The FastJM package implements efficient computation of semi-parametric
joint model of longitudinal and competing risks data. To view a brief
guide on the purpose and use of this package, please refer to our
introductory video.
The FastJM package comes with several simulated datasets. To fit a
joint model, we use jmcs function. In the example below, we are using
the following built-in data sets:
- ydata: longitudinal data for a single biomarker per patient
- cdata: competing risks time-to-event data per patient
require(FastJM)
require(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 0.2743The fitted jmcs object also provides a standard diagnostic plotting
method through plot(). This function displays four panels:
subject-specific residuals for the longitudinal process versus their
corresponding fitted values; a normal Q-Q plot of the residuals of the
standardized subject-specific residuals for the longitudinal process, an
estimate of the marginal survival function for the event process, and an
estimate of the marginal cumulative risk function for the event process.
These plots provide a quick graphical summary of the longitudinal
residual behavior and the estimated event-time process from the fitted
joint model.
plot(fit)
We can further
examine the fitted joint model using diagnostic plots. The timeplot()
function displays the longitudinal biomarker trajectories, the empirical
log residual variance over follow-up time, and the event process. For
competing-risk models, the event plot is shown as cumulative incidence
curves for the specified event types.
The log residual variance plot is intended as an exploratory diagnostic. Apparent changes over time may reflect departures from constant residual variance, but they may also be affected by finite-sample variability, model fit, or changing subject composition over follow-up.
crplot <- timeplot(
object = fit,
biomarker = response,
id_col = ID,
time_col = time,
time_bin_width = 0.5,
fail_code = 1,
cr_code = 2,
censor_code = 0,
primary_event_label = "Event type 1",
competing_event_label = "Event type 2",
x_lab = "Visit time",
event_x_lab = "Survival time",
n.obs = 200
)
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 <- survfitJM(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 0.07432217To assess the prediction accuracy of the fitted joint model, we may run
DynPredAcc to assess the prediction accuracy by calculating all
available evaluation metrics.
res <- DynPredAcc(
object = fit,
landmark.time = 3,
horizon.time = c(3.6, 4, 4.4),
obs.time = "time",
method = "GH",
maxiter = 1000,
n.cv = 3,
quantile.width = 0.25,
metrics = c("AUC", "Cindex", "Brier Score", "MAE", "MAEQ")
)
#> The 1-th validation is done!
#> The 2-th validation is done!
#> The 3-th validation is done!
summary(res, metric = "Brier Score")
#>
#> 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 0.0589 0.0348
#> 2 4.0 0.0889 0.0543
#> 3 4.4 0.1052 0.0687
summary(res, metric = "MAE")
#>
#> Expected mean absolute error at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time MAE1 MAE2
#> 1 3.6 0.1188 0.0699
#> 2 4.0 0.1765 0.1085
#> 3 4.4 0.2090 0.1387
summary(res, metric = "MAEQ")
#>
#> Mean absolute error across quantiles of predicted risk scores at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time MAEQ1 MAEQ2
#> 1 3.6 0.0208 0.0301
#> 2 4.0 0.0446 0.0403
#> 3 4.4 0.0477 0.0379
summary(res, metric = "AUC")
#>
#> Expected AUC at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time AUC1 AUC2
#> 1 3.6 0.7367 0.7097
#> 2 4.0 0.7155 0.6760
#> 3 4.4 0.7337 0.7255
summary(res, metric = "Cindex")
#>
#> Expected Cindex at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time Cindex1 Cindex2
#> 1 3.6 0.6864 0.6773
#> 2 4.0 0.6860 0.6765
#> 3 4.4 0.6862 0.6758Or we can calculate the overall, time-independent Cindex over the entire time period, evaluated by the linear predictor of the (cause-specific) Cox model.
Concord <- Concordance.jmcs(seed = 100, fit, n.cv = 3)
#> The 1 th validation is done!
#> The 2 th validation is done!
