000,  ## Number of trials simulated
##              ## Weight assigned to the informative component for MAP prior
##              weight = 0.5,
##              ## A vector of response rate for the control arm
##              theta    = c(0.3, 0.36),
##              ## A vector of response rate for the treatment arm
##              theta.t  = c(0.3, 0.56))
## OC

## Example of continuous endpoint
## Consider a randomized comparative trial designed to borrow information
## from historical data on the control. We assumed a non-informative prior
## N(0, 1e4) and an informative prior N(0.5, 2) after incorporating
## the historical data. The treatment is regarded as superior to the control
## if Pr(mean.t > mean.c | data) > 0.95, where mean.t and mean.c are mean
##  of the treatment and control, respectively. The operating characteristics
##  were assessed under the scenarios of (mean.c, mean.t) = (0.1, 0.1) and
## (0.5, 1.0).
sigma      <- 2
prior.mean <- 0.5
prior.se   <- sigma/sqrt(100)
## OC <- get_OC(## Informative prior constructed based on historical data
##              if.prior = mixnorm(c(1, prior.mean, prior.se)),
##              ## Non-informative prior used for constructing the SAM prior
##              nf.prior = mixnorm(c(1, 0, 1e4)),
##              delta    = 0.2 * sigma,  ## Clinically significant difference
##              n = 100,                 ## Sample size for the control arm
##              n.t = 200,               ## Sample size for the treatment arm
##              ## Decision rule to compare the whether treatment is superior
##              ## than the control
##              decision = decision2S(0.95, 0, lower.tail=FALSE),
##              ntrial   = 1000,  ## Number of trials simulated
##              ## A vector of mean for the control arm
##              theta    = c(0.1, 0.5),
##              ## A vector of mean for the treatment arm
##              theta.t  = c(0.1, 1.0),
##              sigma = sigma)
## OC

---

Calculating the Propensity Score-Integrated Informative Priors

Description

The PS_prior function is designed to calculate the Propensity Score-Integrated (PS) informative prior constructed based on historical data.

Usage

PS_prior(
  formula,
  data,
  outcome,
  study,
  treat,
  method,
  distance,
  ratio,
  ps.method,
  trim
)

PS_prior.default(
  formula,
  data,
  outcome,
  study,
  treat,
  method,
  distance,
  ratio,
  ps.method,
  trim
)

PS_prior.beta(
  formula,
  data,
  outcome,
  study,
  treat,
  method,
  distance,
  ratio,
  ps.method,
  trim
)

PS_prior.norm(
  formula,
  data,
  outcome,
  study,
  treat,
  method,
  distance,
  ratio,
  ps.method,
  trim
)

Arguments

formula	A two-sided formula object containing the study indicator and covariates to be used in creating the distance measure used in the matching. This formula will be supplied to the functions that estimate the distance measure. For example, the formula should be specified as G ~ X1 + X2 + ... where G represents the name of study indicator and X1 and X2 are covariates.
data	A data frame containing the variables named in formula and possible other arguments.
outcome	The variable name of the outcome.
study	The variable name of the study indicator.
treat	The variable name of the treatment indicator.
method	The matching method to be used. The allowed methods are "nearest" for nearest neighbor matching (on the propensity score by default), "optimal" [method_optimal] for optimal pair matching, "full" [method_full] for optimal full matching, "genetic" [method_genetic] for genetic matching, "cem" [method_cem] for coarsened exact matching, "exact" [method_exact] for exact matching, "cardinality" [method_cardinality] for cardinality and template matching, and "subclass" [method_subclass] for subclassification. When set to "NULL", no matching will occur, but propensity score estimation and common support restrictions will still occur if requested. See the linked pages for each method for more details on what these methods do, how the arguments below are used by each on, and what additional arguments are allowed.
distance	The distance measure to be used. Can be either the name of a method of estimating propensity scores (e.g., "glm"), the name of a method of computing a distance matrix from the covariates (e.g., "mahalanobis"), a vector of already-computed distance measures, or a matrix of pairwise distances. See [distance] for allowable options. The default is "glm" for propensity scores estimated with logistic regression using glm(). Ignored for some methods; see individual methods pages for information on whether and how the distance measure is used.
ratio	For methods that allow it, how many historical control units should be matched to each current control unit in $k:1$ matching. Should be a single integer value. See the individual methods pages for information on whether and how this argument is used. The default is 1 for 1:1 matching.
ps.method	PS method utilize to calculate an informative prior based on historical data. The allowed methods are "Weighting" or "Matching". The default method is "Weighting".
trim	Lower and upper bound of trimming used in "Weighting". The default is [0.1,0.9].

Details

This function aims to calculate informative priors using historical data by incorporating covariate information to enhance borrowing strength and address prior-data conflicts.

