# clmm2

##### Cumulative link mixed models

Fits cumulative link mixed models, i.e. cumulative link models with
random effects via the Laplace approximation or the standard and the
adaptive Gauss-Hermite quadrature approximation. The functionality in
`clm2`

is also implemented here. Currently only a single
random term is allowed in the location-part of the model.

A new implementation is available in `clmm`

that allows
for more than one random effect.

- Keywords
- models

##### Usage

```
clmm2(location, scale, nominal, random, data, weights, start, subset,
na.action, contrasts, Hess = FALSE, model = TRUE, sdFixed,
link = c("logistic", "probit", "cloglog", "loglog",
"cauchit", "Aranda-Ordaz", "log-gamma"), lambda,
doFit = TRUE, control, nAGQ = 1,
threshold = c("flexible", "symmetric", "equidistant"), ...)
```

##### Arguments

- location
as in

`clm2`

.- scale
as in

`clm2`

.- nominal
as in

`clm2`

.- random
a factor for the random effects in the location-part of the model.

- data
as in

`clm2`

.- weights
as in

`clm2`

.- start
initial values for the parameters in the format

`c(alpha, beta, log(zeta), lambda, log(stDev))`

where`stDev`

is the standard deviation of the random effects.- subset
as in

`clm2`

.- na.action
as in

`clm2`

.- contrasts
as in

`clm2`

.- Hess
logical for whether the Hessian (the inverse of the observed information matrix) should be computed. Use

`Hess = TRUE`

if you intend to call`summary`

or`vcov`

on the fit and`Hess = FALSE`

in all other instances to save computing time.- model
as in

`clm2`

.- sdFixed
If

`sdFixed`

is specified (a positive scalar), a model is fitted where the standard deviation for the random term is fixed at the value of`sdFixed`

. If`sdFixed`

is left unspecified, the standard deviation of the random term is estimated from data.- link
as in

`clm2`

.- lambda
as in

`clm2`

.- doFit
as in

`clm2`

although it can also be one of`c("no", "R" "C")`

, where`"R"`

use the R-implementation for fitting,`"C"`

(default) use C-implementation for fitting and`"no"`

behaves as`FALSE`

and returns the environment.- control
a call to

`clmm2.control`

.- threshold
as in

`clm2`

.- nAGQ
the number of quadrature points to be used in the adaptive Gauss-Hermite quadrature approximation to the marginal likelihood. Defaults to

`1`

which leads to the Laplace approximation. An odd number of quadrature points is encouraged and 3, 5 or 7 are usually enough to achive high precision. Negative values give the standard, i.e. non-adaptive Gauss-Hermite quadrature.- …
additional arguments are passed on to

`clm2.control`

and possibly further on to the optimizer, which can lead to surprising error or warning messages when mistyping arguments etc.

##### Details

There are methods for the standard model-fitting functions, including
`summary`

, `vcov`

,
`profile`

,
`plot.profile`

,
`confint`

,
`anova`

, `logLik`

,
`predict`

and an `extractAIC`

method.

A Newton scheme is used to obtain the conditional modes of the random
effects for Laplace and AGQ approximations, and a non-linear
optimization is performed over the fixed parameter set to get the
maximum likelihood estimates.
The Newton
scheme uses the observed Hessian rather than the expected as is done
in e.g. `glmer`

, so results from the Laplace
approximation for binomial fits should in general be more precise -
particularly for other links than the `"logistic"`

.

Core parts of the function are implemented in C-code for speed.

The function calls `clm2`

to up an
environment and to get starting values.

