LDOD (version 1.0)

cfderiv: Auto-constructing Frechet derivative of D-criterion based on general equivalence theorem

Description

Auto-constructs Frechet derivative of D-criterion at $M(\xi, \beta)$ and in direction $M(\xi_x, \beta)$ where $M$ is Fisher information matrix, $\beta$ is vector of parameters, $\xi$ is the interested design and $\xi_x$ is a unique design which has only a point $x$. The constructed Frechet derivative is an Rfunction with argument $x$.

Usage

cfderiv(ymean, yvar, param, points, weights)

Arguments

ymean

a character string, formula of $E(y)$ with specific satndard: characters b1, b2, b3, ...symbolize model parameters and x1, x2, x3, ...symbolize explanatory variables. See 'Exa

yvar

a character string, formula of $Var(y)$ with specific standard as ymean. See 'Details' and 'Examples'.

param

a vector of values of parameters which must correspond to b1, b2, b3, ...in ymean.

points

a vector of points which belong to design $\xi$ . See 'Details'.

weights

a vector of $\xi$ points weights. The sum of weights should be $1$; otherwise they will be normalized.

Value

• fderiv
  a function in which its argument is a vector $x$, an $m$-dimentional design point, and its output is the value of Frechet derivative at $M(\xi, \beta)$ and in direction $M(\xi_x, \beta)$.

Details

If response variables have the same constant variance, for example $\sigma^2$, then yvar must be $1$. Consider design $\xi$ with $n$ $m$-dimensional points. Then, the vector of $\xi$ points is $$(x_1, x_2, \ldots, x_i, \ldots, x_n),$$ where $x_i = (x_{i1}, x_{i2}, \ldots, x_{im})$. Hence the length of vector points is $mn$.

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, 849-879.7.

Examples

## 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"