Chiara Tommasi

Cross-validation methods in principal component analysis: A comparison

In principal component analysis (PCA), it is crucial to know how many principal components (PCs) should be retained in order to account for most of the data variability. A class of “objective” rules for finding this quantity is the class of cross-validation (CV) methods. In this work we compare three CV techniques showing how the performance of these methods depends on the covariance matrix structure. Finally we propose a rule for the choice of the “best” CV method and give an application to real data.

Bayesian optimum designs for discriminating between models with any distribution

The Bayesian KL-optimality criterion is useful for discriminating between any two statistical models in the presence of prior information. If the rival models are not nested then, depending on which model is true, two different Kullback-Leibler distances may be defined. The Bayesian KL-optimality criterion is a convex combination of the expected values of these two possible Kullback-Leibler distances between the competing models. These expectations are taken over the prior distributions of the parameters and the weights of the convex combination are given by the prior probabilities of the models. Concavity of the Bayesian KL-optimality criterion is proved, thus classical results of Optimal Design Theory can be applied. A standardized version of the proposed criterion is also given in order to take into account possible different magnitudes of the two Kullback-Leibler distances. Some illustrative examples are provided.

Optimal Designs for Discriminating among Several Non-Normal Models

Typically T-optimality is used to discriminate among several models with Normal errors. In order to discriminate between two non-Normal models, a criterion based on the Kullback-Liebler distance has been proposed, the so called KL-criterion. In this paper, a generalization of the KL-criterion is proposed to deal with discrimination among several non-Normal models. An example where three logistic regression models are compared is provided.

Optimal estimation for finite population mean in two-phase sampling

Using two-phase sampling scheme, we propose a general class of estimators for finite population mean. This class depends on the sample means and variances of two auxiliary variables. The minimum variance bound for any estimator in the class is provided (up to terms of ordern−1). It is also proved that there exists at least a chain regression type estimator which reaches this minimum. Finally, it is shown that other proposed estimators can reach the minimum variance bound, i.e. the optimal estimator is not unique.

Integral approximations for computing optimum designs in random effects logistic regression models

In the context of nonlinear models, the analytical expression of the Fisher information matrix is essential to compute optimum designs. The Fisher information matrix of the random effects logistic regression model is proved to be equivalent to the information matrix of the linearized model, which depends on some integrals. Some algebraic approximations for these integrals are proposed, which are consistent with numerical integral approximations but much faster to be evaluated. Therefore, these algebraic integral approximations are very useful from a computational point of view. Locally D -, A -, c -optimum designs and the optimum design to estimate a percentile are computed for the univariate logistic regression model with Gaussian random effects. Since locally optimum designs depend on a chosen nominal value for the parameter vector, a Bayesian D -optimum design is also computed. In order to find Bayesian optimum designs it is essential to apply the proposed integral approximations, because the use of numerical approximations makes the computation of these optimum designs very slow.

Max---min optimal discriminating designs for several statistical models

In the literature, different optimality criteria have been considered for model identification. Most of the proposals assume the normal distribution for the response variable and thus they provide optimality criteria for discriminating between regression models. In this paper, a max–min approach is followed to discriminate among competing statistical models (i.e., probability distribution families). More specifically, k different statistical models (plausible for the data) are embedded in a more general model, which includes them as particular cases. The proposed optimal design maximizes the minimum KL-efficiency to discriminate between each rival model and the extended one. An equivalence theorem is proved and an algorithm is derived from it, which is useful to compute max–min KL-efficiency designs. Finally, the algorithm is run on two illustrative examples.

Construction of MV- and SMV-optimum designs for binary response models

Some procedures to construct local MV- and SMV-optimum designs for binary response models are already available in the literature. Explicit formulae of locally optimum designs for all possible values of the parameters are usually of interest. In order to identify locally MV- and SMV-optimum designs for different regions of the parameters of the model some assumptions have to be made and some theoretical results must be proved. The background to producing a computer code for implementing these computations is given. Designs for some models used in practice are also provided as well as the efficiencies for estimating the parameters.

KL-optimum designs: theoretical properties and practical computation

In this paper some new properties and computational tools for finding KL-optimum designs are provided. KL-optimality is a general criterion useful to select the best experimental conditions to discriminate between statistical models. A KL-optimum design is obtained from a minimax optimization problem, which is defined on a infinite-dimensional space. In particular, continuity of the KL-optimality criterion is proved under mild conditions; as a consequence, the first-order algorithm converges to the set of KL-optimum designs for a large class of models. It is also shown that KL-optimum designs are invariant to any scale-position transformation. Some examples are given and discussed, together with some practical implications for numerical computation purposes.

PKL-Optimality Criterion in Copula Models for Efficacy-Toxicity Response

In recent years, there has been an increasing interest in developing dose finding methods incorporating both efficacy and toxicity outcomes. It is reasonable to assume that efficacy and toxicity are associated; therefore, we need to model their stochastic dependence. Copula functions are very useful tools to model different kinds of dependence with arbitrary marginal distributions. We consider a binary efficacy-toxicity response with logit marginal distributions. Since the dose which maximizes the probability of efficacy without toxicity (P-optimal dose) changes depending on different copula functions, we propose a criterion which is useful for choosing between the rival copula models but also protects patients against doses that are far away from the P-optimal dose. The performance of this compromise criterion (called PKL) is illustrated for different choices of the parameter values.

Minimax Designs for a Parametrization of Binary Response Models

Abstract Maximum variance (MV) and Standarized maximum variance (SMV) optimum designs for binary response models have already been studied in the literature. In this work, some theoretical results, useful for researchers working on a specific binary model, are given. The MV- and SMV-optimum designs are also compared with a classical real-life design and with other optimum designs. These comparisons are possible because of explicit formulas for local optimum designs and their efficiencies computed here.