# 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

1 2 3 4 5 |

### Arguments

`fixed` |
two-sided linear formula object for the fixed-effects in the linear mixed model. The response outcome is on the left 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 |

`random` |
optional one-sided formula for the random-effects in the linear mixed model. Covariates with a random-effect are separated by |

`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 |

`ng` |
optional number of latent classes considered. If |

`idiag` |
optional logical for the structure of the variance-covariance matrix of the random-effects. If |

`nwg` |
optional logical indicating if the variance-covariance of the random-effects is class-specific. If |

`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 |

`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 |

`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 |

`V` |
vector containing the upper triangle matrix of variance-covariance estimates of |

`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

### 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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | ```
##### 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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