# 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. The function handles competing risks and Gaussian or non Gaussian (curvilinear) longitudinal outcomes. For curvilinear longitudinal outcomes, normalizing continuous functions (splines or Beta CDF) can be specified as in `lcmm`

.

### Usage

1 2 3 4 5 6 7 8 | ```
Jointlcmm(fixed, mixture, random, subject, classmb, ng=1,
idiag=FALSE, nwg=FALSE, survival,hazard="Weibull",
hazardtype="Specific", hazardnodes=NULL,TimeDepVar=NULL,
link=NULL, intnodes=NULL, epsY=0.5, range=NULL,
cor=NULL,data, B, convB=1e-4, convL=1e-4, convG=1e-4,
maxiter=100, nsim=100, prior,logscale=FALSE,
subset=NULL, na.action=1, posfix=NULL, partialH=FALSE,
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 |

`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 (called subject identifier) 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 |

`survival` |
two-sided formula object. The left side of the formula corresponds to a |

`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 risk function. For these two latter families, the number of nodes and the location of the nodes should be specified as well, separated by |

`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 baseline risk function that is common over classes. In the presence of competing events, a vector of hazardtypes should be given. |

`hazardnodes` |
optional vector containing interior nodes if |

`TimeDepVar` |
optional vector containing an intermediate time corresponding to a change in the risk of event. This time-dependent covariate can only take the form of a time variable with the assumption that there is no effect on the risk before this time and a constant effect on the risk of event after this time (example: initiation of a treatment to account for). |

`link` |
optional family of link functions to estimate. By default, "linear" option specifies a linear link function leading to a standard linear mixed model (homogeneous or heterogeneous as estimated in |

`intnodes` |
optional vector of interior nodes. This argument is only required for a I-splines link function with nodes entered manually. |

`epsY` |
optional definite positive real used to rescale the marker in (0,1) when the beta link function is used. By default, epsY=0.5. |

`range` |
optional vector indicating the range of the outcome (that is the minimum and maximum). By default, the range is defined according to the minimum and maximum observed values of the outcome. The option should be used only for Beta and Splines transformations. |

`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. |

`maxiter` |
optional maximum number of iterations for the Marquardt iterative algorithm. By default, maxiter=150. |

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

`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 O indicates no prior for the subject while a value in 1,...,ng indicates that the subject belongs to the corresponding latent class. |

`logscale` |
optional boolean indicating whether an exponential (logscale=TRUE) or a square (logscale=FALSE -by default) transformation is used to ensure positivity of parameters in the baseline risk functions. See details section |

`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. |

`partialH` |
optional logical for Piecewise and Splines baseline risk functions only. Indicates whether the parameters of the baseline risk functions can be dropped from the Hessian matrix to define convergence criteria. |

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

### Details

A. BASELINE RISK FUNCTIONS

For the baseline risk functions, the following parameterizations were considered. Be careful, parametrisations changed in lcmm_V1.5:

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) if logscale=FALSE and a_0(t) = exp(w_1)*exp(w_2)(t)^(exp(w_2)-1) if logscale=TRUE.

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 if logscale=FALSE and a_0(t) = exp(p_j) for y_j < t =< y_j+1 if logscale=TRUE.

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) if logscale=FALSE and a_0(t) = sum_j exp(s_j) M_j(t) if logscale=TRUE where {M_j} is the basis of cubic M-splines.

Two parametrizations of the baseline risk function are proposed (logscale=TRUE or FALSE) because in some cases, especially when the instantaneous risks are very close to 0, some convergence problems may appear with one parameterization or the other. As a consequence, we recommend to try the alternative parameterization (changing logscale option) when a joint latent class model does not converge (maximum number of iterations reached) where as convergence criteria based on the parameters and likelihood are small.

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 parameters 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. In the presence of competing events, the number of parameters should be adapted to the number of causes of event;
(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`

. In the presence of cause-specific effects, the number of parameters should be multiplied by the number of causes;
(4) for all covariates in `fixed`

, one parameter is required if the covariate is not in `mixture`

, ng parameters 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 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 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 150. 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 when baseline risk functions involve splines (value close to the lower boundary - 0 with logscale=F -infinity with logscale=F) or classmb parameters that are too high or low (perfect classification) or linkfunction parameters. 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. Some problems of convergence may happen when the instantaneous risks of event are very low and "piecewise" or "splines" baseline risk functions are specified. In this case, changing the parameterization of the baseline risk functions with option logscale is recommended (see paragraph A for details).

