lcmm (version 1.4-1)

Jointlcmm: Estimation of joint latent class models for longitudinal and time-to-event data

Description

This function fits joint latent class mixed models for a longitudinal outcome and a right-censored (possibly left-truncated) time-to-event. In this first version, only the Gaussian longitudinal outcome is handled but other types of longitudinal outcomes should be handled soon (continuous -not Gaussian- as well as bounded quantitative, discrete and ordinal) similarly as in lcmm funtion.

Usage

Jointlcmm(fixed, mixture, random, subject, classmb, ng=1, idiag=FALSE,
 nwg=FALSE, survival,hazard="Weibull",hazardtype="Specific", 
hazardnodes=NULL,TimeDepVar=NULL,data, B, convB=1e-4, convL=1e-4, 
convG=1e-4, maxiter=500, nsim=100, prior)

Arguments

fixed

a 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 the ~. By default, an intercept is in

mixture

a 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 reg

random

an 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.

classmb

an 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

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

idiag

optional logical for the variance-covariance structure 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 of class-specific variance-covariance of the random-effects. If FALSE the variance-covariance matrix is common over latent classes (by default). If TRUE a class-specific proportional parameter multiplies the vari

B

an optional vector containing the initial values for the parameters. The order in which the parameters are included is detailed in details section. If no vector is specified, a preliminary analysis involving the estimation of a standard line

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.

maxiter

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

nsim

optional number of points for the predicted survival curves and predicted baseline risk curves. By default, nsim=100.

prior

name of the covariate containing the prior on the latent class membership. The covariate should be an integer with values in 0,1,...,ng. When there is no prior, the value should be 0. When there is a prior for the subject, the value should be the number o

survival

a two-sided formula object. The left side of the formula corresponds to a surv() object of type "counting" for right-censored and left-truncated data (example: Surv(Time,EntryTime,Indicator)) or of type "right" for right-censored

hazard

optional family of hazard function assumed for the survival model. By default, "Weibull" specifies a Weibull baseline risk function. Other possibilities are "piecewise" for a piecewise constant risk function or "splines" for a cubic M-splines baseline ris

hazardtype

optional indicator for the type of baseline risk function when ng>1. By default "Specific" indicates a class-specific baseline risk function. Other possibilities are "PH" for a baseline risk function proportional in each latent class, and "Common" for a b

hazardnodes

optional vector containing internal nodes if splines or piecewise is specified for the baseline hazard function in hazard.

TimeDepVar

optional vector containing an intermediate time corresponding to a change in risk of event. This time-dependent covariate can only take the form of a time variable with the assumption that there is no effect before this time and a constant effect after on

data

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

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
• name.mat.cov
  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 based on the longitudinal data and the time-to-event data
• pprobY
  table of posterior classification and posterior individual class-membership probabilities based only on the longitudinal data
• Xnames
  list of covariates included in the model - for use in function plot.predict.hlme
• 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
• CIstat
  Statistic of the Score Test for the conditional independence assumption of the longitudinal and survival data given the latent class structure. Under the null hypothesis, the statistics is a Chi-square with p degrees of freedom where p indicates the number of random-effects in the longitudinal mixed model. See Jacqmin-Gadda and Proust-Lima (2009) for more details.
• predSurv
  table of predictions giving for the window of times to event (called "time"), the predicted baseline risk function in each latent class (called "RiskFct") and the predicted cumulative baseline risk function in each latent class (called "CumRiskFct").

Details

A. BASELINE RISK FUNCTIONS

For the baseline risk functions, the following parameterizations were considered:

1. With the "Weibull" function: 2 parameters are necessary w_1 and w_2 so that the baseline risk function a_0(t) = w_1^2*w_2^2*(w_1^2*t)^(w_2^2-1).

2. with the "piecewise" step function and nz nodes (y_1,...y_nz), nz-1 parameters are necesssary p_1,...p_nz-1 so that the baseline risk function a_0(t) = p_j^2 for y_j < t =< y_{j+1}.

3. with the "splines" function and nz nodes (y_1,...y_nz), nz+2 parameters are necessary s_1,...s_nz+2 so that the baseline risk function a_0(t) = sum_j s_j^2 M_j(t) where {M_j} is the basis of cubic M-splines.

B. THE VECTOR OF PARAMETERS B

The parameters in the vector of initial values B or 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 if covariates are included in classmb, ng-1 paramaters should be entered for each one; (2) parameters for the baseline risk function: 2 parameters for each Weibull, nz-1 for each piecewise constant risk and nz+2 for each splines risk; this number should be multiplied by ng if specific hazard is specified; otherwise, ng-1 additional proportional effects are expected if PH hazard is specified; otherwise nothing is added if common hazard is specified; (3) for all covariates in survival, ng parameters are required if the covariate is inside a mixture(), otherwise 1 parameter is required. Covariates parameters should be included in the same order as in survival; (4) 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. Parameters should be included in the same order as in fixed; (5) the variance of each random-effect specified in random (including the intercept) if idiag=TRUE and the inferior triangular variance-covariance matrix of all the random-effects if idiag=FALSE; (6) only if nwg=TRUE, ng-1 parameters for class-specific proportional coefficients for the variance covariance matrix of the random-effects; (7) the variance of the residual error.