##### Value

If `doFit = FALSE`

the result is an environment
representing the model ready to be optimized.
If `doFit = TRUE`

the result is an
object of class `"clmm2"`

with the following components:

the standard deviation of the random effects.

the total number of iterations in the Newton updates of the conditional modes of the random effects.

the grouping factor defining the random effects.

the number of quadrature points used in the adaptive Gauss-Hermite Quadrature approximation to the marginal likelihood.

the conditional modes of the random effects, sometimes referred to as "random effect estimates".

the conditional variances of the random effects at their conditional modes.

the parameter estimates of the location part.

the parameter estimates of the scale part on the log
scale; the scale parameter estimates on the original scale are given
by `exp(zeta)`

.

vector or matrix of the threshold parameters.

vector or matrix of the thresholds.

vector of threshold parameters, which, given a
threshold function (e.g. `"equidistant"`

), and possible nominal
effects define the class boundaries, `Theta`

.

the value of lambda if lambda is supplied or estimated, otherwise missing.

the coefficients of the intercepts
(`theta`

), the location (`beta`

), the scale
(`zeta`

), and the link function parameter (`lambda`

).

the number of residual degrees of freedoms, calculated using the weights.

vector of fitted values conditional on the values
of the random effects. Use `predict`

to
get the fitted
values for a random effect of zero. An observation here is taken to
be each of the scalar elements of the multinomial table and not a
multinomial vector.

`TRUE`

if the optimizer terminates wihtout
error and `FALSE`

otherwise.

vector of gradients for the unit-variance random effects at their conditional modes.

list with results from the optimizer. The contents of the list depends on the choice of optimizer.

the log likelihood of the model at optimizer termination.

if the model was fitted with `Hess = TRUE`

, this
is the Hessian matrix of the parameters at the optimum.

`model.frame`

for the scale model.

`model.frame`

for the location model.

`model.frame`

for the nominal model.

the (effective) number of degrees of freedom used by the model.

the starting values.

character, the optimizer.

the response variable.

the names of the levels of the response variable.

the (effective) number of observations, calculated as the sum of the weights.

character, the threshold function used in the model.

`1`

if lambda is estimated in one of the
flexible link functions and `0`

otherwise.

character, the link function used in the model.

the matched call.

contrasts applied to terms in location and scale models.

the function used to filter missing data.

##### References

Agresti, A. (2002) *Categorical Data Analysis.* Second edition. Wiley.

##### Examples

```
# NOT RUN {
options(contrasts = c("contr.treatment", "contr.poly"))
## More manageable data set:
dat <- subset(soup, as.numeric(as.character(RESP)) <= 24)
dat$RESP <- dat$RESP[drop=TRUE]
m1 <- clmm2(SURENESS ~ PROD, random = RESP, data = dat, link="probit",
Hess = TRUE, method="ucminf", threshold = "symmetric")
m1
summary(m1)
logLik(m1)
vcov(m1)
extractAIC(m1)
anova(m1, update(m1, location = SURENESS ~ 1, Hess = FALSE))
anova(m1, update(m1, random = NULL))
## Use adaptive Gauss-Hermite quadrature rather than the Laplace
## approximation:
update(m1, Hess = FALSE, nAGQ = 3)
## Use standard Gauss-Hermite quadrature:
update(m1, Hess = FALSE, nAGQ = -7)
##################################################################
## Binomial example with the cbpp data from the lme4-package:
if(require(lme4)) {
cbpp2 <- rbind(cbpp[,-(2:3)], cbpp[,-(2:3)])
cbpp2 <- within(cbpp2, {
incidence <- as.factor(rep(0:1, each=nrow(cbpp)))
freq <- with(cbpp, c(incidence, size - incidence))
})
## Fit with Laplace approximation:
fm1 <- clmm2(incidence ~ period, random = herd, weights = freq,
data = cbpp2, Hess = 1)
summary(fm1)
## Fit with the adaptive Gauss-Hermite quadrature approximation:
fm2 <- clmm2(incidence ~ period, random = herd, weights = freq,
data = cbpp2, Hess = 1, nAGQ = 7)
summary(fm2)
}
# }
```

*Documentation reproduced from package ordinal, version 2019.12-10, License: GPL (>= 2)*