Estimation of latent class linear mixed models

Description

This function fits linear mixed models and latent class linear mixed models (LCLMM) also known as growth mixture models or heterogeneous linear mixed models. The LCLMM consists in assuming that the population is divided in a finite number of latent classes. Each latent class is characterised by a specific trajectory modelled by a class-specific linear mixed model. Both the latent class membership and the trajectory can be explained according to covariates. This function is limited to a mixture of Gaussian outcomes. For other types of outcomes, please see function lcmm. For multivariate longitudinal outcomes, please see multlcmm.

Usage

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hlme(fixed, mixture, random, subject, classmb, ng = 1,
 idiag = FALSE, nwg = FALSE, cor=NULL, data, B,
 convB=0.0001,convL=0.0001,convG=0.0001,prior,
 maxiter=500, subset=NULL, na.action=1, posfix=NULL,
 verbose=TRUE)

Arguments

fixed

two-sided linear formula object for the fixed-effects in the linear mixed model. The response outcome is on the left of ~ and the covariates are separated by + on the right of ~. By default, an intercept is included. If no intercept, -1 should be the first term included on the right of ~.

mixture

one-sided formula object for the class-specific fixed effects in the linear mixed model (to specify only for a number of latent classes greater than 1). Among the list of covariates included in fixed, the covariates with class-specific regression parameters are entered in mixture separated by +. By default, an intercept is included. If no intercept, -1 should be the first term included.

random

optional one-sided formula for the random-effects in the linear mixed model. Covariates with a random-effect are separated by +. By default, an intercept is included. If no intercept, -1 should be the first term included.

subject

name of the covariate representing the grouping structure specified with ”.

classmb

optional one-sided formula describing the covariates in the class-membership multinomial logistic model. Covariates included are separated by +. No intercept should be included in this formula.

ng

optional number of latent classes considered. If ng=1 (by default) no mixture nor classmb should be specified. If ng>1, mixture is required.

idiag

optional logical for the structure of the variance-covariance matrix of the random-effects. If FALSE, a non structured matrix of variance-covariance is considered (by default). If TRUE a diagonal matrix of variance-covariance is considered.

nwg

optional logical indicating if the variance-covariance of the random-effects is class-specific. If FALSE the variance-covariance matrix is common over latent classes (by default). If TRUE a class-specific proportional parameter multiplies the variance-covariance matrix in each class (the proportional parameter in the last latent class equals 1 to ensure identifiability).

cor

optional brownian motion or autoregressive process modeling the correlation between the observations. "BM" or "AR" should be specified, followed by the time variable between brackets. By default, no correlation is added.

data

optional data frame containing the variables named in fixed, mixture, random, classmb and subject.

B

optional specification for the initial values for the parameters. Three options are allowed: (1) a vector of initial values is entered (the order in which the parameters are included is detailed in details section). (2) nothing is specified. A preliminary analysis involving the estimation of a standard linear mixed model is performed to choose initial values. (3) when ng>1, a hlme object is entered. It should correspond to the exact same structure of model but with ng=1. The program will automatically generate initial values from this model. This specification avoids the preliminary analysis indicated in (2). Note that due to possible local maxima, the B vector should be specified and several different starting points should be tried.

convB

optional threshold for the convergence criterion based on the parameter stability. By default, convB=0.0001.

convL

optional threshold for the convergence criterion based on the log-likelihood stability. By default, convL=0.0001.

convG

optional threshold for the convergence criterion based on the derivatives. By default, convG=0.0001.

prior

optional name of a covariate containing a prior information about the latent class membership. The covariate should be an integer with values in 0,1,...,ng. Value 0 indicates no prior for the subject while a value in 1,...,ng indicates that the subject belongs to the corresponding latent class.

maxiter

optional maximum number of iterations for the Marquardt iterative algorithm. By default, maxiter=500.

subset

a specification of the rows to be used: defaults to all rows. This can be any valid indexing vector for the rows of data or if that is not supplied, a data frame made up of the variable used in formula.

na.action

Integer indicating how NAs are managed. The default is 1 for 'na.omit'. The alternative is 2 for 'na.fail'. Other options such as 'na.pass' or 'na.exclude' are not implemented in the current version.

posfix

Optional vector specifying the indices in vector B of the parameters that should not be estimated. Default to NULL, all parameters are estimated.

verbose

logical indicating if information about computation should be reported. Default to TRUE.

