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Last data update: 2014.03.03

R: Locally D-optimal designs for Michaelis-Menten model

ldmm	R Documentation

Locally D-optimal designs for Michaelis-Menten model

Description

Finds Locally D-optimal designs for Michaelis-Menten model which is defined as E(y) = (ax)/(1+bx) or E(y) = (ax)/(b+x) or E(y) = "x/(a+bx) with Var(y) = σ^2, where a, b and σ are unknown parameters.

Usage

ldmm(a, b, form = 1, lb, ub, user.points = NULL, user.weights = NULL, 
..., n.restarts = 1, n.sim = 1, tol = 1e-8, prec = 53, rseed = NULL)

Arguments

`a`	initial value for paremeter a.
`b`	initial value for paremeter b.
`form`	must be `1` or `2` or `3`. If `form = 1`, then E(y)=(ax)/(1+bx); if `form = 2`, then E(y)=(ax)/(b+x); if `form = 3` then E(y)=x/(a+bx).
`lb`	lower bound of design interval, must be greater than or equal to 0.
`ub`	upper bound of design interval.
`user.points`	(optional) vector of user design points which calculation of its D-efficiency is aimed. Each element of `user.points` must be within the design interval.
`user.weights`	(optional) vector of weights which its elements correspond to `user.points` elements. The sum of weights should be 1; otherwise they will be normalized.
`...`	(optional) additional parameters will be passed to function `curve`.
`prec`	(optional) a number, the maximal precision to be used for D-efficiency calculation, in bite. Must be at least 2 (default 53), see 'Details'.
`n.restarts`	(optional optimization parameter) number of solver restarts required in optimization process (default 1), see 'Details'.
`n.sim`	(optional optimization parameter) number of random parameters to generate for every restart of solver in optimization process (default 1), see 'Details'.
`tol`	(optional optimization parameter) relative tolerance on feasibility and optimality in optimization process (default 1e-8).
`rseed`	(optional optimization parameter) a seed to initiate the random number generator, else system time will be used.

Details

While D-efficiency is NaN, an increase in prec can be beneficial to achieve a numeric value, however, it can slow down the calculation speed.

Values of n.restarts and n.sim should be chosen according to the length of design interval.

Value

plot of derivative function, see 'Note'.

a list containing the following values:

`points`	obtained design points
`weights`	corresponding weights to the obtained design points
`det.value`	value of Fisher information matrix determinant at the obtained design
`user.eff`	D-efficeincy of user design, if `user.design` and `user.weights` are not `NULL`.

Note

To verify optimality of obtained design, derivate function (symmetry of Frechet derivative with respect to the x-axis) will be plotted on the design interval. Based on the equivalence theorem (Kiefer, 1974), a design is optimal if and only if its derivative function are equal or less than 0 on the design interval. The equality must be achieved just at the obtained points.

Author(s)

Ehsan Masoudi, Majid Sarmad and Hooshang Talebi

References

Masoudi, E., Sarmad, M. and Talebi, H. 2012, An Almost General Code in R to Find Optimal Design, In Proceedings of the 1st ISM International Statistical Conference 2012, 292-297.

Dette, H., Melas, V.B., Wong, W.K. (2005). Optimal design for goodness-of-fit of the MichaelisMenten enzyme kinetic function. Journal of the American Statistical Association, 100:1370-1381.

Kiefer, J. C. (1974), General equivalence theory for optimum designs (approximate theory). Ann. Statist., 2, 849-879.

Examples

ldmm(a = 1, b = 2, form = 1, lb = 0, ub =3) # $points: 0.375 3.000

ldmm(a = 1, b = 2, form = 2, lb = 0, ub =3) # $points: 0.8571428 3.0000000

ldmm(a = 1, b = 2, form = 3, lb = 0, ub =3) # $points: 0.375 3.000

## D-effecincy computation:
ldmm(a = 1, b = 2, form = 3, lb = 0, ub =3, user.points = c(.5, 3, 2),
user.weights = rep(.33, 3)) # $user.eff: 0.83174

Results


R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

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Type 'demo()' for some demos, 'help()' for on-line help, or
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> library(LDOD)
Loading required package: Rsolnp
Loading required package: Rmpfr
Loading required package: gmp

Attaching package: 'gmp'

The following objects are masked from 'package:base':

    %*%, apply, crossprod, matrix, tcrossprod

C code of R package 'Rmpfr': GMP using 64 bits per limb


Attaching package: 'Rmpfr'

The following objects are masked from 'package:stats':

    dbinom, dnorm, dpois, pnorm

The following objects are masked from 'package:base':

    cbind, pmax, pmin, rbind

> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/LDOD/ldmm.Rd_%03d_medium.png", width=480, height=480)
> ### Name: ldmm
> ### Title: Locally D-optimal designs for Michaelis-Menten model
> ### Aliases: ldmm
> ### Keywords: optimal design Michaelis-Menten equivalence theorem
> 
> ### ** Examples
> 
> ldmm(a = 1, b = 2, form = 1, lb = 0, ub =3) # $points: 0.375 3.000

Iter: 1 fn: 9.2427	 Pars:  0.37500
Iter: 2 fn: 9.2427	 Pars:  0.37500
solnp--> Completed in 2 iterations
$points
[1] 0.3749999 3.0000000

$weights
[1] 0.5 0.5

$det.value
[1] 9.681871e-05

> 
> ldmm(a = 1, b = 2, form = 2, lb = 0, ub =3) # $points: 0.8571428 3.0000000

Iter: 1 fn: 8.6101	 Pars:  0.85714
Iter: 2 fn: 8.6101	 Pars:  0.85714
solnp--> Completed in 2 iterations
$points
[1] 0.8571428 3.0000000

$weights
[1] 0.5 0.5

$det.value
[1] 0.00018225

> 
> ldmm(a = 1, b = 2, form = 3, lb = 0, ub =3) # $points: 0.375 3.000

Iter: 1 fn: 9.2427	 Pars:  0.37500
Iter: 2 fn: 9.2427	 Pars:  0.37500
solnp--> Completed in 2 iterations
$points
[1] 0.375 3.000

$weights
[1] 0.5 0.5

$det.value
[1] 9.681871e-05

> 
> ## D-effecincy computation:
> ldmm(a = 1, b = 2, form = 3, lb = 0, ub =3, user.points = c(.5, 3, 2),
+ user.weights = rep(.33, 3)) # $user.eff: 0.83174

Iter: 1 fn: 9.2427	 Pars:  0.37500
Iter: 2 fn: 9.2427	 Pars:  0.37500
solnp--> Completed in 2 iterations
$points
[1] 0.375 3.000

$weights
[1] 0.5 0.5

$det.value
[1] 9.681871e-05

$user.eff
[1] 0.83174

> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>