Overview
RetroJM fits a two-stage retrospective joint model for longitudinal
and survival data. The main idea is to analyze the longitudinal outcome
on a retrospective timescale, where time is measured backward from death
rather than forward from baseline.
This perspective is useful when the scientific question concerns how
a longitudinal outcome, such as quality of life, changes as a subject
approaches death.
The longitudinal model is: \[
Y_i(t^*)=\beta_{\mu}(t^*)+A_i\beta_A(t^*)+X_i^T\psi_X+W_i
b_i+\epsilon_i(t^*)
\] where
\(t^*=D-t\), \(D\) is death time and \(t\) is the prospective observation
time.
The survival component is modeled using a Cox proportional hazards
model. The hazard for death: \[
\lambda_i(t)=\exp{\Big(\alpha_0(t)+A_i\alpha_A(t)\Big)},
\]
For subjects with observed death times, the retrospective timescale
is directly available. For censored subjects, death times are not
observed and are multiply imputed from the fitted survival model. The
longitudinal model is then fit on each imputed dataset, and the final
estimates are pooled across imputations.
Package installation
To install the released version from CRAN, run:
To install the development version from GitHub, run:
devtools::install_github("TChang-W/TTM")
Main function: RetroJM()
The input data should be in long format, with repeated longitudinal
measurements for each subject. The dataset should contain subject
identifiers, longitudinal measurements, observation times, treatment
assignment, baseline covariates if applicable, event or censoring times,
and the death indicator.
Arguments
The most important arguments are:
data_long: The dataset used in analysis in long
format. It should contain longitudinal measurement, measurement time, ID
for subjects, covariates of interest, event/censoring time, an indicator
for death status, and random effects by specification.
id: Name of ID variable.
treatment: Name of parameter of interest. Have to be
a binary variable coded as 0/1, such as control/treatment.
outcome: Name of longitudinal measurement.
covariates: Name of covariates. Default is NULL,
meaning no covariates.
os_time: Name of variable for event/censoring
time.
death: Name of indicator variable for death
status.
t_pros: Name of observed time variables. If the
input data is long format, it should be a single character; If the input
is wide format, it should be a character vector. It should have same
length as “outcome”.
random_effects: Names of the random effects to be
controlled.
n_impute: The number of imputations performed per
censored subject.
n_inner_knot: The number of inner knots used to
approximate the trajectories beta_mu and beta_A
Examples
Dataset
The example datasets Retro_data is included in the
package. They could be accessed by:
library(TTM)
data(Retro_data)
The Retro_data is a long format data set. It contains
1000 subjects with repeated measurements of one longitudinal
outcome.
From the data, the name of longitudinal measurement is:
Y, name of measured time are: t_pros, name of
treatment assignment is: A, name of event/censoring time is
OS_time, name of covariates are: X1, X2, name
of ID variable is: id, name of death status is:
death. We want to include a random intercept and a random
slope for A:
Fitting the model
fit <- RetroJM(
data_long = Retro_data,
id = "id",
treatment = "A",
outcome = "Y",
covariates = c("X1", "X2"),
os_time = "OS_time",
death = "death",
t_pros = "t_pros",
random_effects = c("1", "A"),
n_impute = 10,
n_inner_knot = 3
)
Output
The fitted object returned by RetroJM() contains the fitted survival
model, the imputed death times, the spline specification, and the pooled
longitudinal model results.