#> The 3 th validation is done!
summary(Concord)
#> Concordance1 Concordance2
#> 1 0.6721619 0.7037879To fit a joint model with multiple longitudinal outcomes and competing
risks, we can use the mvjmcs function. In the example below, we are
using the following built-in data sets:
- mvydata: longitudinal data for multiple biomarkers per patient
- mvcdata: competing risks time-to-event data per patient
data(mvydata)
data(mvcdata)
mvfit <- mvjmcs(ydata = mvydata, cdata = mvcdata,
long.formula = list(Y1 ~ X11 + X12 + time,
Y2 ~ X11 + X12 + time),
random = list(~ time | ID,
~ 1 | ID),
surv.formula = Surv(survtime, cmprsk) ~ X21 + X22)
mvfit
#>
#> Call:
#> mvjmcs(ydata = mvydata, cdata = mvcdata, long.formula = list(Y1 ~ X11 + X12 + time, Y2 ~ X11 + X12 + time), random = list(~time | ID, ~1 | ID), surv.formula = Surv(survtime, cmprsk) ~ X21 + X22)
#>
#> Data Summary:
#> Number of observations: 5645
#> Number of groups: 800
#>
#> Proportion of competing risks:
#> Risk 1 : 41.62 %
#> Risk 2 : 11.25 %
#>
#> Model Type: joint modeling of multivariate longitudinal continuous and competing risks data
#>
#> Model summary:
#> Runtime: 33.05 seconds
#> Longitudinal process: linear mixed effects model
#> Event process: cause-specific Cox proportional hazard model with non-parametric baseline hazard
#>
#> Fixed effects in the longitudinal sub-model: list(Y1 ~ X11 + X12 + time, Y2 ~ X11 + X12 + time)
#>
#> Estimate SE Z value p-val
#> (Intercept)_bio1 4.97836 0.05389 92.37830 0.0000
#> X11_bio1 1.46373 0.08052 18.17736 0.0000
#> X12_bio1 1.99688 0.01430 139.68697 0.0000
#> time_bio1 0.83770 0.03951 21.20086 0.0000
#> (Intercept)_bio2 9.97514 0.04927 202.47202 0.0000
#> X11_bio2 0.97968 0.07331 13.36384 0.0000
#> X12_bio2 2.00928 0.01309 153.45755 0.0000
#> time_bio2 0.99382 0.00455 218.62849 0.0000
#>
#> Estimate SE Z value p-val
#> sigma^2_bio1 0.49302 0.00018 2729.81717 0.0000
#> sigma^2_bio2 0.49757 0.00965 51.54934 0.0000
#>
#> Fixed effects in the survival sub-model: Surv(survtime, cmprsk) ~ X21 + X22
#>
#> Estimate SE Z value p-val
#> X21_1 0.92687 0.13465 6.88331 0.0000
#> X22_1 0.50892 0.03163 16.08946 0.0000
#> X21_2 -0.22126 0.24916 -0.88800 0.3745
#> X22_2 0.48336 0.05920 8.16466 0.0000
#>
#> Association parameters:
#> Estimate SE Z value p-val
#> (Intercept)_1bio1 0.49735 0.07539 6.59671 0.0000
#> time_1bio1 0.70010 0.08498 8.23809 0.0000
#> (Intercept)_1bio2 -0.54465 0.07972 -6.83163 0.0000
#> (Intercept)_2bio1 0.63098 0.13344 4.72857 0.0000
#> time_2bio1 0.65771 0.16710 3.93611 0.0001
#> (Intercept)_2bio2 -0.48306 0.15877 -3.04259 0.0023
#>
#>
#> Random effects:
#> bio 1 : ~time | ID
#> bio 2 : ~1 | ID
#> Estimate SE Z value p-val
#> Intercept1 1.02128 0.06470 15.78547 0.0000