Details

A. THE VECTOR OF PARAMETERS B

The parameters in the vector of initial values B or equivalently in the vector of maximum likelihood estimates best are included in the following order:

(1) ng-1 parameters are required for intercepts in the latent class membership model, and when covariates are included in classmb, ng-1 paramaters should be entered for each covariate;

(2) for all covariates in fixed, one parameter is required if the covariate is not in mixture, ng paramaters are required if the covariate is also in mixture;

(3) the variance of each random-effect specified in random (including the intercept) when idiag=TRUE, or the inferior triangular variance-covariance matrix of all the random-effects when idiag=FALSE;

(4) only when nwg=TRUE, ng-1 parameters are required for the ng-1 class-specific proportional coefficients in the variance covariance matrix of the random-effects;

(5) when cor is specified, 1 parameter corresponding to the variance of the Brownian motion should be entered with cor=BM and 2 parameters corresponding to the correlation and the variance parameters of the autoregressive process should be entered

(6) the standard error of the residual error.

B. CAUTIONS

Some caution should be made when using the program:

(1) As the log-likelihood of a latent class model can have multiple maxima, a careful choice of the initial values is crucial for ensuring convergence toward the global maximum. The program can be run without entering the vector of initial values (see point 2). However, we recommend to systematically enter initial values in B and try different sets of initial values.

(2) The automatic choice of initial values we provide requires the estimation of a preliminary linear mixed model. The user should be aware that first, this preliminary analysis can take time for large datatsets and second, that the generated initial values can be very not likely and even may converge slowly to a local maximum. This is the reason why several alternatives exist. The vector of initial values can be directly specified in B the initial values can be generated (automatically or randomly) from a model with ng=. Finally, function gridsearch performs an automatic grid search.

(3) Convergence criteria are very strict as they are based on the derivatives of the log-likelihood in addition to the parameter stability and log-likelihood stability. In some cases, the program may not converge and reach the maximum number of iterations fixed at 100. In this case, the user should check that parameter estimates at the last iteration are not on the boundaries of the parameter space. If the parameters are on the boundaries of the parameter space, the identifiability of the model is critical. This may happen especially with splines parameters that may be too close to 0 (lower boundary) or classmb parameters that are too high or low (perfect classification). When identifiability of some parameters is suspected, the program can be run again from the former estimates by fixing the suspected parameters to their value with option posfix. This usually solves the problem. An alternative is to remove the parameters of the Beta of Splines link function from the inverse of the Hessian with option partialH. If not, the program should be run again with other initial values, with a higher maximum number of iterations or less strict convergence tolerances.

Value

The list returned is:

ns

number of grouping units in the dataset

ng

number of latent classes

loglik

log-likelihood of the model

best

vector of parameter estimates in the same order as specified in B and detailed in section details

V

vector containing the upper triangle matrix of variance-covariance estimates of Best with exception for variance-covariance parameters of the random-effects for which V contains the variance-covariance estimates of the Cholesky transformed parameters displayed in cholesky

gconv

vector of convergence criteria: 1. on the parameters, 2. on the likelihood, 3. on the derivatives

conv

status of convergence: =1 if the convergence criteria were satisfied, =2 if the maximum number of iterations was reached, =4 or 5 if a problem occured during optimisation

call

the matched call

niter

number of Marquardt iterations

dataset

dataset

N

internal information used in related functions

idiag

internal information used in related functions

pred

table of individual predictions and residuals; it includes marginal predictions (pred_m), marginal residuals (resid_m), subject-specific predictions (pred_ss) and subject-specific residuals (resid_ss) averaged over classes, the observation (obs) and finally the class-specific marginal and subject-specific predictions (with the number of the latent class: pred_m_1,pred_m_2,...,pred_ss_1,pred_ss_2,...)

