# dcor0: Distribution of the Vanishing Correlation Coefficient (rho=0)... In fdrtool: Estimation of (Local) False Discovery Rates and Higher Criticism

## Description

The above functions describe the distribution of the Pearson correlation coefficient `r` assuming that there is no correlation present (`rho = 0`).

Note that the distribution has only a single parameter: the degree of freedom `kappa`, which is equal to the inverse of the variance of the distribution.

The theoretical value of `kappa` depends both on the sample size `n` and the number `p` of considered variables. If a simple correlation coefficient between two variables (`p=2`) is considered the degree of freedom equals `kappa = n-1`. However, if a partial correlation coefficient is considered (conditioned on `p-2` remaining variables) the degree of freedom is `kappa = n-1-(p-2) = n-p+1`.

## Usage

 ```1 2 3 4``` ```dcor0(x, kappa, log=FALSE) pcor0(q, kappa, lower.tail=TRUE, log.p=FALSE) qcor0(p, kappa, lower.tail=TRUE, log.p=FALSE) rcor0(n, kappa) ```

## Arguments

 `x,q` vector of sample correlations `p` vector of probabilities `kappa` the degree of freedom of the distribution (= inverse variance) `n` number of values to generate. If n is a vector, length(n) values will be generated `log, log.p` logical vector; if TRUE, probabilities p are given as log(p) `lower.tail` logical vector; if TRUE (default), probabilities are P[R <= r], otherwise, P[R > r]

## Details

For density and distribution functions as well as a corresponding random number generator of the correlation coefficient for arbitrary non-vanishing correlation `rho` please refer to the `SuppDists` package by Bob Wheeler bwheeler@echip.com (available on CRAN). Note that the parameter `N` in his `dPearson` function corresponds to `N=kappa+1`.

## Value

`dcor0` gives the density, `pcor0` gives the distribution function, `qcor0` gives the quantile function, and `rcor0` generates random deviates.

## Author(s)

Korbinian Strimmer (https://strimmerlab.github.io).

`cor`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19``` ```# load fdrtool library library("fdrtool") # distribution of r for various degrees of freedom x = seq(-1,1,0.01) y1 = dcor0(x, kappa=7) y2 = dcor0(x, kappa=15) plot(x,y2,type="l", xlab="r", ylab="pdf", xlim=c(-1,1), ylim=c(0,2)) lines(x,y1) # simulated data r = rcor0(1000, kappa=7) hist(r, freq=FALSE, xlim=c(-1,1), ylim=c(0,5)) lines(x,y1,type="l") # distribution function pcor0(-0.2, kappa=15) ```