tpopt: Calculation of optimal discriminating design

Description

Calculates an approximation $\xi^{**}$ of the $T_{\mathrm{P}}$-optimal design $\xi^*$ for discrimination between a given list of models ${\eta_i(x,\theta_i),\; i = 1,\dots,\nu}$. This procedure is based on the algorithms developed by Holger Dette, Viatcheslav B. Melas and Roman Guchenko in [7]. $T_{\mathrm{P}}$-optimal design is a probability measure, which maximizes the functional $$T_{\mathrm{P}}(\xi) = \sum_{i,j=1}^{\nu} p_{i,j} \inf_{\theta_{i,j} \in \Theta_j} \int_{\mathcal{X}} \Big[ \eta_i(x,\overline{\theta}_{i}) - \eta_j(x,\theta_{i,j}) \Big]^2 \xi(dx),$$ where $\xi$ is an arbitrary design on $\mathcal{X}$ (it is presumed here, that $\mathcal{X}$ is an interval from $\mathbf{R}$), $\mathrm{P} = { p_{i,j} }_{i,j = 1}^{\nu}$ is a table of non-negative weights with zeros on the diagonal (comparison table) and $\overline{\theta}_{i}$ are predefined fixed parameters. It was also shown in [7] that calculation of Bayesian $T_{\mathrm{P}}$-optimal design, which maximizes more complicated criterion $$T_{\mathrm{P}}^{\mathrm{B}}(\xi) = \sum_{i,j=1}^{\nu} p_{i,j} \int_{\Theta_i} \inf_{\theta_{i,j} \in \Theta_j} \int_{\mathcal{X}} \Big[ \eta_i(x,\lambda_i) - \eta_j(x,\theta_{i,j}) \Big]^2 \xi(dx) \mathcal{P}_i(d \lambda_i),$$ can be reduced to calculation of ordinary $T_{\mathrm{P}}$-optimal design, when distributions $\mathcal{P}_i$ are discrete. That is why in this case the current function is also suitable for calculation of Bayesian designs.

Usage

tpopt(  x, 
        w = rep(1, length(x)) / length(x), 
        eta, 
        theta.fix, 
        theta.var = NULL, 
        p, 
        x.lb = min(x), 
        x.rb = max(x), 
        opt = list())

Arguments

a numeric vector specifying support points from $\mathcal{X}$ for initial design. Current algorithm operates under the assumption, that $\mathcal{X}$ is an interval from $\mathbf{R}$

a numeric vector specifying weights for initial design. This vector should have the same length as vector of support points. Furthermore, the weights of the design should sum to 1. If this vector is not specified, then the weights are presumed to be equal

eta

a list of models between which proposed optimization should be performed. Every function from this list should be defined in the form of $\eta_i(x,\theta_i)$, where $x$ is one dimensional variable from $\mathcal{X}$ and $\theta_i$ is a vector of correspon

theta.fix

a list of fixed model parameters $\overline{\theta}_{i}$ from the functional $T_{\mathrm{P}}$. This list should have the same length as the list of models.

theta.var

an array with two dimensions specifying initial values for parameter vectors $\theta_{i,j}$. The default value here is NULL, which means that initial guess is calculated automatically.

a $\nu\times\nu$ square table (R-matrix) containing non-negative weights for comparison. The diagonal values of this table should all be zeros. If one want to include comparison of the $i$'th model with fixed parameters against $j$'th model with variable

x.lb

a left bound for support points. If it is not specified, then minimal value from input vector $x$ is taken.

x.rb

a right bound for support points. If it is not specified, then maximal value from input vector $x$ is taken.

opt

a list of options containing such named fields: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Value

Object of class ``tpopt'' which contains the following fields: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Details

Firstly, lets define $$\Psi(x,\xi) = \sum_{i,j=1}^{\nu} p_{i,j} \Big[ \eta_i(x,\overline{\theta}_{i}) - \eta_j(x,\widehat{\theta}_{i,j}) \Big]^2, \widehat{\theta}_{i,j} = \arg\inf_{\theta_{i,j} \in \Theta_j} \int_{\mathcal{X}} \Big[ \eta_i(x,\overline{\theta}_i) - \eta_j(x, \theta_{i,j}) \Big]^2 \xi(dx).$$ The simplified algorithm schema is as follows: Let $\xi_s$ denotes the design obtained on the s'th iteration of the algorithm. Then [object Object],[object Object]

