The Stochastic Process Model (SPM) was developed several decades ago [1,2], and applied for analyses of clinical, demographic, epidemiologic longitudinal data as well as in many other studies that relate stochastic dynamics of repeated measures to the probability of end-points (outcomes). SPM links the dynamic of stochastical variables with a hazard rate as a quadratic function of the state variables [3]. The R-package, “stpm”, is a set of utilities to estimate parameters of stochastic process and modeling survival trajectories and time-to-event outcomes observed from longitudinal studies. It is a general framework for studying and modeling survival (censored) traits depending on random trajectories (stochastic paths) of variables.
require(devtools)
devtools::install_github("izhbannikov/stpm")If you experience errors during installation, please download a binary file from the following url:
Than, execute this command (from R environment):
install.packages("<path to the downloaded r-package stpm>", repos=NULL, type="binary")Data represents a typical longitudinal data in form of two datasets: longitudinal dataset (follow-up studies), in which one record represents a single observation, and vital (survival) statistics, where one record represents all information about the subject. Longitudinal dataset cat contain a subject ID (identification number), status (event(1)/censored(0)), time and measurements across the variables. The stpm can handle an infinite number of variables but in practice, 5-7 variables is enough.
Below there is an example of clinical data that can be used in stpm and we will discuss the fields later.
Longitudinal table:
## ID IndicatorDeath Age DBP BMI
## 1 1 0 30 80.00000 25.00000
## 2 1 0 32 80.51659 26.61245
## 3 1 0 34 77.78412 29.16790
## 4 1 0 36 77.86665 32.40359
## 5 1 0 38 96.55673 31.92014
## 6 1 0 40 94.48616 32.89139Vital statistics table:
## ID IsDead LSmort
## 1 1 1 85.34578
## 2 2 1 80.55053
## 3 3 1 98.07315
## 4 4 1 81.29779
## 5 5 1 89.89829
## 6 6 1 72.47687There are two main SPM types in the package: discrete-time model [4] and continuous-time model [3]. Discrete model assumes equal intervals between follow-up observations. The example of discrete dataset is given below.
library(stpm)
data <- simdata_discr(N=10) # simulate data for 10 individuals
head(data)## id xi t1 t2 y1 y1.next
## [1,] 1 0 30 31 80.00000 79.64956
## [2,] 1 0 31 32 79.64956 86.36129
## [3,] 1 0 32 33 86.36129 97.13298
## [4,] 1 0 33 34 97.13298 98.70302
## [5,] 1 0 34 35 98.70302 96.82858
## [6,] 1 0 35 36 96.82858 104.64651In this case there are equal intervals between \(t_1\) and \(t_2\).
In the continuous-time SPM, in which intervals between observations are not equal (arbitrary or random). The example of such dataset is shown below:
library(stpm)
data <- simdata_cont(N=5) # simulate data for 5 individuals
head(data)## id xi t1 t2 y1 y1.next
## [1,] 0 0 31.08508 32.17657 81.34290 88.86695
## [2,] 0 0 32.17657 33.69339 88.86695 90.77154
## [3,] 0 0 33.69339 35.62579 90.77154 69.65674
## [4,] 0 0 35.62579 37.29229 69.65674 85.05904
## [5,] 0 0 37.29229 38.29349 85.05904 84.99607
## [6,] 0 0 38.29349 40.28417 84.99607 77.42993The discrete model assumes fixed time intervals between consecutive observations. In this model, \(\mathbf{Y}(t)\) (a \(k \times 1\) matrix of the values of covariates, where \(k\) is the number of considered covariates) and \(\mu(t, \mathbf{Y}(t))\) (the hazard rate) have the following form:
\(\mathbf{Y}(t+1) = \mathbf{u} + \mathbf{R} \mathbf{Y}(t) + \mathbf{\epsilon}\)
\(\mu (t, \mathbf{Y}(t)) = [\mu_0 + \mathbf{b} \mathbf{Y}(t) + \mathbf{Y}(t)^* \mathbf{Q} \mathbf{Y}(t)] e^{\theta t}\)
Coefficients \(\mathbf{u}\) (a \(k \times 1\) matrix, where \(k\) is a number of covariates), \(\mathbf{R}\) (a \(k \times k\) matrix), \(\mu_0\), \(\mathbf{b}\) (a \(1 \times k\) matrix), \(\mathbf{Q}\) (a \(k \times k\) matrix) are assumed to be constant in the particular implementation of this model in the R-package stpm. \(\mathbf{\epsilon}\) are normally-distributed random residuals, \(k \times 1\) matrix. A symbol ’*’ denotes transpose operation. \(\theta\) is a parameter to be estimated along with other parameters (\(\mathbf{u}\), \(\mathbf{R}\), \(\mathbf{\mu_0}\), \(\mathbf{b}\), \(\mathbf{Q}\)).
