BJSM_binary {snSMART}

BJSM for snSMART (3 active treatments/placebo and 2 dose level) with binary outcome

Description

This function implements the BJSM (Bayesian Joint Stage Modeling) method which borrows information across both stages to estimate the individual response rate of each treatment/dose level in a snSMART design with binary outcomes.

Usage

BJSM_binary(
  data,
  prior_dist,
  pi_prior,
  normal.par,
  beta_prior,
  n_MCMC_chain,
  n.adapt,
  BURN.IN = 100,
  thin = 1,
  MCMC_SAMPLE,
  ci = 0.95,
  six = TRUE,
  DTR = TRUE,
  jags.model_options = NULL,
  coda.samples_options = NULL,
  verbose = FALSE,
  ...
)

## S3 method for class 'summary.BJSM_binary'
print(x, ...)

## S3 method for class 'BJSM_binary'
print(x, ...)

## S3 method for class 'summary.BJSM_dose_binary'
print(x, ...)

## S3 method for class 'BJSM_dose_binary'
print(x, ...)

Arguments

data

trial data with 4 columns: treatment_stageI, response_stageI, treatment_stageII and response_stageII. Missing data is allowed in stage 2.

prior_dist

for 3 active treatment design: vector of three values ("prior distribution for pi", "prior distribution for beta0", "prior distribution for beta1"). User can choose from "gamma", "beta", "pareto". e.g. prior_dist = c("beta", "beta", "pareto"); for dose level design: vector of two values ("prior distribution for pi_P", "prior distribution for beta")

pi_prior

for 3 active treatment design: vector of six values (a, b, c, d, e, f), where a and b are the parameter a and parameter b of the prior distribution for pi_1A, c and d are the parameter a and parameter b of the prior distribution for pi_1B, and e and f are the parameter a and parameter b of the prior distribution for pi_1C. for dose level design: vector of two values (a, b). a is the parameter a of the prior distribution for pi (response rate) of placebo. b is the parameter b of the prior distribution for pi of placebo. Please check the Details section for more explanation

normal.par

for dose level design: vector of two values (normal.mean, normal.var). our function assumes that the logarithm of treatment effect ratio follows a Gaussian prior distribution N(\mu, \sigma^2), that is log(\pi_L/\pi_P)~N(normal.mean, normal.var), and log(\pi_H/\pi_P)~N(normal.mean, normal.var). normal.mean is the mean of this Gaussian prior. normal.var is the variance of this Gaussian prior distribution

beta_prior

for 3 active treatment design: vector of four values (a, b, c, d). a is the value of parameter a of the prior distribution for linkage parameter beta_0 or beta_0m, b is the value of parameter b of the prior distribution for linkage parameter beta_0 or beta_0m. c is the value of parameter a of the prior distribution for linkage parameter beta_1 or beta_1m. d is the value of parameter b of the prior distribution for linkage parameter beta_1 or beta_1m. for dose level design: vector of two values (a, b). a is the parameter a of the prior distribution for linkage parameter beta. b is the parameter b of the prior distribution for linkage parameter beta. Please check the Details section for more explanation

n_MCMC_chain

number of MCMC chains, default to 1.

n.adapt

the number of iterations for adaptation

BURN.IN

number of burn-in iterations for MCMC

thin

thinning interval for monitors

MCMC_SAMPLE

number of iterations for MCMC

ci

coverage probability for credible intervals, default = 0.95

six

TRUE or FALSE. If TRUE, will run the six beta model (allow for estimating beta_0m and beta_1m values that differ among different treatments m), if FALSE will run the two beta model. default = TRUE. Only need to specify this for 3 active treatment design.

DTR

TRUE or FALSE. If TRUE, will also return the expected response rate of dynamic treatment regimens. default = TRUE. Only need to specify this for 3 active treatment design.

jags.model_options

a list of optional arguments that are passed to jags.model() function.

coda.samples_options

a list of optional arguments that are passed to coda.samples() function.

verbose

TRUE or FALSE. If FALSE, no function message and progress bar will be printed.

...

further arguments. Not currently used.

x

object to summarize.

Details

For gamma distribution, prior.a is the shape parameter r, prior.b is the rate parameter lambda. For beta distribution, prior.a is the shape parameter a, prior.b is the shape parameter b. For pareto distribution, prior.a is the scale parameter alpha, prior.b is the shape parameter c (see jags user manual).