References

[1] Atkinson A.C., Fedorov V.V. (1975) The design of experiments for discriminating between two rival models. Biometrika, vol. 62(1), pp. 57--70. [2] Atkinson A.C., Fedorov V.V. (1975) Optimal design: Experiments for discriminating between several models. Biometrika, vol. 62(2), pp. 289--303. [3] Dette H., Pepelyshev A. (2008) Efficient experimental designs for sigmoidal growth models. Journal of statistical planning and inference, vol. 138, pp. 2--17. [4] Dette H., Melas V.B., Shpilev P. (2013) Robust T-optimal discriminating designs. Annals of Statistics, vol. 41(4), pp. 1693--1715. [5] Braess D., Dette H. (2013) Optimal discriminating designs for several competing regression models. Annals of Statistics, vol. 41(2), pp. 897--922. [6] Braess D., Dette H. (2013) Supplement to ``Optimal discriminating designs for several competing regression models''. Annals of Statistics, online supplementary material. [7] Dette H., Melas V.B., Guchenko R. (2014) Bayesian T-optimal discriminating designs. http://arxiv.org/abs/1412.2548{ArXiv link}.

Examples

Run this code

### Auxiliary libraries for examples
library(mvtnorm)
### EMAX vs MM
#List of models
eta.1 <- function(x, theta.1) 
    theta.1[1] + theta.1[2] * x / (x + theta.1[3])

eta.2 <- function(x, theta.2) 
    theta.2[1] * x / (x + theta.2[2])

eta <- list(eta.1, eta.2)

#List of fixed parameters
theta.1 <- c(1, 1, 1)
theta.2 <- c(1, 1)
theta.fix <- list(theta.1, theta.2)

#Comparison table
p <- matrix(
    c(
        0, 1,
        0, 0
    ), c(2, 2), byrow = TRUE)

#Design estimation
res <- tpopt(x = c(1.2, 1.5, 1.7), eta = eta, theta.fix = theta.fix, p = p, 
    x.lb = 1, x.rb = 2)

plot(res)
summary(res)

### Sigmoidal second
#List of models
eta.1 <- function(x, theta.1)
    theta.1[1] / (1 + theta.1[2] * exp(-theta.1[3] * x)) ^ theta.1[4]
    
eta.2 <- function(x, theta.2)
    theta.2[1] / (1 + theta.2[2] * exp(-theta.2[3] * x))

eta <- list(eta.1, eta.2)

#List of fixed parameters
theta.1 <- c(2, 5, 1, 2)
theta.2 <- c(3, 5, 0.7)
theta.fix <- list(theta.1, theta.2)

#Comparison table
p <- matrix(
    c(  
        0, 1,
        0, 0 
    ), c(2, 2), byrow = TRUE)

#Design estimation
res <- tpopt(x = seq(0, 10), eta = eta, theta.fix = theta.fix, p = p)

plot(res)
summary(res)

### Sigmoidal first
#List of models
eta.1 <- function(x, theta.1)
    theta.1[1] - theta.1[2] * exp(-theta.1[3] * x ^ theta.1[4])

eta.2 <- function(x, theta.2)
    theta.2[1] - theta.2[2] * exp(-theta.2[3] * x)

eta <- list(eta.1, eta.2)

#List of fixed parameters
theta.1 <- c(2, 1, 0.8, 1.5)
theta.2 <- c(2, 1, 1)
theta.fix <- list(theta.1, theta.2)

#Comparision table
p <- matrix(
    c(
        0, 1,
        0, 0
    ), c(2, 2), byrow = TRUE)

#Design estimation
res <- tpopt(x = seq(0, 10), eta = eta, theta.fix = theta.fix, p = p)

plot(res)
summary(res)

