ldrichards: Locally D-optimal designs for Richards model In LDOD: Finding Locally D-optimal optimal designs for some nonlinear and generalized linear models.

Description

Finds Locally D-optimal designs for Richards regression model which is defined as E(y) = a/(1+b\exp(-λ*x))^h with Var(y) = σ^2, where a, b, λ, h and σ are unknown parameters.

Usage

 1 2 ldrichards(a, b, lambda, h, 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. lambda initial value for paremeter λ. h initial value for paremeter h. 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., Pepelyshev, A. (2008), Efficient Experimental Designs for Sigmoidal Growth Models, Statistical Planning and Inference, 138, 2-17.

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