pprob

table of posterior classification and posterior individual class-membership probabilities

Xnames

list of covariates included in the model

predRE

table containing individual predictions of the random-effects : a column per random-effect, a line per subject

cholesky

vector containing the estimates of the Cholesky transformed parameters of the variance-covariance matrix of the random-effects

Author(s)

Cecile Proust-Lima, Benoit Liquet and Viviane Philipps

cecile.proust-lima@inserm.fr

References

Proust-Lima C, Philipps V, Liquet B (2015). Estimation of Extended Mixed Models Using Latent Classes and Latent Processes: the R package lcmm, Arxiv

Verbeke G and Lesaffre E (1996). A linear mixed-effects model with heterogeneity in the random-effects population. Journal of the American Statistical Association 91, 217-21

Muthen B and Shedden K (1999). Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics 55, 463-9

Proust C and Jacqmin-Gadda H (2005). Estimation of linear mixed models with a mixture of distribution for the random-effects. Computer Methods Programs Biomedicine 78, 165-73

See Also

postprob, plot.hlme, summary, predictY

Examples

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##### Example of a latent class model estimated for a varying number
# of latent classes: 
# The model includes a subject- (ID) and class-specific linear 
# trend (intercept and Time in fixed, random and mixture components)
# and a common effect of X1 and its interaction with time over classes 
# (in fixed). 
# The variance of the random intercept and slope are assumed to be equal 
# over classes (nwg=F).
# The covariate X3 predicts the class membership (in classmb).
#
# !CAUTION: initialization of mixed models with latent classes is 
# of most importance because of the problem of multimodality of the likelihood.
# Calls m2a-m2d illustrate the different implementations for the 
# initial values.

### homogeneous linear mixed model (standard linear mixed model) 
### with correlated random-effects
m1<-hlme(Y~Time*X1,random=~Time,subject='ID',ng=1,data=data_hlme)
summary(m1)

### latent class linear mixed model with 2 classes

# a. automatic specification from G=1 model estimates:
m2a<-hlme(Y~Time*X1,mixture=~Time,random=~Time,classmb=~X2+X3,subject='ID',
         ng=2,data=data_hlme,B=m1)
         
# b. vector of initial values provided by the user:
m2b<-hlme(Y~Time*X1,mixture=~Time,random=~Time,classmb=~X2+X3,subject='ID',
         ng=2,data=data_hlme,B=c(0.11,-0.74,-0.07,20.71,
                                 29.39,-1,0.13,2.45,-0.29,4.5,0.36,0.79,0.97))
 
# c. random draws from G = 1 model estimates:
m2c<-hlme(Y~Time*X1,mixture=~Time,random=~Time,classmb=~X2+X3,subject='ID',
          ng=2,data=data_hlme,B=random(m1))

# d. gridsearch with 50 departures and 10 iterations of the algorithm 
#     (see function gridsearch for details)
## Not run: 
m2d <- gridsearch(rep = 50, maxiter = 10, minit = m1, hlme(Y ~ Time * X1, 
mixture =~ Time, random =~ Time, classmb =~ X2 + X3, subject = 'ID', ng = 2, 
data = data_hlme))


## End(Not run)  
          


# summary of the estimation process
summarytable(m1, m2a, m2b, m2c)

# summary of m2a
summary(m2a)

# posterior classification
postprob(m2a)

# plot of predicted trajectories using some newdata
newdata<-data.frame(Time=seq(0,5,length=100),
X1=rep(0,100),X2=rep(0,100),X3=rep(0,100))
plot(predictY(m2a,newdata,var.time="Time"),legend.loc="right",bty="l")

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