How to estimate a latent process mixed model using lcmm function (2024)

The latent process mixed model

The latent process mixed model is introduced in Proust-Lima et al. (2006 - https://doi.org/10.1111/j.1541-0420.2006.00573.x and 2013 - https://doi.org/10.1111/bmsp.12000 ).

The quantity of interest defined as a latent process is modeled according to time using a linear mixed model:

\[\Lambda(t) = X(t) \beta + Z(t)u_i +w_i(t)\]

where:

• \(X(t)\) and \(Z(t)\) are vectors of covariates (\(Z(t)\) is included in \(X(t)\));
• \(\beta\) are the fixed effects (i.e., population mean effects);
• \(u_i\) are the random effects (i.e., individual effects); they are distributed according to a zero-mean multivariate normal distribution with covariance matrix \(B\);
• \((w_i(t))\) is a Gaussian process that might be added in the model to relax the intra-subject correlation structure.

The relationship between the latent process of interest and the observations of the marker \(Y_{ij}\) (for subject \(i\) and occasion \(j\)) is simultaneously defined in an equation of observation:

\[Y_{ij} = H( ~ \Lambda(t_{ij})+\epsilon_{ij} ~ ; \eta)\]

where:

• \(t_{ij}\) is the time of measurement for subject \(i\) and occasion \(j\);
• \(\epsilon_{ij}\) is an independent zero-mean Gaussian error;
• \(H\) is the link function (parameterized by \(\eta\)) that transforms the latent process into the scale and metric of the marker.

Different parametric families are used. When the marker is continuous, \(H^{-1}\) is a parametric family of increasing monotonic functions among:

• the linear transformation: this reduces to the linear mixed model (2 parameters)
• the Beta cumulative distribution family rescaled (4 parameters)
• the basis of quadratic I-splines with m knots (m+2 parameters)

When the marker is discrete (binary or ordinal): \(H\) is a threshold function, that is each level of Y corresponds to an interval of \(\Lambda(t_{ij})+\epsilon_{ij}\) which boundaries are to be estimated.

Identifiability

As in any latent variable model, the metric of the latent variable has to be defined. In lcmm, the variance of the errors is 1 and the mean intercept (in \(\beta\)) is 0.

Model considered:

\[CESD_{ij} = H(~ \beta_{1}age65_{ij}+\beta_{2}male_{i}+\beta_{3}age65_{ij}male_{i} +u_{0i}+u_{1i}age65_{ij}+\epsilon_{ij} ~ ; ~ \eta)\]

Where : \(u_{i} \sim \mathcal{N}(0,B)\) and \(\epsilon_{ij} \sim \mathcal{N}(0,1)\)

The Fixed part is \(\beta_{1}age65_{ij}+\beta_{2}male_{i}+\beta_{3}age65_{ij}male_{i}\) ; the random part is \(u_{0i}+u_{1i}age65_{ij}\).

Estimate the model for different continuous link functions \(H\)

We use the age variable recentered around 65 years old and in decades:

The latent process mixed model can be fitted with different link functions as shown below. This is done with argument link.

mlin <- lcmm(CESD ~ age65*male, random=~ age65, subject='ID', data=paquid) #link= linear

mlin2 <- hlme(CESD ~ age65*male, random=~ age65, subject='ID', data=paquid) #link= linear

mlin$loglik[1] -7056.652mlin2$loglik[1] -7056.652mlin$best age65 male age65:male  0.3596478 -0.7416496 0.1647425  varcov 1 varcov 2 varcov 3  1.7524938 -0.3179336 0.1973436 Linear 1 (intercept) Linear 2 (std err)  6.9846122 5.5489605 mlin2$best intercept age65 male age65:male varcov 1 varcov 2 varcov 3  6.9846116 1.9956717 -4.1153838 0.9141495 53.9609787 -9.7894861 6.0764003  stderr  5.5489602 

mbeta <- lcmm(CESD ~ age65*male, random=~ age65, subject='ID', data=paquid, link='beta')

mspl <- lcmm(CESD ~ age65*male, random=~ age65, subject='ID', data=paquid, link='splines')

mspl5q <- lcmm(CESD ~ age65*male, random=~ age65, subject='ID', data=paquid, link='5-quant-splines')

Select the best model

Objects mlin, mbeta, mspl and mspl5q are latent process mixed models that assume the exact same trajectory for the underlying latent process but different link functions: linear,BetaCDF, I-splines with 5 equidistant knots (default with link=‘splines’) and I-splines with 5 knots at percentiles, respectively. To select the most appropriate link function, one can compare these different models. Usually this is achieved by comparing the models in terms of goodness-of-fit using measures such as AIC or UACV.

The summarytable command gives the AIC (the UACV is in the output of each model):

summarytable(mlin,mbeta,mspl,mspl5q,which = c("loglik", "conv", "npm", "AIC")) loglik conv npm AICmlin -7056.652 1 8 14129.30mbeta -6306.962 1 10 12633.92mspl -6509.681 1 13 13045.36mspl5q -6320.085 1 13 12666.17

In this case, the model with a link function approximated by I-splines with 5 knots placed at the quantiles provides the best fit according to the AIC criterion.

