linkfcn: Calculate the link function for exponential families

In geoBayes: Analysis of Geostatistical Data using Bayes and Empirical Bayes Methods

Calculate the link function for exponential families

Description

Link function for the exponential family.

Usage

linkfcn(mu, linkp, family = "gaussian")

linkinv(z, linkp, family = "gaussian")

Arguments

mu	Numeric. The mean of the response variable.
linkp	The link function parameter. A scalar.
family	The distribution of the response variable from .geoBayes_models. Either an integer or the family name.
z	Numeric. The linear predictor.

Details

linkfcn maps the mean of the response variable mu to the linear predictor z. linkinv is its inverse.

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

z = \frac{sign(\mu)|\mu|^\nu - 1}{\nu}

In the other case, the usual Box-Cox link is used.

For the Poisson and gamma families, if the link parameter is positive, then the link is defined by

z = \frac{sign(w) (e^{\nu |w|}-1)}{\nu}

where w = \log(\mu). In the other case, the usual Box-Cox link is used.

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

\mu = 1 - \exp\{-\max(0, 1 + \nu z)^{\frac{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(\mbox{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.

References

Evangelou, E., & Roy, V. (2019). Estimation and prediction for spatial generalized linear mixed models with parametric links via reparameterized importance sampling. Spatial Statistics, 29, 289-315.

Examples

## Not run: 
mu <- seq(0.1, 0.9, 0.1)
linkfcn(mu, 7, "binomial")       # robit(7) link function
linkfcn(mu, , "binomial.logit")  # 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)

geoBayes documentation built on Aug. 21, 2023, 9:08 a.m.