library(stpm)
#Data simulation (200 individuals)
data <- simdata_discr(N=200)
#Estimation of parameters
pars <- spm_discrete(data)
pars## $Ak2005
## $Ak2005$theta
## [1] 0.078
##
## $Ak2005$mu0
## [1] 0.0002794930949
##
## $Ak2005$b
## [1] -6.821266275e-06
##
## $Ak2005$Q
## [,1]
## [1,] 4.272584586e-08
##
## $Ak2005$u
## [1] 4.155849226
##
## $Ak2005$R
## [1] 0.9468925278
##
## $Ak2005$Sigma
## [1] 5.025340228
##
##
## $Ya2007
## $Ya2007$a
## [,1]
## [1,] -0.05310747221
##
## $Ya2007$f1
## [,1]
## [1,] 78.25356872
##
## $Ya2007$Q
## [,1]
## [1,] 4.272584586e-08
##
## $Ya2007$f
## [,1]
## [1,] 79.82599452
##
## $Ya2007$b
## [,1]
## [1,] 5.025340228
##
## $Ya2007$mu0
## [,1]
## [1,] 7.235912768e-06
##
## $Ya2007$theta
## [1] 0.078
##
##
## attr(,"class")
## [1] "spm.discrete"In the specification of the SPM described in 2007 paper by Yashin and collegaues [3] the stochastic differential equation describing the age dynamics of a covariate is:
\(d\mathbf{Y}(t)= \mathbf{a}(t)(\mathbf{Y}(t) -\mathbf{f}_1(t))dt + \mathbf{b}(t)d\mathbf{W}(t), \mathbf{Y}(t=t_0)\)
In this equation, \(\mathbf{Y}(t)\) (a \(k \times 1\) matrix) is the value of a particular covariate at a time (age) \(t\). \(\mathbf{f}_1(t)\) (a \(k \times 1\) matrix) corresponds to the long-term mean value of the stochastic process \(\mathbf{Y}(t)\), which describes a trajectory of individual covariate influenced by different factors represented by a random Wiener process \(\mathbf{W}(t)\). Coefficient \(\mathbf{a}(t)\) (a \(k \times k\) matrix) is a negative feedback coefficient, which characterizes the rate at which the process reverts to its mean. In the area of research on aging, \(\mathbf{f}_1(t)\) represents the mean allostatic trajectory and \(\mathbf{a}(t)\) represents the adaptive capacity of the organism. Coefficient \(\mathbf{b}(t)\) (a \(k \times 1\) matrix) characterizes a strength of the random disturbances from Wiener process \(\mathbf{W}(t)\).
The following function \(\mu(t, \mathbf{Y}(t))\) represents a hazard rate:
\(\mu(t, \mathbf{Y}(t)) = \mu_0(t) + (\mathbf{Y}(t) - \mathbf{f}(t))^* \mathbf{Q}(t) (\mathbf{Y}(t) - \mathbf{f}(t))\)
here \(\mu_0(t)\) is the baseline hazard, which represents a risk when \(\mathbf{Y}(t)\) follows its optimal trajectory; \(\mathbf{f}(t)\) (a \(k \times 1\) matrix) represents the optimal trajectory that minimizes the risk and \(\mathbf{Q}(t)\) (\(k \times k\) matrix) represents a sensitivity of risk function to deviation from the norm.
