gamlssNP: A function to fit finite mixtures using the gamlss family of...

Description Usage Arguments Details Value Author(s) References See Also Examples

Description

This function will fit a finite (or normal) mixture distribution where the kernel distribution can belong to any gamlss family of distributions using the EM algorithm. The function is based on functions alldist() and allvc of the npmlreg package of Jochen Einbeck, John Hinde and Ross Darnell.

Usage

1
2
3
4
gamlssNP(formula, random = ~1, family = NO(), data = NULL, K = 4, 
          mixture = c("np", "gq"), 
          tol = 0.5, weights, pluginz, control = NP.control(...), 
          g.control = gamlss.control(trace = FALSE, ...), ...)

Arguments

formula

a formula defining the response and the fixed effects for the mu parameters

random

a formula defining the random part of the model

family

a gamlss family object

data

the data frame which for this function is mandatory even if it the data are attached

K

the number of mass points/integretion points (supported values are 1:10,20)

mixture

the mixing distribution, "np" for non-parametric or "gq" for Gaussian Quadrature

tol

the toletance scalar ussualy between zero and one

weights

prior weights

pluginz

optional

control

this sets the control parameters for the EM iterations algorithm. The default setting is the NP.control function

g.control

the gamlss control function, gamlss.control, passed to the gamlss fit

...

for extra arguments

Details

The function gamlssNP() is a modification of the R functions alldist() and allvc created by Jochen Einbeck and John Hinde. Both functions were originally created by Ross Darnell (2002). Here the two functions are merged to one gamlssNP and allows finite mixture from gamlss family of distributions.

The following are comments from the original Einbeck and Hinde documentation.

"The nonparametric maximum likelihood (NPML) approach was introduced in Aitkin (1996) as a tool to fit overdispersed generalized linear models. Aitkin (1999) extended this method to generalized linear models with shared random effects arising through variance component or repeated measures structure. Applications are two-stage sample designs, when firstly the primary sampling units (the upper-level units, e.g. classes) and then the secondary sampling units (lower-level units, e.g. students) are selected, or longitudinal data. Models of this type have also been referred to as multi-level models (Goldstein, 2003). This R function is restricted to 2-level models. The idea of NPML is to approximate the unknown and unspecified distribution of the random effect by a discrete mixture of k exponential family densities, leading to a simple expression of the marginal likelihood, which can then be maximized using a standard EM algorithm. When option 'gq' is set, then Gauss-Hermite masses and mass points are used and considered as fixed, otherwise they serve as starting points for the EM algorithm. The position of the starting points can be concentrated or extended by setting tol smaller or larger than one, respectively. Variance component models with random coefficients (Aitkin, Hinde & Francis, 2005, p. 491) are also possible, in this case the option random.distribution is restricted to the setting 'np' . The weights have to be understood as frequency weights, i.e. setting all weights equal to 2 will duplicate each data point and hence double the disparity and deviance. Warning: There might be some options and circumstances which had not been tested and where the weights do not work." Note that in keeping with the gamlss notation disparity is called global deviance.

Value

The function gamlssNP produces an object of class "gamlssNP". This object contain several components.

family

the name of the gamlss family

type

the type of distribution which in this case is "Mixture"

parameters

the parameters for the kernel gamlss family distribution

call

the call of the gamlssNP function

y

the response variable

bd

the binomial demominator, only for BI and BB models

control

the NP.control settings

weights

the vector of weights of te expanded fit

G.deviance

the global deviance

N

the number of observations in the fit

rqres

a function to calculate the normalized (randomized) quantile residuals of the object (here is the gamlss object rather than gamlssNP and it should change??)

iter

the number of external iterations in the last gamlss fitting (?? do we need this?)

