# Generalized Linear Models with Extra Parameters

### Description

Estimation of generalized linear models with extra parameters, e.g., parametric links, or families with additional parameters (such as negative binomial).

### Usage

1 2 3 4 5 6 |

### Arguments

`formula` |
symbolic description of the model. |

`data, subset, na.action` |
arguments controlling formula processing
via |

`weights` |
optional numeric vector of case weights. |

`offset` |
optional numeric vector(s) with an a priori known component to be included in the linear predictor. |

`family` |
function that returns a |

`xlink` |
link object or a character that can be passed to |

`control` |
a list of control arguments as returned by |

`model, y, x` |
logicals. If |

`...` |
control arguments. |

### Details

The function `glmx`

is a convenience interface that estimates generalized
linear models (GLMs) with extra parameters. Examples would be binary response models
with parametric link functions or count regression using a negative binomial family
(which has one additional parameter).

Hence, `glmx`

needs a `family`

argument which is a family-generating function
depending on one numeric argument for the extra parameters. Then, either profile-likelihood
methods can be used for optimizing the extra parameters or all parameters can be
optimized jointly.

If the generated `family`

contains a list element `loglik.extra`

for the
derivative of the log-likelihood with respect to the extra parameters (i.e., score/gradient
contributions), then this is used in the optimization process. This should be a
`function(y, mu, extra)`

depending on the observed response `y`

, the estimated
mean `mu`

, and the `extra`

parameters.

### Value

`glmx`

returns an object of class `"glmx"`

, i.e., a list with components as follows.
`glmx.fit`

returns an unclassed list with components up to `converged`

.

`coefficients` |
a list with elements |

`residuals` |
a vector of deviance residuals, |

`fitted.values` |
a vector of fitted means, |

`optim` |
list of |

`weights` |
the weights used (if any), |

`offset` |
the list of offset vectors used (if any), |

`n` |
number of observations, |

`nobs` |
number of observations with non-zero weights, |

`df` |
number of estimated parameters, |

`loglik` |
log-likelihood of the fitted model, |

`dispersion` |
estimate of the dispersion parameter (if any), |

`vcov` |
covariance matrix of all parameters in the model, |

`family` |
a list with elements |

`xlink` |
the link object for the extra parameters, |

`control` |
control options used, |

`converged` |
logical indicating successful convergence of |

`call` |
the original function call, |

`formula` |
the formula, |

`terms` |
the terms object for the model, |

`levels` |
the levels of the categorical regressors, |

`contrasts` |
the contrasts corresponding to |

`model` |
the full model frame (if |

`y` |
the response vector (if |

`x` |
the model matrix (if |

### See Also

`glmx.control`

, `hetglm`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | ```
## artificial data from geometric regression
set.seed(1)
d <- data.frame(x = runif(200, -1, 1))
d$y <- rnbinom(200, mu = exp(0 + 3 * d$x), size = 1)
### negative binomial regression ###
## negative binomial regression via glmx
if(require("MASS")) {
m_nb1 <- glmx(y ~ x, data = d,
family = negative.binomial, xlink = "log", xstart = 0)
summary(m_nb1)
## negative binomial regression via MASS::glm.nb
m_nb2 <- glm.nb(y ~ x, data = d)
summary(m_nb2)
## comparison
if(require("lmtest")) {
logLik(m_nb1)
logLik(m_nb2)
coeftest(m_nb1)
coeftest(m_nb2)
exp(coef(m_nb1, model = "extra"))
m_nb2$theta
exp(coef(m_nb1, model = "extra")) * sqrt(vcov(m_nb1, model = "extra"))
m_nb2$SE.theta
}}
## if the score (or gradient) contribution of the extra parameters
## is supplied, then estimation can be speeded up:
negbin <- function(theta) {
fam <- negative.binomial(theta)
fam$loglik.extra <- function(y, mu, theta) digamma(y + theta) - digamma(theta) +
log(theta) + 1 - log(mu + theta) - (y + theta)/(mu + theta)
fam
}
m_nb3 <- glmx(y ~ x, data = d,
family = negbin, xlink = "log", xstart = 0, profile = FALSE)
all.equal(coef(m_nb1), coef(m_nb3))
### censored negative binomial hurdle regression (0 vs. > 0) ###
## negative binomial zero hurdle part via glmx
nbbin <- function(theta) binomial(link = nblogit(theta))
m_hnb1 <- glmx(factor(y > 0) ~ x, data = d,
family = nbbin, xlink = "log", xstart = 0)
summary(m_hnb1)
## negative binomial hurdle regression via pscl::hurdle
## (see only zero hurdle part)
if(require("pscl")) {
m_hnb2 <- hurdle(y ~ x, data = d, dist = "negbin", zero.dist = "negbin")
summary(m_hnb2)
}
``` |

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