Pooled results
The pooled longitudinal results can be accessed by:
fit$pooled_longitudinal_model
#> $fixed_effects
#> (Intercept) A X1 X2 X_spline1 X_spline2
#> 2.1041130 0.9906772 8.5050098 18.6916119 8.0662918 14.7108156
#> X_spline3 X_spline4 AX_spline1 AX_spline2 AX_spline3 AX_spline4
#> 15.7106057 -4.3123072 2.6708466 2.6560630 20.8271074 39.7151475
#>
#> $fixed_effects_vcov
#> 12 x 12 Matrix of class "dsyMatrix"
#> (Intercept) A X1 X2 X_spline1
#> (Intercept) 0.57338869 -0.466627745 -0.078730075 -0.053956812 -0.29015082
#> A -0.46662775 0.936020613 -0.008781805 -0.006307536 0.36436795
#> X1 -0.07873007 -0.008781805 0.190849010 0.001004553 -0.03148702
#> X2 -0.05395681 -0.006307536 0.001004553 0.048436019 -0.01287468
#> X_spline1 -0.29015082 0.364367949 -0.031487017 -0.012874675 0.91216824
#> X_spline2 -0.99182041 0.454598753 0.026968853 0.053049960 -2.41839230
#> X_spline3 -0.42402250 0.806831156 0.494819879 0.153136816 2.95097141
#> X_spline4 2.32349464 -0.985562378 0.938907927 0.244149801 7.72412426
#> AX_spline1 0.31177909 -0.670371673 -0.014942818 -0.008222001 -0.94832920
#> AX_spline2 0.69045389 -1.517104724 0.184841317 0.144208877 1.97269730
#> AX_spline3 2.49392766 0.324407676 -0.877509889 -0.260937782 2.95956790
#> AX_spline4 1.98639839 6.167324309 -1.971490320 -0.641472843 4.40877269
#> X_spline2 X_spline3 X_spline4 AX_spline1 AX_spline2
#> (Intercept) -0.99182041 -0.4240225 2.3234946 0.311779085 0.6904539
#> A 0.45459875 0.8068312 -0.9855624 -0.670371673 -1.5171047
#> X1 0.02696885 0.4948199 0.9389079 -0.014942818 0.1848413
#> X2 0.05304996 0.1531368 0.2441498 -0.008222001 0.1442089
#> X_spline1 -2.41839230 2.9509714 7.7241243 -0.948329205 1.9726973
#> X_spline2 32.68825328 -44.8208939 -125.4151283 2.874907328 -29.9346193
#> X_spline3 -44.82089387 201.8816071 432.7209654 -2.756970184 32.8801597
#> X_spline4 -125.41512835 432.7209654 979.5676397 -8.030669839 99.2403027
#> AX_spline1 2.87490733 -2.7569702 -8.0306698 2.003329373 -5.6610058
#> AX_spline2 -29.93461934 32.8801597 99.2403027 -5.661005757 71.4605799
#> AX_spline3 -11.87287401 -86.1458820 -130.5438748 4.313303342 -110.5845504
#> AX_spline4 10.49885291 -192.7099968 -358.1177468 12.891373369 -293.6549081
#> AX_spline3 AX_spline4
#> (Intercept) 2.4939277 1.9863984
#> A 0.3244077 6.1673243
#> X1 -0.8775099 -1.9714903
#> X2 -0.2609378 -0.6414728
#> X_spline1 2.9595679 4.4087727
#> X_spline2 -11.8728740 10.4988529
#> X_spline3 -86.1458820 -192.7099968
#> X_spline4 -130.5438748 -358.1177468
#> AX_spline1 4.3133033 12.8913734
#> AX_spline2 -110.5845504 -293.6549081
#> AX_spline3 812.3717396 1668.1721319
#> AX_spline4 1668.1721319 3532.5260608
#>
#> $residual_variance
#> [1] 103.695
#>
#> $random_effects_covariance
#> [1] 4.840688
Baseline trajectory
To plot the estimated baseline trajectory \(\beta_\mu(t*)\) with pointwise confidence
intervals:
plot_beta_mu(fit, xlim = c(0, quantile(Retro_data$OS_time, 0.7)), conf.level = 0.95)
To plot the estimated baseline trajectory \(\beta_A(t*)\) with pointwise confidence
intervals:
plot_beta_A(fit, xlim = c(0, quantile(Retro_data$OS_time, 0.7)), conf.level = 0.95)
Interpreting the trajectories
The function \(\beta_\mu(t*)\)
represents the estimated baseline retrospective trajectory, while \(\beta_A(t*)\) represents the estimated
treatment effect as a function of retrospective time.
These curves help describe how the longitudinal outcome changes as
subjects approach death, and how that pattern differs by treatment
group.
Summary
RetroJM provides a practical framework for retrospective joint
modeling when death times are missing for censored subjects. By
combining survival-based imputation with mixed-effects modeling, the
package allows users to estimate and visualize longitudinal trajectories
on the timescale of time before death
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.