title: “Stochastic Process Model for Analysis of Longitudinal and Time-to-Event Outcomes” author: “Ilya Y. Zhbannikov” date: “2016-12-12” output: rmarkdown::html_document vignette: >
%\VignetteIndexEntry{stpm} %\VignetteEngine{knitr::rmarkdown} usepackage[utf8]{inputenc}
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 32.89139
There 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 77.67273
## [2,] 1 0 31 32 77.67273 73.15690
## [3,] 1 0 32 33 73.15690 74.62197
## [4,] 1 0 33 34 74.62197 68.32631
## [5,] 1 0 34 35 68.32631 71.52888
## [6,] 1 0 35 36 71.52888 68.44558
In 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 32.32694 33.45811 80.18027 80.43571
## [2,] 0 0 33.45811 35.32964 80.43571 83.04833
## [3,] 0 0 35.32964 36.80049 83.04833 93.98842
## [4,] 0 0 36.80049 38.31599 93.98842 82.62247
## [5,] 0 0 38.31599 39.61414 82.62247 82.52908
## [6,] 0 0 39.61414 40.89936 82.52908 81.44614
The 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] 0.073
##
## $Ak2005$mu0
## [1] 0.0002857352805
##
## $Ak2005$b
## [1] -6.657235262e-06
##
## $Ak2005$Q
## [,1]
## [1,] 4.115682937e-08
##
## $Ak2005$u
## [1] 4.33341523
##
## $Ak2005$R
## [,1]
## [1,] 0.9460370877
##
## $Ak2005$Sigma
## [1] 5.001615813
##
##
## $Ya2007
## $Ya2007$a
## [,1]
## [1,] -0.05396291233
##
## $Ya2007$f1
## [,1]
## [1,] 80.30358337
##
## $Ya2007$Q
## [,1]
## [1,] 4.115682937e-08
##
## $Ya2007$f
## [,1]
## [1,] 80.87643489
##
## $Ya2007$b
## [,1]
## [1,] 5.001615813
##
## $Ya2007$mu0
## [,1]
## [1,] 1.652855337e-05
##
## $Ya2007$theta
## [1] 0.073
##
##
## 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 36.10163963 38.07038012 80.89662684 89.34557093
## [2,] 0 0 38.07038012 39.41284085 89.34557093 78.38595900
## [3,] 0 0 39.41284085 41.30376811 78.38595900 83.83576015
## [4,] 0 0 41.30376811 42.73212694 83.83576015 83.28455069
## [5,] 0 0 42.73212694 44.66163032 83.28455069 70.52319948
## [6,] 0 0 44.66163032 45.76958044 70.52319948 76.35419178
#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,] -0.05392672042
##
## $f1
## [,1]
## [1,] 81.6548864
##
## $Q
## [,1]
## [1,] 2.198599829e-08
##
## $f
## [,1]
## [1,] 87.998625
##
## $b
## [,1]
## [1,] 4.94928369
##
## $mu0
## [1] 2.19906691e-05
##
## $theta
## [1] 0.08587688842
##
## $status
## [1] 5
##
## $LogLik
## [1] -13660.24062
##
## $objective
## [1] 13660.24062
##
## $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 <- 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]]$a
## [1] -0.04158047269
##
## [[1]]$f1
## [1] 80.13810393
##
## [[1]]$Q
## [1] 1.980836744e-08
##
## [[1]]$f
## [1] 96.44415498
##
## [[1]]$b
## [1] 5.071693561
##
## [[1]]$mu0
## [1] 0.0008072807453
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -8295.438135
