SimSurvNMarker: Simulate Survival time and Markers
The SimSurvNMarker package reasonably fast simulates from a joint survival and marker model. The package uses a combination of Gauss–Legendre quadrature and one dimensional root finding to simulate the event times as described by Crowther and Lambert (2013). Specifically, the package can simulate from the model
\[\begin{align*}\vec Y_{ij}\mid\vec U_i =\vec u_i &\sim N^{(r)}(\vec \mu_i(s_{ij}, \vec u_i), \Sigma)\\\vec\mu_i(s, \vec u)&=\Gamma^\top\vec x_i+B^\top\vec g(s)+U^\top\vec m(s)\\&=\left(I\otimes\vec x_i^\top\right)\text{vec}\Gamma+\left(I\otimes\vec g(s)^\top\right)\text{vec}B+\left(I\otimes\vec m(s)^\top\right)\vec u\\\vec U_i &\sim N^{(K)}(\vec 0,\Psi)\end{align*}\]
\[\begin{align*}h(t\mid\vec u)&=\exp\left(\vec\omega^\top\vec b(t)+\vec z_i^\top\vec\delta+\vec\alpha^\top\vec\mu_i(t, \vec u)\right)\\&=\exp\Bigg(\vec\omega^\top\vec b(t)+\vec z_i^\top\vec\delta+\vec 1^\top\left(\text{diag}(\vec\alpha)\otimes\vec x_i^\top\right)\text{vec}\Gamma+\vec 1^\top\left(\text{diag}(\vec\alpha)\otimes\vec g(t)^\top\right)\text{vec} B\\&\hspace{50pt}+\vec 1^\top\left(\text{diag}(\vec\alpha)\otimes\vec m(t)^\top\right)\vec u\Bigg)\end{align*}\]
where \(\vec Y_{ij}\in\mathbb R^{n_y}\) is individual \(i\)’s \(j\)th observed marker at time \(s_{ij}\), \(U_i\in\mathbb R^K\) is individual \(i\)’s random effect, and \(h\) is the instantaneous hazard rate for the time-to-event outcome. \(\vec\alpha\) is the so-called association parameter. It shows the strength of the relation between the latent mean function, \(\vec\mu_i(t,\vec u)\), and the log of the instantaneous rate, \(h(t\mid \vec u)\). \(\vec m(t)\), \(\vec g(t)\) and \(\vec b(t)\) are basis expansions of time. As an example, these can be a polynomial, a B-spline, or a natural cubic spline. The expansion for the baseline hazard, \(\vec b(t)\), is typically made on \(\log t\) instead of \(t\). One reason is that the model reduces to a Weibull distribution when a first polynomial is used and \(\vec\alpha = \vec 0\). \(\vec x_i\) and \(\vec z_i\) are individual specific known time-invariant covariates.
We provide an example of how to use the package here, in inst/test-data directory on Github, and at rpubs.com/boennecd/SimSurvNMarker-ex where we show that we simulate from the correct model by using a simulation study. The former examples includes
- using polynomials as the basis functions.
- using polynomials with derivatives in the log hazard as the basis functions.
- using natural cubic splines as the basis functions.
The purpose/goal of this package is to
- allow the user to specify all components in R including the basis functions.
- simulate reasonably fast from the model given this requirement.
Installation
The package can be installed from Github using theremotes package:
stopifnot(require(remotes)) # needs the remotes package
install_github("boennecd/SimSurvNMarker")It can also be installed from CRAN using install.packages:
install.packages("SimSurvNMarker")Example with Polynomials
We start with an example where we use polynomials as the basis functions. First, we assign the polynomial functions we will use.
b_func <- function(x){
x <- x - 1
cbind(x^3, x^2, x)
}
g_func <- function(x){
x <- x - 1
cbind(x^3, x^2, x)
}
m_func <- function(x){
x <- x - 1
cbind(x^2, x, 1)
}We use a third order polynomial for the two fixed terms, \(\vec b\) and \(\vec g\), and a second order random polynomial for the random term, \(U_i^\top \vec m(s)\), in the latent mean function. We choose the following parameters for the baseline hazard.
