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.
install.packages("stpm")require(devtools)
devtools::install_github("izhbannikov/stpm")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 32.89139There 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 84.24353
## [2,] 1 0 31 32 84.24353 79.03284
## [3,] 1 0 32 33 79.03284 76.42617
## [4,] 1 0 33 34 76.42617 73.97344
## [5,] 1 0 34 35 73.97344 83.42408
## [6,] 1 0 35 36 83.42408 86.01618In 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 34.98552 36.20709 78.60190 76.39118
## [2,] 0 0 36.20709 37.97197 76.39118 75.72559
## [3,] 0 0 37.97197 39.09253 75.72559 79.51251
## [4,] 0 0 39.09253 40.12665 79.51251 83.27112
## [5,] 0 0 40.12665 41.58184 83.27112 75.16489
## [6,] 0 0 41.58184 43.17803 75.16489 82.74287The 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=100)
#Estimation of parameters
pars <- spm_discrete(data)
pars## $Ak2005
## $Ak2005$theta
## [1] 0.068
##
## $Ak2005$mu0
## [1] 0.0004499861356
##
## $Ak2005$b
## [1] -1.013524064e-05
##
## $Ak2005$Q
## [,1]
## [1,] 6.117270622e-08
##
## $Ak2005$u
## [1] 3.612360187
##
## $Ak2005$R
## [,1]
## [1,] 0.952813853
##
## $Ak2005$Sigma
## [1] 5.031195885
##
##
## $Ya2007
## $Ya2007$a
## [,1]
## [1,] -0.04718614699
##
## $Ya2007$f1
## [,1]
## [1,] 76.55552355
##
## $Ya2007$Q
## [,1]
## [1,] 6.117270622e-08
##
## $Ya2007$f
## [,1]
## [1,] 82.84119884
##
## $Ya2007$b
## [,1]
## [1,] 5.031195885
##
## $Ya2007$mu0
## [,1]
## [1,] 3.017839308e-05
##
## $Ya2007$theta
## [1] 0.068
##
##
## 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 50 individuals
data <- simdata_cont(N=50)
head(data)## id xi t1 t2 y1 y1.next
## [1,] 0 0 36.65871887 37.81465271 79.95944105 84.61537383
## [2,] 0 0 37.81465271 39.08069130 84.61537383 73.25167251
## [3,] 0 0 39.08069130 40.55323362 73.25167251 78.08056355
## [4,] 0 0 40.55323362 42.28593127 78.08056355 83.59968160
## [5,] 0 0 42.28593127 43.35636208 83.59968160 86.33725714
## [6,] 0 0 43.35636208 45.32297420 86.33725714 88.69132001#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)## Parameter f achieved lower/upper bound.
## 88pars## $a
## [,1]
## [1,] -0.05477426863
##
## $f1
## [,1]
## [1,] 79.06298754
##
## $Q
## [,1]
## [1,] 2.197290751e-08
##
## $f
## [,1]
## [1,] 88
##
## $b
## [,1]
## [1,] 5.044968905
##
## $mu0
## [1] 2.19995064e-05
##
## $theta
## [1] 0.08437629393
##
## $status
## [1] 5
##
## $LogLik
## [1] -6864.37787
##
## $objective
## [1] 6864.37405
##
## $message
## [1] "NLOPT_MAXEVAL_REACHED: Optimization stopped because maxeval (above) was reached."
##
## $limit
## [1] TRUE
##
## 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 <- 10
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]]$a
## [1] -0.06179885086
##
## [[1]]$f1
## [1] 79.15334446
##
## [[1]]$Q
## [1] 2.472435371e-08
##
## [[1]]$f
## [1] 99.9614111
##
## [[1]]$b
## [1] 4.895185383
##
## [[1]]$mu0
## [1] 0.0007653187682
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -1677.809469
##
## [[1]]$objective
## [1] 1677.808346
##
## [[1]]$message
## [1] "NLOPT_MAXEVAL_REACHED: Optimization stopped because maxeval (above) was reached."Lower and upper boundaries can be set up with parameters \(lb\) and \(ub\), which represents simple numeric vectors. Note: lengths of \(lb\) and \(ub\) must be the same as the total length of the parameters. Lower and upper boundaries can be set for continuous-time and time-dependent models only.