### Sigmoidal first --- Bayes

#List of fixed parameters
sigma <- sqrt(0.3)
theta.1.sigma <- matrix(
    c(
        sigma^2, 0,
        0, sigma^2
    ), c(2, 2), byrow = TRUE)
grid <- expand.grid(
    theta.1[1],
    theta.1[2],
    seq(theta.1[3] - sigma, theta.1[3] + sigma, length.out = 5),
    seq(theta.1[4] - sigma, theta.1[4] + sigma, length.out = 5)
    )

eta <- c(replicate(length(grid[,1]), eta.1, simplify = FALSE), eta.2)

theta.fix <- list()
for(i in 1:length(grid[,1]))
    theta.fix[[length(theta.fix) + 1]] <- as.numeric(grid[i,])
theta.fix[[length(theta.fix) + 1]] <- theta.2

density.on.grid <- dmvnorm(grid[,3:4], mean = theta.1[3:4], sigma = theta.1.sigma)
density.on.grid <- density.on.grid / sum(density.on.grid)

#Comparison table
p <- rep(0,length(eta))
for(i in 1:length(grid[,1]))
    p <- rbind(p, c(rep(0,length(eta) - 1), density.on.grid[i]))
p <- rbind(p, rep(0,length(eta)))
p <- p[-1,]

res <- tpopt(x = seq(0, 10), eta = eta, theta.fix = theta.fix, p = p)

plot(res)
summary(res)

### Dose response study 
#List of models
eta.1 <- function(x, theta.1)
    theta.1[1] + theta.1[2] * x

eta.2 <- function(x, theta.2)
    theta.2[1] + theta.2[2] * x * (theta.2[3] - x)

eta.3 <- function(x, theta.3)
    theta.3[1] + theta.3[2] * x / (theta.3[3] + x)

eta.4 <- function(x, theta.4)
    theta.4[1] + theta.4[2] / (1 + exp((theta.4[3] - x) / theta.4[4]))

eta <- list(eta.1, eta.2, eta.3, eta.4)

#List of fixed parameters
theta.1 <- c(60, 0.56)
theta.2 <- c(60, 7/2250, 600)
theta.3 <- c(60, 294, 25)
theta.4 <- c(49.62, 290.51, 150, 45.51)

theta.fix <- list(theta.1, theta.2, theta.3, theta.4)

#Comparison table
p <- matrix(
    c(
        0, 0, 0, 0,
        1, 0, 0, 0,
        1, 1, 0, 0,
        1, 1, 1 ,0
    ), c(4, 4), byrow = TRUE)

#Design estimation
res <- tpopt(x = seq(0, 500, 100), eta = eta, theta.fix = theta.fix, p = p)

plot(res)
summary(res)

### Dose response study --- Bayes

#List of fixed parameters
sigma <- 37
theta.4.sigma <- matrix(
    c(
        sigma^2, 0, 0, 0,
        0, sigma^2, 0, 0,
        0, 0, sigma^2, 0,
        0, 0, 0, sigma^2
    ), c(4, 4), byrow = TRUE)
grid <- expand.grid(
    seq(theta.4[1] - sigma, theta.4[1] + sigma, length.out = 3),
    seq(theta.4[2] - sigma, theta.4[2] + sigma, length.out = 3),
    seq(theta.4[3] - sigma, theta.4[3] + sigma, length.out = 3),
    seq(theta.4[4] - sigma, theta.4[4] + sigma, length.out = 3)
    )

eta <- c(eta.1, eta.2, eta.3, replicate(length(grid[,1]), eta.4, simplify = FALSE))

theta.fix <- list(theta.1, theta.2, theta.3)
for(i in 1:length(grid[,1]))
    theta.fix[[length(theta.fix) + 1]] <- as.numeric(grid[i,])

density.on.grid <- dmvnorm(grid, mean = theta.4, sigma = theta.4.sigma)
density.on.grid <- density.on.grid / sum(density.on.grid)

#Comparison table
p <- rbind(
    rep(0, length(eta)), 
    c(1, rep(0, length(eta) - 1)), 
    c(1, 1, rep(0,length(eta) - 2))
    )
for(i in 1:length(grid[,1]))
    p <- rbind(p, c(rep(density.on.grid[i], 3), rep(0, length(eta) - 3)))

#Design estimation
res <- tpopt(x = seq(0, 500, 100), eta = eta, theta.fix = theta.fix, p = p)

plot(res)
summary(res)

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