#> time1 0.91442 0.05819 15.71410 0.0000
#> Intercept2 0.88199 0.05324 16.56595 0.0000
#> Intercept1:time1 -0.09442 0.04533 -2.08271 0.0373
#> Intercept1:Intercept2 0.04355 0.04051 1.07499 0.2824
#> time1:Intercept2 -0.06492 0.04223 -1.53739 0.1242We can extract the components of the model as follows:
# Longitudinal fixed effects
fixef(mvfit, process = "Longitudinal")
#> (Intercept)_bio1 X11_bio1 X12_bio1 time_bio1 (Intercept)_bio2 X11_bio2 X12_bio2
#> 4.9783622 1.4637306 1.9968810 0.8377000 9.9751421 0.9796767 2.0092771
#> time_bio2
#> 0.9938159
summary(mvfit, process = "Longitudinal")
#> Longitudinal coef SE 95%Lower 95%Upper p-values
#> 1 (Intercept)_bio1 4.9784 0.0539 4.8727 5.0840 0
#> 2 X11_bio1 1.4637 0.0805 1.3059 1.6216 0
#> 3 X12_bio1 1.9969 0.0143 1.9689 2.0249 0
#> 4 time_bio1 0.8377 0.0395 0.7603 0.9151 0
#> 5 (Intercept)_bio2 9.9751 0.0493 9.8786 10.0717 0
#> 6 X11_bio2 0.9797 0.0733 0.8360 1.1234 0
#> 7 X12_bio2 2.0093 0.0131 1.9836 2.0349 0
#> 8 time_bio2 0.9938 0.0045 0.9849 1.0027 0
#> 9 sigma^2_bio1 0.4930 0.0002 0.4927 0.4934 0
#> 10 sigma^2_bio2 0.4976 0.0097 0.4787 0.5165 0
# Survival fixed effects
fixef(mvfit, process = "Event")
#> $Risk1
#> X21_1 X22_1
#> 0.9268691 0.5089241
#>
#> $Risk2
#> X21_2 X22_2
#> -0.2212556 0.4833562
summary(mvfit, process = "Event")
#> Survival coef exp(coef) SE(coef) 95%Lower 95%Upper 95%exp(Lower) 95%exp(Upper) p-values
#> 1 X21_1 0.9269 2.5266 0.1347 0.6629 1.1908 1.9405 3.2897 0.0000
#> 2 X22_1 0.5089 1.6635 0.0316 0.4469 0.5709 1.5635 1.7699 0.0000
#> 3 X21_2 -0.2213 0.8015 0.2492 -0.7096 0.2671 0.4918 1.3062 0.3745
#> 4 X22_2 0.4834 1.6215 0.0592 0.3673 0.5994 1.4439 1.8210 0.0000
#> 5 (Intercept)_1bio1 0.4973 1.6444 0.0754 0.3496 0.6451 1.4185 1.9062 0.0000
#> 6 time_1bio1 0.7001 2.0139 0.0850 0.5335 0.8667 1.7049 2.3790 0.0000
#> 7 (Intercept)_1bio2 -0.5446 0.5800 0.0797 -0.7009 -0.3884 0.4961 0.6781 0.0000
#> 8 (Intercept)_2bio1 0.6310 1.8794 0.1334 0.3694 0.8925 1.4469 2.4413 0.0000
#> 9 time_2bio1 0.6577 1.9304 0.1671 0.3302 0.9852 1.3912 2.6784 0.0001
#> 10 (Intercept)_2bio2 -0.4831 0.6169 0.1588 -0.7942 -0.1719 0.4519 0.8421 0.0023
# Random effects for first few subjects
head(ranef(mvfit))
#> (Intercept)_bio1 time_bio1 (Intercept)_bio2
#> 1 1.2319610 -0.52584340 -1.20371719
#> 2 -0.5308272 -0.32954734 1.56026919
#> 3 -1.1627322 0.33104170 0.17052091
#> 4 -1.4296389 -1.93508778 -0.09598218
#> 5 0.2379468 -1.94922572 0.02283060
#> 6 -0.1229965 -0.02043454 0.06420000The FastJM package can now make dynamic prediction in the presence of
multiple longitudinal outcomes. Below is a toy example for competing
risks data. Conditional cumulative incidence probabilities for each
failure will be presented.
