Table of Contents

• Package
• Installation
• Usage
  • LatentClassJM
  • create.data
• Help
• Contact
• Reference

Package

LatentClassJM is a package that performs the EM algorithm to compute the sieve non-parametric maximum likelihood estimator (NPMLE) for a semiparametric latent-class joint model, using the method proposed by Wong et al. (2022). LatentClassJM relies on the R-packages survival and nnet, which are hosted on CRAN.

Installation

LatentClassJM can be installed from Github directly:

install.packages("devtools")
library(devtools)
install_github("alexwky/LatentClassJM")
library(LatentClassJM)

Usage

The package contains 2 functions:

Function	Description
LatentClassJM	Performs the EM algorithm to compute the sieve NPMLE for a semiparametric latent-class joint model
create.data	Creates a dataset based on the setting from Wong et al. (2022)

LatentClassJM

LatentClassJM(Y, X, W, Time, D, ni, Z, G, nknots, knots = NA, degree, covar = "ind",
like.diff1 = FALSE, like.diff2 = TRUE, accelEM = TRUE, bound = 5, qq = 10, epsilon = 0.001, epsilon2 = 1e-06, 
init.param = NULL, qq2 = 10, cal.inf = FALSE, max.iter = 5000, seed = 1)

This function performs the (accelerated) EM algorithm to compute the sieve NPMLE. The algorithm starts with the standard EM algorithm. Once the difference between the log-likelihood values or the parameter values of consecutive iterations becomes smaller than a certain threshold, an accelerated EM algorithm (Vardhan and Roland 2008) will be adopted until convergence.

The details of the model can be found in Wong et al. (2022) and the help manual named "LatentClassJM_0.1.0.pdf".

Input

The description of the input arguments is as follows:

Argument	Description	Optional	Default
Y	An (n × J × m) array of longitudinal outcome measurements, where n is the sample size, J is the number of longitudinal measurement types, and m is the maximum number of measurement times. It can contain NA values if the number of measurements for a subject is fewer than the maximum number of measurements. The (i; j; k)th element corresponds to the kth measurement of the jth type of longitudinal outcome for the ith subject	✖	-
X	An (n × J × m × pX) array of covariates (excluding intercept) of the longitudinal outcome model, where n is the sample size, J is the number of longitudinal measurement types, m is the number of measurement times, and pX is the number of covariates. The (i; j; k; l)th element corresponds to the lth covariate for the kth measurement of the jth type of longitudinal outcome for the ith subject	✖	-
W	An (n × pW) matrix of covariates for the latent class regression model, where n is the sample size, and pW is the number of covariate. The (i; l)th element corresponds to the lth covariate for the ith subject	✖	-
Time	An n-vector of observed event or censoring times	✖	-
D	An n-vector of event indicators	✖	-
ni	An (n × J) matrix of numbers of measurements for the longitudinal outcomes	✖	-
Z	An (n × pZ) matrix of time-independent covariates for the survival model, where n is the sample size, and pZ is the number of covariates. The (i; l)th element corresponds to the lth convariate for the ith subject	✖	-
G	Number of latent classes	✖
nknots	Number of interior knots for the B-spline basis functions	✖	-
knots	An optional vector of interior knot positions. If not supplied, then the interior knots will be selected based on quantiles of the observed event time.	✔	NA
degree	The degree of the B-spline basis functions	✖	-
covar	Covariance structure for Y. For covar = ind, repeated longitudinal measurements are independent conditional on the random effect $b$ and latent class $C$ $(\sigma_{gj2} = 0)$ For covar = exchange, repeated longitudinal measurements have an exchangeable covariance matrix conditional on the random effect $b$ and latent class $C$ $(\sigma_{gj2} \neq 0)$	✔	ind
like.diff1	Logical; If TRUE, then convergence of the standard EM algorithm is based on the difference between log-likelihood values of consecutive iterations; otherwise, convergence is based on the maximum difference between parameter values	✔	FALSE
like.diff2	Logical; If TRUE, then convergence of the accelerated EM algorithm is based on the difference between log-likelihood values of consecutive iterations; otherwise, convergence is based on the maximum difference between parameter values	✔	FALSE
accelEM	Logical; The iteration begins with standard EM algorithm. If TRUE, then the accelerated EM algorithm will be adopted after the end of the standard EM algorithm; otherwise, the program terminates after the standard EM algorithm	✔	TRUE
bound	The upper bound of the absolute value of the parameter estimates	✔	5
h	The number of abscissas for the Gauss-Hermite quadrature in the E-step	✔	10
h2	The number of abscissas for the Gauss-Hermite quadrature in the calculation of the log-likelihood	✔	10
epsilon	Threshold for convergence of standard EM algorithm	✔	1e-3
epsilon2	Threshold for convergence of accelerated EM algorithm	✔	1e-6
init.param	A named list of user-input initial values of model parameters, including alpha, beta, sigma2, xi, eta, gamma and haz. Please find the format of init.param below	✔	NULL
cal.inf	Logical; if TRUE, then the information matrix will be calculated	✔	FALSE
max.iter	Maximum number of iterations	✔	5000
seed	Seed used for parameter initialization	✔	1

