Simulation Examples
Bayesian inference
Consider a clinical trial designed to enroll 100 patients over a three-year period and last up to 5 years. Details of the patient enrollment simulation are outlined below:
- Total number of patients: 100
- Accrual period: 3 years
- Number of groups: 5
- Length of follow-up: 2 years
- Trial duration: 5 years
- Hazard rate: 0.03
- Event rate: 0.1
- Patients per group: 20
data <- Simulate_Enroll(n = 100,
lambda = 0.03,
event = 0.1,
M = 1,
group = 5,
maxt = 5,
accrual = 3,
censor = 0.9,
followup = 2,
partition = "Even")In the function Bayes_test, the Bayesian survival model
uses the Gamma prior function Gamma. When users set
type = "Posterior", the Bayesian inference can conduct
statistical tests based on MCMC sampling outputs. With the specific
threshold value, this function can test if the posterior
estimation of \(lambda\) is less than
this value by using test = "less" and provide the
corresponding probability \(P(\hat{\lambda}
< \text{threshold})\). For example, the Bayesian survival
model utilizes Gamma prior Gamma(alpha = 3, beta = 20), and
the simulated data is used to test if the estimated hazard rate is less
than 0.03. The outputs can be summarized through summary
function, which shows that the estimation results and probability of
\(lambda\) are less than 0.03.
test <- Bayes_test(data,
alpha = 3,
beta = 20,
test = "less",
threshold = 0.03,
type = "Posterior",
diagnosis = FALSE)
summary(test)##
## Bayesian inference can conclude that:
## Posterior mean is 0.0334 and standard deviation is 0.3845
## Credible interval and summary statistics is given by
## 2.5% 5% 25% 50% 75% 95% 97.5%
## 0.0178 0.0198 0.0268 0.0326 0.0392 0.0499 0.0539
##
## Posterior probability P( hr^ < 0.03 ) = 0.3845
## Standardized z score is 0.3717Users can change test = "greater" or
test = "two_sided to test if \(lambda\) is greater than or not equal to
0.03.
test <- Bayes_test(data,
alpha = 3,
beta = 20,
test = "greater",
pred = 2,
threshold = 0.03,
type = "Posterior",
diagnosis = FALSE)
summary(test)##
## Bayesian inference can conclude that:
## Posterior mean is 0.0334 and standard deviation is 0.6139
## Credible interval and summary statistics is given by
## 2.5% 5% 25% 50% 75% 95% 97.5%
## 0.0178 0.0198 0.0268 0.0326 0.0391 0.0499 0.0539
##
## Posterior probability P( hr^ >= 0.03 ) = 0.6139
## Standardized z score is 0.369In those examples, the standardized z score is calculated as \[ \frac{\hat{\lambda} - \lambda_0}{SD( \hat{\lambda} )}.\] This value can be compared with a certain significance level.
In addition, when setting type = "Predictive", the
function can test if the predicted survival probability of specific time
point pred is greater than, less than, or equal to the
threshold value. For example, the predictive probability of a 2-year
survival rate greater than 90% can be estimated by using inputs
test = "greater", pred = 2, and
threshold = 0.9.
test <- Bayes_test(data,
alpha = 3,
beta = 60,
test = "greater",
pred = 2,
threshold = 0.9,
type = "Predictive",
diagnosis = TRUE)
summary(test)##
## Bayesian inference can conclude that:
## Predictive survival mean is 0.9413 and standard deviation is 0.0157
## Credible interval and summary statistics is given by
## 2.5% 5% 25% 50% 75% 95% 97.5%
## 90.71 91.33 93.15 94.26 95.26 96.48 96.82
##
## Predictive probability P( S^ >= 0.9 ) = 0.989
## Standardized z score is 2.6272
## The Bayes factor of chosen prior is 5.3974 compared with non-informative prior.The standardized z score is calculated by \[ z = \frac{\hat{S} - S_0}{SD( \hat{S} )},\] and the Bayes factor is estimated based on Jeffrey’s prior as \(\pi \propto 1/\lambda\).
Interim analysis
The function BayesAT is designed for a Bayesian adaptive
trial that allows assessing multiple-stage interim outcomes.