##
## [[1]]$objective
## [1] 8295.437621
##
## [[1]]$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=100, 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,] -0.1044492519
##
## $f1
## [,1]
## [1,] 80.03364219
##
## $Q
## [,1]
## [1,] 4.41179567e-06
##
## $f
## [,1]
## [1,] 119.6458285
##
## $b
## [,1]
## [1,] 1.013923425
##
## $mu0
## [1] 4.083620741e-05
##
## $theta
## [1] 0.1370560284
##
## $status
## [1] 5
##
## $LogLik
## [1] -6800.22092
##
## $objective
## [1] 6799.794669
##
## $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=100,
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,] -0.150856346702 0.001661573818
## [2,] 0.001971840251 -0.148663547174
##
## $f1
## [,1]
## [1,] 106.8058205
## [2,] 193.6376038
##
## $Q
## [,1] [,2]
## [1,] 1.310659714e-06 1.313518226e-07
## [2,] 1.313101575e-07 1.301185543e-06
##
## $f
## [,1]
## [1,] 108.5860771
## [2,] 211.8438959
##
## $b
## [,1]
## [1,] 1.094669999
## [2,] 1.897739657
##
## $mu0
## [1] 0.0001117494201
##
## $theta
## [1] 0.06647904233
##
## $status
## [1] 5
##
## $LogLik
## [1] 14291.18272
##
## $objective
## [1] -19290.18666
##
## $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] -0.06635529237
##
## [[1]]$f1
## [1] 79.03406417
##
## [[1]]$Q
## [1] 3.058138724e-08
##
## [[1]]$f
## [1] 105.2440569
##
## [[1]]$b
## [1] 5.159369644
##
## [[1]]$mu0
## [1] 2.66832555e-05
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -1697.659335
##
## [[1]]$objective
## [1] 1697.658416
##
## [[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]]$a
## [1] -0.05769819069
##
## [[1]]$f1
## [1] 80.28823344
##
## [[1]]$Q
## [1] 2.313582339e-08
##
## [[1]]$b
## [1] 60
##
## [[1]]$mu0
## [1] 3.75
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -4392.632811
##
## [[1]]$objective
## [1] 4392.632811
##
## [[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]]$a
## [1] -0.05427045765
##
## [[1]]$f1
## [1] 79.77237051
##
## [[1]]$Q
## [1] 3.001632522e-08
##
## [[1]]$b
## [1] 5.410960193
##
## [[1]]$mu0
## [1] 0.001297190599
##
## [[1]]$status
## [1] 5
##
## [[1]]$LogLik
## t2
## -1546.859746
##
## [[1]]$objective
## [1] 1546.859516
##
## [[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 84.74114423
## [2,] 1 0 31 32 84.74114423 80.86820531
## [3,] 1 0 32 33 80.86820531 84.03292402
## [4,] 1 0 33 34 84.03292402 79.81371424
## [5,] 1 0 34 35 79.81371424 84.10072758
## [6,] 1 0 35 36 84.10072758 81.30214116
The 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 31.47511530 32.67594431 81.03459455 84.01087838
## [2,] 0 0 32.67594431 34.05016265 84.01087838 82.07392080
## [3,] 0 0 34.05016265 35.57808316 82.07392080 76.76822173
## [4,] 0 0 35.57808316 37.53798147 76.76822173 81.75756804
## [5,] 0 0 37.53798147 38.84543722 81.75756804 76.12327313
## [6,] 0 0 38.84543722 39.91622931 76.12327313 75.41875042
Stochastic 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 39.26171116 40.34239024 0 78.67500184 74.12791015
## 2 0 0 40.34239024 41.39102424 0 74.12791015 67.12993324
## 3 0 0 41.39102424 42.33062314 0 67.12993324 61.37691805
## 4 0 0 42.33062314 43.31260522 0 61.37691805 57.91698972
## 5 0 0 43.31260522 44.34166935 0 57.91698972 57.28446863
## 6 0 0 44.34166935 45.36536747 0 57.28446863 50.32451129
In 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 68.25265811 69.18946907 0 79.56514122 80.51161049
