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 86.25020
## 2 1 0 31 32 86.25020 81.79907
## 3 1 0 32 33 81.79907 81.45211
## 4 1 0 33 34 81.45211 86.94463
## 5 1 0 34 35 86.94463 88.89067
## 6 1 0 35 36 88.89067 86.45167In 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 35.81822 37.53857 80.06986 87.09513
## 2 0 0 37.53857 39.33472 87.09513 87.04549
## 3 0 0 39.33472 40.96465 87.04549 81.40690
## 4 0 0 40.96465 42.66117 81.40690 79.10515
## 5 0 0 42.66117 44.48431 79.10515 77.24869
## 6 0 0 44.48431 46.11514 77.24869 68.53075The 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] 0.087
##
## $Ak2005$mu0
## [1] 0.0001043099398
##
## $Ak2005$b
## [1] -2.501312732e-06
##
## $Ak2005$Q
## [,1]
## [1,] 1.562087221e-08
##
## $Ak2005$u
## [1] 3.673849
##
## $Ak2005$R
## [,1]
## [1,] 0.9545481853
##
## $Ak2005$Sigma
## [1] 5.002397651
##
##
## $Ya2007
## $Ya2007$a
## [,1]
## [1,] -0.04545181473
##
## $Ya2007$f1
## [,1]
## [1,] 80.8295339
##
## $Ya2007$Q
## [,1]
## [1,] 1.562087221e-08
##
## $Ya2007$f
## [,1]
## [1,] 80.06315839
##
## $Ya2007$b
## [,1]
## [1,] 5.002397651
##
## $Ya2007$mu0
## [,1]
## [1,] 4.178441102e-06
##
## $Ya2007$theta
## [1] 0.087
##
##
## 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 37.18830965 38.38532357 80.84701326 78.85409477
## 2 0 0 38.38532357 39.89515512 78.85409477 80.95643321
## 3 0 0 39.89515512 41.14905240 80.95643321 81.68487288
## 4 0 0 41.14905240 42.78137833 81.68487288 77.45321377
## 5 0 0 42.78137833 44.23262189 77.45321377 75.62343636
## 6 0 0 44.23262189 45.34381438 75.62343636 76.18384956#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,] -0.04134913982
##
## $f1
## [,1]
## [1,] 83.03493302
##
## $Q
## [,1]
## [1,] 2.180789757e-08
##
## $f
## [,1]
## [1,] 87.60257588
##
## $b
## [,1]
## [1,] 5.07019509
##
## $mu0
## [1] 2.197999562e-05
##
## $theta
## [1] 0.08792592377
##
## $status
## [1] 5
##
## $LogLik
## [1] -6821.144615
##
## $objective
## [1] 6821.022466
##
## $message
## [1] "NLOPT_MAXEVAL_REACHED: Optimization stopped because maxeval (above) was reached."
##
## $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 <- 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]]$a
## [1] -0.04569345759
##
## [[1]]$f1
## [1] 81.46354224
##
## [[1]]$Q
## [1] 2.388485903e-08
##
## [[1]]$f
## [1] 85.44560317
##
## [[1]]$b
## [1] 5.258932856
##
## [[1]]$mu0
## [1] 0.0007501468909
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -1697.804192
##
## [[1]]$objective
## [1] 1697.803648
##
## [[1]]$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,] -0.1305867752
##
## $f1
## [,1]
## [1,] 79.91904904
##
## $Q
## [,1]
## [1,] 4.823056787e-06
##
## $f
## [,1]
## [1,] 122.2001375
##
## $b
## [,1]
## [1,] 1.051350317
##
## $mu0
## [1] 4.32436139e-05
##
## $theta
## [1] 0.1502390857
##
## $status
## [1] 5
##
## $LogLik
## [1] -716.8290115
##
## $objective
## [1] 716.7670665
##
## $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,] -0.150267163523 0.001692834917
## [2,] 0.001979205983 -0.148012155129
##
## $f1
## [,1]
## [1,] 105.2777868
## [2,] 195.2059583
##
## $Q
## [,1] [,2]
## [1,] 1.302225923e-06 1.305236854e-07
## [2,] 1.299338548e-07 1.285194449e-06
##
## $f
## [,1]
## [1,] 107.4474372
## [2,] 210.4295117
##
## $b
## [,1]
## [1,] 1.119337586
## [2,] 1.958657445
##
## $mu0
## [1] 0.0001124533236
##
## $theta
## [1] 0.073700401
##
## $status
## [1] 5
##
## $LogLik
## [1] 1723.230971
##
## $objective
## [1] -2214.634527
##
## $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] -0.04704987341
##
## [[1]]$f1
## [1] 81.98070463
##
## [[1]]$Q
## [1] 4.97815002e-08
##
## [[1]]$f
## [1] 128.8455244
##
## [[1]]$b
## [1] 4.87244349
##
## [[1]]$mu0
## [1] 0.005279955427
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -1274.779292
##
## [[1]]$objective
## [1] 1274.779114
##
## [[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]]$a
## [1] -0.0375
##
## [[1]]$f1
## [1] 83.68943277
##
## [[1]]$Q
## [1] 1.5e-08
##
## [[1]]$b
## [1] 60
##
## [[1]]$mu0
## [1] 3.75
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -5106.12866
