Parameters

We use artificial data contained in the library. These are 500 longitudinal data points of 0 and 1 values generated by the parameters below:

3 groups ;
group probabilities are \(\pi_1=0.22\), \(\pi_2=0.44\) and \(\pi_3=0.34\) ;
period is 10 time points ;
we use 3 polynomial shapes:
- degree 3 with \(\beta_1=(-2.266,-0.109,0.14,-0.011)\) ;
- degree 4 with \(\beta_2=(1.291,0.811,-0.554,0.077,-0.003)\) ;
- degree 0 with \(\beta_{3}=(-1.99243)\).

The data contains the time-dependent variable \(Y_i\) in data[,2:11], the time variable in data[,12:21], a time-dependent covariate in data[,24:33], and a time-invariant covariate that influences the group membership probability in data[,48:49].

data[,2:11] is a matrix with 0 and 1 values.
data[,12:21] is a matrix with time 1 to 10.
data[,24:33] is a time-dependent covariate that influences the shape of the trajectory. It is a matrix with 0 and 1 values, representing the presence or absence of a characteristic for the individual.
data[,48:49] is a matrix with 0 and 1 values, representing a time-invariant covariate.

library(trajeR)

data(data_LOGIT)

matplot(
  t(data_LOGIT[, 12:21]),t(data_LOGIT[, 2:11]),
  pch = 1, type='b', col="black", lty=1,
  xlab = "Times", ylab = "Values", main = "Plot of the individual's trajectories")

We use the each method to fit the model. For all method, we specify the number of group of our model, ng=3, the degree of the polynomial shape of the trajectories. Here we choice a line parallel to abscissa axis, a cubic polynomial and a quadric polynomial. So degree is vector \((0,3,4)\).

We specify hessian=TRUE to ask the calculus of the hessian matrix.

Itermax is set to 300 to assure a good approximation of the parameters.

For the Likelihood method we call trajeR with option Method ="L".

solL <- trajeR(
  Y = data_LOGIT[,2:11], A = data_LOGIT[,12:21],
  ng = 3, degre = c(0,3,4),
  Model = "LOGIT", Method = "L",
  hessian = TRUE
  )

solL
#> Call TrajeR with 3 groups and a 0,3,4 degrees of polynomial shape of trajectory.
#> Model : Logit
#> Method : Likelihood 
#> Convergence : Successful (Code 4)
#> 
#>   group   Parameter   Estimate   Std. Error   T for H0:   Prob>|T|  
#>                                                param.=0             
#> --------------------------------------------------------------------
#>       1   Intercept   -1.94514      0.11179   -17.40035          0  
#>                                                                     
#>       2   Intercept   -1.78181      1.22449    -1.45514    0.14569  
#>              Linear     -0.426      0.64633    -0.65911    0.50986  
#>           Quadratic     0.1834      0.10933     1.67757    0.09349  
#>               Cubic   -0.01211      0.00586    -2.06705    0.03878  
#>                                                                     
#>       3   Intercept    1.74118      0.77057     2.25959    0.02389  
#>              Linear     0.6527      0.78256     0.83406    0.40429  
#>           Quadratic   -0.55852      0.25858    -2.15991    0.03083  
#>               Cubic    0.08247      0.03335      2.4728    0.01344  
#>             Quartic   -0.00338      0.00147    -2.29819    0.02159  
#> --------------------------------------------------------------------
#>       1         pi1    0.41814      0.03818           0          0  
#>       2         pi2    0.21874      0.04173   -15.52842          0  
#>       3         pi3    0.36312      0.03402    -4.14709          0  
#> --------------------------------------------------------------------
#> Likelihood : -2852.746

For the EM method, we use the same parameters except Method = EM or EMIRLS. We specifiy itermax = 300 to ensure convergence.

#EM
solEM <- trajeR(Y = data_LOGIT[,2:11], A = data_LOGIT[,12:21],
               ng = 3, degre = c(0,3,4), 
               Model = "LOGIT", Method = "EM", hessian = TRUE, itermax = 300) 
#EMIRLS
solEMIRLS <- trajeR(Y = data_LOGIT[,2:11], A = data_LOGIT[,12:21],
                    ng = 3, degre=c(0,3,4),
                    Model = "LOGIT", Method = "EMIRLS", hessian = TRUE,
                    itermax = 300)