## 2 0 0 69.18946907 70.16611261 0 80.51161049 85.74824838
## 3 0 0 70.16611261 71.08461371 0 85.74824838 83.45648640
## 4 0 0 71.08461371 72.01378825 0 83.45648640 83.10359438
## 5 0 0 72.01378825 73.05611209 0 83.10359438 86.58179962
## 6 0 0 73.05611209 74.11670421 0 86.58179962 84.71717523
#Parameters estimation:
pars <- spm_pobs(x=data)
pars
## $aH
## [,1]
## [1,] -0.05417412961
##
## $aL
## [,1]
## [1,] -0.009164836038
##
## $f1H
## [,1]
## [1,] 65.57376475
##
## $f1L
## [,1]
## [1,] 72.40791989
##
## $QH
## [,1]
## [1,] 1.822605408e-08
##
## $QL
## [,1]
## [1,] 2.646974576e-08
##
## $fH
## [,1]
## [1,] 55.33537797
##
## $fL
## [,1]
## [1,] 82.93168863
##
## $bH
## [,1]
## [1,] 4.007416033
##
## $bL
## [,1]
## [1,] 4.930243885
##
## $mu0H
## [1] 7.425669662e-06
##
## $mu0L
## [1] 9.073984215e-06
##
## $thetaH
## [1] 0.07272048227
##
## $thetaL
## [1] 0.09003043705
##
## $p
## [1] 0.2384296368
##
## $limit
## [1] FALSE
##
## attr(,"class")
## [1] "pobs.spm"
Here \textbf{H} and \textbf{L} represents parameters when \(Z\) = 1 (H) and 0 (L).
###Joint analysis of two datasets: first dataset with genetic and second dataset with non-genetic component
library(stpm)
data.genetic <- sim_pobs(N=10, mode='observed')
head(data.genetic)
## id xi t1 t2 Z y1 y1.next
## 1 0 0 41.46503559 42.38610105 0 79.59021026 78.14830103
## 2 0 0 42.38610105 43.47827185 0 78.14830103 77.83083679
## 3 0 0 43.47827185 44.48899124 0 77.83083679 70.93158529
## 4 0 0 44.48899124 45.48480193 0 70.93158529 72.09847307
## 5 0 0 45.48480193 46.46117736 0 72.09847307 70.29964304
## 6 0 0 46.46117736 47.54258204 0 70.29964304 66.38264433
data.nongenetic <- sim_pobs(N=50, mode='unobserved')
head(data.nongenetic)
## id xi t1 t2 y1 y1.next
## 1 0 0 76.85700432 77.81526649 79.31219431 73.79383497
## 2 0 0 77.81526649 78.91301679 73.79383497 73.68143663
## 3 0 0 78.91301679 79.93980558 73.68143663 74.40748434
## 4 0 0 79.93980558 81.00496575 74.40748434 74.73499637
## 5 0 0 81.00496575 82.02716004 74.73499637 74.53775670
## 6 0 0 82.02716004 82.94169391 74.53775670 66.83216177
#Parameters estimation:
pars <- spm_pobs(x=data.genetic, y = data.nongenetic, mode='combined')
## Parameter mu0L achieved lower/upper bound.
## 9e-06
pars
## $aH
## [,1]
## [1,] -0.05192291143
##
## $aL
## [,1]
## [1,] -0.01024625962
##
## $f1H
## [,1]
## [1,] 62.16918344
##
## $f1L
## [,1]
## [1,] 76.32588519
##
## $QH
## [,1]
## [1,] 2.058443562e-08
##
## $QL
## [,1]
## [1,] 2.252926471e-08
##
## $fH
## [,1]
## [1,] 64.9824248
##
## $fL
## [,1]
## [1,] 76.15573939
##
## $bH
## [,1]
## [1,] 4.281274764
##
## $bL
## [,1]
## [1,] 4.93461931
##
## $mu0H
## [1] 8.144217926e-06
##
## $mu0L
## [1] 9e-06
##
## $thetaH
## [1] 0.07230152972
##
## $thetaL
## [1] 0.09004305572
##
## $p
## [1] 0.274548463
##
## $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.