The individual response rate is regarded as a permanent feature of the treatment. The second stage outcome is modeled conditionally on the first stage results linking the first and second stage response probabilities through linkage parameters. The first stage response rate is denoted as \pi_m for treatment m. In the two \beta model, the second stage response rate for first stage responders is equal to \beta_1\pi_m. For nonresponders to treatment m in the first stage who receive treatment m' in the second the stage, the second stage response rate in the second stage is equal to \beta_0\pi_{m'}. In the six \beta model, the second stage response rate of the first stage responders to treatment m is denoted by \beta_{1m}\pi_m, and the second stage response rate of the non-responders to first stage treatment $m$ who receive treatment m' in the second stage is denoted by \beta_{0m}\pi_{m'}. All the \betas are linkage parameters.

Please refer to the paper listed under reference section for standard snSMART trial design and detailed definition of parameters.

Note that this package does not include the JAGS library, users need to install JAGS separately. Please check this page for more details: https://sourceforge.net/projects/mcmc-jags/

Value

posterior_sample

an mcmc.list object generated through the coda.samples() function, which includes posterior samples of the link parameters and response rates generated through the MCMC process

pi_hat_bjsm

estimate of response rate/treatment effect

se_hat_bjsm

standard error of the response rate

ci_pi_A(P), ci_pi_B(L), ci_pi_C(H)

x% credible intervals for treatment A(P), B(L), C(H)

diff_AB(PL), diff_BC(LH). diff_AC(PH)

estimate of differences between treatments A(P) and B(L), B(L) and C(H), A(P) and C(H)

ci_diff_AB(PL), ci_diff_BC(LH), ci_diff_AC(PH)

x% credible intervals for the estimated differences between treatments A(P) and B(L), B(L) and C(H), A(P) and C(H)

se_AB(PL), se_BC(LH), se_AC(PH)

standard error for the estimated differences between treatments A(P) and B(L), B(L) and C(H), A(P) and C(H)

beta0_hat, beta1_hat

linkage parameter beta0 and beta1 estimates

se_beta0_hat, se_beta1_hat

standard error of the estimated value of linkage parameter beta0 and beta1

ci_beta0_hat, ci_beta1_hat

linkage parameter beta0 and beta1 credible interval

pi_DTR_est

expected response rate of dynamic treatment regimens (DTRs)

pi_DTR_se

standard error for the estimated DTR response rate

ci_pi_AB, ci_pi_AC, ci_pi_BA, ci_pi_BC, ci_pi_CA, ci_pi_CB

x% credible intervals for the estimated DTR response rate

References

Wei, B., Braun, T.M., Tamura, R.N. and Kidwell, K.M., 2018. A Bayesian analysis of small n sequential multiple assignment randomized trials (snSMARTs). Statistics in medicine, 37(26), pp.3723-3732.

Chao, Y.C., Trachtman, H., Gipson, D.S., Spino, C., Braun, T.M. and Kidwell, K.M., 2020. Dynamic treatment regimens in small n, sequential, multiple assignment, randomized trials: An application in focal segmental glomerulosclerosis. Contemporary clinical trials, 92, p.105989.

Fang, F., Hochstedler, K.A., Tamura, R.N., Braun, T.M. and Kidwell, K.M., 2021. Bayesian methods to compare dose levels with placebo in a small n, sequential, multiple assignment, randomized trial. Statistics in Medicine, 40(4), pp.963-977.

See Also

LPJSM_binary
sample_size

Examples

mydata <- data_binary

BJSM_result <- BJSM_binary(
  data = mydata, prior_dist = c("beta", "beta", "pareto"),
  pi_prior = c(0.4, 1.6, 0.4, 1.6, 0.4, 1.6), beta_prior = c(1.6, 0.4, 3, 1),
  n_MCMC_chain = 1, n.adapt = 1000, MCMC_SAMPLE = 2000, ci = 0.95,
  six = TRUE, DTR = TRUE, verbose = FALSE
)


BJSM_result2 <- BJSM_binary(
  data = mydata, prior_dist = c("beta", "beta", "pareto"),
  pi_prior = c(0.4, 1.6, 0.4, 1.6, 0.4, 1.6), beta_prior = c(1.6, 0.4, 3, 1),
  n_MCMC_chain = 1, n.adapt = 10000, MCMC_SAMPLE = 60000, ci = 0.95,
  six = FALSE, DTR = FALSE, verbose = FALSE
)

summary(BJSM_result)
summary(BJSM_result2)


data <- data_dose
BJSM_dose_result <- BJSM_binary(
  data = data_dose, prior_dist = c("beta", "gamma"),
  pi_prior = c(3, 17), normal.par = c(0.2, 100), beta_prior = c(2, 2),
  n_MCMC_chain = 2, n.adapt = 1000, MCMC_SAMPLE = 6000, ci = 0.95, verbose = FALSE
)

summary(BJSM_dose_result)