solEM
#> Call TrajeR with 3 groups and a 0,3,4 degrees of polynomial shape of trajectory.
#> Model : Logit
#> Method : Expectation-maximization 
#> Convergence : Successful (Code 1)
#> 
#>   group   Parameter   Estimate   Std. Error   T for H0:   Prob>|T|  
#>                                                param.=0             
#> --------------------------------------------------------------------
#>       1   Intercept   -1.94766      0.09785   -19.90467          0  
#>                                                                     
#>       2   Intercept   -1.34092      0.81565      -1.644    0.10024  
#>              Linear   -0.61492      0.49088    -1.25268    0.21038  
#>           Quadratic    0.20769      0.09134     2.27389    0.02301  
#>               Cubic   -0.01308      0.00517    -2.53057    0.01142  
#>                                                                     
#>       3   Intercept    1.75157      0.80598     2.17321    0.02981  
#>              Linear     0.7902      0.81429     0.97042    0.33189  
#>           Quadratic   -0.61231      0.26769     -2.2874    0.02221  
#>               Cubic    0.08893      0.03451     2.57704    0.00999  
#>             Quartic   -0.00362      0.00152    -2.38186    0.01726  
#> --------------------------------------------------------------------
#>       1         pi1    0.41732      0.02973    14.03545          0  
#>       2         pi2    0.23236      0.02801     8.29681          0  
#>       3         pi3    0.35032      0.04085     8.57664          0  
#> --------------------------------------------------------------------
#> Likelihood : -2852.915
solEMIRLS
#> Call TrajeR with 3 groups and a 0,3,4 degrees of polynomial shape of trajectory.
#> Model : Logit
#> Method : Expectation-maximization with IWRLS
#> Convergence : Successful (Code 1)
#> 
#>   group   Parameter   Estimate   Std. Error   T for H0:   Prob>|T|  
#>                                                param.=0             
#> --------------------------------------------------------------------
#>       1   Intercept   -1.94514      0.09673   -20.10856          0  
#>                                                                     
#>       2   Intercept    -1.7818      1.01371    -1.75769    0.07886  
#>              Linear     -0.426       0.5851    -0.72809    0.46659  
#>           Quadratic     0.1834      0.10483     1.74952    0.08026  
#>               Cubic   -0.01211      0.00578    -2.09538    0.03619  
#>                                                                     
#>       3   Intercept    1.74118      0.76282     2.28256     0.0225  
#>              Linear    0.65271      0.77462     0.84262    0.39948  
#>           Quadratic   -0.55852      0.25545    -2.18637    0.02883  
#>               Cubic    0.08247      0.03302     2.49736    0.01254  
#>             Quartic   -0.00338      0.00146    -2.31562    0.02062  
#> --------------------------------------------------------------------
#>       1         pi1    0.41814      0.02941    14.21838          0  
#>       2         pi2    0.21874      0.02786     7.85256          0  
#>       3         pi3    0.36312      0.04051     8.96429          0  
#> --------------------------------------------------------------------
#> Likelihood : -2852.746

We can add risk covariate that influence the belonging probability. By default, the effect of risk variable are compared to the first group that is reference group but we can change this setting by the option refgr.

solLRisk <- trajeR(Y = data_LOGIT[,2:11], A = data_LOGIT[,12:21], Risk = data_LOGIT[,22:23],
                  ng = 3, degre = c(0,3,4),
                  Model = "LOGIT", Method = "L", hessian = TRUE,
                  itermax = 300)
#> 
#> ── Fitting the trajeR model ────────────────────────────────────────────────────
#> ℹ In progress...
#> ✔ In progress... [598ms]
#> 
#> ── Result ──────────────────────────────────────────────────────────────────────

solLRisk
#> Call TrajeR with 3 groups and a 0,3,4 degrees of polynomial shape of trajectory.
#> Model : Logit
#> Method : Likelihood 
#> Convergence : Successful (Code 4)
#> 
#>   group   Parameter   Estimate   Std. Error   T for H0:   Prob>|T|  
#>                                                param.=0             
#> --------------------------------------------------------------------
#>       1   Intercept   -1.93481      0.10745   -18.00706          0  
#>                                                                     
#>       2   Intercept   -4.48562      4.53071    -0.99005     0.3222  
#>              Linear    0.88044      2.24444     0.39228    0.69487  
#>           Quadratic    -0.0103      0.34419    -0.02993    0.97612  
#>               Cubic    -0.0031      0.01665    -0.18607     0.8524  
#>                                                                     
#>       3   Intercept    1.65457      0.71553     2.31238     0.0208  
#>              Linear    0.49525      0.69794     0.70959    0.47799  
#>           Quadratic     -0.492      0.23359    -2.10624    0.03523  
#>               Cubic    0.07471      0.03065     2.43755    0.01482  
#>             Quartic   -0.00311      0.00137    -2.26743    0.02341  
#> --------------------------------------------------------------------
#>       1    Baseline          0           NA          NA         NA  
#>                                                                     
#>       2   Intercept   -0.99839       0.4274    -2.33595    0.01953  
#>                  X1   -0.07154      0.32883    -0.21756    0.82778  
#>                  X2    0.36804      0.34428       1.069    0.28512  
#>                                                                     
#>       3   Intercept    0.22836       0.2132     1.07111    0.28417  
#>                  X1   -0.24236       0.2262    -1.07143    0.28403  
#>                  X2   -0.38481      0.23448    -1.64113    0.10083  
#> --------------------------------------------------------------------
#> Likelihood : -2849.035

