Composite Likelihood Information Criterion
Description
Calculates the composite likelihood information criterion for a latent autoregressive count model fitted through maximum pairwise likelihood.
Usage
CLIC(object, ...)
Arguments
object
a fitted model object of class "lacm".
...
optional arguments.
Details
Function CLIC computes the composite likelihood information criterion (Varin and Vidoni, 2005) for a latent autoregressive count model estimated by maximum pairwise likelihood. See Pedeli and Varin (2020) for details.
When comparing models fitted by maximum pairwise likelihood to the same data, the smaller the CLIC, the better the fit.
Value
a numeric value with the corresponding CLIC.
Author(s)
Xanthi Pedeli and Cristiano Varin.
References
Pedeli, X. and Varin, C. (2020). Pairwise likelihood estimation of latent autoregressive count models. Statistical Methods in Medical Research.doi:10.1177/0962280220924068.
Varin, C. and Vidoni, P. (2005). A note on composite likelihood inference and model selection. Biometrika, 92, 519–528.
See Also
lacm .
Examples
data("polio", package = "lacm")
## model components
trend <- 1:length(polio)
sin.term <- sin(2 * pi * trend / 12)
cos.term <- cos(2 * pi * trend / 12)
sin2.term <- sin(2 * pi * trend / 6)
cos2.term <- cos(2 * pi * trend / 6)
## fit model with pairwise likelihood of order 1
mod1 <- lacm(polio ~ I(trend * 10^(-3)) + sin.term + cos.term + sin2.term + cos2.term)
CLIC(mod1)
Fitting Latent Autoregressive Count Models by Maximum Pairwise Likelihood
Description
Fits latent autoregressive count models by maximum pairwise likelihood.
Usage
lacm(formula, data, subset, offset, contrasts = NULL, start.theta = NULL, fixed, d = 1,
kernel.type = c("Rectangular", "Trapezoidal"), fit = TRUE, gh.num = 20,
reltol.opt = 1e-06, opt.method = c("BFGS", "Nelder-Mead"), maxit.opt = 1000,
sandwich.lag = NULL, bread.method = c("Outer-product", "Hessian"), ...)
Arguments
formula
an object of class "formula" (or one that
can be coerced to that class): a symbolic description of the
model to be fitted. The details of model specification are given
under ‘Details’.
data
an optional data frame, list or environment (or object
coercible by as.data.frame to a data frame) containing
the variables in the model. If not found in data, the
variables are taken from environment(formula),
typically the environment from which lacm is called.
subset
an optional vector specifying a subset of observations to be used in the fitting process.
offset
this can be used to specify an a priori known
component to be included in the linear predictor during fitting.
This should be NULL or a numeric vector of length equal to
the number of cases. One or more offset terms can be
included in the formula instead or as well, and if more than one is
specified their sum is used. See model.offset .
contrasts
an optional list. See the contrasts.arg
of model.matrix.default .
start.theta
optional numeric vector with starting values for the model parameters.
fixed
optional numeric vector of the same length as the total number of parameters. If supplied, only NA entries in fixed will be varied.
d
order of the pairwise likelihood. Defaults to 1.
kernel.type
one of "Rectangular", "Trapezoidal" indicating the type of kernel weights to be used in the weighted pairwise likelihood specification. The default "Rectangular" corresponds to equal contribution from all pairs of observations that are distant apart up to lag d. Can be abbreviated.
fit
a logical value indicating whether to compute the maximum pairwise likelihood estimates or not.
gh.num
number of the Gauss-Hermite quadrature nodes. Defaults to 20.
reltol.opt
relative convergence tolerance to be passed to optim . Defaults to 1e-6.
opt.method
one of "BFGS" or "Nelder-Mead" indicating the optimization method to be passed to optim . Can be abbreviated. See optim for details.
maxit.opt
the maximum number of iterations to be passed to optim . Defaults to 1000.
sandwich.lag
the lag length used for computing the bandwith of the sandwich variance. See ‘Details’.
bread.method
one of "Outer-product" or "Hessian" indicating whether the bread matrix of the sandwich variance is estimated with the outer-product of the individual gradients or with a numerical approximation of the Hessian matrix. Can be abbreviated. See ‘Details’.
...
further arguments passed to or from other methods.
Details
Function lacm performs maximum pairwise likelihood estimation in latent autoressive count models, see Pedeli and Varin (2020) for details.
Evaluation of the pairwise likelihood is performed through double Gauss-Hermite quadrature with the gh.num nodes and weights calculated by gauss.quad .
Standard formula y ~ x1 + x2 indicates that the mean response is modelled as a function of covariates x1 and x2 through a log link function.
Starting values supplied by the user can be specified through start.theta. If start.theta=NULL, then starting values are obtained using the method-of-moments as discussed in Ng et al. (2011).
Sandwich standard errors that are robust to heteroschedasticity and autocorrelation (HAC-type) are computed. The "meat matrix" is estimated using a lag length equal to sandwich.lag. If sandwich.lag is NULL, then L is set equal to 10 * log10(n), where n is the time series length and d is the pairwise likelihood order. The "bread matrix" is computed with the average of outer products of the individual grandients (bread.matrix = "Outer-product") or with a numerical approximation of the Hessian (bread.method = "Hessian"). Details are given in Pedeli and Varin (2020).