Let $G$ be the study indicator, where $G = 1$ indicate patient is from current control study, and $G = 0$ indicate patient is from historical control study. Given the covariates data $X$, the propensity score is defined as follows,

$e(X) = \Pr(G = 1 | X),$

where distance allows different methods to estimate the propensity scores.

Calculate informative prior through PS matching is to identify a subset of historical data ($D_h^*$) that have similar PS as current control data ($D$). Various algorithms are available for PS matching, please refer to method. The informative prior can then be calculated based on the matched historical dataset.

Alternative, we can utilize the inverse probability of treatment weighting (IPTW) to adjust the distribution of $X$ in historical data $D_h$, making it similar to that in $D$. Specifically, for the $i$th subject, we assign a weight $\alpha_i$ to the outcome $y_i$ in $D_h$ based on its PS $e(X_i)$ and a fixed weight $\alpha_i = 1$ to $X_i$ in $D$, as follows:

$\alpha_i = G_1 + (1 - G_i) \frac{e(X_i)}{1 - e(X_i)}.$

To avoid extremely large weights that may compromise IPTW, symmetric trimming rule can be used to trim the tails of the PS distribution by input trim with default [0.1,0.9], that is to trim observations whose estimated PS is outside of this range.

To standardized $\alpha$, we compute the effective sample size (ESS), which approximately reflects the level of precision or equivalently its sample size, retained in the sample after weight as $n^{*}_h = (\sum \alpha_i)^2 / \sum{\alpha_i^2}$. The standardized weight is given by

$\alpha_i^{*} = G_i + (1 - G_i)\frac{G_i}{\sum{\alpha_i} / n_h^{*}}.$

For binary endpoint $Y \sim Ber(\theta)$, the informative prior $\pi_1(\theta)$ can be constructed as follows,

$\pi_1(\theta) \propto L(\theta | D_h, \alpha^{*}) \pi_0(\theta) = Beta(a + \sum \alpha_i^{*}y_i, b + n_h^* - \sum \alpha_i^{*}y_i )\},$

where $\pi_0(\theta)$ is a non-informative prior, a natural choice is $Beta(a, b)$, with $a = b = 1$.

For continuous endpoint $Y \sim N(0, \sigma^2)$, suppose $\sigma^2$ is unknown, with non-informative prior $p(\theta, \sigma^2) \propto 1/\sigma^2$, $\pi_1(\theta)$ follows a student-$t$ distribution with degree of freedom $n_h^{*} - 1$. Given that $n_h^{*}$ is moderate and large, it can be approximated by a normal distribution $N(\bar{y}^{*}, {s^{*}}^2 / n_h^{*})$ with

$\bar{y}^{*} = \sum \alpha_i^* y_i / \alpha_i^*, ~~ {s^{*}}^2 = \sum \alpha_i^* (y_i - \bar{y}^{*})^2 / (n_h^{*} - 1).$

Value

Displays the informative prior calculated from historical data based on the selected PS method.

Functions

• PS_prior.default(): The function calculates the Propensity Score-Integrated informative prior based on historical data for binary and continuous endpoint.

• PS_prior.beta(): The function calculates the Propensity Score-Integrated informative prior based on historical data for binary endpoint.

• PS_prior.norm(): The function calculates the Propensity Score-Integrated informative prior based on historical data for continuous endpoint.

References

Zhao Y, Laird G, Chen J, Yuan Y. PS-SAM: doubly robust propensity-score-integrated self-adapting mixture prior to dynamically borrow information from historical data.

See Also

matchit

Examples

## Load example data
data('PS_SAM_data')
## Subset the data to contain historical data and current control
dat <- PS_SAM_data[PS_SAM_data$A == 0, ]
str(dat)

## Examples for binary endpoints
## Generate the informative prior based on historical data using PS Matching
summary(PS_prior(formula = 'G ~ X_1 + X_2 + X_3',
                 data = dat, ps.method = 'Matching', method = 'nearest',
                 outcome = 'Y_binary', study = 'G', treat = 'A'))

## Generate the informative prior based on historical data using PS Weighting
summary(PS_prior(formula = 'G ~ X_1 + X_2 + X_3',
                 data = dat, ps.method = 'Weighting',
                 outcome = 'Y_binary', study = 'G', treat = 'A'))

## Examples for continuous endpoints
## Generate the informative prior based on historical data using PS Matching
summary(PS_prior(formula = 'G ~ X_1 + X_2 + X_3',
                 data = dat, ps.method = 'Matching', method = 'nearest',
                 outcome = 'Y_continuous', study = 'G', treat = 'A'))

## Generate the informative prior based on historical data using PS Weighting
summary(PS_prior(formula = 'G ~ X_1 + X_2 + X_3',
                 data = dat, ps.method = 'Weighting',
                 outcome = 'Y_continuous', study = 'G', treat = 'A'))

---

Simulated Data for the Construction of Propensity Score-Integrated Informative Priors

Description

This dataset demonstrates the construction of a Propensity Score-Integrated (PS) SAM prior. It simulates a two-arm randomized clinical trial (RCT) with a 2:1 randomization ratio between treatment and control arms, considering both binary and continuous endpoints.

