# Binomial Outcome In bayesCT: Simulation and Analysis of Adaptive Bayesian Clinical Trials

knitr::opts_chunk$set(comment = "#>", collapse = TRUE) suppressWarnings(RNGversion("3.5.0")) set.seed(28999)  library(bayesCT)  # Introduction The purpose of this vignette is to introduce how to simulate and analyze an adaptive Bayesian clinical trial for binomial outcomes. The simulation section compromises the design of the trial itself which provides type I error rate and power at each interim look. We use the beta-binomial conjugate prior for the estimation of posterior probabilities. Available historical data can be used as an informative prior; we use the bayesDP package as the engine for incorporating the historical data. By default, the model uses a non-informative prior of$Beta(a = 1, b = 1)$with or without the incorporation of historical data. Instead of using traditional R function, we use pipes to input our parameters. # Estimation of Treatment Difference Let$y$and$N$denote the number of events and the sample size, respectively. Let$a_0$and$b_0$denote the rate parameters of a Beta distribution. Then, the posterior distribution of the event rate under vague (flat) priors is $$\tilde{\theta}\mid y,N \; \sim \; \mathcal{B}eta\left(y+a_0,\,N-y+b_0 \right).$$ When historical data is present,$y_0$and$N_0$denote the number of events and sample size of the historical data. The weight of the historical data included in the study design and analysis is denoted by$\hat\alpha$. For more details on the computation of$\hat{\alpha}$, please refer to the vignette of binomial counts available at https://CRAN.R-project.org/package=bayesDP. The posterior distribution of the event rate with historical data incorporated under vague (flat) priors is $$\tilde{\theta} \mid y,N, y_0, N_0 \; \sim \; \mathcal{B}eta\left(y+y_0\hat{\alpha}+a_0,\, N-y+\hat{\alpha}(N_0-y_0)+b_0 \right),$$ Since there is no closed-form solution for the difference in beta distributed random variables, we use Monte Carlo simulations to estimate the posterior of the treatment difference. The estimation of the difference in proportions is$\tilde{\theta_T} - \tilde{\theta_C}$, where$\theta_T$is the posterior of the event rates in the treatment group and$\theta_C$is the posterior of the event rates in the control group. The following section lays out each of the functions and inputs for carrying out simulations and analyses of Bayesian adaptive trials. # Wrapper Function for Design and Analysis Unlike traditional R functions, the bayesCT package depends on pipe inputs with different wrapper functions. All the wrapper functions are illustrated below along with details on the arguments for the simulation and analysis. • binomial_outcome - wrapper function for specifying event rates. Inputs: • p_treatment - required input; proportion of events for a single-arm or double-arm trials for the treatment arm, where$0 < p_{treatment} < 1$. • p_control - NULL as default for single-arm trials; for two-arm trials p_control specifies the proportion of events for the control arm, where$0 < p_{control} < 1$. • study_details - wrapper function for specifying sample size, study length, interim looks, and loss to follow-up. Inputs: • total_sample_size - sample size required to run the simulation • study_period - length of the study, i.e., the follow-up period • interim_look - enrollment values where interim looks are carried out. Specify where the trials are stopped to check for early success or futility. For more details on the early stopping for futility or success, please see https://thevaachandereng.github.io/bayesCT/articles/bayesCT.html#early-stopping-for-futility-or-success. Each interim_look value must be smaller than total_sample_size. • prop_loss_to_followup - proportion of subjects loss to follow-up. Must be a value be 0 and 1. Default value is 0.10. • enrollment_rate - wrapper function for specifying enrollment. Inputs: • hypothesis - wrapper function for both simulation and analysis. Inputs: • delta - threshold difference between treatment and control groups or the threshold for single-arm trial (the default is set to 0). For non-inferiority trials, the difference must be < 0. • prob_accept_ha - probability of accepting alternative hypothesis. Default value is 0.95. • futility_prob - futility rate, i.e.,$\omega$in https://thevaachandereng.github.io/bayesCT/articles/bayesCT.html#futility. The default is set to 0.10. To prevent the trial for stopping for futility, set the futility_prob to 0. • expected_success_prob - controls stopping early for success rate, i.e.,$\Delta$in https://thevaachandereng.github.io/bayesCT/articles/bayesCT.html#success](https://thevaachandereng.github.io/bayesCT/articles/bayesCT.html#success). The default value is 0.90. To prevent the trial for stopping for early success, use 1. • alternative - sign of the alternative hypothesis. Character string specifying one of "greater"$p_{treatment} > p_{control}$(default), "less"$p_{treatment} < p_{control}$or "two-sided"$p_{treatment} \neq p_{control}$. • randomize - wrapper function for specifying the randomization scheme. Inputs: • impute - wrapper function specifying imputation of outcomes at the interim looks. Inputs: • no_of_impute - number of imputations. Used to impute outcomes for subjects potentially loss to follow-up when estimating the probability of stopping for early success and/or futility. In the case of the futility calculation, subjects not yet enrolled or imputed as well. Default value is 10 so that the vignette can be generated quickly. However, much larger values should be used in practice, e.g., 10000. • number_mcmc - number of Monte Carlo iterations for sampling from posterior distributions during imputation. Default value is 10000. • beta_prior - wrapper function for specifying the non-informative prior. Inputs: • a_0 - shape value of the beta distribution. Default is 1. To completely avoid