Description
Semi-Parametric Joint Modeling of Longitudinal and Survival Data.
Description
Maximum likelihood estimation for the semi-parametric joint modeling of competing risks and longitudinal data applying customized linear scan algorithms, proposed by Li and colleagues (2022) <doi:10.1155/2022/1362913>. The time-to-event data is modelled using a (cause-specific) Cox proportional hazards regression model with time-fixed covariates. The longitudinal outcome is modelled using a linear mixed effects model. The association is captured by shared random effects. The model is estimated using an Expectation Maximization algorithm.
README.md
FastJM
The FastJM package implement efficient computation of semi-parametric joint model of longitudinal and competing risks data.
Example
The FastJM package comes with several simulated datasets. To fit a joint model, we use jmcs function.
require(FastJM)
#> Loading required package: FastJM
#> Loading required package: statmod
#> Loading required package: MASS
require(survival)
#> Loading required package: survival
data(ydata)
data(cdata)
fit <- jmcs(ydata = ydata, cdata = cdata,
long.formula = response ~ time + gender + x1 + race,
surv.formula = Surv(surv, failure_type) ~ x1 + gender + x2 + race,
random = ~ time| ID)
fit
#>
#> Call:
#> jmcs(ydata = ydata, cdata = cdata, long.formula = response ~ time + gender + x1 + race, random = ~time | ID, surv.formula = Surv(surv, failure_type) ~ x1 + gender + x2 + race)
#>
#> Data Summary:
#> Number of observations: 3067
#> Number of groups: 1000
#>
#> Proportion of competing risks:
#> Risk 1 : 34.9 %
#> Risk 2 : 29.8 %
#>
#> Numerical intergration:
#> Method: pseudo-adaptive Guass-Hermite quadrature
#> Number of quadrature points: 6
#>
#> Model Type: joint modeling of longitudinal continuous and competing risks data
#>
#> Model summary:
#> Longitudinal process: linear mixed effects model
#> Event process: cause-specific Cox proportional hazard model with non-parametric baseline hazard
#>
#> Loglikelihood: -8989.389
#>
#> Fixed effects in the longitudinal sub-model: response ~ time + gender + x1 + race
#>
#> Estimate SE Z value p-val
#> (Intercept) 2.01853 0.05704 35.38803 0.0000
#> time 0.98292 0.03147 31.22885 0.0000
#> genderMale -0.07766 0.05860 -1.32527 0.1851
#> x1 -1.47810 0.05851 -25.26356 0.0000
#> raceWhite 0.04527 0.05911 0.76581 0.4438
#>
#> Estimate SE Z value p-val
#> sigma^2 0.49182 0.01793 27.43751 0.0000
#>
#> Fixed effects in the survival sub-model: Surv(surv, failure_type) ~ x1 + gender + x2 + race
#>
#> Estimate SE Z value p-val
#> x1_1 0.54672 0.18540 2.94892 0.0032
#> genderMale_1 -0.18781 0.11935 -1.57359 0.1156
#> x2_1 -1.10450 0.12731 -8.67602 0.0000
#> raceWhite_1 -0.10027 0.11802 -0.84960 0.3955
#> x1_2 0.62986 0.20064 3.13927 0.0017
#> genderMale_2 0.10834 0.13065 0.82926 0.4070
#> x2_2 -1.76738 0.15245 -11.59296 0.0000
#> raceWhite_2 0.03194 0.13049 0.24479 0.8066
#>
#> Association parameters:
#> Estimate SE Z value p-val
#> (Intercept)_1 0.93973 0.12160 7.72809 0.0000
#> time_1 0.31691 0.19318 1.64051 0.1009
#> (Intercept)_2 0.96486 0.13646 7.07090 0.0000
#> time_2 0.03772 0.24137 0.15629 0.8758
#>
#>
#> Random effects:
#> Formula: ~time | ID
#> Estimate SE Z value p-val
#> (Intercept) 0.52981 0.03933 13.47048 0.0000
#> time 0.25885 0.02262 11.44217 0.0000
#> (Intercept):time -0.02765 0.02529 -1.09330 0.2743
The FastJM package can make dynamic prediction given the longitudinal history information. Below is a toy example for competing risks data. Conditional cumulative incidence probabilities for each failure will be presented.