Usage

PS_SAM_data

Format

A data frame with 600 observations.

• "A" is the treatment assignment (1 = treated, 0 = control).

• "G" is the study indicator (1 = current, 0 = historical).

• "$X_1$" is a binary covariate.

• "$X_2$" is a continuous covariate.

• "$X_3$" is a continuous covariate.

• "$Y_{binary}$" is binary outcome.

• "$Y_{continuous}$" is continuous outcome.

Details

The dataset includes:

• Sample size for treatment arm: $n_t = 200$.

• Sample size for control arm: $n_c = 100$.

• Sample size for historical control study: $n_h = 300$.

Covariates for the control arm were generated from

$X_1 \sim Ber(0.5), ~~ X_2 \sim N(0, 1), ~~ X_3 \sim N(0.5, 1),$

where $Ber(\cdot)$ stands for Bernoulli distribution. Covariates for the historical controls were generated from a mixture distribution, with half were generated the same as for the control arm, while the other half were drawn from

$X_1 \sim Ber(0.8), ~~ X_2 \sim N(-0.4, 1), ~~ X_3 \sim N(-0.2, 1).$

For the binary endpoint, $y_i$ were generated from the logit model:

$logit(\Pr(y_i = 1 | X_{1i}, X_{2i}, X_{3i}, A_i)) = -1.4 - 0.5 X_{1i} + X_{2i} + 2 X_{3i} + \lambda A_i,$

where $\lambda$ is the treatment effect size, and we let $\lambda = 0.9$ to generate a moderate treatment effect size so that they study has a reasonable power.

For the continuous endpoint, $y_i$ were generated from the following normal model:

$y_i = 1.8 X_{1i} + 0.9 X_{2i} - 2 X_{3i} + \lambda A_i + \epsilon_i,$

where we let $\lambda = 1$, and $\epsilon_i \sim N(0, 3.5^2)$.

This dataset enables evaluation of the PS-SAM prior's performance in addressing heterogeneity between the RCT control arm and historical controls.

Examples

# Load the dataset
data(PS_SAM_data)

# View the structure
str(PS_SAM_data)

---

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; 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)

---

Calculating Mixture Weight of SAM Priors

Description

The SAM_weight function is designed to calculate the mixture weight of the SAM priors according to the degree of prior-data conflicts (Yang, et al., 2023).

Usage

SAM_weight(if.prior, theta.h, method.w, prior.odds, data, delta, ...)

## S3 method for class 'betaMix'
SAM_weight(if.prior, theta.h, method.w, prior.odds, data, delta, n, r, ...)

## S3 method for class 'normMix'
SAM_weight(
  if.prior,
  theta.h,
  method.w,
  prior.odds,
  data,
  delta,
  m,
  n,
  sigma,
  ...
)

## S3 method for class 'gammaMix'
SAM_weight(if.prior, theta.h, method.w, prior.odds, data, delta, u, w, ...)

Arguments

if.prior	Informative prior constructed based on historical data, represented (approximately) as a mixture of conjugate distributions.
theta.h	Estimate of the treatment effect based on historical data. If missing, the default value is set to be the posterior mean estimate from if.prior.
method.w	Methods used to determine the mixture weight for SAM priors. The default method is "LRT" (Likelihood Ratio Test), the alternative option is "PPR" (Posterior Probability Ratio). See Details section for more information.
prior.odds	The prior probability of $H_0$ being true compared to the prior probability of $H_1$ being true using PPR method. The default value is 1. See Details section for more information.
data	Data of the control arm from the current trial, see Methods section for more details.
delta	Clinically significant difference used for the SAM prior.
...	Additional parameters required for different endpoints.
n	Number of subjects in the control arm for continuous endpoint.
r	Number of responses in the control arm for binary endpoint.
m	Mean estimate in the control arm for continuous endpoint.
sigma	Standard deviation in the control arm for continuous endpoint.
u	Number of events in the control arm for time-to-event endpoint.
w	Total observed time in the control arm for time-to-event endpoint.

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

The mixture weight of the SAM priors.

Methods (by class)

• SAM_weight(betaMix): The function calculates the mixture weight of SAM priors for beta mixture distribution. The input data can be patient-level data (i.e., a vector of 0 and 1 representing the response status of each patient) or summary statistics (i.e., the number of patients and the number of responses).