library(SimSurvNMarker)omega <- c(1.4, -1.2, -2.1)
delta <- -.5 # the intercept
# quadrature nodes
gl_dat <- get_gl_rule(30L)
# hazard function without marker
par(mar = c(5, 5, 1, 1), mfcol = c(1, 2))
plot(function(x) exp(drop(b_func(x) %*% omega) + delta),
xlim = c(0, 2), ylim = c(0, 1.5), xlab = "Time",
ylab = "Hazard (no marker)", xaxs = "i", yaxs = "i", bty = "l")
grid()
# survival function without marker
plot(function(x) eval_surv_base_fun(x, omega = omega, b_func = b_func,
gl_dat = gl_dat, delta = delta),
xlim = c(1e-4, 2.5),
xlab = "Time", ylab = "Survival probability (no marker)", xaxs = "i",
yaxs = "i", bty = "l", ylim = c(0, 1.01))
grid()
Then we set the following parameters for the random effect, \(\vec U_i\), and the parameters for the marker process. We also simulate a number of latent markers’ mean curves and observed marker values and plot the result. The dashed curve is the mean, \(\vec\mu_i(s, \vec 0)\), the fully drawn curve is the individual specific curve, \(\vec\mu_i(s, \vec U_i)\), the shaded areas are a pointwise 95% interval for each mean curve, and the points are observed markers, \(\vec y_{ij}\).
r_n_marker <- function(id)
# the number of markers is Poisson distributed
rpois(1, 10) + 1L
r_obs_time <- function(id, n_markes)
# the observations times are uniform distributed
sort(runif(n_markes, 0, 2))
Psi <- structure(c(0.18, 0.05, -0.05, 0.1, -0.02, 0.06, 0.05, 0.34, -0.25,
-0.06, -0.03, 0.29, -0.05, -0.25, 0.24, 0.04, 0.04,
-0.12, 0.1, -0.06, 0.04, 0.34, 0, -0.04, -0.02, -0.03,
0.04, 0, 0.1, -0.08, 0.06, 0.29, -0.12, -0.04, -0.08,
0.51), .Dim = c(6L, 6L))
B <- structure(c(-0.57, 0.17, -0.48, 0.58, 1, 0.86), .Dim = 3:2)
sig <- diag(c(.6, .3)^2)
# function to simulate a given number of individuals' markers' latent means
# and observed values
show_mark_mean <- function(B, Psi, sigma, m_func, g_func, ymax){
tis <- seq(0, ymax, length.out = 100)
Psi_chol <- chol(Psi)
par_old <- par(no.readonly = TRUE)
on.exit(par(par_old))
par(mar = c(4, 3, 1, 1), mfcol = c(4, 4))
sigma_chol <- chol(sigma)
n_y <- NCOL(sigma_chol)
for(i in 1:16){
U <- draw_U(Psi_chol, n_y = n_y)
y_non_rng <- t(eval_marker(tis, B = B, g_func = g_func, U = NULL,
offset = NULL, m_func = m_func))
y_rng <- t(eval_marker(tis, B = B, g_func = g_func, U = U,
offset = NULL, m_func = m_func))
sds <- sapply(tis, function(ti){
M <- (diag(n_y) %x% m_func(ti))
G <- (diag(n_y) %x% g_func(ti))
sds <- sqrt(diag(tcrossprod(M %*% Psi, M)))
cbind(drop(G %*% c(B)) - 1.96 * sds,
drop(G %*% c(B)) + 1.96 * sds)
}, simplify = "array")
lbs <- t(sds[, 1, ])
ubs <- t(sds[, 2, ])
y_obs <- sim_marker(B = B, U = U, sigma_chol = sigma_chol,
m_func = m_func, r_n_marker = r_n_marker,
r_obs_time = r_obs_time, g_func = g_func,
offset = NULL)
matplot(tis, y_non_rng, type = "l", lty = 2, ylab = "", xlab = "Time",
ylim = range(y_non_rng, y_rng, lbs, ubs, y_obs$y_obs))
matplot(tis, y_rng , type = "l", lty = 1, add = TRUE)
matplot(y_obs$obs_time, y_obs$y_obs, type = "p", add = TRUE, pch = 3:4)
polygon(c(tis, rev(tis)), c(lbs[, 1], rev(ubs[, 1])), border = NA,
col = rgb(0, 0, 0, .1))
polygon(c(tis, rev(tis)), c(lbs[, 2], rev(ubs[, 2])), border = NA,
col = rgb(1, 0, 0, .1))
}
invisible()
}
set.seed(1)
show_mark_mean(B = B, Psi = Psi, sigma = sig, m_func = m_func,
g_func = g_func, ymax = 2)
As an example, we simulate the random effects and plot the conditional hazards and survival functions. We start by assigning the association parameter, \(\vec\alpha\).
