It plots the sensitivity (derivative) function of the robust criterion at a given approximate (continuous) design and also calculates its efficiency lower bound (ELB) with respect to the optimality criterion. For an approximate (continuous) design, when the design space is one or two-dimensional, the user can visually verify the optimality of the design by observing the sensitivity plot. Furthermore, the approximity of the design to the optimal design can be measured by the ELB without knowing the latter. See, for more details, Masoudi et al. (2017).
sensrobust(formula, predvars, parvars, family = gaussian(), x, w, lx, ux,
prob, parset, fimfunc = NULL, sens.minimax.control = list(),
calculate_criterion = TRUE, plot_3d = c("lattice", "rgl"),
plot_sens = TRUE, npar = dim(parset)[2], silent = FALSE)A vector of characters. Denotes the predictors in the formula.
A vector of characters. Denotes the unknown parameters in the formula.
A description of the response distribution and the link function to be used in the model.
This can be a family function, a call to a family function or a character string naming the family.
Every family function has a link argument allowing to specify the link function to be applied on the response variable.
If not specified, default links are used. For details see family.
By default, a linear gaussian model gaussian() is applied.
Vector of the design (support) points. See 'Details' of sensminimax for models with more than one predictors.
Vector of the corresponding design weights for x.
Vector of lower bounds for the predictors. Should be in the same order as predvars.
Vector of upper bounds for the predictors. Should be in the same order as predvars.
A vector of the probability measure \(\pi\) associated with each row of parset.
A matrix that provides the vector of initial estimates for the model parameters, i.e. support of \(\pi\).
Every row is one vector (nrow(parset) == length(prob)). See 'Details'.
A function. Returns the FIM as a matrix. Required when formula is missing. See 'Details' of minimax.
Control parameters to verify the general equivalence theorem. For details, see the function sens.minimax.control.
Calculate the optimality criterion? See 'Details' of sensminimax.
Which package should be used to plot the sensitivity (derivative) function for models with two predictors.
Either "rgl" or "lattice" (default).
Plot the sensitivity (derivative) function? Defaults to TRUE.
Number of model parameters. Used when fimfunc is given instead of formula to specify the number of model parameters.
If not given, the sensitivity plot may be shifted below the y-axis.
When NULL, it is set to length(inipars).
Do not print anything? Defaults to FALSE.
an object of class sensminimax that is a list with the following elements:
typeArgument type that is required for print methods.
optimaA matrix that stores all the local optima over the parameter space.
The cost (criterion) values are stored in a column named Criterion_Value.
The last column (Answering_Set)
shows if the optimum belongs to the answering set (1) or not (0). See 'Details' of sens.minimax.control.
Only applicable for minimax or standardized maximin designs.
muProbability measure on the answering set.
Corresponds to the rows of optima for which the associated row in column Answering_Set is equal to 1.
Only applicable for minimax or standardized maximin designs.
max_derivGlobal maximum of the sensitivity (derivative) function (\(\epsilon\) in 'Details').
ELBD-efficiency lower bound. Can not be larger than 1. If negative, see 'Note' in sensminimax or sens.minimax.control.
merge_tolMerging tolerance to create the answering set from the set of all local optima. See 'Details' in sens.minimax.control.
Only applicable for minimax or standardized maximin designs.
crtvalCriterion value. Compare it with the column Crtiterion_Value in optima for minimax and standardized maximin designs.
timeUsed CPU time (rough approximation).
Let \(\Theta\) be the set initial estimates for the model parameters and \(\pi\) be a probability measure having support in \(\Theta\). A design \(\xi^*\) is robust with respect to \(\pi\) if the following inequality holds for all \(\boldsymbol{x} \in \chi\): $$c(\boldsymbol{x}, \pi, \xi^*) = \int_{\pi} tr M^{-1}(\xi^*, \theta)I(\boldsymbol{x}, \theta)\pi(\theta) d(\theta)-p \leq 0,$$ with equality at all support points of \(\xi^*\). Here, \(p\) is the number of model parameters.
ELB is a measure of proximity of a design to the optimal design without knowing the latter.
Given a design, let \(\epsilon\) be the global maximum
of the sensitivity (derivative) function over \(x \in \chi\).
ELB is given by $$ELB = p/(p + \epsilon),$$
where \(p\) is the number of model parameters. Obviously,
calculating ELB requires finding \(\epsilon\) and
another optimization problem to be solved.
The tuning parameters of this optimization can be regulated via the argument sens.minimax.control.
# NOT RUN {
# Verifying a robust design for the two-parameter logistic model
sensrobust(formula = ~1/(1 + exp(-b *(x - a))),
predvars = c("x"),
parvars = c("a", "b"),
family = binomial(),
prob = rep(1/4, 4),
parset = matrix(c(0.5, 1.5, 0.5, 1.5, 4.0, 4.0, 5.0, 5.0), 4, 2),
x = c(0.260, 1, 1.739), w = c(0.275, 0.449, 0.275),
lx = -5, ux = 5)
# }
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