ND <- ydata[ydata$ID %in% c(419, 218), ]
ID <- unique(ND$ID)
NDc <- cdata[cdata$ID %in% ID, ]
survfit <- survfitjmcs(fit,
ynewdata = ND,
cnewdata = NDc,
u = seq(3, 4.8, by = 0.2),
method = "GH",
obs.time = "time")
survfit
#>
#> Prediction of Conditional Probabilities of Event
#> based on the pseudo-adaptive Guass-Hermite quadrature rule with 6 quadrature points
#> $`218`
#> times CIF1 CIF2
#> 1 2.441634 0.00000000 0.0000000
#> 2 3.000000 0.09629588 0.1110072
#> 3 3.200000 0.11862304 0.1369133
#> 4 3.400000 0.15142590 0.1679708
#> 5 3.600000 0.18413127 0.1839693
#> 6 3.800000 0.21269800 0.2096528
#> 7 4.000000 0.23043413 0.2249182
#> 8 4.200000 0.25459317 0.2500146
#> 9 4.400000 0.25811390 0.2599361
#> 10 4.600000 0.28856883 0.2896654
#> 11 4.800000 0.30829095 0.3134531
#>
#> $`419`
#> times CIF1 CIF2
#> 1 2.432155 0.00000000 0.00000000
#> 2 3.000000 0.02972511 0.02073398
#> 3 3.200000 0.03757608 0.02601222
#> 4 3.400000 0.05003929 0.03270990
#> 5 3.600000 0.06332292 0.03635232
#> 6 3.800000 0.07563241 0.04273814
#> 7 4.000000 0.08376596 0.04677029
#> 8 4.200000 0.09564633 0.05378957
#> 9 4.400000 0.09743720 0.05674168
#> 10 4.600000 0.11449841 0.06602758
#> 11 4.800000 0.12639379 0.07432217
To assess the prediction accuracy of the fitted joint model, we may run PEjmcs to calculate the Brier score.
## evaluate prediction accuracy of fitted joint model using cross-validated Brier Score
PE <- PEjmcs(fit, seed = 100, landmark.time = 3, horizon.time = c(3.6, 4, 4.4),
obs.time = "time", method = "GH",
quadpoint = NULL, maxiter = 1000, n.cv = 3,
survinitial = TRUE)
#> The 1 th validation is done!
#> The 2 th validation is done!
#> The 3 th validation is done!
summary(PE, error = "Brier")
#>
#> Expected Brier Score at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time Brier Score 1 Brier Score 2
#> 1 3.6 0.06214519 0.03483533
#> 2 4.0 0.09404153 0.05420762
#> 3 4.4 0.11043488 0.06861870
An alternative to assess the prediction accuracy is to run MAEQjmcs to calculate the prediction error by comparing the predicted and empirical risks stratified on different risk groups based on quantile of the predicted risks.
## evaluate prediction accuracy of fitted joint model using cross-validated mean absolute prediction error
MAEQ <- MAEQjmcs(fit, seed = 100, landmark.time = 3, horizon.time = c(3.6, 4, 4.4),
obs.time = "time", method = "GH",
quadpoint = NULL, maxiter = 1000, n.cv = 3,
survinitial = TRUE)
#> The 1 th validation is done!
#> The 2 th validation is done!
#> The 3 th validation is done!
summary(MAEQ, digits = 3)
#>
#> Sum of absolute error across quintiles of predicted risk scores at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time CIF1 CIF2
#> 1 3.6 0.249 0.079
#> 2 4.0 0.292 0.106
#> 3 4.4 0.293 0.147
We may also calculate the area under the ROC curve (AUC) to assess the discrimination measure of joint models.
## evaluate prediction accuracy of fitted joint model using cross-validated mean AUC
AUC <- AUCjmcs(fit, seed = 100, landmark.time = 3, horizon.time = c(3.6, 4, 4.4),
obs.time = "time", method = "GH",
quadpoint = NULL, maxiter = 1000, n.cv = 3)
#> The 1 th validation is done!
#> The 2 th validation is done!
#> The 3 th validation is done!
summary(AUC, digits = 3)
#>
#> Expected AUC at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time AUC1 AUC2
#> 1 3.6 0.7172788 0.6561398
#> 2 4.0 0.6914785 0.6538589
#> 3 4.4 0.6966017 0.6933577