boxcoxtype {boxcoxmix}

Box-Cox-type link function for logistic mixed-effects Models

Description

The boxcoxtype() performs a grid search over the parameter Lambda for logistic mixed-effects models and then optimizes over this grid, to calculate the maximum likelihood estimator of the transformation.

Usage

boxcoxtype(
  formula,
  random = ~1,
  k = 3,
  trials = 1,
  data,
  find.in.range = c(-2, 2),
  s = 20,
  plot.opt = 1,
  random.distribution = "np",
  ...
)

boxcoxpower(Lambda = 0)

binomial(link = boxcoxpower(0))

Arguments

formula

a formula describing the transformed response and the fixed effect model (e.g. y ~ x).

random

a formula defining the random model. Set random= ~1 to model logistic-type overdispersion model. For a two-level logistic-type model, set random= ~1|groups, where groups are at the upper level.

k

the number of mass points.

trials

optional prior weights for the data. For Bernoulli distribution, set trials=1.

data

a data frame containing variables used in the fixed and random effect models.

find.in.range

search in a range of Lambda, with default (-2,2) in step of 0.1.

s

number of points in the grid search of Lambda.

plot.opt

Set plot.opt=1, to plot the profile log-likelihood against Lambda. if plot.opt=0, no plot is printed.

random.distribution

the mixing distribution, Gaussian Quadrature (gq) or NPML (np) can be set.

...

extra arguments will be ignored.

Lambda

the power of the transformation

link

the link function to be used.

Details

The Box-Cox transformation (Box & Cox, 1964) is applied to the logistic mixed-effects models with an unspecified mixing distribution. The NPML estimate of the mixing distribution is known to be a discrete distribution involving a finite number of mass-points and corresponding masses (Aitkin et al., 2009). An Expectation-Maximization (EM) algorithm is used for fitting the finite mixture distribution, one needs to specify the number of components k of the finite mixture in advance. This algorithm can be implemented using the npmlreg function alldist for the logistic-type overdispersion model and the npmlreg function allvc for the two-level logistic-type model, setting family = binomial(link = boxcoxpower(Lambda)) where Lambda is the value of the power transformation. When k=1, the npmlreg function alldist() fits the logistic regression model without random effects.

boxcoxtype() performs a grid search over the parameter Lambda and then optimizes over this grid, to calculate the maximum likelihood estimator of the transformation. It produces a plot of the profile likelihood function that summarises information concerning Lambda, including a vertical line indicating the best value of Lambda that maximizes the profile log-likelihood.

Value

List with class boxcoxmix containing:

Maximum

the best estimate of Lambda found.

objective

the value of the profile log-likelihood corresponding to Maximum.

coef

the vector of coefficients.

profile.loglik

the profile log-likelihood of the fitted regression model.

fit

the fitted alldist object from the last EM iteration.

aic

the Akaike information criterion of the fitted regression model.

bic

the Bayesian information criterion of the fitted regression model.

The other outcomes are not relevant to users and they are intended for internal use only.

Author(s)

Amani Almohaimeed and Jochen Einbeck

References

Box G. and Cox D. (1964). An analysis of transformations. Journal of the Royal Statistical Society. Series B (Methodological), pages 211-252.

Aitkin, M. A., Francis, B., Hinde, J., and Darnell, R. (2009). Statistical modelling in R. Oxford University Press Oxford.

Jochen Einbeck, Ross Darnell and John Hinde (2014). npmlreg: Nonparametric maximum likelihood estimation for random effect models. R package version 0.46-1.

See Also

np.boxcoxmix, optim.boxcox, tolfind.boxcox, Kfind.boxcox.

Examples

#Beta blockers data
data("betablocker", package = "flexmix")
library(npmlreg)
betavc <-allvc(cbind(Deaths, Total - Deaths) ~ Treatment, data = betablocker,random=~1|Center,
 k=3,random.distribution='np',family = binomial(link = boxcoxpower(0)))
betavc$disparity
#[1] 318.7211
betavc3 <-boxcoxtype(cbind(Deaths, Total - Deaths) ~ Treatment,random=~1|Center, 
data = betablocker, find.in.range = c(-2,0.4),s=40,k=3,random.distribution='np')
#Maximum Profile Log-likelihood: -158.6025 at lambda= -0.56
betavc3$fit$disparity
#[1] 317.2049
betavc3$aic
#[1] 331.2049
betavc3$bic
#[1] 343.6942