MSOpt {multiDoE}R Documentation

Experimental setup

Description

The MSOpt function allows the user to define the structure of the experiment, the set of optimization criteria and the a priori model to be considered. The output is a list containing all information about the settings of the experiment. According to the declared criteria, the list also contains the basic matrices for their implementation, such as information matrix, matrix of moments and matrix of weights. The returned list is argument of the Score and MSSearch functions of the multiDoE package.

Usage

MSOpt(facts, units, levels, etas, criteria, model)

Arguments

facts

A list of vectors representing the distribution of factors across strata. Each item in the list represents a stratum and the first item is the highest stratum of the multi-stratum structure of the experiment. Within the vectors, experimental factors are indicated by progressive integer from 1 (the first factor of the highest stratum) to the total number of experimental factors (the last factor of the lowest stratum). Blocking factors are denoted by empty vectors.

units

A list whose i-th element is the number of experimental units within each unit at the previous stratum i-1. The first item in the list, n_1, represents the number of experimental units in the stratum 0, defined as the entire experiment (so that n_0 = 1).

levels

A vector containing the number of available levels for each experimental factor in facts (blocking factors are excluded). If all experimental factors share the number of levels one integer is sufficient.

etas

A list specifying the ratios of error variance between subsequent strata. It follows that length(etas) must be equal to length(facts)-1.

criteria

A list specifying the criteria to be optimized. It can contain any combination of:

  • “I" : I-optimality

  • “Id" : Id-optimality

  • “D" : D-optimality

  • “A" : Ds-optimality

  • “Ds" : A-optimality

  • “As" : As-optimality

More detailed information on the available criteria is given under Details.

model

A string which indicates the type of model, among “main", “interaction" and “quadratic".

Details

A little notation is introduced to show the criteria that can be used in the multi-objective approach of the multiDoE package.

For an experiment with N runs and s strata, with stratum i having n_i units within each unit at stratum i-1 and stratum 0 being defined as the entire experiment (n_0 = 1), the general form of the model can be written as:

y = X\beta + \sum\limits_{i = 1}^{s} Z_i\varepsilon_i

where y is an N-dimensional vector of responses (N = \prod_{j = 1}^{s}n_j), X is an N by p model matrix, \beta is a p-dimensional vector containing the p fixed model parameters, Z_i is an N by b_i indicator matrix of 0 and 1 for the units in stratum i (i.e. the (k,l)th element of Z_i is 1 if the kth run belongs to the lth block in stratum i and 0 otherwise) and b_i = \prod_{j = 1}^{i}n_j. Finally, the vector \varepsilon_i \sim N(0,\sigma_i^2I_{b_i}) is a b_i-dimensional vector containing the random effects, which are all uncorrelated. The variance components \sigma^{2}_{i} (i = 1, \dots, s) have to be estimated and this is usually done using the REML (REstricted Maximum Likelihood) method.

The best linear unbiased estimator for the parameter vector \beta is the generalized least square estimator:

\hat{\beta}_{GLS} = (X'V^{-1}X)^{-1}X'V^{-1}y

This estimator has variance-covariance matrix:

Var(\hat{\beta}_{GLS}) = \sigma^{2}(X'V^{-1}X)^{-1}

where V = \sum\limits_{i = 1}^{s}\eta_i Z_i'Zi, \eta_i = \frac{\sigma_i^{2}}{\sigma^{2}} and \sigma^{2} = \sigma^{2}_{s}.

Let M = (X' V^{-1} X) be the information matrix about \hat{\beta} and let \eta be the vector of the variance components.

Value

MSOpt returns a list containing the following components:

References

M. Borrotti and F. Sambo and K. Mylona and S. Gilmour. A multi-objective coordinate-exchange two-phase local search algorithm for multi-stratum experiments. Statistics & Computing, 2017.

S. G. Gilmour, J. M. Pardo, L. A. Trinca, K. Niranjan, D.S. Mottram. A split-plot response surface design for improving aroma retention in freeze dried coffee. In: Proceedings of the 6th. European conference on Food-Industry Statist, 2000.

Examples

## This example uses MSOpt to setup a split-plot design with
## 1 whole-plot factor and 4 subplot factors, which in the \code{facts}
## element appear numbered from 2 to 5.
## The experiment must be structured as follows: 6 whole plots and 5 subplots
## per whole plot, for a total of 30 runs.
## Each experimental factor has 3 different levels.

## To check the number of digits to be printed.
backup_options <- options()
options(digits = 10)

facts <- list(1, 2:5)
units <- list(6, 5)
levels <- 3
etas <- list(1)
criteria <- c('I', 'D', 'A')
model <- "quadratic"

msopt <- MSOpt(facts, units, levels, etas, criteria, model)

options(backup_options)


[Package multiDoE version 0.9.4 Index]