Below we show the example of setting up \(lb\) and \(ub\) when we have a single covariate:
library(stpm)
data <- simdata_cont(N=10, ystart = 80, a = -0.1, Q = 1e-06, mu0 = 1e-5, theta = 0.08, f1 = 80, f=80, b=1, dt=1, sd0=5)
ans <- spm_continuous(dat=data,
a = -0.1,
f1 = 82,
Q = 1.4e-6,
f = 77,
b = 1,
mu0 = 1.6e-5,
theta = 0.1,
stopifbound = FALSE,
lb=c(-0.2, 60, 0.1e-6, 60, 0.1, 0.1e-5, 0.01),
ub=c(0, 140, 5e-06, 140, 3, 5e-5, 0.20))
ans## $a
## [,1]
## [1,] -0.08173015597
##
## $f1
## [,1]
## [1,] 79.48830866
##
## $Q
## [,1]
## [1,] 3.843041665e-06
##
## $f
## [,1]
## [1,] 124.8731961
##
## $b
## [,1]
## [1,] 0.9583799199
##
## $mu0
## [1] 4.810394523e-05
##
## $theta
## [1] 0.164603153
##
## $status
## [1] 5
##
## $LogLik
## [1] -648.4149134
##
## $objective
## [1] 648.4037285
##
## $message
## [1] "NLOPT_MAXEVAL_REACHED: Optimization stopped because maxeval (above) was reached."
##
## $limit
## [1] FALSE
##
## attr(,"class")
## [1] "spm.continuous"This is an example for two physiological variables (covariates).
library(stpm)
data <- simdata_cont(N=10,
a=matrix(c(-0.1, 0.001, 0.001, -0.1), nrow = 2, ncol = 2, byrow = T),
f1=t(matrix(c(100, 200), nrow = 2, ncol = 1, byrow = F)),
Q=matrix(c(1e-06, 1e-7, 1e-7, 1e-06), nrow = 2, ncol = 2, byrow = T),
f=t(matrix(c(100, 200), nrow = 2, ncol = 1, byrow = F)),
b=matrix(c(1, 2), nrow = 2, ncol = 1, byrow = F),
mu0=1e-4,
theta=0.08,
ystart = c(100,200), sd0=c(5, 10), dt=1)
a.d <- matrix(c(-0.15, 0.002, 0.002, -0.15), nrow = 2, ncol = 2, byrow = T)
f1.d <- t(matrix(c(95, 195), nrow = 2, ncol = 1, byrow = F))
Q.d <- matrix(c(1.2e-06, 1.2e-7, 1.2e-7, 1.2e-06), nrow = 2, ncol = 2, byrow = T)
f.d <- t(matrix(c(105, 205), nrow = 2, ncol = 1, byrow = F))
b.d <- matrix(c(1, 2), nrow = 2, ncol = 1, byrow = F)
mu0.d <- 1.1e-4
theta.d <- 0.07
ans <- spm_continuous(dat=data,
a = a.d,
f1 = f1.d,
Q = Q.d,
f = f.d,
b = b.d,
mu0 = mu0.d,
theta = theta.d,
lb=c(-0.5, ifelse(a.d[2,1] > 0, a.d[2,1]-0.5*a.d[2,1], a.d[2,1]+0.5*a.d[2,1]), ifelse(a.d[1,2] > 0, a.d[1,2]-0.5*a.d[1,2], a.d[1,2]+0.5*a.d[1,2]), -0.5,
80, 100,
Q.d[1,1]-0.5*Q.d[1,1], ifelse(Q.d[2,1] > 0, Q.d[2,1]-0.5*Q.d[2,1], Q.d[2,1]+0.5*Q.d[2,1]), ifelse(Q.d[1,2] > 0, Q.d[1,2]-0.5*Q.d[1,2], Q.d[1,2]+0.5*Q.d[1,2]), Q.d[2,2]-0.5*Q.d[2,2],
80, 100,
0.1, 0.5,
0.1e-4,
0.01),
ub=c(-0.08, 0.002, 0.002, -0.08,
110, 220,
Q.d[1,1]+0.1*Q.d[1,1], ifelse(Q.d[2,1] > 0, Q.d[2,1]+0.1*Q.d[2,1], Q.d[2,1]-0.1*Q.d[2,1]), ifelse(Q.d[1,2] > 0, Q.d[1,2]+0.1*Q.d[1,2], Q.d[1,2]-0.1*Q.d[1,2]), Q.d[2,2]+0.1*Q.d[2,2],
110, 220,
1.5, 2.5,
1.2e-4,
0.10))
