# Calculate the link function for exponential families

### Description

Link function for the exponential family.

### Usage

1 2 3 4 5 |

### Arguments

`mu` |
Numeric. The mean of the response variable. |

`linkp` |
The link function parameter. A scalar but for the binomial family is also allowed to have the character values "logit" or "probit". |

`family` |
The distribution of the response variable. |

`z` |
Numeric. The linear predictor. |

### Details

`linkfcn`

maps the mean of the response variable `mu`

to
the linear predictor `z`

. `linkinv`

is its inverse.

Note that the logit link for the binomial family is defined as the quantile of the logistic distribution with scale 0.6458.

For the Gaussian family, if the link parameter is positive, then the extended link is used, defined by

*z = (sign(mu)*abs(mu)^nu -
1)/nu*

In the other case, the link function is the same as for the Poisson and gamma families.

For the Poisson and gamma families, the Box-Cox transformation is used, defined by

*z = (mu^nu -
1)/nu*

For the GEV binomial family, the link function is defined by

*mu =
1 - exp[-max(0, 1 + nu z)^(1/nu)]*

for any real *nu*. At
*nu = 0* it reduces to the complementary log-log
link.

The Wallace binomial family is a fast approximation to the robit family. It is defined as

*mu =
Phi(sign(z) c(nu) sqrt{nu log(1 + z^2/nu)})*

where *c(nu) = (8*nu+1)/(8*nu+3)*

### Value

A numeric array of the same dimension as the function's first argument.

### See Also

`comparebinlinks`

### Examples

1 2 3 4 5 6 7 8 9 10 11 | ```
## Not run:
mu <- seq(0.1, 0.9, 0.1)
linkfcn(mu, 7, "binomial") # robit(7) link function
linkfcn(mu, "logit", "binomial") # logit link function
mu <- seq(-3, 3, 1)
linkfcn(mu, 0.5, "gaussian") # sqrt transformation
linkinv(linkfcn(mu, 0.5, "gaussian"), 0.5, "gaussian")
curve(linkfcn(x, 0.5, "gaussian"), -3, 3)
## End(Not run)
``` |