alpha <- c(.5, .9)
# function to plot simulated conditional hazards and survival
# functions
sim_surv_curves <- function(sig, Psi, delta, omega, alpha, B, m_func,
g_func, b_func, ymax) {
par_old <- par(no.readonly = TRUE)
on.exit(par(par_old))
par(mfcol = c(1, 2), mar = c(5, 5, 1, 1))
# hazard functions
tis <- seq(0, ymax, length.out = 50)
n_y <- NCOL(sig)
Us <- replicate(100, draw_U(chol(Psi), n_y = n_y),
simplify = "array")
hz <- apply(Us, 3L, function(U)
vapply(tis, function(x)
exp(drop(delta + b_func(x) %*% omega +
alpha %*% eval_marker(ti = x, B = B, m_func = m_func,
g_func = g_func, U = U,
offset = NULL))),
FUN.VALUE = numeric(1L)))
matplot(tis, hz, lty = 1, type = "l",
col = rgb(0, 0, 0, .1), xaxs = "i", bty = "l", yaxs = "i",
ylim = c(0, max(hz, na.rm = TRUE)), xlab = "time",
ylab = "Hazard")
grid()
# survival functions
ys <- apply(Us, 3L, surv_func_joint,
ti = tis, B = B, omega = omega, delta = delta,
alpha = alpha, b_func = b_func, m_func = m_func,
gl_dat = gl_dat, g_func = g_func, offset = NULL)
matplot(tis, ys, lty = 1, type = "l", col = rgb(0, 0, 0, .1),
xaxs = "i", bty = "l", yaxs = "i", ylim = c(0, 1),
xlab = "time", ylab = "Survival probability")
grid()
}
set.seed(1)
sim_surv_curves(sig = sig, Psi = Psi, delta = delta, omega = omega,
alpha = alpha, B = B, m_func = m_func, g_func = g_func,
b_func = b_func, ymax = 2)
We end by assigning the functions to get the covariates, coefficients for the fixed effects, the left-truncation function, and right-censoring function.
r_z <- function(id)
# returns a design matrix for a dummy setup
cbind(1, (id %% 3) == 1, (id %% 3) == 2)
r_z(1:6) # covariates for the first six individuals
#> [,1] [,2] [,3]
#> [1,] 1 1 0
#> [2,] 1 0 1
#> [3,] 1 0 0
#> [4,] 1 1 0
#> [5,] 1 0 1
#> [6,] 1 0 0
# same covariates for the fixed time-invariant effects for the marker
r_x <- r_z
r_left_trunc <- function(id)
# no left-truncation
0
r_right_cens <- function(id)
# right-censoring time is exponentially distributed
rexp(1, rate = .5)
# fixed effect coefficients in the hazard
delta_vec <- c(delta, -.5, .5)
# fixed time-invariant effect coefficients in the marker process
gamma <- matrix(c(.25, .5, 0, -.4, 0, .3), 3, 2)A full data set can now be simulated as follows. We consider the time it takes in seconds by using the system.time function.
set.seed(70483614)
system.time(dat <- sim_joint_data_set(
n_obs = 1000L, B = B, Psi = Psi, omega = omega, delta = delta_vec,
alpha = alpha, sigma = sig, b_func = b_func, g_func = g_func,
m_func = m_func, gl_dat = gl_dat, r_z = r_z, r_left_trunc = r_left_trunc,
r_right_cens = r_right_cens, r_n_marker = r_n_marker,
r_obs_time = r_obs_time, y_max = 2, gamma = gamma, r_x = r_x))
#> user system elapsed
#> 0.464 0.047 0.511The first entries of the survival data and the observed markers looks as follows.