require(dplyr)
#> Loading required package: dplyr
#> Warning: package 'dplyr' was built under R version 4.5.2
#>
#> Attaching package: 'dplyr'
#> The following object is masked from 'package:MASS':
#>
#> select
#> The following objects are masked from 'package:stats':
#>
#> filter, lag
#> The following objects are masked from 'package:base':
#>
#> intersect, setdiff, setequal, union
set.seed(08252025)
sampleID <- sample(mvcdata$ID, 5, replace = FALSE)
subcdata <- mvcdata %>%
dplyr::filter(ID %in% sampleID)
subydata <- mvydata %>%
dplyr::filter(ID %in% sampleID)
### Set up a landmark time of 4.75 and make predictions at time u
survmvfit <- survfitJM(mvfit, seed = 100, ynewdata = subydata, cnewdata = subcdata,
u = c(7, 8, 9), Last.time = 4.75, obs.time = "time")
survmvfit
#>
#> Prediction of Conditional Probabilities of Event
#> based on the first order approximation
#> $`177`
#> times CIF1 CIF2
#> 1 4.75 0.00000000 0.000000000
#> 2 7.00 0.01861973 0.003167409
#> 3 8.00 0.02670221 0.004780677
#> 4 9.00 0.02981999 0.006005652
#>
#> $`182`
#> times CIF1 CIF2
#> 1 4.75 0.0000000 0.00000000
#> 2 7.00 0.2460705 0.03392616
#> 3 8.00 0.3322085 0.04804826
#> 4 9.00 0.3626926 0.05807446
#>
#> $`260`
#> times CIF1 CIF2
#> 1 4.75 0.00000000 0.00000000
#> 2 7.00 0.03340964 0.01753985
#> 3 8.00 0.04761368 0.02629538
#> 4 9.00 0.05303220 0.03288541
#>
#> $`305`
#> times CIF1 CIF2
#> 1 4.75 0.00000000 0.00000000
#> 2 7.00 0.02891562 0.01545646
#> 3 8.00 0.04126797 0.02320727
#> 4 9.00 0.04599079 0.02905144
#>
#> $`800`
#> times CIF1 CIF2
#> 1 4.75 0.00000000 0.000000000
#> 2 7.00 0.01309319 0.002113995
#> 3 8.00 0.01880334 0.003195467
#> 4 9.00 0.02101032 0.004017890To assess the prediction accuracy of the fitted joint model, we may run
DynPredAcc to assess the prediction accuracy by calculating all
available evaluation metrics.
res <- DynPredAcc(
object = mvfit,
landmark.time = 3,
horizon.time = c(3.6, 4, 4.4),
obs.time = "time",
maxiter = 1000,
n.cv = 3,
quantile.width = 0.25,
metrics = c("AUC", "Cindex", "Brier Score", "MAE", "MAEQ")
)
#> The 1-th validation is done!
#> The 2-th validation is done!
#> The 3-th validation is done!
summary(res, metric = "Brier Score")
#>
#> Expected Brier Score at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time Brier Score 1 Brier Score 2
#> 3.6 3.6 0.0395 0.0101
#> 4 4.0 0.0555 0.0143
#> 4.4 4.4 0.0669 0.0266
summary(res, metric = "MAE")
#>
#> Expected mean absolute error at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time MAE1 MAE2
#> 3.6 3.6 0.0793 0.0204
#> 4 4.0 0.1122 0.0284
#> 4.4 4.4 0.1349 0.0515
summary(res, metric = "MAEQ")
#>
#> Mean absolute error across quantiles of predicted risk scores at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time MAEQ1 MAEQ2
#> 1 3.6 0.0202 0.0102
#> 2 4.0 0.0299 0.0172
#> 3 4.4 0.0381 0.0285
summary(res, metric = "AUC")
#>
#> Expected AUC at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time AUC1 AUC2
#> 3.6 3.6 0.7981 0.7850
#> 4 4.0 0.8290 0.7786
#> 4.4 4.4 0.8313 0.7210
summary(res, metric = "Cindex")
#>
#> Expected Cindex at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time Cindex1 Cindex2
#> 3.6 3.6 0.8257 0.6914
#> 4 4.0 0.8256 0.6922
#> 4.4 4.4 0.8256 0.6927- ydatah: longitudinal data for a single biomarker per patient
- cdatah: competing risks time-to-event data per patient
data(ydatah)
data(cdatah)
## fit a joint model
fit <- JMMLSM(cdata = cdatah, ydata = ydatah,
long.formula = Y ~ Z1 + Z2 + Z3 + time,
surv.formula = Surv(survtime, cmprsk) ~ var1 + var2 + var3,
