BirthDeathUrn: Birth and Death Urn

In grouprar: Group Response Adaptive Randomization for Clinical Trials

Birth and Death Urn

Description

Simulating the birth and death urn procedure (number of arms \ge 2) with two-sided hypothesis testing in a clinical trial context.

Usage

BirthDeathUrn(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 number of treatment balls of each type in the urn. One immigration ball is added automatically. For instance, if Y0 = c(1, 1, 1), the urn starts with one ball of each treatment and one immigration ball. 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

The birth and death urn works as follows. Initially the urn contains balls of K treatment types and an immigration ball. A ball is drawn at random with replacement. If it is the immigration ball, one ball of each treatment type is added to the urn, no patient is treated, and the next ball is drawn. This is repeated until a type i ball (i = 1, \ldots, K) is drawn, and then the patient is assigned to treatment i. After a success a type i ball is added to the urn, and after a failure a type i ball is removed (Hu and Rosenberger, 2006). More details can be found in Ivanova et al. (2000).

When the best success rate is at least 1/2 the urn concentrates on that arm, and the other arms can end with very few patients. The asymptotic tests can then be well above their nominal level under the null hypothesis, especially for K \ge 3. Use typeI = TRUE to check the type I error.

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

Hu, F. and Rosenberger, W. F. (2006). The Theory of Response-Adaptive Randomization in Clinical Trials. John Wiley & Sons.

Ivanova, A., Rosenberger, W. F., Durham, S. D. and Flournoy, N. (2000). A birth and death urn for randomized clinical trials: asymptotic methods. Sankhya, Series B, 62(1), 104-118.

Examples

## a simple use
bd.res <- BirthDeathUrn(k = 3, p = c(0.6, 0.7, 0.6), ssn = 200, Y0 = NULL,
                        nsim = 100, alpha = 0.05)

## view the output
bd.res

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

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

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