Suppose there are four clinical trials simulated with the following settings:
- Total number of patients: 120
- Accrual period: 3 years
- Number of groups: 6
- Length of follow-up: 1 years
- Trial duration: 4 years
- Hazard rate: 0.025
- Event rate: 0.1
- Number of patients in each group: 17, 23, 16, 24, 15, and 25 patients
data <- Simulate_Enroll(n = c(17,23,16,24,15,25),
lambda = 0.025,
event = 0.05,
M = 4,
group = 6,
maxt = 4,
accrual = 3,
censor = 0.95,
followup = 1,
partition = "Uneven")
head(data[[1]])## Time Censor Enroll
## 1 0.8163012 1 0.20743724
## 2 2.2753054 0 0.25532639
## 3 2.6890483 0 0.24341465
## 4 2.8542002 0 0.27123046
## 5 3.2998083 0 0.05095704
## 6 4.0000000 0 0.06977653The function BayesAT can conduct multiple-stage interim
analysis. The duration of interim analysis D needs to be
set with matching length of enrollment, so the function can allocate
patients into each stage based on the enrollment time. For example, the
interim analysis can be arranged as follows,
start = 1.5Interim analysis starts at 18 monthsobjective = 2Analysis targets a 2-year survival rate for assessmentthreshold = 0.89The threshold of a 2-year survival rate is 89%- Gamma prior function has parameters
alpha = 2andbeta = 20
Each stage of interim analysis applies the Bayesian test to assess efficacy or futility. Default stopping criteria are based on the normal quantile for a 5% significance level, divided by the number of stages for efficacy assessments, and 95% divided by the number of stages for determining futility.
IA <- BayesAT(data,
D = 3,
stage = 7,
threshold = 0.89,
start = 1.5,
objective = 2,
alpha = 2,
beta = 20)
summary(IA)##
## Interim analysis results:
##
## Trial 1 :
## Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Stage 7
## Upper bound 4.3 2.394 2.128 1.96 1.8339 1.7317 1.6449
## Lower bound -4.3 -1.0013 -0.477 -0.0627 0.3407 0.8122 1.6449
## Z score -0.3982 0.74 0.7153 1.1556 2.9559 3.6052 3.5884
## Efficacy Prob 0 0.0026 0.0461 0.2236 0.8681 0.9532 0.9585
## Futility Prob 0.0014 0.0623 0.1231 0.12 0.0157 0.0109 0.0419
## Efficacy - - - - + + +
## Futility - - - - - - -
##
## Trial 2 :
## Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Stage 7
## Upper bound 4.3 2.394 2.128 1.96 1.8339 1.7317 1.6449
## Lower bound -4.3 -1.0013 -0.477 -0.0627 0.3407 0.8122 1.6449
## Z score 0.2248 0.0969 1.1869 3.4447 3.2251 5.5358 4.4056
## Efficacy Prob 0 0 0.1684 0.917 0.907 0.9975 0.9888
## Futility Prob 8e-04 0.1381 0.067 0.0054 0.0107 5e-04 0.0118
## Efficacy - - - + + + +
## Futility - - - - - - -
##
## Trial 3 :
## Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Stage 7
## Upper bound 4.3 2.394 2.128 1.96 1.8339 1.7317 1.6449
## Lower bound -4.3 -1.0013 -0.477 -0.0627 0.3407 0.8122 1.6449
## Z score -0.4452 1.7253 1.901 2.4628 3.1593 3.045 4.2755
## Efficacy Prob 0 0.2868 0.4797 0.7308 0.8987 0.8966 0.9864
## Futility Prob 0.0016 0.0192 0.0258 0.02 0.0112 0.0261 0.0136
## Efficacy - - - + + + +
## Futility - - - - - - -
##
## Trial 4 :
## Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Stage 7
## Upper bound 4.3 2.394 2.128 1.96 1.8339 1.7317 1.6449
## Lower bound -4.3 -1.0013 -0.477 -0.0627 0.3407 0.8122 1.6449
## Z score -0.5082 0.2107 1.144 1.1756 3.006 3.0442 4.2305
## Efficacy Prob 0 0 0.1537 0.2291 0.8727 0.8963 0.9858
## Futility Prob 0.0018 0.1195 0.0712 0.1163 0.0148 0.0263 0.0145
## Efficacy - - - - + + +
## Futility - - - - - - -
The results include the boundaries, the standardized z score, and
predictive power for efficacy and futility. Users can use
summary function to show the table of interim analysis
results for each trial, and use plot to visualize the
results.