Value
An object of class "lacm" with the following components:
nobs
the number of observations.
p
the number of regressors, including the intercept.
d
the order of the pairwise likelihood.
npar
the number of parameters.
Y
the response used.
X
the model matrix used for the mean response.
offset
the offset used.
sandwich.lag
the lag length used for the bandwith of the HAC-type standard errors.
fit
logical. Was the model fitted or not?
gh.num
number of Gauss-Hermite nodes used.
call
the matched call.
terms
the terms object used.
latent
logical. Does the model include the latent part?
fixed
the numeric vector indicating which parameters are constants.
ifree
indeces of the free parameters.
kweights
the kernel weights used.
start.theta
the starting values.
objfun
function computing the logarithm of the pairwise likelihood of order d.
grad
function computing the gradient of the pairwise likelihood of order d.
gh
Gauss-Hermite nodes and weights used.
opt.method
a character string specifying the method argument passed to optim. The default optimization routine is the quasi-Newton algorithm BFGS. See optim for details.
convergence
an integer code indicating convergence of the optimizer. See link{optim} for details.
gh
a list with components the Gauss-Hermite nodes and the weights used for approximating the pairwise likelihood.
plik
the maximum pairwise likelihood value.
theta
the maximum pairwise likelihood estimate.
jacobian
the jacobian of the individual pairwise likelihood terms.
outer-product
logical. Was the bread matrix of the sandwich variance computed with the outer product of the individual scores?
H
the bread matrix.
J
the meat matrix.
vcov
the variance-covariance matrix of the maximum pairwise likelihood estimate.
CLIC
the composite likelihood information criterion.
Functions summary.lacm , coefficients and vcov.lacm can be used to obtain or print a summary of the results, extract coefficients and their estimated variance-covariance matrix of the model fitted by lacm .
Author(s)
Xanthi Pedeli and Cristiano Varin.
References
Ng, C., Joe, H., Karlis, D., and Liu, J. (2011). Composite likelihood for time series models with a latent autoregressive process. Statistica Sinica, 21, 279–305.
Pedeli, X. and Varin, C. (2020). Pairwise likelihood estimation of latent autoregressive count models. Statistical Methods in Medical Research.doi:10.1177/0962280220924068.
See Also
CLIC .
Examples
data("polio", package = "lacm")
## model components
trend <- 1:length(polio)
sin.term <- sin(2 * pi * trend / 12)
cos.term <- cos(2 * pi * trend / 12)
sin2.term <- sin(2 * pi * trend / 6)
cos2.term <- cos(2 * pi * trend / 6)
## fit model with pairwise likelihood of order 1
mod1 <- lacm(polio ~ I(trend * 10^(-3)) + sin.term + cos.term + sin2.term + cos2.term)
mod1
summary(mod1)
## refit with d = 3
mod3 <- update(mod1, d = 3)
summary(mod3)
Polio Time Series
Description
Time series of Polio incidences in USA from 1970 to 1983.
Usage
data(polio)Format
Time series of monthly Polio cases in USA from January 1970 to December 1983.
Source
Zeger, S.L. (1988). A regression model for time series of counts. Biometrika 75, 822–835.
Examples
data(polio, package = "lacm")
Methods for lacm Objects
Description
Methods for fitted latent autoregressive count model objects of class "lacm"
Usage
## S3 method for class 'lacm'
summary(object, ...)
## S3 method for class 'lacm'
print(x, digits = max(3L, getOption("digits") - 3L), ...)
## S3 method for class 'lacm'
coef(object, ...)
## S3 method for class 'lacm'
vcov(object, ...)
## S3 method for class 'lacm'
simulate(object, nsim = 1, seed = NULL, ...)
Arguments
digits
the number of significant digits to use when printing.
nsim
number of response vectors to simulate. Defaults to 1.
seed
an object specifying if and how the random number generator should be initialized ('seeded'). See simulate .
...
additional optional arguments.
Value
The function summary.lacm returns an object of class "summary.lacm", a list of some components of the "lacm" object, plus
coefficients
a summary of the parameter estimates, standard errors, z-values and corresponding p-values.
clic
the composite likelihood information criterion.
The function simulate.lacm returns a list of simulated responses.
The function print returns the call and coefficients, coef returns the estimated coefficients and vcov the corresponding variance-covariance matrix.
Author(s)
Xanthi Pedeli and Cristiano Varin.
References
Pedeli, X. and Varin, C. (2020). Pairwise likelihood estimation of latent autoregressive count models. Statistical Methods in Medical Research.doi:10.1177/0962280220924068.
See Also
CLIC .
Examples
data("polio", package = "lacm")
## model components
trend <- 1:length(polio)
sin.term <- sin(2 * pi * trend / 12)
cos.term <- cos(2 * pi * trend / 12)
sin2.term <- sin(2 * pi * trend / 6)
cos2.term <- cos(2 * pi * trend / 6)
## fit model with pairwise likelihood of order 1
mod1 <- lacm(polio ~ I(trend * 10^(-3)) + sin.term + cos.term + sin2.term + cos2.term)
mod1
summary(mod1)
## refit with d = 3
mod3 <- update(mod1, d = 3)
summary(mod3)