C. CAUTION

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 that 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 a reason why specification of initial values in B should be preferred.

(4) Convergence criteria are very strict as they are based on derivatives of the log-likelihood in addition to the parameter and log-likelihood stability. In some cases, the program may not converge and reach the maximum number of iterations fixed at 70. 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 should be assessed. If not, the program should be run again with other initial values.

References

Lin, H., Turnbull, B. W., McCulloch, C. E. and Slate, E. H. (2002). Latent class models for joint analysis of longitudinal biomarker and event process data: application to longitudinal prostate-specific antigen readings and prostate cancer. Journal of the American Statistical Association 97, 53-65.

Proust-Lima, C. and Taylor, J. (2009). Development and validation of a dynamic prognostic tool for prostate cancer recurrence using repeated measures of post-treatment PSA: a joint modelling approach. Biostatistics 10, 535-49.

Jacqmin-Gadda, H. and Proust-Lima, C. (2010). Score test for conditional independence between longitudinal outcome and time-to-event given the classes in the joint latent class model. Biometrics 66(1), 11-9

See Also

postprob, plot.survival, plot.baselinerisk, plot.predict

Examples

##### Example of a joint latent class model estimated for a varying number
# of latent classes: 
# The linear mixed 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 (X1_Time) over classes 
# (in fixed).
# The variance of the random intercept and slopes are assumed to be equal 
# over classes (nwg=F).
# The covariate X3 predicts the  class membership (in classmb). 
# The baseline hazard function is modelled with cubic M-splines -5 
# equidistant nodes- (in hazard) and a proportional hazard over classes is 
# assumed (in hazardtype). Covariates X1 and X2 predict the risk of event 
# (in survival) with a common effect over classes for X1 and a class-specific
# effect of X2.
# !CAUTION: for illustration, only default initial values where used but 
# other sets of initial values should be tried to ensure convergence towards
# the global maximum.



#### data loading
data(data_Jointlcmm)

#### estimation with 1 latent class (ng=1): independent models for the 
# longitudinal outcome and the time of event
m1 <- Jointlcmm(fixed= Ydep1~Time+X1+X1_Time,random=~Time,subject='ID'
,survival = Surv(Tevent,Event)~ X1+X2 ,hazard="5-equi-splines"
,hazardtype="PH",ng=1,data=data_Jointlcmm)
#Goodness-of-fit statistics for m1:
#     maximum log-likelihood: -3938.91; AIC: 7911.82; BIC: 7974.79

#### estimation with 2 latent classes (ng=2)
m2 <- Jointlcmm(fixed= Ydep1~Time+X1+X1_Time,mixture=~Time,random=~Time,
classmb=~X3,subject='ID',nwg=F,survival = Surv(Tevent,Event)~X1+mixture(X2)
,hazard="5-equi-splines",hazardtype="PH",ng=2,data=data_Jointlcmm)
summary(m2)
#Goodness-of-fit statistics for m2:
#     maximum log-likelihood: -3913.96;  AIC: 7873.93; BIC: 7959.12

#### estimation with 3 latent classes (ng=3)
m3 <- Jointlcmm(fixed= Ydep1~Time+X1+X1_Time,mixture=~Time,random=~Time,
classmb=~X3,subject='ID',survival = Surv(Tevent,Event)~ mixture(X1)+X2
,hazard="5-equi-splines",hazardtype="PH",ng=3,data=data_Jointlcmm)
summary(m3)
#Goodness-of-fit statistics for m3:
#     maximum log-likelihood: -3885.94 ;  AIC: 7829.89; BIC: 7937.29

#### estimation with 4 latent classes (ng=4)
m4 <- Jointlcmm(fixed= Ydep1~Time+X1+X1_Time,mixture=~Time,random=~Time,
classmb=~X3,subject='ID',survival = Surv(Tevent,Event)~mixture(X1)+X2,
hazard="5-equi-splines",hazardtype="PH",ng=4,data=data_Jointlcmm)
summary(m4)
#Goodness-of-fit statistics for m4:
#     maximum log-likelihood: -3882.38; AIC: 7834.75; BIC: 7964.38

##### The model with 3 latent classes is retained according to the BIC  
##### and the conditional independence assumption is not rejected at
##### the 5# posterior classification
postprob(m3)
# Class-specific predicted baseline risk & survival functions in the 
# 3-class model retained (for the reference value of the covariates) 
plot.baselinerisk(m3)
plot.baselinerisk(m3,ylim=c(0,5))
plot.survival(m3)
# class-specific predicted trajectories in the 3-class model retained 
# (with characteristics of subject ID=193)
data <- data_Jointlcmm[data_Jointlcmm$ID==193,]
plot.predict(m3,var.time="Time",newdata=data)
############## end of example ##############