library(stpm)
#Simulate some data for 100 individuals
data <- simdata_cont(N=100)
head(data)## id xi t1 t2 y1 y1.next
## [1,] 0 0 35.44212551 36.63518729 79.72934447 81.95768546
## [2,] 0 0 36.63518729 38.01784896 81.95768546 82.25371387
## [3,] 0 0 38.01784896 39.67961635 82.25371387 80.92134836
## [4,] 0 0 39.67961635 41.20955527 80.92134836 84.33144654
## [5,] 0 0 41.20955527 42.63565476 84.33144654 74.49085661
## [6,] 0 0 42.63565476 44.35604600 74.49085661 65.89954272#Estimate parameters
# a=-0.05, f1=80, Q=2e-8, f=80, b=5, mu0=2e-5, theta=0.08 are starting values for estimation procedure
pars <- spm_continuous(dat=data,a=-0.05, f1=80, Q=2e-8, f=80, b=5, mu0=2e-5, theta=0.08)
pars## $a
## [,1]
## [1,] -0.05007861842
##
## $f1
## [,1]
## [1,] 78.91036593
##
## $Q
## [,1]
## [1,] 2.08060158e-08
##
## $f
## [,1]
## [1,] 81.47941595
##
## $b
## [,1]
## [1,] 5.01193387
##
## $mu0
## [1] 1.950645797e-05
##
## $theta
## [1] 0.08095031041
##
## $limit
## [1] FALSE
##
## attr(,"class")
## [1] "spm.continuous"The coefficient conversion between continuous- and discrete-time models is as follows (‘c’ and ‘d’ denote continuous- and discrete-time models respectively; note: these equations can be used if intervals between consecutive observations of discrete- and continuous-time models are equal; it also required that matrices \(\mathbf{a}_c\) and \(\mathbf{Q}_{c,d}\) must be full-rank matrices):
\(\mathbf{Q}_c = \mathbf{Q}_d\)
\(\mathbf{a}_c = \mathbf{R}_d - I(k)\)
\(\mathbf{b}_c = \mathbf{\Sigma}\)
\({\mathbf{f}_1}_c = -\mathbf{a}_c^{-1} \times \mathbf{u}_d\)
\(\mathbf{f}_c = -0.5 \mathbf{b}_d \times \mathbf{Q}^{-1}_d\)
\({\mu_0}_c = {\mu _0}_d - \mathbf{f}_c \times \mathbf{Q_c} \times \mathbf{f}_c^*\)
\(\theta_c = \theta_d\)
where \(k\) is a number of covariates, which is equal to model’s dimension and ’*’ denotes transpose operation; \(\mathbf{\Sigma}\) is a \(k \times 1\) matrix which contains s.d.s of corresponding residuals (residuals of a linear regression \(\mathbf{Y}(t+1) = \mathbf{u} + \mathbf{R}\mathbf{Y}(t) + \mathbf{\epsilon}\); s.d. is a standard deviation), \(I(k)\) is an identity \(k \times k\) matrix.
In previous models, we assumed that coefficients is sort of time-dependant: we multiplied them on to \(e^{\theta t}\). In general, this may not be the case [5]. We extend this to a general case, i.e. (we consider one-dimensional case):
\(\mathbf{a(t)} = \mathbf{par}_1 t + \mathbf{par}_2\) - linear function.
The corresponding equations will be equivalent to one-dimensional continuous case described above.
library(stpm)
#Data preparation:
n <- 50
data <- simdata_time_dep(N=n)
# Estimation:
opt.par <- spm_time_dep(data,
start = list(a = -0.05, f1 = 80, Q = 2e-08, f = 80, b = 5, mu0 = 0.001),
frm = list(at = "a", f1t = "f1", Qt = "Q", ft = "f", bt = "b", mu0t= "mu0"))
opt.par## [[1]]
## [[1]]$a
## [1] -0.05740929497
##
## [[1]]$f1
## [1] 80.25586796
##
## [[1]]$Q
## [1] 2.269684109e-08
##
## [[1]]$f
## [1] 85.47037859
##
## [[1]]$b
## [1] 5.068353201
##
## [[1]]$mu0
## [1] 0.0008671811525
##
## [[1]]$status
## [1] 3
##
## [[1]]$LogLik
## t2
## -8040.740093
##
## [[1]]$objective
## [1] 8040.736959
##
## [[1]]$message
## [1] "NLOPT_FTOL_REACHED: Optimization stopped because ftol_rel or ftol_abs (above) was reached."We added one- and multi- dimensional simulation to be able to generate test data for hyphotesis testing. Data, which can be simulated can be discrete (equal intervals between observations) and continuous (with arbitrary intervals).
