pdf {CorrBin} | R Documentation |
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.
betabin.pdf(p, rho, n)
qpower.pdf(p, rho, n)
p |
numeric, the probability of success. |
rho |
numeric between 0 and 1 inclusive, the within-cluster correlation. |
n |
integer, cluster size. |
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}
.
a numeric vector of length n+1
giving the value of P(X=x)
for x=0,\ldots,n
.
Aniko Szabo, aszabo@mcw.edu
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.
ran.CBData
for generating an entire dataset using
these functions
#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")