using a non-informative prior, users can set this value to 0. • b_0 - shape value of the beta distribution. Default is 1. To completely avoid using a non-informative prior, users can set this value to 0. • historical_binomial - wrapper function for specifying historical data, if available. This function should only be used when historical data is incorporated in the analysis or simulation. For more details on the method and computation, please see https://CRAN.R-project.org/package=bayesDP. Inputs: • y0_treatment - number of events in the historical treatment group. • N0_treatment - sample size for the historical treatment group. • y0_control - number of events in the historical control group. • N0_control - sample size for the historical control group. • discount_function - discount function to use for controlling the weight given to the historical data. Currently supports weibull, scaledweibull, and identity. The discount function scaledweibull scales the output of the Weibull CDF to have a max value of 1. The identity discount function uses the posterior probability directly as the discount weight. Default value is "identity". • alpha_max - maximum weight the discount function can apply. Default is 1. For a two-arm trial, users may specify a vector of two values where the first value is used to weight the historical treatment group and the second value is used to weight the historical control group. • fix_alpha - Should alpha be set to alpha_max? Default value is FALSE. • weibull_scale - scale parameter of the Weibull discount function used to compute alpha, the weight parameter of the historical data. Default value is 0.135. For a two-arm trial, users may specify a vector of two values where the first value is used to estimate the weight of the historical treatment group and the second value is used to estimate the weight of the historical control group. Not used when discount_function = "identity". • weibull_shape - shape parameter of the Weibull discount function used to compute alpha, the weight parameter of the historical data. Default value is 3. For a two-arm trial, users may specify a vector of two values where the first value is used to estimate the weight of the historical treatment group and the second value is used to estimate the weight of the historical control group. Not used when discount_function = "identity". • method - analysis method with respect to estimation of the weight paramter alpha. Default method mc estimates alpha for each Monte Carlo iteration. Alternate value fixed estimates alpha once and holds it fixed throughout the analysis. See the the bdpsurvival vignette vignette("bdpbinomial-vignette", package="bayesDP" for more details. • data_binomial - wrapper function for inputting data with a binomial outcome. Inputs: • treatment - assignment for patients in the trial, 1 for treatment group and 0 for control group. Input should be a vector. • outcome - outcome of the trial, 1 for response (success or failure), 0 for no response. Input should be a vector with same length as treatment. • complete - indicator for whether outcome is observed; 1 for complete outcome, 0 for loss to follow up or not yet observed. If a value is 0, the outcome of the patient is ignored. If not provided, the dataset is assumed to be complete. • analysis - wrapper function for analyzing a trial. Inputs: • input - list of input to analyze the adaptive trial. • type - outcome type of the trial. Use type="binomial" for a trial with binomial outcome. • simulate - wrapper function for simulating trials. Should be used as the terminal end of the pipe. Inputs: • input - list of inputs specifying the adaptive trial simulation set-up. • no_of_sim - number of trials to simulate. Default value is 10000. # Design of Adaptive Trials In the following section, we will discuss the design of adaptive trials using bayesCT for binomial outcomes. We illustrate an example for one-arm trial and two-arm trials using the wrapper functions described above. ## One-arm Trial In the example below, we will illustrate how to compute power, type 1 error, and other characteristics for an objective performance criterion (OPC) trial with an observed proportion of events and hypothesis described as follows, $$H_0: \theta_{treatment} \geq 0.08 \qquad H_A:\theta_{treatment} < 0.08.$$ The most important wrapper functions are study_details and binomial_outcome (especially since there are no default values). Binomial events are simulated using an event rate of 0.08. The total sample size is 900 with a study length of 50 days. A 10% loss to follow-up as assumed. Based on this information, the adaptive trials are simulated 2 times to obtain the following output (NOTE: for the purpose of reproducing the vignette quickly, we reduce the number of simulations to 5, you should use a much larger value, e.g., 10000). The aforementioned inputs were chosen for illustration purposes only. value <- binomial_outcome(p_treatment = 0.08) %>% study_details(total_sample_size = 900, study_period = 50, interim_look = NULL, prop_loss_to_followup = 0.10) # Simulate 2 trials output <- value %>% simulate(no_of_sim = 2) # Structure of the simulation output str(output)  To allow for early stopping for success or futility, we can add interim looks to the design. We'll check for success or futility at the enrollment of the 600th, 700th and 800th subject. Upon adding this interim look requirement, the trial is simulated 2 times to obtain the output. # Adding interim looks value <- value %>% study_details(total_sample_size = 900, study_period = 50, interim_look = c(600, 700, 800), prop_loss_to_followup = 0.10) # Simulate 2 trials output <- value %>% simulate(no_of_sim = 2) # Structure of the simulation output str(output)  Patient enrollment is assumed to follow a Poisson process. The default enrollment rate is 0.3 patients per day. In this simulation we'll introduce a step-wise Poisson