variance.formula = ~ Z1 + Z2 + Z3 + time,
random = ~ 1|ID)
fit
#>
#> Call:
#> JMMLSM(cdata = cdatah, ydata = ydatah, long.formula = Y ~ Z1 + Z2 + Z3 + time, surv.formula = Surv(survtime, cmprsk) ~ var1 + var2 + var3, variance.formula = ~Z1 + Z2 + Z3 + time, random = ~1 | ID)
#>
#> Data Summary:
#> Number of observations: 1353
#> Number of groups: 200
#>
#> Proportion of competing risks:
#> Risk 1 : 45.5 %
#> Risk 2 : 32.5 %
#>
#> Numerical intergration:
#> Method: adaptive Guass-Hermite quadrature
#> Number of quadrature points: 6
#>
#> Model Type: joint modeling of longitudinal continuous and competing risks data with the presence of intra-individual variability
#>
#> Model summary:
#> Longitudinal process: Mixed effects location scale model
#> Event process: cause-specific Cox proportional hazard model with non-parametric baseline hazard
#>
#> Loglikelihood: -3621.603
#>
#> Fixed effects in mean of longitudinal submodel: Y ~ Z1 + Z2 + Z3 + time
#>
#> Estimate SE Z value p-val
#> (Intercept) 4.85342 0.12451 38.97918 0.0000
#> Z1 1.55235 0.16535 9.38841 0.0000
#> Z2 1.93774 0.14598 13.27409 0.0000
#> Z3 1.09289 0.05321 20.53796 0.0000
#> time 4.01129 0.02978 134.71376 0.0000
#>
#> Fixed effects in variance of longitudinal submodel: log(sigma^2) ~ Z1 + Z2 + Z3 + time
#>
#> Estimate SE Z value p-val
#> (Intercept) 0.50745 0.12838 3.95260 0.0001
#> Z1 0.50509 0.16005 3.15590 0.0016
#> Z2 -0.42508 0.13781 -3.08463 0.0020
#> Z3 0.14405 0.04494 3.20563 0.0013
#> time 0.09050 0.02422 3.73720 0.0002
#>
#> Survival sub-model fixed effects: Surv(survtime, cmprsk) ~ var1 + var2 + var3
#>
#> Estimate SE Z value p-val
#> var1_1 1.09710 0.32647 3.36051 0.0008
#> var2_1 0.19237 0.26154 0.73553 0.4620
#> var3_1 0.49611 0.08908 5.56951 0.0000
#>
#> var1_2 -0.88311 0.33702 -2.62037 0.0088
#> var2_2 0.80905 0.30127 2.68549 0.0072
#> var3_2 0.20871 0.09312 2.24143 0.0250
#>
#> Association parameters:
#> Estimate SE Z value p-val
#> (Intercept)_1 0.97480 0.62808 1.55202 0.1207
#> (Intercept)_2 -0.18580 0.47949 -0.38750 0.6984
#> var_(Intercept)_1 0.50030 0.58190 0.85977 0.3899
#> var_(Intercept)_2 -0.84481 0.52520 -1.60857 0.1077
#>
#>
#> Random effects:
#> Formula: ~1 | ID
#> Estimate SE Z value p-val
#> (Intercept) 0.49542 0.11339 4.36913 0.0000
#> var_(Intercept) 0.45581 0.11129 4.09578 0.0000
#> (Intercept):var_(Intercept) 0.26738 0.07854 3.40429 0.0007cnewdata <- cdatah[cdatah$ID %in% c(122, 152), ]
ynewdata <- ydatah[ydatah$ID %in% c(122, 152), ]
survfit <- survfitJM(fit, seed = 100, ynewdata = ynewdata, cnewdata = cnewdata,
u = seq(5.2, 7.2, by = 0.5), Last.time = "survtime",
obs.time = "time", method = "GH")
survfit
#>
#> Prediction of Conditional Probabilities of Event
#> based on the adaptive Guass-Hermite quadrature rule with 6 quadrature points
#> $`122`
#> times CIF1 CIF2
#> 1 5.069089 0.00000000 0.0000000
#> 2 5.200000 0.05596021 0.0000000
#> 3 5.700000 0.14584944 0.0000000
#> 4 6.200000 0.33882152 0.0000000
#> 5 6.700000 0.33882152 0.0000000
#> 6 7.200000 0.33882152 0.2171424
#>
#> $`152`
#> times CIF1 CIF2
#> 1 5.133665 0.0000000 0.00000000
#> 2 5.200000 0.0517717 0.00000000
#> 3 5.700000 0.1357406 0.00000000
#> 4 6.200000 0.3195265 0.00000000
#> 5 6.700000 0.3195265 0.00000000
#> 6 7.200000 0.3195265 0.06007945
oldpar <- par(mfrow = c(2, 2), mar = c(5, 4, 4, 4))
plot(survfit, include.y = TRUE)
par(oldpar)To assess the prediction accuracy of the fitted joint model, we may run
DynPredAcc to assess the prediction accuracy by calculating all
available evaluation metrics.
