P.Mean: Bayesian inference on the geometric distribution (created 2011-05-22).
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I am interested in modelling accrual patterns in clinical trials. I have developed models for an exponential waiting time, and will develop a comparable model for a Poisson count. Another distribution that could be used for accrual is the geometric distribution.
The geometric distribution is quite simple. It assigns probability to non-negative integers (0, 1, 2, ...) using the following formula.
P[X=k] = p (1-p)^k
It can be thought of as a version of the exponential distribution where the value generated by the exponential distribution is rounded down to an integer value. If Y is an exponential variable, then
P[ k <= Y < (k+1)] = (1-exp(-lambda*(k+1)) - (1-exp(-labmda*k))
= exp(-lambda*k) - exp(-lambda*(k+1))
= exp(-lambda*k) (1 - exp(-lambda))
= q^k (1-q)
where
q = exp(-lambda)
The gamma distribution is a common choice for the prior distribution on lambda. the distribution for q is going to be
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