# survival data
head(dat$survival_data)
#> Z1 Z2 Z3 left_trunc y event id
#> 1 1 1 0 0 0.869 TRUE 1
#> 2 1 0 1 0 0.320 TRUE 2
#> 3 1 0 0 0 1.535 TRUE 3
#> 4 1 1 0 0 0.129 TRUE 4
#> 5 1 0 1 0 2.000 FALSE 5
#> 6 1 0 0 0 0.369 FALSE 6
# marker data
head(dat$marker_data, 10)
#> obs_time Y1 Y2 X1 X2 X3 id
#> 1 0.4392 1.984 0.8920 1 1 0 1
#> 2 0.6009 1.651 0.2385 1 1 0 1
#> 3 0.7747 0.819 0.1867 1 1 0 1
#> 4 0.0702 3.796 -0.5575 1 0 1 2
#> 5 0.1186 3.122 0.2074 1 0 1 2
#> 6 0.2737 2.977 -0.2365 1 0 1 2
#> 7 0.2829 2.506 -0.2323 1 0 1 2
#> 8 0.1121 2.211 0.1531 1 0 0 3
#> 9 0.3337 1.595 0.0196 1 0 0 3
#> 10 0.3430 0.294 -0.1353 1 0 0 3To illustrate that we simulate from the correct model, we can estimate a linear mixed models for the markers as follows.
library(lme4)
# estimate the linear mixed model (skip this if you want and look at the
# estimates in the end)
local({
m_dat <- dat$marker_data
n_y <- NCOL(gamma)
d_x <- NROW(gamma)
d_g <- NROW(B)
d_m <- NROW(Psi) / n_y
Y_names <- paste0("Y", 1:n_y)
id_vars <- c("id", "obs_time")
if(d_x > 0)
id_vars <- c(id_vars, paste0("X", seq_len(d_x)))
lme_dat <- melt(m_dat, id.vars = id_vars, measure.vars = Y_names,
variable.name = "XXTHEVARIABLEXX",
value.name = "XXTHEVALUEXX")
if(length(alpha) > 1){
if(length(B) > 0L)
frm <- substitute(
XXTHEVALUEXX ~
XXTHEVARIABLEXX : g_func(ti) - 1L +
(XXTHEVARIABLEXX : m_func(ti) - 1L | i),
list(ti = as.name("obs_time"), i = as.name("id")))
else
frm <- substitute(
XXTHEVALUEXX ~
(XXTHEVARIABLEXX : m_func(ti, m_ks) - 1L | i),
list(ti = as.name("obs_time"), i = as.name("id")))
frm <- eval(frm)
if(d_x > 0)
for(i in rev(seq_len(d_x))){
frm_call <- substitute(
update(frm, . ~ XXTHEVARIABLEXX : x_var + .),
list(x_var = as.name(paste0("X", i))))
frm <- eval(frm_call)
}
} else {
if(length(B) > 0L)
frm <- substitute(
XXTHEVALUEXX ~
ns_func(ti, g_ks) - 1L +
(ns_func(ti, m_ks) - 1L | i),
list(ti = as.name("obs_time"), i = as.name("id"),
g_ks = as.name("g_ks"), m_ks = as.name("m_ks")))
else
frm <- substitute(
XXTHEVALUEXX ~
(ns_func(ti, m_ks) - 1L | i),
list(ti = as.name("obs_time"), i = as.name("id"),
m_ks = as.name("m_ks")))
frm <- eval(frm)
if(d_x > 0)
for(i in rev(seq_len(d_x))){
frm_call <- substitute(
update(frm, . ~ x_var + .),
list(x_var = as.name(paste0("X", i))))
frm <- eval(frm_call)
}
}
fit <- lmer(frm, lme_dat, control = lmerControl(
check.conv.grad = .makeCC("ignore", tol = 1e-3, relTol = NULL)))
gamma <- t(matrix(fixef(fit)[seq_len(d_x * n_y)], nr = n_y))
B <- t(matrix(fixef(fit)[seq_len(d_g * n_y) + (d_x * n_y)], nr = n_y))
vc <- VarCorr(fit)
Psi <- vc$id
attr(Psi, "correlation") <- attr(Psi, "stddev") <- NULL
dimnames(Psi) <- NULL
K <- SimSurvNMarker:::get_commutation(n_y, d_m)
Psi <- tcrossprod(K %*% Psi, K)
Sigma <- diag(attr(vc, "sc")^2, n_y)
list(gamma = gamma, B = B, Psi = Psi, Sigma = Sigma)
})
#> $gamma
#> [,1] [,2]
#> [1,] 0.1684 -0.384104
#> [2,] 0.5224 0.000604
#> [3,] -0.0767 0.219848
#>
#> $B
#> [,1] [,2]
#> [1,] -0.569 0.484
#> [2,] 0.282 0.988
#> [3,] -0.466 0.893
#>
#> $Psi
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 0.5247 0.0687 -0.16088 0.12023 -0.0285 0.1125
#> [2,] 0.0687 0.3572 -0.24080 -0.03208 -0.0450 0.2571
#> [3,] -0.1609 -0.2408 0.25861 -0.00238 0.0464 -0.1342
#> [4,] 0.1202 -0.0321 -0.00238 0.16060 -0.0057 0.0470
#> [5,] -0.0285 -0.0450 0.04644 -0.00570 0.0533 -0.0832
#> [6,] 0.1125 0.2571 -0.13418 0.04703 -0.0832 0.4165
#>
#> $Sigma
#> [,1] [,2]
#> [1,] 0.225 0.000
#> [2,] 0.000 0.225Although we assume equal noise variance, the estimates are close to the true values.
