CorrBin (version 1.3)

qpower.pdf: Parametric distributions for correlated binary data

Description

qpower.pdf and betabin.pdf calculate the probability distribution function for the number of responses in a cluster of the q-power and beta-binomial distributions, respectively.

Usage

qpower.pdf(p, rho, n)
betabin.pdf(p, rho, n)

Arguments

p

numeric, the probability of success.

rho

numeric between 0 and 1 inclusive, the within-cluster correlation.

n

integer, cluster size.

Value

• a numeric vector of length $n+1$ giving the value of $P(X=x)$ for $x=0,\ldots,n$.

Details

The pdf of the q-power distribution is $$P(X=x) = {{n}\choose{x}}\sum_{k=0}^x (-1)^k{{x}\choose{k}}q^{(n-x+k)^\gamma},$$ $x=0,\ldots,n$, where $q=1-p$, and the intra-cluster correlation $$\rho = \frac{q^{2^\gamma}-q^2}{q(1-q)}.$$ The pdf of the beta-binomial distribution is $$P(X=x) = {{n}\choose{x}} \frac{B(\alpha+x, n+\beta-x)}{B(\alpha,\beta)},$$ $x=0,\ldots,n$, where $\alpha= p\frac{1-\rho}{\rho}$, and $\alpha= (1-p)\frac{1-\rho}{\rho}$.

References

Kuk, A. A (2004) litter-based approach to risk assessement in developmental toxicity studies via a power family of completely monotone functions Applied Statistics, 52, 51-61. Williams, D. A. (1975) The Analysis of Binary Responses from Toxicological Experiments Involving Reproduction and Teratogenicity Biometrics, 31, 949-952.

See Also

ran.CBData for generating an entire dataset using these functions

Examples

#the distributions have quite different shapes
#with q-power assigning more weight to the "all affected" event than other distributions
plot(0:10, betabin.pdf(0.3, 0.4, 10), type="o", ylim=c(0,0.34), 
   ylab="Density", xlab="Number of responses out of 10")
lines(0:10, qpower.pdf(0.3, 0.4, 10), type="o", col="red")