Bai.Hu.Shen.Urn: Bai Hu Shen's Urn

In grouprar: Group Response Adaptive Randomization for Clinical Trials

Bai Hu Shen's Urn

Description

Bai, Hu, and Shen (2002) proposed an adaptive design for multi-arm clinical trials. The allocation probabilities adapt to the performance of the patients already treated: a success on a treatment increases the chance that the next patient is assigned to it, and a failure moves probability to the other treatments in proportion to their estimated success rates. This function simulates the Bai, Hu, and Shen urn with two-sided hypothesis testing in a clinical trial context.

Usage

Bai.Hu.Shen.Urn(k, p, ssn, Y0 = NULL, nsim = 2000, alpha = 0.05,
                test.fun = NULL, typeI = FALSE, seed = NULL)

Arguments

k	A positive integer. The number of treatment groups in the trial (k \ge 2).
p	A vector of length k with values between 0 and 1. The true success rates of the treatments, used to generate data for the simulations.
ssn	A positive integer. The total number of participants in each simulated trial.
Y0	A vector of length k giving the initial urn composition (number of balls of each treatment type). For instance, if Y0 = c(1, 1, 1), the first patient is assigned to each treatment with probability Y0 / sum(Y0). If Y0 is NULL (default), it is set to a vector of k ones.
nsim	A positive integer. The number of simulated trials, with a default value of 2000.
alpha	A number between 0 and 1. The significance level of the two-sided test, with a default value of 0.05.
test.fun	An optional function function(outcome, assignment) that returns the p-value of a test for one simulated trial, where outcome holds the observed responses and assignment the treatment labels (integers 1 to k); missing responses are removed first. The null hypothesis is rejected when the p-value is at most alpha. The default NULL uses the built-in two-sided tests: the t-test for two arms and the Wald chi-squared test of equal means for more than two arms (for binary responses, if some arms have no variability, the variances are computed from (S_k + 1)/(n_k + 2) as in Agresti and Caffo, 2000).
typeI	Logical. If TRUE, the simulation is repeated under the null hypothesis, with every arm's success rate set to the average of p, and the rejection rate is reported as type I error. The default is FALSE.
seed	An optional integer passed to set.seed before simulating, so that the results can be reproduced. The default NULL leaves the random number generator unchanged.

Details

Bai, Hu and Shen's urn can be described as follows. An urn initially contains balls of K types, where balls of types 1, 2, \ldots, K represent treatments 1, 2, \ldots, K. A ball is drawn at random from the urn and, if it is of type k, the next patient is assigned to treatment k. After the response is observed, the urn composition is updated. A success on treatment k adds one ball of type k to the urn. A failure on treatment k adds \hat p_j/(\hat M - \hat p_k) balls of each other type j \ne k, where \hat p_j = (S_j + 1)/(N_j + 1) is the current estimate of the success rate of treatment j (S_j successes among N_j patients) and \hat M = \hat p_1 + \cdots + \hat p_K. The estimates use the responses of the previous patients only. This is adaptive design 3 of Bai, Hu and Shen (2002), the design proposed in the paper. (Versions of grouprar before 0.2.0 used the true success rates instead, which is their design 2.)

Value

An object of class "grouprar", a list that is printed as a short summary (see print.grouprar), with the following elements.

method	The name of the procedure.
sample size	The total sample size.
parameter	The true success rates used in the simulations, named pA, pB, ...
propotion	The mean allocation proportion of each arm over the simulations, named treatment A, treatment B, ...
sd of propotion	The standard deviation of the allocation proportion of each arm over the simulations.
failure rate	The mean failure rate over the simulations.
sd of failure rate	The standard deviation of the failure rate (or mean response) over the simulations.
power	The proportion of simulated trials that reject the null hypothesis of equal success rates. Simulations in which the test cannot be computed are dropped.
data: failureRate	The failure rate (or mean response) of each simulated trial.
data: test	The test decision of each simulated trial (1 = reject).
data: assignment	The treatment assignments of the last simulated trial.
data: propotion	A data frame with the allocation proportions of each simulated trial.
data: allocation	An nsim by ssn matrix with the treatment assignments of every simulated trial.
type I error	Only if typeI = TRUE. The rejection rate under the null hypothesis.

References

Bai, Z. D., Hu, F. and Shen, L. (2002). An adaptive design for multi-arm clinical trials. Journal of Multivariate Analysis, 81(1), 1-18. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1006/jmva.2001.1987")}

Examples

## a simple use
bhs.res <- Bai.Hu.Shen.Urn(k = 3,
                           p = c(0.7, 0.8, 0.6),
                           ssn = 200,
                           Y0 = NULL,
                           nsim = 100,
                           alpha = 0.05)

## view the output
bhs.res

  ## view all simulation settings
  bhs.res[["method"]]
  bhs.res[["parameter"]]

  ## view the simulation results
  bhs.res[["propotion"]]
  bhs.res[["failure rate"]]
  bhs.res[["power"]]
  bhs.res[["data: assignment"]]
  

grouprar documentation built on Oct. 9, 2026, 9:07 a.m.