process with rate$\lambda$as follows: [ \lambda = \left{ \begin{array}{ll} 0.3 & \text(time) \in [0, 25) \ 1 & \text(time) \in [25, \infty) \ \end{array} \right. ] This enrollment scheme is illustrated below. value <- value %>% enrollment_rate(lambda = c(0.3, 1), time = 25) output <- value %>% simulate(no_of_sim = 2) str(output)  The hypothesis is an important wrapper function which controls the probability of futility, probability of accepting the alternative hypothesis, probability of early success, the alternative hypothesis, and the treatment difference margin. Since, in an OPC trial, the proportion of events in the treatment group are simulated using the input provided, delta controls the maximum threshold allowed for the trial to succeed/fail. The default value of delta is 0. Here, we'll use delta = -0.03 (i.e$\hat\theta_{treatment} - 0.08 > -0.03$). We'll further set the futility probability to 0.05, the expected success probability for early stopping to 0.90, and the final probability of accepting the alternative to 0.95. The alternative is "less" due to the hypothesis function specified above. value <- value %>% hypothesis(delta = -0.03, futility_prob = 0.05, prob_accept_ha = 0.95, expected_success_prob = 0.90, alternative = "less") output <- value %>% simulate(no_of_sim = 2) str(output)  Next, we'll illustrate imputations for imputing outcomes for subjects loss to follow up. We'll carry out 5 imputations and draw 1000 values from the posterior of each imputation. value <- value %>% impute(no_of_impute = 5, number_mcmc = 1000) output <- value %>% simulate(no_of_sim = 10) str(output)  The default non-informative beta prior used in the simulation is$\mathcal{B}eta(1, 1)$. In our OPC trial simulation, we'll change the default to$\mathcal{B}eta(5,5)$. This will increase the weight of the non-informative prior in the simulation. This non-informative beta prior is implemented using beta_prior wrapper function. value <- value %>% beta_prior(a0 = 5, b0 = 5) output <- value %>% simulate(no_of_sim = 2) str(output)  Historical data is not required to compute the simulation. However, if historical data is available, it can be incorporated into the analysis using the discount prior approach as implemented in the bayesDP R package. In our OPC trial, we'll illustrate historical data incorporation. We'll assume that the historical data had 5 events in 55 subjects. We'll incorporate this historical data using the identity discount function. For more details on the historical data incorporation method and computation, please see https://CRAN.R-project.org/package=bayesDP. value <- value %>% historical_binomial(y0_treatment = 5, N0_treatment = 55, discount_function = "identity", y0_control = NULL, N0_control = NULL, alpha_max = 1, fix_alpha = FALSE, weibull_scale = 0.135, weibull_shape = 3, method = "fixed") output <- value %>% simulate(no_of_sim = 2) str(output)  The above flow was for illustrative purposes. Instead of inputting parameters step by step, the trial parameters can be filled in all at once as illustrated below. The pipe function connects all inputs together and the trial is simulated 2 times to obtain results. value <- binomial_outcome(p_treatment = 0.08) %>% enrollment_rate(lambda = c(0.3, 1), time = 25) %>% study_details(total_sample_size = 900, study_period = 50, interim_look = c(600, 700, 800), prop_loss_to_followup = 0.10) %>% hypothesis(delta = -0.03, futility_prob = 0.05, prob_accept_ha = 0.95, expected_success_prob = 0.90, alternative = "less") %>% impute(no_of_impute = 25, number_mcmc = 1000) %>% beta_prior(a0 = 5, b0 = 5) %>% historical_binomial(y0_treatment = 5, N0_treatment = 55, discount_function = "identity", y0_control = NULL, N0_control = NULL, alpha_max = 1, fix_alpha = FALSE, weibull_scale = 0.135, weibull_shape = 3, method = "fixed") %>% simulate(no_of_sim = 2) str(value)  ## Two-arm trial In this section, we will illustrate how to perform the design of a two-arm trial without incorporating historical data. The example will compute the type 1 error, power, and other outputs for a superiority trial. The study hypothesis is $$H_0: \theta_{treatment} - \theta_{control} \leq 0 \qquad H_A: \theta_{treatment} - \theta_{control} > 0.$$ The binomial events are simulated using an event rate of 0.15 for the treatment group and 0.12 for the control group. The total sample size is 400, with a study length of 30 days. A 15% loss to follow up is assumed. Further, we will illustrate block randomization. The following code simulates a trial 2 times using the piping procedure. value <- binomial_outcome(p_treatment = 0.15, p_control = 0.12) %>% study_details(total_sample_size = 400, study_period = 30, interim_look = 350, prop_loss_to_followup = 0.15) %>% hypothesis(delta = 0, futility_prob = 0.10, prob_accept_ha = 0.975, expected_success_prob = 1, alternative = "greater") %>% randomize(block_size = 9, randomization_ratio = c(4, 5)) %>% impute(no_of_impute = 5, number_mcmc = 5000) %>% beta_prior(a0 = 0, b0 = 0) %>% simulate(no_of_sim = 2) str(value)  # Analysis In this section, we will demonstrate how to run an adaptive Bayesian trial using bayesCT. A sample dataset is provided in the package. The dataset binomialdata contains the results of 300 subjects from a two-arm trial with binomial outcome. The complete column indicates whether the outcome was observed, i.e., loss to follow-up. data(binomialdata) head(binomialdata)  The minimum input needed to run an adaptive Bayesian trial is the data itself. The data_binomial input allows the input of the data. The treatment group (0 for control, 1 for treatment) and outcome input are essential for the analysis. However, if the complete input is not provided, the function assumes the outcome data is complete. A default analysis is carried out below. input <- data_binomial(treatment = binomialdata$treatment,
outcome   = binomialdata$outcome, complete = binomialdata$complete)