The corresponding function is (k - a number of variables(covariates), equal to model’s dimension):
simdata_discr(N=100, a=-0.05, f1=80, Q=2e-8, f=80, b=5, mu0=1e-5, theta=0.08, ystart=80, tstart=30, tend=105, dt=1)
Here:
N - Number of individuals
a - A matrix of kxk, which characterize the rate of the adaptive response
f1 - A particular state, which if a deviation from the normal (or optimal). This is a vector with length of k
Q - A matrix of k by k, which is a non-negative-definite symmetric matrix
f - A vector-function (with length k) of the normal (or optimal) state
b - A diffusion coefficient, k by k matrix
mu0 - mortality at start period of time (baseline hazard)
theta - A displacement coefficient of the Gompertz function
ystart - A vector with length equal to number of dimensions used, defines starting values of covariates
tstart - A number that defines a start time (30 by default). Can be a number (30 by default) or a vector of two numbers: c(a, b) - in this case, starting value of time is simulated via uniform(a,b) distribution.
tend - A number, defines a final time (105 by default)
dt - A time interval between observations.
This function returns a table with simulated data, as shown in example below:
library(stpm)
data <- simdata_discr(N=10)
head(data)## id xi t1 t2 y1 y1.next
## [1,] 1 0 30 31 80.00000000 71.05661530
## [2,] 1 0 31 32 71.05661530 66.27099977
## [3,] 1 0 32 33 66.27099977 65.73137314
## [4,] 1 0 33 34 65.73137314 70.53494657
## [5,] 1 0 34 35 70.53494657 73.37986659
## [6,] 1 0 35 36 73.37986659 75.05224487The corresponding function is (k - a number of variables(covariates), equal to model’s dimension):
simdata_cont(N=100, a=-0.05, f1=80, Q=2e-07, f=80, b=5, mu0=2e-05, theta=0.08, ystart=80, tstart=c(30,50), tend=105)
Here:
N - Number of individuals
a - A matrix of kxk, which characterize the rate of the adaptive response
f1 - A particular state, which if a deviation from the normal (or optimal). This is a vector with length of k
Q - A matrix of k by k, which is a non-negative-definite symmetric matrix
f - A vector-function (with length k) of the normal (or optimal) state
b - A diffusion coefficient, k by k matrix
mu0 - mortality at start period of time (baseline hazard)
theta - A displacement coefficient of the Gompertz function
ystart - A vector with length equal to number of dimensions used, defines starting values of covariates
tstart - A number that defines a start time (30 by default). Can be a number (30 by default) or a vector of two numbers: c(a, b) - in this case, starting value of time is simulated via uniform(a,b) distribution.
tend - A number, defines a final time (105 by default)
This function returns a table with simulated data, as shown in example below:
library(stpm)
data <- simdata_cont(N=10)
head(data)## id xi t1 t2 y1 y1.next
## [1,] 0 0 38.17902850 39.84507074 80.03167612 76.34472382
## [2,] 0 0 39.84507074 40.94297748 76.34472382 74.83343846
## [3,] 0 0 40.94297748 42.21895547 74.83343846 78.52860191
## [4,] 0 0 42.21895547 43.58787575 78.52860191 86.02490234
## [5,] 0 0 43.58787575 45.31921660 86.02490234 89.41061743
## [6,] 0 0 45.31921660 47.22382956 89.41061743 83.76220725Stochastic Process Model has many applications in analysis of longitudinal biodemographic data. Such data contain various physiological variables (known as covariates). Data can also potentially contain genetic information available for all or a part of participants. Taking advantage from both genetic and non-genetic information can provide future insights into a broad range of processes describing aging-related changes in the organism.
In this package, SPM with partially observed covariates is implemented in form of GenSPM (Genetic SPM), presented in 2009 by Arbeev at al [6] and further advanced in [7,8], further elaborates the basic stochastic process model conception by introducing a categorical variable, \(Z\), which may be a specific value of a genetic marker or, in general, any categorical variable. Currently, \(Z\) has two gradations: 0 or 1 in a genetic group of interest, assuming that \(P(Z=1) = p\), \(p \in [0, 1]\), were \(p\) is the proportion of carriers and non-carriers of an allele in a population. Example of longitudinal data with genetic component \(Z\) is provided below.