ans## $a
## [,1] [,2]
## [1,] -0.15028031830 0.001692321558
## [2,] 0.00197897407 -0.148024049463
##
## $f1
## [,1]
## [1,] 105.3020459
## [2,] 195.1978728
##
## $Q
## [,1] [,2]
## [1,] 1.302256686e-06 1.305255974e-07
## [2,] 1.299346324e-07 1.285262354e-06
##
## $f
## [,1]
## [1,] 107.4483907
## [2,] 210.4038072
##
## $b
## [,1]
## [1,] 1.117474561
## [2,] 1.958029591
##
## $mu0
## [1] 0.0001124070742
##
## $theta
## [1] 0.07345124042
##
## $status
## [1] 5
##
## $LogLik
## [1] 1216.83246
##
## $objective
## [1] -1569.443099
##
## $message
## [1] "NLOPT_MAXEVAL_REACHED: Optimization stopped because maxeval (above) was reached."
##
## $limit
## [1] FALSE
##
## attr(,"class")
## [1] "spm.continuous"This model uses only one covariate, therefore setting-up model parameters is easy:
n <- 10
data <- simdata_time_dep(N=n)
# Estimation:
opt.par <- spm_time_dep(data, start=list(a=-0.05, f1=80, Q=2e-08, f=80, b=5, mu0=0.001),
lb=c(-1, 30, 1e-8, 30, 1, 1e-6), ub=c(0, 120, 5e-8, 130, 10, 1e-2))
opt.par## [[1]]
## [[1]]$a
## [1] -0.06006110872
##
## [[1]]$f1
## [1] 78.89943309
##
## [[1]]$Q
## [1] 1.815306343e-08
##
## [[1]]$f
## [1] 105.9918715
##
## [[1]]$b
## [1] 4.628929519
##
## [[1]]$mu0
## [1] 0.001423309162
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -1580.621949
##
## [[1]]$objective
## [1] 1580.618315
##
## [[1]]$message
## [1] "NLOPT_MAXEVAL_REACHED: Optimization stopped because maxeval (above) was reached."Imagine a situation when one parameter function you want to be equal to zero: \(f=0\). Let’s emulate this case:
library(stpm)
n <- 10
data <- simdata_time_dep(N=n)
# Estimation:
opt.par <- spm_time_dep(data, frm = list(at="a", f1t="f1", Qt="Q", ft="0", bt="b", mu0t="mu0"))
opt.par## [[1]]
## [[1]]$a
## [1] -0.06237635947
##
## [[1]]$f1
## [1] 81.43693942
##
## [[1]]$Q
## [1] 1.504868462e-08
##
## [[1]]$b
## [1] 60
##
## [[1]]$mu0
## [1] 3.75
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -4854.839382
##
## [[1]]$objective
## [1] 4854.839382
##
## [[1]]$message
## [1] "NLOPT_MAXEVAL_REACHED: Optimization stopped because maxeval (above) was reached."As you can see, there is no parameter \(f\) in \(opt.par\). This because we set \(f=0\) in \(frm\)!
Then, is you want to set the constraints, you must not specify the starting value (parameter \(start\)) and \(lb\)/\(ub\) for the parameter \(f\) (otherwise, the function raises an error):
n <- 10
data <- simdata_time_dep(N=n)
# Estimation:
opt.par <- spm_time_dep(data, frm = list(at="a", f1t="f1", Qt="Q", ft="0", bt="b", mu0t="mu0"),
start=list(a=-0.05, f1=80, Q=2e-08, b=5, mu0=0.001),
lb=c(-1, 30, 1e-8, 1, 1e-6), ub=c(0, 120, 5e-8, 10, 1e-2))
opt.par## [[1]]
## [[1]]$a
## [1] -0.04644455244
##
## [[1]]$f1
## [1] 79.87837657
##
## [[1]]$Q
## [1] 1e-08
##
## [[1]]$b
## [1] 5.368190996
##
## [[1]]$mu0
## [1] 1e-06
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -1701.155226
##
## [[1]]$objective
## [1] 1701.155226
##
## [[1]]$message
## [1] "NLOPT_MAXEVAL_REACHED: Optimization stopped because maxeval (above) was reached."You can do the same manner if you want two or more parameters to be equal to zero.
