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 80.07890
## [2,] 1 0 31 32 80.07890 78.17307
## [3,] 1 0 32 33 78.17307 82.72450
## [4,] 1 0 33 34 82.72450 87.27957
## [5,] 1 0 34 35 87.27957 94.99048
## [6,] 1 0 35 36 94.99048 97.00163In 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 39.33457 41.09516 80.95667 81.19574
## [2,] 0 0 41.09516 42.27438 81.19574 71.11206
## [3,] 0 0 42.27438 44.02813 71.11206 66.61501
## [4,] 0 0 44.02813 45.77484 66.61501 60.46489
## [5,] 0 0 45.77484 46.86583 60.46489 65.67223
## [6,] 0 0 46.86583 47.92258 65.67223 70.84383The 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] 0.079
##
## $Ak2005$mu0
## [1] 8.583980901e-05
##
## $Ak2005$b
## [1] -1.97077126e-06
##
## $Ak2005$Q
## [,1]
## [1,] 1.322736595e-08
##
## $Ak2005$u
## [1] 3.951049548
##
## $Ak2005$R
## [,1]
## [1,] 0.9494759988
##
## $Ak2005$Sigma
## [1] 5.067412504
##
##
## $Ya2007
## $Ya2007$a
## [,1]
## [1,] -0.05052400119
##
## $Ya2007$f1
## [,1]
## [1,] 78.20143803
##
## $Ya2007$Q
## [,1]
## [1,] 1.322736595e-08
##
## $Ya2007$f
## [,1]
## [1,] 74.49598309
##
## $Ya2007$b
## [,1]
## [1,] 5.067412504
##
## $Ya2007$mu0
## [,1]
## [1,] 1.243253777e-05
##
## $Ya2007$theta
## [1] 0.079
##
##
## 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 37.90784651 39.03449079 82.95780714 80.39496502
## [2,] 0 0 39.03449079 40.77151522 80.39496502 77.94821149
## [3,] 0 0 40.77151522 42.01514290 77.94821149 80.45867575
## [4,] 0 0 42.01514290 43.47118576 80.45867575 74.56386550
## [5,] 0 0 43.47118576 44.83565998 74.56386550 70.21005016
## [6,] 0 0 44.83565998 46.77426172 70.21005016 76.07907843#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.
## 72pars## $a
## [,1]
## [1,] -0.05104116715
##
## $f1
## [,1]
## [1,] 79.28265455
##
## $Q
## [,1]
## [1,] 2.191391509e-08
##
## $f
## [,1]
## [1,] 72
##
## $b
## [,1]
## [1,] 5.010311196
##
## $mu0
## [1] 2.198934107e-05
##
## $theta
## [1] 0.08738369417
##
## $status
## [1] 5
##
## $LogLik
## [1] -6743.892448
##
## $objective
## [1] 6743.879501
##
## $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]]$a
## [1] -0.04467112825
##
## [[1]]$f1
## [1] 79.88096088
##
## [[1]]$Q
## [1] 2.021713349e-08
##
## [[1]]$f
## [1] 60.02940724
##
## [[1]]$b
## [1] 5.285945632
##
## [[1]]$mu0
## [1] 0.001242835718
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -1624.298056
##
## [[1]]$objective
## [1] 1624.29527
##
## [[1]]$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,] -0.1008039144
##
## $f1
## [,1]
## [1,] 80.13077022
##
## $Q
## [,1]
## [1,] 4.244926566e-06
##
## $f
## [,1]
## [1,] 123.2182974
##
## $b
## [,1]
## [1,] 0.9484193453
##
## $mu0
## [1] 4.781522803e-05
##
## $theta
## [1] 0.1699525424
##
## $status
## [1] 5
##
## $LogLik
## [1] -626.0603428
##
## $objective
## [1] 626.0562718
##
## $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,] -0.150797307775 0.001661918661
## [2,] 0.001972372463 -0.148678425730
##
## $f1
## [,1]
## [1,] 106.9843262
## [2,] 193.9944177
##
## $Q
## [,1] [,2]
## [1,] 1.309828456e-06 1.310580263e-07
## [2,] 1.312322432e-07 1.304841253e-06
##
## $f
## [,1]
## [1,] 108.6928395
## [2,] 212.6677976
##
## $b
## [,1]
## [1,] 1.113434783
## [2,] 1.912557351
##
## $mu0
## [1] 0.0001116954406
##
## $theta
## [1] 0.0668841698
##
## $status
## [1] 5
##
## $LogLik
## [1] 1601.535636
##
## $objective
## [1] -2232.961198
##
## $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] -0.03053466992
##
## [[1]]$f1
## [1] 79.94221641
##
## [[1]]$Q
## [1] 2.906831506e-08
##
## [[1]]$f
## [1] 102.9875
##
## [[1]]$b
## [1] 5.151942992
##
## [[1]]$mu0
## [1] 0.00137112597
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -1641.773561
##
## [[1]]$objective
## [1] 1641.773536
##
## [[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]]$a
## [1] -0.05745271213
##
## [[1]]$f1
## [1] 76.38144454
##
## [[1]]$Q
## [1] 2.45741694e-08
##
## [[1]]$b
## [1] 60
##
## [[1]]$mu0
## [1] 3.75
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -4855.209345