gamma
#> [,1] [,2]
#> [1,] 0.25 -0.4
#> [2,] 0.50 0.0
#> [3,] 0.00 0.3
B
#> [,1] [,2]
#> [1,] -0.57 0.58
#> [2,] 0.17 1.00
#> [3,] -0.48 0.86
Psi
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 0.18 0.05 -0.05 0.10 -0.02 0.06
#> [2,] 0.05 0.34 -0.25 -0.06 -0.03 0.29
#> [3,] -0.05 -0.25 0.24 0.04 0.04 -0.12
#> [4,] 0.10 -0.06 0.04 0.34 0.00 -0.04
#> [5,] -0.02 -0.03 0.04 0.00 0.10 -0.08
#> [6,] 0.06 0.29 -0.12 -0.04 -0.08 0.51
sig
#> [,1] [,2]
#> [1,] 0.36 0.00
#> [2,] 0.00 0.09We then fit a Cox model with only the observed markers (likely biased but gives us an idea about whether we are using the correct model).
local({
library(survival)
tdat <- tmerge(dat$survival_data, dat$survival_data, id = id,
tstart = left_trunc, tstop = y, ev = event(y, event))
for(i in seq_along(alpha)){
new_call <- substitute(tmerge(
tdat, dat$marker_data, id = id, tdc(obs_time, YVAR)),
list(YVAR = as.name(paste0("Y", i))))
names(new_call)[length(new_call)] <- paste0("Y", i)
tdat <- eval(new_call)
}
tdat <- na.omit(tdat)
sformula <- Surv(left_trunc, y, ev) ~ 1
for(i in seq_along(delta_vec)){
new_call <- substitute(update(sformula, . ~ . + XVAR),
list(XVAR = as.name(paste0("Z", i))))
sformula <- eval(new_call)
}
for(i in seq_along(alpha)){
new_call <- substitute(update(sformula, . ~ . + XVAR),
list(XVAR = as.name(paste0("Y", i))))
sformula <- eval(new_call)
}
fit <- coxph(sformula, tdat)
print(summary(fit))
invisible(fit)
})
#> Call:
#> coxph(formula = sformula, data = tdat)
#>
#> n= 4042, number of events= 535
#>
#> coef exp(coef) se(coef) z Pr(>|z|)
#> Z1 NA NA 0.0000 NA NA
#> Z2 -0.8688 0.4194 0.1168 -7.44 1.0e-13 ***
#> Z3 0.4566 1.5787 0.1007 4.53 5.8e-06 ***
#> Y1 0.5474 1.7287 0.0419 13.06 < 2e-16 ***
#> Y2 0.4519 1.5712 0.0489 9.23 < 2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> exp(coef) exp(-coef) lower .95 upper .95
#> Z1 NA NA NA NA
#> Z2 0.419 2.384 0.334 0.527
#> Z3 1.579 0.633 1.296 1.923
#> Y1 1.729 0.578 1.592 1.877
#> Y2 1.571 0.636 1.428 1.729
#>
#> Concordance= 0.693 (se = 0.012 )
#> Likelihood ratio test= 270 on 4 df, p=<2e-16
#> Wald test = 271 on 4 df, p=<2e-16
#> Score (logrank) test = 278 on 4 df, p=<2e-16This is close-ish to the true values.