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 79.06243863
## [2,] 1 0 31 32 79.06243863 75.81040404
## [3,] 1 0 32 33 75.81040404 78.14520714
## [4,] 1 0 33 34 78.14520714 88.09476992
## [5,] 1 0 34 35 88.09476992 85.92314838
## [6,] 1 0 35 36 85.92314838 82.52189811The 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 31.34892635 33.33825735 78.00063388 93.58460947
## [2,] 0 0 33.33825735 34.98283208 93.58460947 90.84349364
## [3,] 0 0 34.98283208 36.49559593 90.84349364 80.67501511
## [4,] 0 0 36.49559593 38.33948544 80.67501511 78.20360710
## [5,] 0 0 38.33948544 39.55744914 78.20360710 77.55578555
## [6,] 0 0 39.55744914 40.93644301 77.55578555 78.69533712Stochastic 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 78.34925245 79.34535257 0 78.88442909 72.18693213
## 2 0 0 79.34535257 80.38214218 0 72.18693213 72.34691604
## 3 0 0 80.38214218 81.31966875 0 72.34691604 78.80325292
## 4 0 0 81.31966875 82.29883287 0 78.80325292 74.85552971
## 5 0 0 82.29883287 83.29472501 0 74.85552971 70.27606301
## 6 0 0 83.29472501 84.30189499 0 70.27606301 66.17626196In 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 35.21221978 36.26340582 1 80.68019654 77.75189912
## 2 0 0 36.26340582 37.27937793 1 77.75189912 82.17704836
## 3 0 0 37.27937793 38.30757011 1 82.17704836 85.65196308
## 4 0 0 38.30757011 39.29134767 1 85.65196308 79.30313233
## 5 0 0 39.29134767 40.29299718 1 79.30313233 74.19480508
## 6 0 0 40.29299718 41.35501294 1 74.19480508 77.61789345#Parameters estimation:
pars <- spm_pobs(x=data)## Parameter thetaL achieved lower/upper bound.
## 0.09pars## $aH
## [,1]
## [1,] -0.03258525373
##
## $aL
## [,1]
## [1,] -0.005048219202
##
## $f1H
## [,1]
## [1,] 65.63590692
##
## $f1L
## [,1]
## [1,] 86.23767447
##
## $QH
## [,1]
## [1,] 1.886958361e-08
##
## $QL
## [,1]
## [1,] 2.459964796e-08
##
## $fH
## [,1]
## [1,] 64.69427555
##
## $fL
## [,1]
## [1,] 84.8718174
##
## $bH
## [,1]
## [1,] 3.968609091
##
## $bL
## [,1]
## [1,] 4.709011386
##
## $mu0H
## [1] 8.501697117e-06
##
## $mu0L
## [1] 9.020504479e-06
##
## $thetaH
## [1] 0.07240873446
##
## $thetaL
## [1] 0.09
##
## $p
## [1] 0.2326078687
##
## $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=5, mode='observed')
head(data.genetic)## id xi t1 t2 Z y1 y1.next
## 1 0 0 40.50991234 41.45436305 0 78.25720326 75.69911316
## 2 0 0 41.45436305 42.48432957 0 75.69911316 76.33847804
## 3 0 0 42.48432957 43.38617619 0 76.33847804 78.89219065
## 4 0 0 43.38617619 44.38079712 0 78.89219065 83.15940763
## 5 0 0 44.38079712 45.28706211 0 83.15940763 82.21072388
## 6 0 0 45.28706211 46.19252020 0 82.21072388 81.13478287data.nongenetic <- sim_pobs(N=10, mode='unobserved')
head(data.nongenetic)## id xi t1 t2 y1 y1.next
## 1 0 0 51.92838679 52.89245973 80.01619153 81.71556359
## 2 0 0 52.89245973 53.86704947 81.71556359 84.14562667
## 3 0 0 53.86704947 54.79321676 84.14562667 83.02397698
## 4 0 0 54.79321676 55.84553766 83.02397698 86.26956399
## 5 0 0 55.84553766 56.90015506 86.26956399 85.05385945
## 6 0 0 56.90015506 57.96068411 85.05385945 79.09773970#Parameters estimation:
pars <- spm_pobs(x=data.genetic, y = data.nongenetic, mode='combined')
pars## $aH
## [,1]
## [1,] -0.03731125489
##
## $aL
## [,1]
## [1,] -0.0109116872
##
## $f1H
## [,1]
## [1,] 64.92456312
##
## $f1L
## [,1]
## [1,] 75.53260554
##
## $QH
## [,1]
## [1,] 1.784927785e-08
##
## $QL
## [,1]
## [1,] 1.898980753e-08
##
## $fH
## [,1]
## [1,] 60.02210963
##
## $fL
## [,1]
## [1,] 83.99757534
##
## $bH
## [,1]
## [1,] 4.395263396
##
## $bL
## [,1]
## [1,] 4.921127419
##
## $mu0H
## [1] 7.398269398e-06
##
## $mu0L
## [1] 9.00518042e-06
##
## $thetaH
## [1] 0.0724133025
##
## $thetaL
## [1] 0.0900532744
##
## $p
## [1] 0.2741033316
##
## $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
[8] Arbeev K., Arbeeva L., Akushevich I., Kulminski A., Ukraintseva S., Yashin A., Latent Class and Genetic Stochastic Process Models: Implications for Analyses of Longitudinal Data on Aging, Health, and Longevity, JSM-2015, Seattle, WA.