##
## [[1]]$objective
## [1] 4855.208135
##
## [[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]]$a
## [1] -0.08275801972
##
## [[1]]$f1
## [1] 79.86841526
##
## [[1]]$Q
## [1] 2.754916937e-08
##
## [[1]]$b
## [1] 4.597659126
##
## [[1]]$mu0
## [1] 2.221535721e-06
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -1650.386201
##
## [[1]]$objective
## [1] 1650.352064
##
## [[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 73.49219161
## [2,] 1 0 31 32 73.49219161 78.64326137
## [3,] 1 0 32 33 78.64326137 68.41033233
## [4,] 1 0 33 34 68.41033233 72.20068392
## [5,] 1 0 34 35 72.20068392 81.51849402
## [6,] 1 0 35 36 81.51849402 81.68657864The 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 33.54797696 35.10568162 78.54938222 73.39914175
## [2,] 0 0 35.10568162 36.57621642 73.39914175 64.77459905
## [3,] 0 0 36.57621642 37.71246754 64.77459905 62.83400178
## [4,] 0 0 37.71246754 39.24469619 62.83400178 64.35110502
## [5,] 0 0 39.24469619 40.61724460 64.35110502 74.01363216
## [6,] 0 0 40.61724460 42.21174220 74.01363216 74.15673456Stochastic 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 46.86622513 47.81871230 0 79.04493203 78.41484354
## 2 0 0 47.81871230 48.85389645 0 78.41484354 79.36297120
## 3 0 0 48.85389645 49.75673386 0 79.36297120 81.48793422
## 4 0 0 49.75673386 50.71200057 0 81.48793422 83.93943579
## 5 0 0 50.71200057 51.68551922 0 83.93943579 79.02191328
## 6 0 0 51.68551922 52.72393703 0 79.02191328 71.33418878In 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 0 101.78776843 102.88122954 0 79.05238037 84.82981906
## 2 0 0 102.88122954 103.96255668 0 84.82981906 83.86967160
## 3 0 0 103.96255668 104.91013620 0 83.86967160 76.28445493
## 4 1 0 93.84339597 94.76025014 0 78.96509929 85.61205779
## 5 1 0 94.76025014 95.74026045 0 85.61205779 87.03231792
## 6 1 0 95.74026045 96.82829450 0 87.03231792 88.97240967#Parameters estimation:
pars <- spm_pobs(x=data)
pars## $aH
## [,1]
## [1,] -0.05466884631
##
## $aL
## [,1]
## [1,] -0.01092603056
##
## $f1H
## [,1]
## [1,] 54.18661883
##
## $f1L
## [,1]
## [1,] 87.18047855
##
## $QH
## [,1]
## [1,] 1.932777382e-08
##
## $QL
## [,1]
## [1,] 2.707584684e-08
##
## $fH
## [,1]
## [1,] 55.12711397
##
## $fL
## [,1]
## [1,] 82.54747546
##
## $bH
## [,1]
## [1,] 4.032798339
##
## $bL
## [,1]
## [1,] 5.347547527
##
## $mu0H
## [1] 8.419591082e-06
##
## $mu0L
## [1] 9.005010938e-06
##
## $thetaH
## [1] 0.07549981856
##
## $thetaL
## [1] 0.0900344867
##
## $p
## [1] 0.2352760825
##
## $limit
## [1] FALSE
##
## 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 52.12857449 53.22171627 0 80.57555212 86.02113248
## 2 0 0 53.22171627 54.20361552 0 86.02113248 95.41612757
## 3 0 0 54.20361552 55.24457420 0 95.41612757 92.31169408
## 4 0 0 55.24457420 56.28939001 0 92.31169408 86.99988870
## 5 0 0 56.28939001 57.19770695 0 86.99988870 83.60179412
## 6 0 0 57.19770695 58.19208258 0 83.60179412 84.55784742data.nongenetic <- sim_pobs(N=10, mode='unobserved')
head(data.nongenetic)## id xi t1 t2 y1 y1.next
## 1 0 0 46.54322848 47.61021596 78.15793491 78.33414191
## 2 0 0 47.61021596 48.62625814 78.33414191 79.09417087
## 3 0 0 48.62625814 49.64797864 79.09417087 83.23957866
## 4 0 0 49.64797864 50.69471113 83.23957866 80.29735080
## 5 0 0 50.69471113 51.72536679 80.29735080 84.52822099
## 6 0 0 51.72536679 52.82126260 84.52822099 89.28185097#Parameters estimation:
pars <- spm_pobs(x=data.genetic, y = data.nongenetic, mode='combined')
pars## $aH
## [,1]
## [1,] -0.04417742442
##
## $aL
## [,1]
## [1,] -0.0108712587
##
## $f1H
## [,1]
## [1,] 65.8480937
##
## $f1L
## [,1]
## [1,] 72.1710221
##
## $QH
## [,1]
## [1,] 2.15209848e-08
##
## $QL
## [,1]
## [1,] 1.674950437e-08
##
## $fH
## [,1]
## [1,] 65.90762214
##
## $fL
## [,1]
## [1,] 87.51566531
##
## $bH
## [,1]
## [1,] 3.961271449
##
## $bL
## [,1]
## [1,] 5.086892772
##
## $mu0H
## [1] 7.356031578e-06
##
## $mu0L
## [1] 9.005668534e-06
##
## $thetaH
## [1] 0.07203704885
##
## $thetaL
## [1] 0.09004777585
##
## $p
## [1] 0.27309831
##
## $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.