library(stpm)
data <- sim_pobs(N=10)
head(data)## id xi t1 t2 Z y1 y1.next
## 1 0 0 99.46889485 100.5443216 0 81.77743575 79.07220059
## 2 0 0 100.54432164 101.5045688 0 79.07220059 74.42370537
## 3 0 0 101.50456877 102.4385202 0 74.42370537 77.29910322
## 4 0 0 102.43852020 103.4828045 0 77.29910322 80.66806776
## 5 0 0 103.48280449 104.5317042 0 80.66806776 86.53013222
## 6 1 0 61.12878076 62.0845993 0 80.37739011 74.53147414In the specification of the SPM described in 2007 paper by Yashin and colleagues [3] the stochastic differential equation describing the age dynamics of a physiological variable (a dynamic component of the model) is:
\(dY(t) = a(Z, t)(Y(t) - f1(Z, t))dt + b(Z, t)dW(t), Y(t = t_0)\)
Here in this equation, \(Y(t)\) is a \(k \times 1\) matrix, where \(k\) is a number of covariates, which is a model dimension) describing the value of a physiological variable at a time (e.g. age) t. \(f_1(Z,t)\) is a \(k \times 1\) matrix that corresponds to the long-term average value of the stochastic process \(Y(t)\), which describes a trajectory of individual variable influenced by different factors represented by a random Wiener process \(W(t)\). The negative feedback coefficient \(a(Z,t)\) (\(k \times k\) matrix) characterizes the rate at which the stochastic process goes to its mean. In research on aging and well-being, \(f_1(Z,t)\) represents the average allostatic trajectory and \(a(t)\) in this case represents the adaptive capacity of the organism. Coefficient \(b(Z,t)\) (\(k \times 1\) matrix) characterizes a strength of the random disturbances from Wiener process \(W(t)\). All of these parameters depend on \(Z\) (a genetic marker having values 1 or 0). The following function \(\mu(t,Y(t))\) represents a hazard rate:
\(\mu(t,Y(t)) = \mu_0(t) + (Y(t) - f(Z, t))^*Q(Z, t)(Y(t) - f(Z, t))\)
In this equation: \(\mu_0(t)\) is the baseline hazard, which represents a risk when \(Y(t)\) follows its optimal trajectory; f(t) (\(k \times 1\) matrix) represents the optimal trajectory that minimizes the risk and \(Q(Z, t)\) (\(k \times k\) matrix) represents a sensitivity of risk function to deviation from the norm. In general, model coefficients \(a(Z, t)\), \(f1(Z, t)\), \(Q(Z, t)\), \(f(Z, t)\), \(b(Z, t)\) and \(\mu_0(t)\) are time(age)-dependent. Once we have data, we then can run analysis, i.e. estimate coefficients (they are assumed to be time-independent and data here is simulated):
library(stpm)
#Generating data:
data <- sim_pobs(N=10)
head(data)## id xi t1 t2 Z y1 y1.next
## 1 0 0 86.71362594 87.66960140 0 79.94674627 85.75194409
## 2 0 0 87.66960140 88.74631640 0 85.75194409 79.77500017
## 3 0 0 88.74631640 89.67148405 0 79.77500017 82.91212493
## 4 0 0 89.67148405 90.72023040 0 82.91212493 78.56597243
## 5 0 0 90.72023040 91.65991436 0 78.56597243 80.77661900
## 6 0 0 91.65991436 92.69503672 0 80.77661900 87.17250316#Parameters estimation:
pars <- spm_pobs(x=data)## Parameter f1H achieved lower/upper bound.
