LDOD (version 1.0)

ldiq: 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 $Var(y) = \sigma^2$, where $a$, $b$, $c$ and $\sigma$ 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

a

initial value for paremeter $a$, see 'Details'.

b

initial value for parameter $b$, see 'Details'.

c

initial value for parameter $c$, see 'Details'.

form

must be 1 or 2. If form = 1, then $E(y)=ax/(b+x+cx^2)$; if form = 2, then $E(y)=x/(a+bx+cx^2)$.

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.

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.

Details

For each form of Inverse Quadratic model, the parameters must satisfy specific conditions: if form = 1 $$a,b,c>0, 2\sqrt(bc)>1,$$ if form = 2 $$a,c>0, |b|<\sqrt(ac),$$ 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.

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