# ldpoisson: Locally D-optimal designs for Poisson model In LDOD: Finding Locally D-optimal optimal designs for some nonlinear and generalized linear models.

## 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) = a\exp(-bx) with Var(y) = E(y), respectively, where a and b are unknown parameters.

## Usage

 ```1 2``` ```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)=a\exp(-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.

## See Also

`cfisher`, `cfderiv` and `eff`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19``` ```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 ```

LDOD documentation built on May 2, 2019, 3:26 a.m.