## 54
## Parameter QL achieved lower/upper bound.
## 2.75e-08
## Parameter mu0H achieved lower/upper bound.
## 7.2e-06
## Parameter thetaL achieved lower/upper bound.
## 0.09pars## $aH
## [,1]
## [1,] -0.05415998541
##
## $aL
## [,1]
## [1,] -0.009754778486
##
## $f1H
## [,1]
## [1,] 54
##
## $f1L
## [,1]
## [1,] 79.71328702
##
## $QH
## [,1]
## [1,] 2.188148156e-08
##
## $QL
## [,1]
## [1,] 2.75e-08
##
## $fH
## [,1]
## [1,] 65.92442671
##
## $fL
## [,1]
## [1,] 74.72696149
##
## $bH
## [,1]
## [1,] 3.618414945
##
## $bL
## [,1]
## [1,] 5.063820344
##
## $mu0H
## [1] 7.2e-06
##
## $mu0L
## [1] 9.003957339e-06
##
## $thetaH
## [1] 0.07315909157
##
## $thetaL
## [1] 0.09
##
## $p
## [1] 0.2721164912
##
## $limit
## [1] TRUE
##
## attr(,"class")
## [1] "pobs.spm"Here and represents parameters when \(Z\) = 1 (H) and 0 (L).
library(stpm)
data.genetic <- sim_pobs(N=10, mode='observed')
head(data.genetic)## id xi t1 t2 Z y1 y1.next
## 1 0 0 69.13140783 70.07921545 0 78.07307815 84.89852680
## 2 0 0 70.07921545 71.16321540 0 84.89852680 79.03626042
## 3 0 0 71.16321540 72.08940085 0 79.03626042 81.35377142
## 4 0 0 72.08940085 73.12641991 0 81.35377142 81.46236846
## 5 0 0 73.12641991 74.12835031 0 81.46236846 66.09969380
## 6 0 0 74.12835031 75.02855590 0 66.09969380 65.08279920data.nongenetic <- sim_pobs(N=50, mode='unobserved')
head(data.nongenetic)## id xi t1 t2 y1 y1.next
## 1 0 0 90.55443227 91.65328813 81.32287704 78.32438920
## 2 0 0 91.65328813 92.60998921 78.32438920 80.18359039
## 3 0 0 92.60998921 93.63949841 80.18359039 89.76796461
## 4 0 0 93.63949841 94.73374603 89.76796461 90.73317902
## 5 0 0 94.73374603 95.75006616 90.73317902 84.50176161
## 6 0 0 95.75006616 96.67227923 84.50176161 81.55706134#Parameters estimation:
pars <- spm_pobs(x=data.genetic, y = data.nongenetic, mode='combined')
pars## $aH
## [,1]
## [1,] -0.04222454925
##
## $aL
## [,1]
## [1,] -0.01070928209
##
## $f1H
## [,1]
## [1,] 56.38280885
##
## $f1L
## [,1]
## [1,] 80.53083853
##
## $QH
## [,1]
## [1,] 2.198883637e-08
##
## $QL
## [,1]
## [1,] 2.359188823e-08
##
## $fH
## [,1]
## [1,] 56.69390069
##
## $fL
## [,1]
## [1,] 77.95009252
##
## $bH
## [,1]
## [1,] 4.150744746
##
## $bL
## [,1]
## [1,] 4.999278913
##
## $mu0H
## [1] 7.32536981e-06
##
## $mu0L
## [1] 9.002457382e-06
##
## $thetaH
## [1] 0.07208770662
##
## $thetaL
## [1] 0.09002035774
##
## $p
## [1] 0.2734764324
##
## $limit
## [1] FALSE
##
## attr(,"class")
## [1] "pobs.spm"Here mode ‘observed’ is used for simlation of data with genetic component \(Z\) and ‘unobserved’ - without genetic component.
[1] Woodbury M.A., Manton K.G., Random-Walk of Human Mortality and Aging. Theoretical Population Biology, 1977 11:37-48.
[2] Yashin, A.I., Manton K.G., Vaupel J.W. Mortality and aging in a heterogeneous population: a stochastic process model with observed and unobserved varia-bles. Theor Pop Biology, 1985 27.
[3] Yashin, A.I. et al. Stochastic model for analysis of longitudinal data on aging and mortality. Mathematical Biosciences, 2007 208(2) 538-551.
[4] Akushevich I., Kulminski A. and Manton K.: Life tables with covariates: Dynamic model for Nonlinear Analysis of Longitudinal Data. 2005. Mathematical Popu-lation Studies, 12(2), pp.: 51-80.
[5] Yashin, A. et al. Health decline, aging and mortality: how are they related? Biogerontology, 2007 8(3), 291-302.
[6] Arbeev, K.G., Akushevich, I., Kulminski, A.M., Arbeeva, L.S., Akushevich, L., Ukraintseva, S.V., Culminskaya, I.V., Yashin, A.I.: Genetic model for longitudinal studies of aging, health, and longevity and its potential application to incomplete data. Journal of Theoretical Biology 258(1), 103{111 (2009).
[7] Arbeev K.G, Akushevich I., Kulminski A.M., Ukraintseva S.V., Yashin A.I., Joint Analyses of Longitudinal and Time-to-Event Data in Research on Aging: Implications for Predicting Health and Survival, Front Public Health. 2014 Nov 6;2:228. doi: 10.3389/fpubh.2014.00228
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