### Value

The list returned is:

`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 |

`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 |

`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 |

`scoretest` |
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"). |

`hazard` |
internal information about the hazard specification used in related functions |

### Author(s)

Cecile Proust Lima, Amadou Diakite 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

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

Proust-Lima, Sene, Taylor and Jacqmin-Gadda (2014). Joint latent class models of longitudinal and time-to-event data: a review. Statistical Methods in Medical Research 23, 74-90.

### See Also

`postprob`

, `plot.Jointlcmm`

,
`plot.predict`

, `epoce`

### 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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | ```
#### 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 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 -3
# nodes at the quantiles- (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.
## Not run:
#### estimation with 1 latent class (ng=1): independent models for the
# longitudinal outcome and the time of event
m1 <- Jointlcmm(fixed= Ydep1~X1*Time,random=~Time,subject='ID'
,survival = Surv(Tevent,Event)~ X1+X2 ,hazard="3-quant-splines"
,hazardtype="PH",ng=1,data=data_lcmm)
summary(m1)
#Goodness-of-fit statistics for m1:
# maximum log-likelihood: -3944.77 ; AIC: 7919.54 ; BIC: 7975.09
## End(Not run)
#### estimation with 2 latent classes (ng=2)
m2 <- Jointlcmm(fixed= Ydep1~Time*X1,mixture=~Time,random=~Time,
classmb=~X3,subject='ID',survival = Surv(Tevent,Event)~X1+mixture(X2),
hazard="3-quant-splines",hazardtype="PH",ng=2,data=data_lcmm,
B=c(0.64,-0.62,0,0,0.52,0.81,0.41,0.78,0.1,0.77,-0.05,10.43,11.3,-2.6,
-0.52,1.41,-0.05,0.91,0.05,0.21,1.5))
summary(m2)
#Goodness-of-fit statistics for m2:
# maximum log-likelihood: -3921.27; AIC: 7884.54; BIC: 7962.32
## Not run:
#### estimation with 3 latent classes (ng=3)
m3 <- Jointlcmm(fixed= Ydep1~Time*X1,mixture=~Time,random=~Time,
classmb=~X3,subject='ID',survival = Surv(Tevent,Event)~ X1+mixture(X2),
hazard="3-quant-splines",hazardtype="PH",ng=3,data=data_lcmm,
B=c(0.77,0.4,-0.82,-0.27,0,0,0,0.3,0.62,2.62,5.31,-0.03,1.36,0.82,
-13.5,10.17,10.24,11.51,-2.62,-0.43,-0.61,1.47,-0.04,0.85,0.04,0.26,1.5))
summary(m3)
#Goodness-of-fit statistics for m3:
# maximum log-likelihood: -3890.26 ; AIC: 7834.53; BIC: 7934.53
#### estimation with 4 latent classes (ng=4)
m4 <- Jointlcmm(fixed= Ydep1~Time*X1,mixture=~Time,random=~Time,
classmb=~X3,subject='ID',survival = Surv(Tevent,Event)~ X1+mixture(X2),
hazard="3-quant-splines",hazardtype="PH",ng=4,data=data_lcmm,
B=c(0.54,-0.42,0.36,-0.94,-0.64,-0.28,0,0,0,0.34,0.59,2.6,2.56,5.26,
-0.1,1.27,1.34,0.7,-5.72,10.54,9.02,10.2,11.58,-2.47,-2.78,-0.28,-0.57,
1.48,-0.06,0.61,-0.07,0.31,1.5))
summary(m4)
#Goodness-of-fit statistics for m4:
# maximum log-likelihood: -3886.93 ; AIC: 7839.86; BIC: 7962.09
##### The model with 3 latent classes is retained according to the BIC
##### and the conditional independence assumption is not rejected at
##### the 5% level.
# posterior classification
plot(m3,which="postprob")
# Class-specific predicted baseline risk & survival functions in the
# 3-class model retained (for the reference value of the covariates)
plot(m3,which="baselinerisk",bty="l")
plot(m3,which="baselinerisk",ylim=c(0,5),bty="l")
plot(m3,which="survival",bty="l")
# class-specific predicted trajectories in the 3-class model retained
# (with characteristics of subject ID=193)
data <- data_lcmm[data_lcmm$ID==193,]
plot(predictY(m3,var.time="Time",newdata=data,bty="l")
# predictive accuracy of the model evaluated with EPOCE
vect <- 1:15
cvpl <- epoce(m3,var.time="Time",pred.times=vect)
summary(cvpl)
plot(cvpl,bty="l",ylim=c(0,2))
############## end of example ##############
## End(Not run)
``` |

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