type

the type of the distribution or the response variable here set to "Mixture"

method

which algorithm is used for the gamlss fit, RS(), CG() or mixed()

contrasts

the type of contrasts use in the fit

converged

whether the gamlss fit has converged

residuals

the normalized (randomized) quantile residuals of the model

mu.fv

the fitted values of the extended mu model, also sigma.fv, nu.fv, tau.fv for the other parameters if present

mu.lp

the linear predictor of the extended mu model, also sigma.lp, nu.lp, tau.lp for the other parameters if present

mu.wv

the working variable of the extended mu model, also sigma.wv, nu.wv, tau.wv for the other parameters if present

mu.wt

the working weights of the mu model, also sigma.wt, nu.wt, tau.wt for the other parameters if present

mu.link

the link function for the mu model, also sigma.link, nu.link, tau.link for the other parameters if present

mu.terms

the terms for the mu model, also sigma.terms, nu.terms, tau.terms for the other parameters if present

mu.x

the design matrix for the mu, also sigma.x, nu.x, tau.x for the other parameters if present

mu.qr

the QR decomposition of the mu model, also sigma.qr, nu.qr, tau.qr for the other parameters if present

mu.coefficients

the linear coefficients of the mu model, also sigma.coefficients, nu.coefficients, tau.coefficients for the other parameters if present

mu.formula

the formula for the mu model, also sigma.formula, nu.formula, tau.formula for the other parameters if present

mu.df

the mu degrees of freedom also sigma.df, nu.df, tau.df for the other parameters if present

mu.nl.df

the non linear degrees of freedom, also sigma.nl.df, nu.nl.df, tau.nl.df for the other parameters if present

df.fit

the total degrees of freedom use by the model

df.residual

the residual degrees of freedom left after the model is fitted

data

the original data set

EMiter

the number of EM iterations

EMconverged

whether the EM has converged

allresiduals

the residuas for the long fit

mass.points

the estimates mass point (if "np" mixture is used)

K

the number of mass points used

post.prob

contains a matrix of posteriori probabilities,

prob

the estimated mixture probalilities

aic

the Akaike information criterion

sbc

the Bayesian information criterion

formula

the formula used in the expanded fit

random

the random effect formula

pweights

prior weights

ebp

the Empirical Bayes Predictions (Aitkin, 1996b) on the scale of the linear predictor

Note that in case of Gaussian quadrature, the coefficient given at 'z' in coefficients corresponds to the standard deviation of the mixing distribution.

As a by-product, gamlssNP produces a plot showing the global deviance against the iteration number. Further, a plot with the EM trajectories is given. The x-axis corresponds to the iteration number, and the y-axis to the value of the mass points at a particular iteration. This plot is not produced when mixture is set to "gq"

Author(s)

Mikis Stasinopoulos based on function created by Jochen Einbeck John Hinde and Ross Darnell

References

Aitkin, M. and Francis, B. (1995). Fitting overdispersed generalized linear models by nonparametric maximum likelihood. GLIM Newsletter 25 , 37-45.

Aitkin, M. (1996a). A general maximum likelihood analysis of overdispersion in generalized linear models. Statistics and Computing 6 , 251-262.

Aitkin, M. (1996b). Empirical Bayes shrinkage using posterior random effect means from nonparametric maximum likelihood estimation in general random effect models. Statistical Modelling: Proceedings of the 11th IWSM 1996 , 87-94.

Aitkin, M., Francis, B. and Hinde, J. (2005) Statistical Modelling in GLIM 4. Second Edition, Oxford Statistical Science Series, Oxford, UK.

Einbeck, J. & Hinde, J. (2005). A note on NPML estimation for exponential family regression models with unspecified dispersion parameter. Technical Report IRL-GLWY-2005-04, National University of Ireland, Galway.

Einbeck, J. Darnell R. and Hinde J. (2006) npmlreg: Nonparametric maximum likelihood estimation for random effect models, R package version 0.34

Hinde, J. (1982). Compound Poisson regression models. Lecture Notes in Statistics 14 ,109-121.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape, (with discussion), Appl. Statist., 54, part 3, pp 507-554.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2003) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also http://www.gamlss.org/).

See Also

gamlss, gamlss.family

Examples

1
2
3
4
5
data(enzyme)
# equivalent model using gamlssNP
mmNP1 <- gamlssNP(act~1, data=enzyme, random=~1,family=NO, K=2)
mmNP2 <- gamlssNP(act~1, data=enzyme, random=~1, sigma.fo=~MASS, family=NO, K=2)
AIC(mmNP1, mmNP2)

mstasinopoulos/GAMLSS-Finite-Mixtures documentation built on May 27, 2019, 4:51 a.m.