R: General Nonlinear Mixture of Curves.

moc {moc}

R Documentation

General Nonlinear Mixture of Curves.

Description

moc fits user-specified mixture of curves models to one,two and three parameters distributions. The likelihood for the vector of observation over time or repeated measurements of a subject takes the form

f( Y[i] = y[i] | Z[i] = z[i], W[i] = w[i]) = Sum_k P( C[i] = k | z[i]) h( y[i] | C[i] = k, w[i])

The C[i] represents the mixture groups ( or latent classes ) and h() the conditional joint density of Y[i] given the covariates. The user supplies either the joint or marginal conditional density(ies) of the components of Y[i]. In the latter case, the joint conditional density is constructed by taking the product of the marginal densities. Censoring should be handled by the supplied density. The user can also define the mixture probability function.

The procedure minimizes the resulting -log likelihood without constraints, the parameters are all assumed to be real numbers. Thus the user should supply appropriate link functions and parameterize the density accordingly ( see the examples ).

The printed output includes -2 log likelihood, the corresponding df, AIC and BIC, mean mixture probabilities, mean expected and observed latent curves, the maximum likelihood estimates and standard errors.

The deviance residuals, fitted values and posterior probabilities are obtained through the use of the methods residuals,fitted and post. The user has the option of weighting the residuals according to the posterior mixture probabilities.

Usage


       moc(y, density=NULL, joint=FALSE, groups=1,
       gmu=NULL, gshape=NULL, gextra=NULL, gmixture=glogit, expected = NULL,
       pgmu=NULL, pgshape=NULL, pgextra=NULL, pgmix=NULL, wt=1,
       ndigit=10, gradtol=0.0001, steptol=gradtol,iterlim=100,...)

Arguments

`y`	A matrix giving the vector of observations for each subject.
`density`	A function giving the conditional joint or marginal density of the observations and calling the location, shape and extra functions.
`joint`	Specify if the density gives the joint or common marginal density of the vector of observations.
`groups`	Number of mixtures.
`gmu`	A user-specified function of `pgmu', giving the regression equation of the location parameter for each time and latent groups. The function should return a vector of length times*groups when the locations are the same across individuals or a matrix of n such vectors.
`gshape`	A user-specified function of `pgshape', giving the regression equation of the dispersion or shape parameter for each time and latent groups. The function should return a vector of length times*groups or a matrix of n such vectors. This function is usually parametrized such that the parameters are the log of the dispersion.
`gextra`	A user-specified function of `pgextra', giving the regression equation of the extra parameter for each time and latent groups. The function should return a vector of length times*groups or a matrix of n such vectors.
`gmixture`	A user-specified function `pgmix', giving the regression function of the mixture probabilities. The function should return a vector of length groups or a matrix of n such vectors. The default is the the inverse generalized logit.
`expected`	A function returning the expected response value with respect to the parameters. This function will be used to compute the fitted values and response residuals (not deviance). By default, `gmu' will be taken. It is especially useful for cases where the location function dosn't correspond to the expected value as for censored normal or zero inflated poisson distributions.
`pgmu`	Vector of initial estimates for parameters of the location function.
`pgshape`	Vector of initial estimates for parameters of the shape function.
`pgextra`	Vector of initial estimates for parameters of the extra function.
`pgmix`	Vector of initial estimates for parameters of the mixture function.
`wt`	Weight vector.
`ndigit,gradtol,steptol,iterlim,...`	Arguments controlling `nlm'.

Value

A list of class `moc' is returned that contains all of the relevant information calculated, including error codes.

Author(s)

Bernard Boulerice <Bernard.Boulerice@umontreal.ca>

References

McLachlan, Geoffrey and Peel, David (2000) Finite mixture models,Wiley-Interscience, New York.

Roeder, K., Lynch, K. and Nagin, D. (1999) Modeling Uncertainty in Latent Class Membership: A Case Study in Criminology, J. Amer. Statist. Assoc., 94, pp. 766–776.

Lindsay, Bruce G. and Roeder, K. (1992) Residual diagnostics for mixture models, J. Amer. Statist. Assoc., 87, pp. 785–794.