delta_vec
#> [1] -0.5 -0.5 0.5
alpha
#> [1] 0.5 0.9Using Derivatives
It is possible to use derivatives of the latent mean, \(\vec\mu_i(s, \vec u)\), with respect to time in the hazard. As an example, we consider the first-order derivative below and plot conditional hazards and survival functions simulated from the new model.
g_func_surv <- function(x){
x <- x - 1
cbind(3 * x^2, 2 * x, 1)
}
m_func_surv <- function(x){
x <- x - 1
cbind(2 * x, 1, 0)
}
set.seed(1)
sim_surv_curves(sig = sig, Psi = Psi, delta = delta, omega = omega,
alpha = alpha, B = B, m_func = m_func_surv,
g_func = g_func_surv, b_func = b_func, ymax = 2)
A new data set can now be simulated as follows.
set.seed(70483614)
system.time(dat <- sim_joint_data_set(
n_obs = 1000L, B = B, Psi = Psi, omega = omega, delta = delta_vec,
alpha = alpha, sigma = sig, b_func = b_func, g_func = g_func,
m_func = m_func, gl_dat = gl_dat, r_z = r_z, r_left_trunc = r_left_trunc,
r_right_cens = r_right_cens, r_n_marker = r_n_marker,
r_obs_time = r_obs_time, y_max = 2, gamma = gamma, r_x = r_x,
# the additions
m_func_surv = m_func_surv, g_func_surv = g_func_surv,
use_fixed_latent = FALSE))
#> user system elapsed
#> 0.567 0.035 0.601The first entries of the new data looks as follows.
# survival data
head(dat$survival_data)
#> Z1 Z2 Z3 left_trunc y event id
#> 1 1 1 0 0 1.038 FALSE 1
#> 2 1 0 1 0 1.704 TRUE 2
#> 3 1 0 0 0 1.752 TRUE 3
#> 4 1 1 0 0 1.170 TRUE 4
#> 5 1 0 1 0 1.343 TRUE 5
#> 6 1 0 0 0 0.369 FALSE 6
# marker data
head(dat$marker_data, 10)
#> obs_time Y1 Y2 X1 X2 X3 id
#> 1 0.4392 1.984 0.892 1 1 0 1
#> 2 0.6009 1.651 0.239 1 1 0 1
#> 3 0.7747 0.819 0.187 1 1 0 1
#> 4 0.8931 0.853 0.521 1 1 0 1
#> 5 0.9442 1.522 -0.216 1 1 0 1
#> 6 0.0702 3.796 -0.558 1 0 1 2
#> 7 0.1186 3.122 0.207 1 0 1 2
#> 8 0.2737 2.977 -0.237 1 0 1 2
#> 9 0.2829 2.506 -0.232 1 0 1 2
#> 10 0.4253 2.098 -0.124 1 0 1 2Example with Natural Cubic Splines
In this section, we will use natural cubic splines for the time-varying basis functions. We start by assigning all the variables that we will pass to the functions in the package.
# quadrature nodes
gl_dat <- get_gl_rule(30L)
# knots for the spline functions
b_ks <- seq(log(1), log(10), length.out = 4)
m_ks <- seq(0, 10, length.out = 3)
g_ks <- m_ks
# simulation functions
r_n_marker <- function(id)
rpois(1, 10) + 1L
r_obs_time <- function(id, n_markes)
sort(runif(n_markes, 0, 10))
r_z <- function(id)
as.numeric(runif(1L) > .5)
r_x <- function(id)
numeric()
r_left_trunc <- function(id)
rbeta(1, 1, 2) * 3
r_right_cens <- function(id)
rbeta(1, 2, 1) * 6 + 4
# model parameters
omega <- c(-0.96, -2.26, -3.04, .45)
Psi <- structure(c(1.08, 0.12, -0.36, -0.48, 0.36, -0.12, 0.12, 0.36,
0, 0.12, 0, -0.12, -0.36, 0, 0.84, 0.12, 0.12, 0.12, -0.48, 0.12,
0.12, 0.84, -0.12, 0.24, 0.36, 0, 0.12, -0.12, 0.84, -0.12, -0.12,
-0.12, 0.12, 0.24, -0.12, 0.6), .Dim = c(6L, 6L))
B <- structure(c(0.97, 0.01, -0.07, -0.78, -0.86, 0.98), .Dim = 3:2)
sig <- diag(c(.2, .1)^2)
alpha <- c(0.7, 0.6)
delta <- 0.
gamma <- numeric()# spline functions
b_func <- get_ns_spline(b_ks, do_log = TRUE)
m_func <- get_ns_spline(m_ks, do_log = FALSE)
g_func <- get_ns_spline(g_ks, do_log = FALSE)Notice that we use the get_ns_spline functions which reduces the computation time a lot. We show the baseline hazard function and the survival function without the marker below.