res <- DynPredAcc(
object = fit,
landmark.time = 3, horizon.time = c(4:6),
obs.time = "time",
method = "GH",
maxiter = 1000,
n.cv = 3,
quantile.width = 0.25,
metrics = c("AUC", "Cindex", "Brier Score", "MAE", "MAEQ")
)
#> The 1-th validation is done!
#> The 2-th validation is done!
#> The 3-th validation is done!
summary(res, metric = "Brier Score")
#>
#> Expected Brier Score at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time Brier Score 1 Brier Score 2
#> 4 4 0.0637 0.0619
#> 5 5 0.1084 0.1105
#> 6 6 0.2019 0.1161
summary(res, metric = "MAE")
#>
#> Expected mean absolute error at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time MAE1 MAE2
#> 4 4 0.1314 0.1258
#> 5 5 0.2192 0.2195
#> 6 6 0.3818 0.2291
summary(res, metric = "MAEQ")
#>
#> Mean absolute error across quantiles of predicted risk scores at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time MAEQ1 MAEQ2
#> 1 4 0.0965 0.0656
#> 2 5 0.1190 0.0864
#> 3 6 0.1139 0.1074
summary(res, metric = "AUC")
#>
#> Expected AUC at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time AUC1 AUC2
#> 4 4 0.5502 0.6839
#> 5 5 0.6182 0.6524
#> 6 6 0.6104 0.7066
summary(res, metric = "Cindex")
#>
#> Expected Cindex at the landmark time of 3
#> based on 3 fold cross validation
#> Horizon Time Cindex1 Cindex2
#> 4 4 0.6154 0.6510
#> 5 5 0.6133 0.6463
#> 6 6 0.6126 0.6463Or we can calculate the overall, time-independent Cindex over the entire time period, evaluated by the linear predictor of the (cause-specific) Cox model.
Concord <- Concordance.JMMLSM(seed = 100, fit, n.cv = 3)
#> The 1 th validation is done!
#> The 2 th validation is done!
#> The 3 th validation is done!
summary(Concord)
#> Concordance1 Concordance2
#> 1 0.7943508 0.7413602In order to create simulated data for mvjmcs, we can use the
simmvJMdata function, which creates longitudinal and survival data as
a nested list (which are unpacked the this example). When first calling
the function, it provides censoring and risk rates.
# Simulate data
sim <- simmvJMdata(seed = 100, N = 50) # returns list of cdata and ydata for a sample size of 50
#> The censoring rate is: 44%
#> The risk 1 rate is: 48%
#> The risk 2 rate is: 8%
c_data <- sim$mvcdata # survival-side data, one row per ID
y_data <- sim$mvydata # longitudinal measurements (multiple rows per ID)Below is the simulated longitudinal data for multiple biomarkers, wherein Y1 and Y2 represent our biomarkers and X11 and X12 represent measurement-level predictors for the longitudinal submodel.
head(y_data)
#> ID time Y1 Y2 X11 X12
#> 1 1 0.0 2.325975 3.493627 0 -2.347892
#> 2 1 0.7 2.328122 4.649502 0 -2.347892
#> 3 1 1.4 2.793674 6.112850 0 -2.347892
#> 4 1 2.1 2.221392 5.375753 0 -2.347892
#> 5 1 2.8 1.864348 4.481401 0 -2.347892
#> 6 1 3.5 3.988955 5.496069 0 -2.347892Below is the simulated survival data wherein X21 and X22 represent patient-level predictors for the survival model.
head(c_data)
#> ID survtime cmprsk X21 X22
#> 1 1 6.10116281 0 0 -2.3478921
#> 2 2 0.05456028 1 1 0.1826885
#> 3 3 6.52978656 0 1 2.3791087
#> 4 4 0.04942950 1 1 2.7961091
#> 5 5 6.96785721 0 0 -3.8530560
#> 6 6 7.20378227 0 0 1.1237335