equivalence_multiple: Checking the optimality of a given design with respect to the multi-objective criterion for the 4-parameter logistic model.

Description

The equivalence theorem states that for a given vector of weights $\lambda = (\lambda_1, \lambda_2, \lambda_3)$, the design $\xi_\lambda$ is the multi-objective optimal design if and only if for all does $x$ in the dose range $\chi$ (design space) $$d(x, \xi_\lambda) \leq 0$$ with equality when $x$ is a dose level of design $\xi_\lambda$. See Eq. 6 of Hyun and Wong (2015) for the details.

Usage

equivalence_multiple(x, w, lx, ux, param, lambda, delta,
  maxeval_equivalence = 6000, plot_sensitivity = TRUE)

Arguments

a vector of design points. If the model has $n$ explanatory variables, let $x_{ij}$ be the $j$th component of the $$i$th design point. The argument x is $x = (x_{11}, x_{21},..., x_{k1},..., x_{1n}, x_{2n},... x_{kn})$. See "Examples" on how to set this argument when the design space does not have one dimension, e.g. is of two-dimension.

a vector of design weights For the locally D-optimal design,lp and up must be fixed to the same values, i.e. lp = up.

lower bound of the design space $\chi$

upper bound of the design space $\chi$

param

a vector of parameters $\bold{\theta} = (\theta_1, \theta_2, \theta_3, \theta_4)$.

lambda

the relative importance of each of the 3 criteria, i.e. $\lambda = (\lambda_1, \lambda_2, \lambda_3)$. Here $0 < \lambda_i < 1$ and the sum of $\lambda_1$, $\lambda_2$ and $\lambda_3$ must be one.

delta

predetermined meaningful value of the minimum effective dose MED.

maxeval_equivalence

maximum number of evaluations (maxeval) that will be passed to optimization function directL to find the maximum of the sensitivity function required for calculating ELB. See "Details" of equivalence.

plot_sensitivity

logical, if TRUE, the sensitivity function will be plotted.

Value

an object of class 'equivalence' that is a list contains:

max_deriv: maximum of the sensitivity function
ELB: Efficiency lower bound If it is negative, then the value of maxeval_equivalence should be increased to find the global maximum.
crtval: criterion value

Details

When $\lambda_2 = 1$, the function checks the equivalence theorem with respect to the c-optimality criterion for estimating ED50. When $\lambda_3 = 1$, it checks the equivalence theorem with respect to the c-optimality criterion for estimating MED. In both cases, due to the tolerance issue for computing the generalized inverse, the results may not be true. Therefore, this function should only be used for multiple-objective optimal design $\lambda_1 \neq 0$ and $\lambda_2 \neq 0.$

The tolerance for finding the general inverse is set to .Machine$double.xmin.

Examples

Run this code

# NOT RUN {
## verfying the design in Table 2 of Hyun and Wong (2015), first row, fisrt column.
Theta1 <- c(1.563, 1.790, 8.442, 0.137)
equivalence_multiple (x = c(log(.001), -5.21, -4.08, log(1000)),
                     w = c(.25, .25, .25, .25),
                     lx = log(.001), ux = log(1000),
                     param = Theta1,
                     lambda = c(1, 0, 0),
                     delta = -1)

# }