# hazard function without marker
par(mar = c(5, 5, 1, 1), mfcol = c(1, 2))
plot(function(x) exp(drop(b_func(x) %*% omega)),
xlim = c(1e-8, 10), ylim = c(0, .61), xlab = "Time",
ylab = "Hazard (no marker)", xaxs = "i", bty = "l")
grid()
# survival function without marker
plot(function(x) eval_surv_base_fun(x, omega = omega, b_func = b_func,
gl_dat = gl_dat, delta = delta),
xlim = c(1e-4, 10),
xlab = "Time", ylab = "Survival probability (no marker)", xaxs = "i",
yaxs = "i", bty = "l", ylim = c(0, 1.01))
grid()
Next, we simulate individual specific markers. Each plot is for a given individual.
set.seed(1)
show_mark_mean(B = B, Psi = Psi, sigma = sig, m_func = m_func,
g_func = g_func, ymax = 10)
We sample a number of random effects and plot the hazard curves and survival functions given these random effects below.
set.seed(1)
sim_surv_curves(sig = sig, Psi = Psi, delta = delta, omega = omega,
alpha = alpha, B = B, m_func = m_func, g_func = g_func,
b_func = b_func, ymax = 10)
We end by drawing a data set.
set.seed(70483614)
delta_vec <- 1
system.time(dat <- sim_joint_data_set(
n_obs = 1000L, B = B, Psi = Psi, omega = omega, delta = delta_vec,
alpha = alpha, sigma = sig, b_func = b_func, g_func = g_func,
m_func = m_func, gl_dat = gl_dat, r_z = r_z, r_left_trunc = r_left_trunc,
r_right_cens = r_right_cens, r_n_marker = r_n_marker,
r_obs_time = r_obs_time, y_max = 10, gamma = gamma, r_x = r_x))
#> user system elapsed
#> 0.593 0.039 0.632Finally, we show a few of the first rows along with some summary statistics.
# survival data
head(dat$survival_data)
#> Z1 left_trunc y event id
#> 1 0 0.3830 7.85 FALSE 1
#> 2 1 1.0779 2.10 TRUE 2
#> 3 0 0.8923 1.37 TRUE 3
#> 4 0 0.9055 8.04 FALSE 4
#> 5 1 0.8378 6.22 TRUE 5
#> 6 1 0.0649 1.84 TRUE 6
# marker data
head(dat$marker_data, 10)
#> obs_time Y1 Y2 id
#> 1 1.32 0.824 -2.165 1
#> 2 1.85 0.163 -2.132 1
#> 3 2.20 1.052 -1.914 1
#> 4 3.00 0.831 -1.941 1
#> 5 3.59 1.303 -1.840 1
#> 6 4.91 1.061 -1.660 1
#> 7 4.92 0.803 -1.770 1
#> 8 5.74 0.595 -1.520 1
#> 9 5.87 0.432 -1.364 1
#> 10 1.77 1.285 -0.497 2
# rate of observed events
mean(dat$survival_data$event)
#> [1] 0.742
# mean event time
mean(subset(dat$survival_data, event )$y)
#> [1] 3.49
# mean event time for the two group
mean(subset(dat$survival_data, event & Z1 == 1L)$y)
#> [1] 3.08
mean(subset(dat$survival_data, event & Z1 == 0L)$y)
#> [1] 4.16
# quantiles of the event time
quantile(subset(dat$survival_data, event)$y)
#> 0% 25% 50% 75% 100%
#> 0.0707 1.6318 2.7531 5.2749 9.5271
# fraction of observed markers per individual
NROW(dat$marker_data) / NROW(dat$survival_data)
#> [1] 4.08References
Crowther, Michael J., and Paul C. Lambert. 2013. “Simulating Biologically Plausible Complex Survival Data.” Statistics in Medicine 32 (23): 4118–34. https://doi.org/10.1002/sim.5823.