Description Usage Arguments Details Value Note Author(s) References Examples

Auto-constructs Frechet derivative of D-criterion at *M(ξ, β)* and in direction *M(ξ_x, β)* where *M* is Fisher information matrix, *β* is vector of parameters, *ξ* is the interested design and *ξ_x* is a unique design which has only a point *x*. The constructed Frechet derivative is an **R** function with argument *x*.

1 |

`ymean` |
a character string,
formula of |

`yvar` |
a character string, formula of |

`param` |
a vector of values of parameters which must correspond to |

`points` |
a vector of points which belong to design |

`weights` |
a vector of |

If response variables have the same constant variance, for example *σ^2*, then `yvar`

must be *1*.

Consider design *ξ* with *n* *m*-dimensional points. Then, the vector of *ξ* points is

*(x_1, x_2, …, x_i, …, x_n),*

where *x_i = (x_{i1}, x_{i2}, …, x_{im})*. Hence the length of vector points is *mn*.

`fderiv` |
a function in which its argument is a vector |

A design *ξ* is D-optimal if and only if Frechet derivative at *M(ξ, β)* and in direction *M(ξ_x, β)*is greater than or equal to *0* on the design space. The equality must be achieved just at *ξ* points. Here, *x* is an arbitrary point on design space.

This function is applicable for models that can be written as *E(Y_i) = f(x_i,β)*
where *y_i* is the *ith* response variable, *x_i* is the observation vector of the *ith* explanatory variables, *β* is the vector of parameters and *f* is a continuous and differentiable function with respect to *β*.
In addition, response variables must be independent with distributions that belong to the Natural exponential family. Logistic,Poisson, Negative Binomial, Exponential, Richards, Weibull, Log-linear, Inverse Quadratic and Michaelis-Menten are examples of these models.

Ehsan Masoudi, Majid Sarmad and Hooshang Talebi

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, 849-879.7.

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## Logistic dose response model:
ymean <- "(1/(exp(-b2 * (x1 - b1)) + 1))"
yvar <- "(1/(exp(-b2 * (x1 - b1)) + 1))*(1 - (1/(exp(-b2 * (x1 - b1)) + 1)))"
func <- cfderiv(ymean, yvar, param = c(.9, .8), points = c(-1.029256, 2.829256),
weights = c(.5, .5))
## plot func on the design interval to verify the optimality of the given design
x <- seq(-5, 5, by = .1)
plot(x, -func(x), type = "l")
## Inverse Quadratic model
ymean <- "x1/(b1 + b2 * x1 + b3 * (x1)^2)"
yvar <- "1"
func <- cfderiv(ymean, yvar, param = c(17, 15, 9), points = c(0.33, 1.37, 5.62),
weights = rep(.33, 3))
## plot func on the design interval to verify the optimality of the given design
x <- seq(0, 15, by = .1)
plot(x, -func(x), type = "l")
#####################################################################
## In the following, ymean and yvar for some famous models are given:
## Inverse Quadratic model (another form):
ymean <- "(b1 * x1)/(b2 + x1 + b3 * (x1)^2)"
yvar <- "1"
## Logistic dose response model:
ymean <- "(1/(exp(-b2 * (x1 - b1)) + 1))"
yvar <- "(1/(exp(-b2 * (x1 - b1)) + 1)) * (1 - (1/(exp(-b2 * (x1 - b1)) + 1)))"
## Logistic model:
ymean <- "1/(exp(-b1 - b2 * x1) + 1)"
yvar <- "(1/(exp(-b1 - b2 * x1) + 1)) * (1 - (1/(exp(-b1 - b2 * x1) + 1)))"
## Poisson model:
ymean <- yvar <- "exp(b1 + b2 * x1)"
## Poisson dose response model:
ymean <- yvar <- "b1 * exp(-b2 * x1)"
## Weibull model:
ymean <- "b1 - b2 * exp(-b3 * x1^b4)"
yvar <- "1"
## Richards model:
ymean <- "b1/(1 + b2 * exp(-b3 * x1))^b4"
yvar <- "1"
## Michaelis-Menten model:
ymean <- "(b1 * x1)/(1 + b2 * x1)"
yvar <- "1"
#
ymean <- "(b1 * x1)/(b2 + x1)"
yvar <= "1"
#
ymean <- "x1/(b1 + b2 * x1)"
yvar <- "1"
## log-linear model:
ymean <- "b1 + b2 * log(x1 + b3)"
yvar <- "1"
## Exponential model:
ymean <- "b1 + b2 * exp(x1/b3)"
yvar <- "1"
## Emax model:
ymean <- "b1 + (b2 * x1)/(x1 + b3)"
yvar <- "1"
## Negative binomial model Y ~ NB(E(Y), theta) where E(Y) = b1*exp(-b2*x1):
theta = 5
ymean <- "b1 * exp(-b2 * x1)"
yvar <- paste ("b1 * exp(-b2 * x1) * (1 + (1/", theta, ") * b1 * exp(-b2 * x1))" , sep = "")
## Linear regression model:
ymean <- "b1 + b2 * x1 + b3 * x2 + b4 * x1 * x2"
yvar = "1"
``` |

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