##
## [[1]]$objective
## [1] 5106.121231
##
## [[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]]$a
## [1] -0.05227796834
##
## [[1]]$f1
## [1] 80.84791871
##
## [[1]]$Q
## [1] 1e-08
##
## [[1]]$b
## [1] 4.925471984
##
## [[1]]$mu0
## [1] 1e-06
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -1676.055363
##
## [[1]]$objective
## [1] 1676.055363
##
## [[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 83.60481970
## 2 1 0 31 32 83.60481970 80.70189128
## 3 1 0 32 33 80.70189128 86.43841255
## 4 1 0 33 34 86.43841255 89.76982544
## 5 1 0 34 35 89.76982544 92.57696671
## 6 1 0 35 36 92.57696671 94.83275796The 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 31.72682358 32.92908683 77.91414488 80.19649527
## 2 0 0 32.92908683 34.17246146 80.19649527 76.85946341
## 3 0 0 34.17246146 35.52961685 76.85946341 77.66892449
## 4 0 0 35.52961685 36.57491690 77.66892449 69.92476658
## 5 0 0 36.57491690 37.73835833 69.92476658 73.76061612
## 6 0 0 37.73835833 38.90755467 73.76061612 84.54401686Stochastic 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 63.17332371 64.25203432 0 81.20122238 74.78828948
## 2 0 0 64.25203432 65.21982323 0 74.78828948 75.95224649
## 3 0 0 65.21982323 66.28305836 0 75.95224649 76.82903555
## 4 0 0 66.28305836 67.25979431 0 76.82903555 77.59126462
## 5 0 0 67.25979431 68.24780096 0 77.59126462 73.13709182
## 6 0 0 68.24780096 69.29910890 0 73.13709182 68.67616157In 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 57.64869006 58.56494888 1 79.73098639 78.05253305
## 2 0 0 58.56494888 59.47474439 1 78.05253305 80.93624524
## 3 0 0 59.47474439 60.52790416 1 80.93624524 80.33744347
## 4 0 0 60.52790416 61.58339471 1 80.33744347 83.47784620
## 5 0 0 61.58339471 62.62502567 1 83.47784620 83.84680540
## 6 0 0 62.62502567 63.54213952 1 83.84680540 79.30699126#Parameters estimation:
pars <- spm_pobs(x=data)## Parameter QL achieved lower/upper bound.
## 2.75e-08
## Parameter mu0H achieved lower/upper bound.
## 7.2e-06pars## $aH
## [,1]
## [1,] -0.0507758089
##
## $aL
## [,1]
## [1,] -0.004669682719
##
## $f1H
## [,1]
## [1,] 54.6663515
##
## $f1L
## [,1]
## [1,] 79.49806084
##
## $QH
## [,1]
## [1,] 2.193619395e-08
##
## $QL
## [,1]
## [1,] 2.75e-08
##
## $fH
## [,1]
## [1,] 54.54986408
##
## $fL
## [,1]
## [1,] 73.38983074
##
## $bH
## [,1]
## [1,] 3.635395175
##
## $bL
## [,1]
## [1,] 5.18425761
##
## $mu0H
## [1] 7.2e-06
##
## $mu0L
## [1] 9.022154094e-06
##
## $thetaH
## [1] 0.07221750705
##
## $thetaL
## [1] 0.09020004911
##
## $p
## [1] 0.2392009353
##
## $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 92.66108468 93.69565832 0 80.34160166 86.67819150
## 2 0 0 93.69565832 94.79162208 0 86.67819150 86.47413785
## 3 0 0 94.79162208 95.81302883 0 86.47413785 82.07365994
## 4 0 0 95.81302883 96.72653687 0 82.07365994 76.52074214
## 5 0 0 96.72653687 97.74321552 0 76.52074214 80.21985675
## 6 0 0 97.74321552 98.75837721 0 80.21985675 77.72977970data.nongenetic <- sim_pobs(N=10, mode='unobserved')
head(data.nongenetic)## id xi t1 t2 y1 y1.next
## 1 0 0 35.30842026 36.34793744 78.29696947 72.51570672
## 2 0 0 36.34793744 37.32976760 72.51570672 71.41804763
## 3 0 0 37.32976760 38.31125489 71.41804763 70.57451791
## 4 0 0 38.31125489 39.23408802 70.57451791 66.93686876
## 5 0 0 39.23408802 40.20640220 66.93686876 68.47615941
## 6 0 0 40.20640220 41.24148660 68.47615941 73.81358886#Parameters estimation:
pars <- spm_pobs(x=data.genetic, y = data.nongenetic, mode='combined')## Parameter fH achieved lower/upper bound.
## 66pars## $aH
## [,1]
## [1,] -0.01614742708
##
## $aL
## [,1]
## [1,] -0.01063397354
##
## $f1H
## [,1]
## [1,] 65.98974079
##
## $f1L
## [,1]
## [1,] 74.24100358
##
## $QH
## [,1]
## [1,] 2.165875725e-08
##
## $QL
## [,1]
## [1,] 2.687057485e-08
##
## $fH
## [,1]
## [1,] 66
##
## $fL
## [,1]
## [1,] 87.37895703
##
## $bH
## [,1]
## [1,] 4.392504816
##
## $bL
## [,1]
## [1,] 5.198136774
##
## $mu0H
## [1] 8.614204916e-06
##
## $mu0L
## [1] 9.035658957e-06
##
## $thetaH
## [1] 0.07262852807
##
## $thetaL
## [1] 0.09002842496
##
## $p
## [1] 0.2749116176
##
## $limit
## [1] TRUE
##
## 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.