pdf {CorrBin} | R Documentation |

## 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

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

### 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}
```

.

### Value

a numeric vector of length `n+1`

giving the value of `P(X=x)`

for `x=0,\ldots,n`

.

### Author(s)

Aniko Szabo, aszabo@mcw.edu

### References

Kuk, A. A (2004) Litter-based approach to risk assessment 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")
```

*CorrBin*version 1.6.1 Index]