Plot

We plot only the trajectory on a graph. We have just to use the trajeR plot function. It use the class of the object to draw the polynomial shape. The values in abscissa are the first row of the time variable.

plotrajeR(solLRisk)

We can add longitudinal data to this graph. If we want them on the plot we have to specify Y and A in the function plot(). For more visibility, we have enlarge the 0 and 1 value on the graph and plot the data with a little and random shift, control by the index dec in the function plot.

By default colors are gray scale, but we can specify colors we want.

# colour's defintion
trans <- "70"
col1 <- "#034569"
col1.1 <- paste0("#64AAD0", trans)
col2 <- "#750062"
col2.1 <- paste0("#D962C7", trans)
col3 <- "#A68900"
col3.1 <- paste0("#FFE773", trans)
cols1 <- c(col1.1, col2.1, col3.1)
cols2 <- c(col1, col2, col3)
vcol <- c(cols1, cols2)

plotrajeR(solLRisk, Y = data_LOGIT[,2:11], A = data_LOGIT[,12:21], dec = 5, col = vcol)

A time-dependent covariate can directly influence the shape of the trajectories. In the data, we have a time-dependent covariate composed of 0 and 1 values. We can account for its effects by using the TCOV option in the trajeR command. Any of the three methods above can be used.

solLTCOV <- trajeR(
  Y = data_LOGIT[,2:11], A =data_LOGIT[,12:21], TCOV = data_LOGIT[,24:33],
  ng = 3, degre = c(0,3,4),
  Model = "LOGIT", Method = "L", hessian = TRUE,
  itermax = 300
  )
#> 
#> ── Fitting the trajeR model ────────────────────────────────────────────────────
#> ℹ In progress...
#> ✔ In progress... [8.6s]
#> 
#> ── Result ──────────────────────────────────────────────────────────────────────

solEMTCOV <- trajeR(
  Y = data_LOGIT[,2:11], A = data_LOGIT[,12:21], TCOV = data_LOGIT[,24:33],
  ng = 3, degre = c(0,3,4),
  Model = "LOGIT", Method = "EM", hessian = TRUE,
  itermax = 300
  )
#> 
#> ── Fitting the trajeR model ────────────────────────────────────────────────────
#> ℹ In progress...
#> ✔ In progress... [6.9s]
#> 
#> ── Result ──────────────────────────────────────────────────────────────────────

solEMIRLSTCOV <- trajeR(
  Y = data_LOGIT[,2:11], A = data_LOGIT[,12:21], TCOV = data_LOGIT[,24:33],
  ng = 3, degre = c(0,3,4),
  Model = "LOGIT", Method = "EMIRLS", hessian = FALSE,
  itermax = 300
  )
#> 
#> ── Fitting the trajeR model ────────────────────────────────────────────────────
#> ℹ In progress...
#> ✔ In progress... [1.8s]
#> 
#> ── Result ──────────────────────────────────────────────────────────────────────

