Locally D-optimal designs for Inverse Quadratic model

Description

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

Usage

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

Arguments

Details

For each form of Inverse Quadratic model, the parameters must satisfy specific conditions:

if form = 1

a,b,c>0, 2√(bc)>1,

if form = 2

a,c>0, |b|<√(ac),

for more details see Dette and Kiss (2009).

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:

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., Kiss, C., (2009), Optimal experimental designs for Inverse Quadratic Regression models, Statistica Sinica, 19, 1567-1586.

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

See Also

cfisher, cfderiv and eff.

Examples

ldiq(a = 17 , b = 15, c = 9, form = 1, lb = 0, ub = 15) 
# $points: 0.4141466 1.2909896 4.0242083

## D-effecincy computation
ldiq(a = 17 , b = 15, c = 9, form = 2, lb = 0, ub = 15, user.points = c(10,2,4),
user.weights = c(.33, .33, .33)) # $user.eff: 0.18099

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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> 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/ldiq.Rd_%03d_medium.png", width=480, height=480)
> ### Name: ldiq
> ### Title: Locally D-optimal designs for Inverse Quadratic model
> ### Aliases: ldiq
> ### Keywords: optimal design Invesre Quadratic equivalence theorem
> 
> ### ** Examples
> 
> ldiq(a = 17 , b = 15, c = 9, form = 1, lb = 0, ub = 15) 

Iter: 1 fn: 31.6295	 Pars:  1.29106 4.02466 0.41416
Iter: 2 fn: 31.6295	 Pars:  1.29100 4.02429 0.41415
solnp--> Completed in 2 iterations
$points
[1] 0.4141496 1.2910030 4.0242887

$weights
[1] 0.3333333 0.3333333 0.3333333

$det.value
[1] 1.834341e-14

> # $points: 0.4141466 1.2909896 4.0242083
> 
> ## D-effecincy computation
> ldiq(a = 17 , b = 15, c = 9, form = 2, lb = 0, ub = 15, user.points = c(10,2,4),
+ user.weights = c(.33, .33, .33)) # $user.eff: 0.18099

Iter: 1 fn: 47.4638	 Pars:  1.37432 5.62444 0.33582
Iter: 2 fn: 47.4638	 Pars:  1.37437 5.62470 0.33582
solnp--> Completed in 2 iterations
$points
[1] 0.3358194 1.3743703 5.6247017

$weights
[1] 0.3333333 0.3333333 0.3333333

$det.value
[1] 2.436331e-21

$user.eff
[1] 0.18099

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