Example

dataset <- create.data(n=1000)
result <- LatentClassJM(Y=dataset$Y,X=dataset$X,W=dataset$W,Time=dataset$Time,D=dataset$D,ni=dataset$ni,
Z=dataset$Z,G=4,nknots=2,degree=1,cal.inf=TRUE,init.param=NULL,bound=10,h=20,h2=20,covar="exchange")

The input list init.param should be in the following format:

Variable	Dimension
alpha	A (G × pW) matrix
beta	A (G × J × pX) array
sigma2	If covar = exchange, then dimension equals (G × J × 2); If covar = ind, then dimension equals (G × J)
xi	A G-vector
eta	A G-vector
gamma	A (G × q) matrix, where q is the number of B-splines functions; q = nknots+degree+1
haz	A vector with the same lenght of Time

Output

The output list consists of the following 14 elements:

Variable	Description
alpha	A matrix of (G × pW) regression parameters for the multinomial regression. The gth row is the parameter vector for the gth latent class; the last row must be zero
beta	A matrix of (G × J × pX) regression parameters. The (g; j)th row is the lth parameter vector for the gth latent class at jth measurement type
sigma2	sigma2 is the variance of the error terms of the longitudinal measurements. If covar = exchange, then sigma2 is a (G × J × 2) array. The (g; j; 1)th element is $\sigma_{gj1}$, and the (g; j; 2)th element is $\sigma_{gj2}$; If covar = ind, then sigma2 is a (G × J) matrix. The (g; j)th element is $\sigma_{gj1}$. In this case, $\sigma_{gj2}$ is fixed to be zero
xi	A G-vector of class-specific variances of the latent variable
gamma	gamma is a (G × (pZ + q)) matrix of class-specific regression parameters, consisting of 2 parts, The first pZ columns correspond to regression parameters of the covariates Z, and the last q columns correspond to regression parameters of the spline functions, where q = nknots+degree+1; that is, the gth row of gamma is $(\boldsymbol{\gamma}_g^T ; \boldsymbol{\alpha}^T_g )$. The defines can be found in Wong et al. (2022) or under Details from the help manual LatentClassJM_0.1.0.pdf
eta	A G-vector of class-specific regression parameters of the random effect in the survival model
Tt	A vector of ordered unique observed event times
Haz	A (t × q) matrix of all estimated latent class-specific cumulative hazard function values at Tt, where t is the length of Tt
Bmat	A (t × q) matrix of B-spline basis function values at Tt
post.prob	Subject-specific posterior group probabilities
gridb	An (n × G × qq) array of grid for the adaptive Gauss-Hermite quadrature The (n; g)th row corresponds to the grid for the ith subject under the gth latent class
weightb	An (n × G × qq) array of weight for the adaptive Gauss-Hermite quadrature The (n; g)th row corresponds to the weight for the ith subject of the gth latent class
Information	Information matrix; NA when cal.inf = FALSE
loglike	The log-likelihood value

createdata

create.data(n, seed=1)

This function is to create a dataset based on the setting from the simulation studies in Wong et al. (2022)

Input

The description of the input arguments as below:

Argument	Description	Optional	Default
n	Sample size	✖	-
seed	Seed of the random generator	✔	1

Example

dataset <- create.data(n=1000)

Output

The output list consists of following variables:

Variable	Dimension
Y	n × J × m
X	n × J × m × pX
W	n × pW
Time	n
D	n
ni	n
Z	n × pZ

Based on the setting in the simulation studies in Wong et al. (2022), we fix J = 2, m = 10, pX = 3, pW = 2, and pZ = 2.

Help

Details about the package can be found in the user manual:

?LatentClassJM

Contact

Wong Kin Yau, Alex <kin-yau.wong@polyu.edu.hk>

Reference

Vardhan, R. & Roland, C. (2008). Simple and globally convergent methods for accelerating the convergence of any EM algorithm. Scandinavian Journal of Statistics. 35: 335–353.

Wong, K. Y., Zeng, D., & Lin, D. Y. (2022). Semiparametric latent-class models for multivariate longitudinal and survival data. The Annals of Statistics. 50: 487–510.