Type:	Package
Title:	Self-Adapting Mixture (SAM) Priors
Version:	2.0.0
Maintainer:	Peng Yang <py11@rice.edu>
Description:	Implementation of the SAM prior and generation of its operating characteristics for dynamically borrowing information from historical data. For details, please refer to Yang et al. (2023) <doi:10.1111/biom.13927>.
Depends:	R (≥ 3.5.0), RBesT, MatchIt
Imports:	Metrics, assertthat, checkmate, ggplot2
Suggests:	rmarkdown, knitr, testthat (≥ 2.0.0), foreach, purrr, rstanarm (≥ 2.17.2), scales, tools, broom, tidyr, parallel
VignetteBuilder:	knitr
License:	GPL (≥ 3)
Encoding:	UTF-8
LazyData:	true
RoxygenNote:	7.3.1
NeedsCompilation:	no
Packaged:	2025-01-16 18:58:06 UTC; yangpeng
Author:	Peng Yang [aut, cre], Ying Yuan [aut]
Repository:	CRAN
Date/Publication:	2025-01-17 14:50:06 UTC

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 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

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 ith 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'))

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

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

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)

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

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

Support of Distributions

Description

Returns the support of a distribution.

Usage

mixlink(mix, x)

Details

takes x and transforms it according to the defined link function of the mixture