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R: Locally D-optimal designs for Poisson model

ldpoisson

R Documentation

Locally D-optimal designs for Poisson model

Description

Finds Locally D-optimal designs for Poisson and Poisson dose-response models which are defined as E(y) = exp(a+bx) and E(y) = aexp(-bx) with Var(y) = E(y), respectively, where a and b are unknown parameters.

Usage

ldpoisson(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`. If `form = 1`, then E(y)=exp(a+bx); if `form = 2`, then E(y)=aexp(-bx).
`lb`	lower bound of design interval.
`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.

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

Examples

ldpoisson(a = .9, b = .8, form = 1, lb = -5, ub = 5) # $points: 2.5 5.0

ldpoisson(a = .9, b = .8, form = 2, lb = -5, ub = 5) # $points: -5.0 -2.5

## D-effecincy computation
ldpoisson(a = .9 , b = .8, lb = -5, ub = 5, user.points = c(3, 4), 
          user.weights = c(.5, .5)) # $user.eff: 0.32749

## usage of n.sim and n.restars
# Various responses for different values of rseed

ldpoisson(a = 22 , b = 16, lb = 9, ub = 12, rseed = 12) 
# $points: 9.208083 11.467731

ldpoisson(a = 22 , b = 16, lb = 9, ub = 12, rseed = 10) 
# $points: 10.05836 11.80563

ldpoisson(a = 22 , b = 16, lb = 9, ub = 12, n.restarts = 10, n.sim = 10) 
# (valid respnse) $points: 11.875, 12.000

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)

R is free software and comes with ABSOLUTELY NO WARRANTY.
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Type 'license()' or 'licence()' for distribution details.

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Type 'contributors()' for more information and
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Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> 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/ldpoisson.Rd_%03d_medium.png", width=480, height=480)
> ### Name: ldpoisson
> ### Title: Locally D-optimal designs for Poisson model
> ### Aliases: ldpoisson
> ### Keywords: optimal design Poisson equivalence theorem
> 
> ### ** Examples
> 
> ldpoisson(a = .9, b = .8, form = 1, lb = -5, ub = 5) # $points: 2.5 5.0

Iter: 1 fn: -8.2463	 Pars:  5.00000 2.50000
Iter: 2 fn: -8.2463	 Pars:  5.00000 2.50000
solnp--> Completed in 2 iterations
$points
[1] 2.5 5.0

$weights
[1] 0.5 0.5

$det.value
[1] 3813.441

> 
> ldpoisson(a = .9, b = .8, form = 2, lb = -5, ub = 5) # $points: -5.0 -2.5

Iter: 1 fn: -6.4463	 Pars:  -2.50000
Iter: 2 fn: -6.4463	 Pars:  -2.50000
solnp--> Completed in 2 iterations
$points
[1] -5.0 -2.5

$weights
[1] 0.5 0.5

$det.value
[1] 630.3575

> 
> ## D-effecincy computation
> ldpoisson(a = .9 , b = .8, lb = -5, ub = 5, user.points = c(3, 4), 
+           user.weights = c(.5, .5)) # $user.eff: 0.32749

Iter: 1 fn: -8.2463	 Pars:  5.00000 2.50000
Iter: 2 fn: -8.2463	 Pars:  5.00000 2.50000
solnp--> Completed in 2 iterations
$points
[1] 2.5 5.0

$weights
[1] 0.5 0.5

$det.value
[1] 3813.441

$user.eff
[1] 0.32749

> 
> ## usage of n.sim and n.restars
> # Various responses for different values of rseed
> 
> ldpoisson(a = 22 , b = 16, lb = 9, ub = 12, rseed = 12) 

Iter: 1 fn: -378.8429	 Pars:   9.20808 11.46773
Iter: 2 fn: -378.8429	 Pars:   9.20808 11.46773
solnp--> Completed in 2 iterations
$points
[1]  9.208083 11.467731

$weights
[1] 0.5 0.5

$det.value
[1] 3.383639e+164

> # $points: 9.208083 11.467731
> 
> ldpoisson(a = 22 , b = 16, lb = 9, ub = 12, rseed = 10) 

Iter: 1 fn: -393.5690	 Pars:  11.80563 10.05836
Iter: 2 fn: -393.5690	 Pars:  11.80563 10.05836
solnp--> Completed in 2 iterations
$points
[1] 10.05836 11.80563

$weights
[1] 0.5 0.5

$det.value
[1] 8.410956e+170

> # $points: 10.05836 11.80563
> 
> ldpoisson(a = 22 , b = 16, lb = 9, ub = 12, n.restarts = 10, n.sim = 10) 

Iter: 1 fn: -420.4548	 Pars:  11.87500 12.00000
Iter: 2 fn: -420.4548	 Pars:  11.87500 12.00000
solnp--> Completed in 2 iterations

Iter: 1 fn: -420.4548	 Pars:  11.87500 12.00000
Iter: 2 fn: -420.4548	 Pars:  11.87500 12.00000
solnp--> Completed in 2 iterations

Iter: 1 fn: -420.4548	 Pars:  11.87500 12.00000
Iter: 2 fn: -420.4548	 Pars:  11.87500 12.00000
solnp--> Completed in 2 iterations

Iter: 1 fn: -420.4548	 Pars:  12.00000 11.87500
Iter: 2 fn: -420.4548	 Pars:  12.00000 11.87500
solnp--> Completed in 2 iterations

Iter: 1 fn: -405.7007	 Pars:  10.77000 11.90690
solnp--> Completed in 1 iterations

Iter: 1 fn: -420.4548	 Pars:  12.00000 11.87500
Iter: 2 fn: -420.4548	 Pars:  12.00000 11.87500
solnp--> Completed in 2 iterations

Iter: 1 fn: -420.4548	 Pars:  11.87500 12.00000
Iter: 2 fn: -420.4548	 Pars:  11.87500 12.00000
solnp--> Completed in 2 iterations

Iter: 1 fn: -420.4548	 Pars:  11.87500 12.00000
Iter: 2 fn: -420.4548	 Pars:  11.87500 12.00000
solnp--> Completed in 2 iterations

Iter: 1 fn: -420.4548	 Pars:  12.00000 11.87500
Iter: 2 fn: -420.4548	 Pars:  12.00000 11.87500
solnp--> Completed in 2 iterations

Iter: 1 fn: -420.4548	 Pars:  12.00000 11.87500
Iter: 2 fn: -420.4548	 Pars:  12.00000 11.87500
solnp--> Completed in 2 iterations
$points
[1] 11.875 12.000

$weights
[1] 0.5 0.5

$det.value
[1] 3.992172e+182

> # (valid respnse) $points: 11.875, 12.000
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>