• SAM_weight(normMix): The function calculates the mixture weight of SAM priors for normal mixture distribution. The input data should be a vector of patient-level observations. The input data can be patient-level data (i.e., a vector of continuous response of each patient) or summary statistics (i.e., the mean estimate, number of subjects, and the standard deviation in the control arm).

• SAM_weight(gammaMix): The function calculates the mixture weight of SAM priors for gamma mixture distribution. The input data can be patient-level data (i.e., a matrix with the first row as the censoring indicator and the second row recording the observed time) or summary statistics (i.e., the number of uncensored observations u and total observed time w).

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.

Examples

set.seed(123)
## Examples for binary endpoints
## Example 1: no prior-data conflict
## Suppose that the informative prior constructed based on historical data is
## beta(40, 60)
prior.historical <- mixbeta(c(1, 40, 60))
## Data of 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
                   )
print(wSAM)

## Example 2: in the presence of prior-data conflict, where the current data
## has 12 responses in 60 patients
wSAM <- SAM_weight(if.prior = prior.historical,
                   delta = 0.15,    ## Clinically significant difference
                   ## Methods to determine mixture weight for the SAM priors
                   ## by Posterior Probability Ratio
                   method.w = 'PPR',
                   ## Prior odds of favoring no prior-data conflicts to
                   ## the presence of prior-data conflict
                   prior.odd = 1/9,
                   n = 60,          ## Number of patients in the control arm
                   r = 12           ## Number of responses in the control arm
                   )
print(wSAM)

## Example 3: in the presence of prior-data conflict, where the current data
## has 12 responses in 60 patients
wSAM <- SAM_weight(if.prior = prior.historical,
                   delta = 0.15, ## Clinically significant difference
                   n = 60,       ## Number of patients in the control arm
                   r = 12        ## Number of responses in the control arm
                   )
print(wSAM)

## Examples for continuous endpoints
## Example 1: no prior-data conflict
## Suppose that the informative prior constructed from 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)
wSAM <- SAM_weight(if.prior = prior.historical,
                   delta = 0.3 * sigma,    ## Clinically significant difference
                   data = data.control     ## Control arm data
                   )
print(wSAM)

## Example 2: in the presence of prior-data conflict, where the current data
## has mean of 0.5
data.control     <- rnorm(80, mean = 1, sd = sigma)
wSAM  <- SAM_weight(if.prior = prior.historical,
                    delta = 0.3 * sigma,    ## Clinically significant difference
                    data = data.control     ## Control arm data
                    )
print(wSAM)

## Examples for survival endpoints
## Example 1: no prior-data conflict
## Suppose the survival times from historical data follows exp(1) distribution
## with random censoring time follows U(0.5, 5) distribution
T_hi <- rexp(100, rate = 1)
C_hi <- runif(100, min = 0.5, max = 5)
## Indicators of the uncensored events
delta_hi <- as.numeric(T_hi < C_hi)
## Observed survival times from historical data
U_hi     <- T_hi
U_hi[delta_hi == 0] <- C_hi[delta_hi == 0]
## Construct the informative prior based on simulated historical data
prior.historical <- mixgamma(c(1, sum(delta_hi), sum(U_hi)),
                             param = 'ab', likelihood = 'exp')
## Suppose the survival times from control data follows exp(0.95) distribution
## with random censoring time follows U(0.5, 5) distribution
T_ci <- rexp(100, rate = 0.95)
C_ci <- runif(100, min = 0.5, max = 5)
## Indicators of the uncensored events
delta_ci <- as.numeric(T_ci < C_ci)
## Observed survival times from control data
U_ci     <- T_ci
U_ci[delta_ci == 0] <- C_ci[delta_ci == 0]
## Data of the control arm
data.control     <- rbind(sum(delta_ci), sum(U_ci))
wSAM <- SAM_weight(if.prior = prior.historical,
                   delta = 0.2,            ## Clinically significant difference
                   data = data.control     ## Control arm data
                   )
print(wSAM)

## Example 2: in the presence of prior-data conflict, where the current survival
## times follows exp(2) distribution with random censoring time follows U(0.5, 5)
## distribution
T_ci <- rexp(100, rate = 2)
C_ci <- runif(100, min = 0.5, max = 5)
## Indicators of the uncensored events
delta_ci <- as.numeric(T_ci < C_ci)
## Observed survival times from control data
U_ci     <- T_ci
U_ci[delta_ci == 0] <- C_ci[delta_ci == 0]
## Data of the control arm
data.control     <- rbind(sum(delta_ci), sum(U_ci))
wSAM  <- SAM_weight(if.prior = prior.historical,
                    delta = 0.2,            ## Clinically significant difference
                    data = data.control     ## Control arm data
                    )
print(wSAM)

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