The different estimated link functions can be compared in a plot:

col <- rainbow(5)plot(mlin, which="linkfunction", bty='l', ylab="CES-D", col=col[1], lwd=2, xlab="underlying latent process")plot(mbeta, which="linkfunction", add=TRUE, col=col[2], lwd=2)plot(mspl, which="linkfunction", add=TRUE, col=col[3], lwd=2)plot(mspl5q, which="linkfunction", add=TRUE, col=col[4], lwd=2)legend(x="topleft", legend=c("linear", "beta","splines (5equidistant)","splines (5 at quantiles)"), lty=1, col=col, bty="n", lwd=2)

We see that the 2 splines transformations are very close. The linear model does not seem to be appropriate, as shown by the gap betwwen the linear curve and the splines curves. The beta transformation departs from the splines only in the high values of the latent process.

Confidence bands of the transformations can be obtained by the Monte Carlo method :

linkspl5q <- predictlink(mspl5q,ndraws=2000)plot(linkspl5q, col=col[4], lty=2, shades=TRUE)legend(x="left", legend=c("95% confidence bands","for splines at quantiles"),lty=c(2,NA), col=c(col[4],NA), bty="n", lwd=1, cex=0.8)

Estimate the model with a discrete link function \(H\)

Sometimes, with markers that have only a restricted number of different levels, continuous link functions are not appropriate and the ordinal nature of the marker has to be handled. lcmm function handles such a case by considering threshold link function. However, one has to know that numerical complexity of the model with threshold link function is much more important (due to a numerical integration over the random effect distribution). This has to be kept in mind when fitting this model and the number of random effects is to be chosen parcimoniously.

Note that this model becomes a cumulative probit mixed model.

Here is an example with \(HIER\) variable (4 levels) as considering a threshold link function for CESD would involve too many parameters given the range in 0-52 (e.g., 52 threshold parameters).

mthresholds <- lcmm(HIER ~ age65*male, random=~ age65, subject='ID', data=paquid, link='thresholds')

Postfit outputs

summary(mspl5q) General latent class mixed model  fitted by maximum likelihood method  lcmm(fixed = CESD ~ age65 * male, random = ~age65, subject = "ID",  link = "5-quant-splines", data = paquid) Statistical Model:  Dataset: paquid  Number of subjects: 500  Number of observations: 2104  Number of observations deleted: 146  Number of latent classes: 1  Number of parameters: 13  Link function: Quadratic I-splines with nodes 0 2 6 12 52  Iteration process:  Convergence criteria satisfied  Number of iterations: 19  Convergence criteria: parameters= 1.3e-08  : likelihood= 1.5e-07  : second derivatives= 3.1e-14  Goodness-of-fit statistics:  maximum log-likelihood: -6320.08  AIC: 12666.17  BIC: 12720.96   Discrete posterior log-likelihood: -6309.09  Discrete AIC: 12644.18   Mean discrete AIC per subject: 12.6442  Mean UACV per subject: 12.6439  Mean discrete LL per subject: -12.6182  Maximum Likelihood Estimates:  Fixed effects in the longitudinal model: coef Se Wald p-valueintercept (not estimated) 0 age65 0.42421 0.06279 6.756 0.00000male -0.83140 0.19742 -4.211 0.00003age65:male 0.23371 0.10300 2.269 0.02327Variance-covariance matrix of the random-effects: intercept age65intercept 1.89911 age65 -0.39567 0.1711Residual standard error (not estimated) = 1Parameters of the link function: coef Se Wald p-valueI-splines1 -2.03816 0.13469 -15.132 0.00000I-splines2 1.04627 0.02461 42.510 0.00000I-splines3 0.74190 0.03773 19.665 0.00000I-splines4 0.98399 0.03237 30.400 0.00000I-splines5 1.55606 0.04480 34.735 0.00000I-splines6 0.93273 0.16614 5.614 0.00000I-splines7 1.38790 0.17687 7.847 0.00000

datnew <- data.frame(age=seq(65,95,length=100))datnew$age65 <- (datnew$age - 65)/10datnew$male <- 0women <- predictY(mspl5q, newdata=datnew, var.time="age", draws=TRUE)datnew$male <- 1men <- predictY(mspl5q, newdata=datnew, var.time="age", draws=TRUE)

plot(women, lwd=c(2,1), type="l", col=6, ylim=c(0,20), xlab="age in year",ylab="CES-D",bty="l", legend=NULL, shades = TRUE)plot(men, add=TRUE, col=4, lwd=c(2,1), shades=TRUE)legend(x="topleft", bty="n", ncol=2, lty=c(1,1,2,2), col=c(6,4,6,4), legend=c("women","men", "95% CI", "95% CI"), lwd=c(2,2,1,1)) 

plot(mspl5q, cex.main=0.9)

Goodness of fit 2: plot of predictions versus observations

The mean predictions and observations can be plotted according to time. Note that the predictions and observations are in the scale of the latent process (observations are transformed with the estimated link function):

plot(mspl5q, which="fit", var.time="age65", bty="l", xlab="(age-65)/10", break.times=8, ylab="latent process", lwd=2, marg=FALSE, ylim=c(-1,2), shades=TRUE, col=2)

heterogeneous profiles of trajectories

The latent process mixed model extends to the heterogeneous case with latent classes. The same strategy as explained with hlme (see vignette ) can be used.

joint analysis of a time to event

The latent process mixed model extends to the case of a joint model. This is done in Jointlcmm and mpjlcmm. See the Jointlcmm vignette.

multiple markers of the same latent process

In some cases, several markers of the same underlying latent process may be measured. The latent process mixed model extends to that case. This is the purpose of multlcmm (see the vignette for continuous and ordinal outcomes.