PSM (version 0.8-6)

PSM.estimate: Estimate population parameters

Description

Estimates population parameters in a linear or non-linear mixed effects model based on stochastic differential equations by use of maximum likelihood and the Kalman filter.

Usage

PSM.estimate(Model, Data, Par, CI = FALSE, trace = 0, control=NULL, fast=TRUE)

Arguments

Model

A list containing the following elements:

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Data

An unnamed list where each element contains data for one individual. Each element in Data is a list containing:

[object Object],[object Object],[object Object],[object Object]

Par

A list containing the following elements:

[object Object],[object Object]

CI

Boolean. If true, the program estimates 95% confidence intervals, standard deviation and correlation matrix for the parameter estimates based on the Hessian of the likelihood function. The Hessian is estimated by hessian in the

trace

Non-negative integer. If positive, tracing information on the progress of the optimization is produced. Higher values produces more tracing information.

control

A list of control parameters for the optimization of the likelihood function. The list has one required component, namely: [object Object] The remaining components in the list are given as the control argument for the chosen optimizer. See

fast

Boolean. Use compiled Fortran code for faster estimation.

Value

• A list containing the following elements:
• NegLogL
  Value of the negative log-likelihood function at optimum.
• THETA
  Population parameters at optimum
• CI
  95% confidence interval for the estimated parameters
• SD
  Standard deviation for the estimated parameters
• COR
  Correlation matrix for the estimated parameters
• sec
  Time for the estimation in seconds
• opt
  Raw output from optim

Numerical Jacobians of f and g

Automatic numerical approximations of the Jacobians of f and g can be used in PSM. In the folliwing, the name of the model object is assumed to be MyModel.

First define the functions MyModel$Functions$f and MyModel$Functions$g. When these are defined in MyModel the functions df and dg can be added to the model object by writing as below:

MyModel$Functions$df = function(x,u,time,phi) { jacobian(MyModel$Functions$f,x=x,u=u,time=time,phi=phi) } MyModel$Functions$dg = function(x,u,time,phi) { jacobian(MyModel$Functions$g,x=x,u=u,time=time,phi=phi) }

This way of defining df and dg forces a numerical evaluation of the Jacobians using the numDeriv package. It may be usefull in some cases, but it should be stressed that it will probably give at least a ten-fold increase in estimation times.

Details

The first stage model describing intra-individual variations is for linear models defined as

$$dx_t = (A(\phi_i)x_t + B(\phi_i)u_t)dt + \sigma(\phi_i) d\omega_t$$ $$y_{ij} = C(\phi_i)x_{ij} + D(\phi_i)u_{ij} + e_{ij}$$

and for non-linear models as

$$dx_t = f(x_t,u_t,t,\phi_i)dt + \sigma(u_t,t,\phi_i) d\omega_t$$ $$y_{ij} = g(x_{ij},u_{ij},t_{ij},\phi_i) + e_{ij}$$

where $e_{ij} \sim N(0,S(u_{ij},t_{ij},\phi_i))$ and $\omega_t$ is a standard Brownian motion.

The second stage model describing inter-individual variations is defined as:

$$\phi_i = h(\eta_i,\theta,Z_i)$$

where $\eta_i \sim N(0,\Omega)$, $\theta$ are the fixed effect parameters and $Z_i$ are covariates for individual i. In a model without random-effects the function $h$ is only used to include possible covariates in the model.

References

Please visit http://www.imm.dtu.dk/psm or refer to the main help page for PSM.

See Also

PSM, PSM.smooth, PSM.simulate, PSM.plot, PSM.template

Examples

cat("Examples are included in the package vignette.
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