out <- input %>%
analysis(type = "binomial")

str(out)


We'll now illustrate using piping to carry out the complete analysis. First, we'll assume the following hypothesis: $$H_0:\theta_{treatment} - \theta_{control} <= 0.02 \quad H_A: \theta_{treatment} - \theta_{control} > 0.02$$ The delta and alternative used to analyze the trial is 0.02 and "greater" respectively. The probability of accepting the alternative is 0.95, the probability of stopping for futility is 0.05, and the probability of stopping for success is 0.90. We will carry out imputations on subjects loss to follow up. Additionally, we will incorporate historical data on the treatment arm.

out <- data_binomial(treatment = binomialdata$treatment, outcome = binomialdata$outcome,
complete  = binomialdata\$complete) %>%
hypothesis(delta                 = 0.02,
futility_prob         = 0.05,
prob_accept_ha        = 0.95,
expected_success_prob = 0.90,
alternative           = "greater") %>%
impute(no_of_impute = 50,
number_mcmc  = 10000) %>%
beta_prior(a0 = 3,
b0 = 3) %>%
historical_binomial(y0_treatment      = 12,
N0_treatment      = 100,
y0_control        = NULL,
N0_control        = NULL,
discount_function = "weibull",
alpha_max         = 1,
fix_alpha         = FALSE,
weibull_scale     = 0.135,
weibull_shape     = 3,
method            = "fixed") %>%
analysis(type = "binomial")

str(out)


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bayesCT documentation built on July 2, 2020, 2:34 a.m.