Examples


data(hyp)

# Censored Normal

cnorm<-function(x,mu,sig,min,max)
{mi<-(x==min)*1
ma<-(x==max)*1
mi*pnorm((min-mu)/sig)+ma*(1-pnorm((max-mu)/sig))+(1-mi-ma)*dnorm((x-mu)/sig)/sig}

cmean<-function(mu,sig,min,max) {
max-(max-mu)*pnorm((max-mu)/sig)+(min-mu)*pnorm((min-mu)/sig)-sig*(dnorm((max-mu)/sig)-dnorm((min-mu)/sig)) }

cnorm1<-function(x,mu,sig,...) {cnorm(x,mu,sig,0,10)}
gmu1<- function(p) {rbind(c(rep(p[1],3),rep(p[2],3),rep(p[3],3)))}
# Homogeneous variances

gshape1<- function(p) {rbind(c(rep(exp(p[1]),9)))}
cmean1<-function(p) { cmean(gmu1(p[1:3]),gshape1(p[4]),0,10) }

moc1<-
moc(hyp[,2:4],density=cnorm1,groups=3,gmu=gmu1,gshape=gshape1,expected=cmean1,
pgmu=c(2.1, 4.5, 7.8),pgshape=c(0.56),pgmix=c(0.6, -1.0), gradtol=1E-4)


# Heterogeneous variances across latent groups

gshape2<-function(p) {rbind(c(rep(exp(p[1]),3),rep(exp(p[2]),3),rep(exp(p[3]),3)))}

moc2<-
moc(hyp[,2:4],density=cnorm1,groups=3,gmu=gmu1,gshape=gshape2,
pgmu=moc1$coef[1:3],pgshape=c(rep(moc1$coef[4],3)),pgmix=moc1$coef[5:6],gradtol=1E-4)

# Heterogeneous variances across time

gshape3<-function(p) {rbind(c(rep(c(exp(p[1]),exp(p[2]),exp(p[3])),3)))}

moc3<-
moc(hyp[,2:4],density=cnorm1,groups=3,gmu=gmu1,gshape=gshape3,
pgmu=moc1$coef[1:3],pgshape=c(rep(moc1$coef[4],3)),pgmix=moc1$coef[5:6],gradtol=1E-4)

times<-c(1.7,3,4.2)


# Last class is a linear function of time

gmu1t<-function(p) {rbind(c(rep(p[1],3),rep(p[2],3),p[3]+p[4]*times))}

moc4<-
moc(hyp[,2:4],density=cnorm1,groups=3,gmu=gmu1t,gshape=gshape1,
pgmu=c(moc1$coeff[1:3],0),pgshape=c(moc1$coef[4]),pgmix=moc1$coef[5:6],gradtol=1E-4)

# Zero inflated Poisson log-linear in time for the third class

zip<- function(x,la,shape=1,extra)
{ mix<- exp(extra)/(1+exp(extra))
  mix*(x==0)+(1-mix)*dpois(x,la) }

gmup<-function(p) {rbind(c(rep(exp(p[1]),3),rep(exp(p[2]),3),exp(p[3]+p[4]*times)))}

zipfit<-function(p) { gmup(p)/(1+exp(p[5]))}

gextrap<-function(p) {rbind(c(rep(p[1],9)))}

moc5<-
moc(hyp[,2:4],density=zip,groups=3,gmu=gmup,gextra=gextrap,expected = zipfit,
pgmu=c(0.6, 1.5, 1.9, 0),pgextra=c(-9),pgmix=c(1, -0.5), gradtol=1E-4)


# Standard Poisson with mixture depending on time independant covariate

dumm<-hyp[,1]-1
gmixt<-function(pm)
    {mix<-cbind(pm[1]+pm[2]*dumm,pm[3]+pm[4]*dumm)
cbind(1,exp(mix))/(1+apply(exp(mix),1,sum))}

poiss<-function(x,la,...) {dpois(x,la)}

moc6<-
moc(hyp[,2:4],density=poiss,groups=3,gmu=gmup,gmixture=gmixt,
pgmu=c(0.6, 1.5, 1.9, 0),pgmix=c(1.6,-1,0.4,-1),gradtol=1E-4)

[Package Contents]