P.Mean: BUGS model for the simple Poisson accrual model (created 2012-04-18).

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I have been working on various extensions to the Bayesian model for patient accrual. Most of these extensions would require the use of the program BUGS. The first step to developing these extensions is to program simple models in BUGS, models where there is a closed form analytical solution. Here is an example of using BUGS to model the simple Poisson accrual model.

ac.model <- function() {
 k <- n*P
 V <- T*P
 lambda ~ dgamma(k,V)
 theta <- 1 / lambda
 for (i in 1:tm) {
  count[i] ~ dpois(lambda)
 t.rem <- T-tm
 lambda.t <- lambda*t.rem
 n.rem ~ dpois(lambda.t)
 n.tot <- sum(count[])+n.rem

  ac.data <- list(n=350,T=1095,P=0.5,tm=240,
ac.init <- list(list(lambda=1/3,n.rem=300))
m <- sum(count)
k <- m+n*P
V <- tm+T*P
p <- c(0.025,0.5,0.975)
c1 <- 350-41
c2 <- 350*0.5+41
c3 <- 1095*0.5+239
bpct <- qbeta(p,c1,c2)

Here are some comments on the program.

It's easy to derive the closed form solutions for the Poisson accrual case. Let n represent the number of patients that you hope to recruit and T represent the time in days that you hope the recruitment will take.Assume that the number of patients recruited on day i (i=1,...,m) follows a Poisson distribution with rate parameter lambda.

Poisson distribution

Place a gamma prior on lambda.

Gamma prior

The strength of the prior belief is controlled by P, a number between zero and one. Let k=nP and V=TP. The posterior distribution for lambda is proportional to the product of the prior distribution and the likelihood.

Posterior distribution

which simplifies to

Posterior distribution

Once you figure out the right normalizing constant, you will establish that the posterior distribution is gamma with k=nP+m and V=TP+tm. Now you want the posterior predictive distribution for the number of patients recruited during the remaining T-tm days in the trial. If lambda were known, then the number of patients recruited for the remaining T=tm would be Poisson, but you have to account for the uncertainty in the rate parameter. You do this by integrating the product of the Poisson probability and the posterior.

Posterior predictive distribution

Note the extra multiplier of (T-tm) in the Poisson distribution. You need to predict not a the Poisson count for a single day but the sum of Poisson counts across the remaining T-tm days in the study. Pull everything out of the integral that does not depend on lambda.

Posterior predictive distribution

When you find the right normalizing constants, this simplifies to

Posterior predictive distribution

which is a negative binomial distribution. To see this, set




Then you get

Negative binomial distribution

Creative Commons License This page was written by Steve Simon and is licensed under the Creative Commons Attribution 3.0 United States License. Need more information? I have a page with general help resources. You can also browse for pages similar to this one at Accrual Problems.