solLTCOV
#> Call TrajeR with 3 groups and a 0,3,4 degrees of polynomial shape of trajectory.
#> Model : Logit
#> Method : Likelihood 
#> Convergence : Successful (Code 4)
#> 
#>   group   Parameter   Estimate   Std. Error   T for H0:   Prob>|T|  
#>                                                param.=0             
#> --------------------------------------------------------------------
#>       1   Intercept   -2.01725      0.15322   -13.16592          0  
#>               TCOV1    0.14188      0.17483     0.81155    0.41709  
#>                                                                     
#>       2   Intercept   -1.84958      1.26722    -1.45956    0.14447  
#>              Linear   -0.42661      0.66024    -0.64615    0.51822  
#>           Quadratic    0.18422      0.11027     1.67068    0.09485  
#>               Cubic   -0.01217      0.00586    -2.07782    0.03778  
#>               TCOV1    0.11965      0.19879     0.60186    0.54729  
#>                                                                     
#>       3   Intercept    1.72138      0.70961     2.42582    0.01531  
#>              Linear    0.66141      0.73626     0.89834    0.36905  
#>           Quadratic    -0.5608      0.24817    -2.25973    0.02388  
#>               Cubic     0.0827      0.03236     2.55529    0.01064  
#>             Quartic   -0.00339      0.00144    -2.35862    0.01838  
#>               TCOV1    0.01868      0.11451     0.16312    0.87043  
#> --------------------------------------------------------------------
#>       1         pi1     0.4187      0.04103           0          0  
#>       2         pi2    0.21781      0.04439    -14.7225          0  
#>       3         pi3     0.3635      0.03459     -4.0878          0  
#> --------------------------------------------------------------------
#> Likelihood : -2851.937
solEMTCOV
#> Call TrajeR with 3 groups and a 0,3,4 degrees of polynomial shape of trajectory.
#> Model : Logit
#> Method : Expectation-maximization 
#> Convergence : Successful (Code 1)
#> 
#>   group   Parameter   Estimate   Std. Error   T for H0:   Prob>|T|  
#>                                                param.=0             
#> --------------------------------------------------------------------
#>       1   Intercept   -2.01997      0.13963   -14.46618          0  
#>               TCOV1    0.14232      0.17354     0.82009    0.41221  
#>                                                                     
#>       2   Intercept   -1.60997      0.90539    -1.77821    0.07543  
#>              Linear   -0.53054      0.53194    -0.99737    0.31863  
#>           Quadratic    0.19795      0.09748     2.03061    0.04235  
#>               Cubic   -0.01274      0.00546    -2.33407    0.01963  
#>               TCOV1    0.11604       0.1947     0.59599    0.55121  
#>                                                                     
#>       3   Intercept    1.72741      0.77988     2.21498    0.02681  
#>              Linear    0.71915      0.78989     0.91044    0.36263  
#>           Quadratic   -0.58334      0.26019    -2.24194    0.02501  
#>               Cubic    0.08538       0.0336     2.54086    0.01109  
#>             Quartic   -0.00349      0.00148    -2.34995    0.01881  
#>               TCOV1    0.01399      0.11608     0.12049     0.9041  
#> --------------------------------------------------------------------
#>       1         pi1    0.41789      0.02964    14.09934          0  
#>       2         pi2    0.22445      0.02794     8.03456          0  
#>       3         pi3    0.35766      0.04073     8.78159          0  
#> --------------------------------------------------------------------
#> Likelihood : -2851.971
solEMIRLSTCOV
#> Call TrajeR with 3 groups and a 0,3,4 degrees of polynomial shape of trajectory.
#> Model : Logit
#> Method : Expectation-maximization with IWRLS
#> Convergence : Successful (Code 1)
#> 
#>   group   Parameter   Estimate   Std. Error   T for H0:   Prob>|T|  
#>                                                param.=0             
#> --------------------------------------------------------------------
#>       1   Intercept   -2.01725           NA          NA         NA  
#>               TCOV1    0.14188           NA          NA         NA  
#>                                                                     
#>       2   Intercept   -1.84956           NA          NA         NA  
#>              Linear   -0.42662           NA          NA         NA  
#>           Quadratic    0.18422           NA          NA         NA  
#>               Cubic   -0.01217           NA          NA         NA  
#>               TCOV1    0.11965           NA          NA         NA  
#>                                                                     
#>       3   Intercept    1.72138           NA          NA         NA  
#>              Linear    0.66141           NA          NA         NA  
#>           Quadratic    -0.5608           NA          NA         NA  
#>               Cubic     0.0827           NA          NA         NA  
#>             Quartic   -0.00339           NA          NA         NA  
#>               TCOV1    0.01868           NA          NA         NA  
#> --------------------------------------------------------------------
#>       1         pi1     0.4187           NA          NA         NA  
#>       2         pi2    0.21781           NA          NA         NA  
#>       3         pi3     0.3635           NA          NA         NA  
#> --------------------------------------------------------------------
#> Likelihood : -2851.937

plotrajeR(solLTCOV, col = vcol)

If we want show the impact of a particular value of the time covariate in the trajectory, we can add this to the plot by plotcov option. The fill line is the trajectory with the time covariate matrix and the dashed one show the impact on this trajectory of a particular value.

plotrajeR(solLTCOV, col = vcol, plotcov = c(0,0,0,0,0,1,1,1,1,1))

Logit model

Cédric NOEL - Jang SHILTZ

2026-08-03

Introduction

Parameters

Plot