Factor Analysis of Data Matrices: New Theoretical and Computational Aspects With Applications

The classical fitting problem in exploratory factor analysis (EFA) is to find estimates for the factor loadings matrix and the matrix of unique factor variances which give the best fit to the sample covariance or correlation matrix with respect to some goodness-of-fit criterion. Predicted factor scores can be obtained as a function of these estimates and the data. In this thesis, the EFA model is considered as a specific data matrix decomposition with fixed unknown matrix parameters. Fitting the EFA model directly to the data yields simultaneous solutions for both loadings and factor scores. Several new algorithms are introduced for the least squares and weighted least squares estimation of all EFA model unknowns. The numerical procedures are based on the singular value decomposition, facilitate the estimation of both common and unique factor scores, and work equally well when the number of variables exceeds the number of available observations. Like EFA, noisy independent component analysis (ICA) is a technique for reduction of the data dimensionality in which the interrelationships among the observed variables are explained in terms of a much smaller number of latent factors. The key difference between EFA and noisy ICA is that in the latter model the common factors are assumed to be both independent and non-normal. In contrast to EFA, there is no rotational indeterminacy in noisy ICA. In this thesis, noisy ICA is viewed as a method of factor rotation in EFA. Starting from an initial EFA solution, an orthogonal rotation matrix is sought that minimizes the dependence between the common factors. The idea of rotating the scores towards independence is also employed in three-mode factor analysis to analyze data sets having a three-way structure. The new theoretical and computational aspects contained in this thesis are illustrated by means of several examples with real and artificial data

On the Procrustean analogue of individual differences scaling (INDSCAL)

In this paper, individual differences scaling (INDSCAL) is revisited, considering INDSCAL as being embedded within a hierarchy of individual difference scaling models. We explore the members of this family, distinguishing (i) models, (ii) the role of identification and substantive constraints, (iii) criteria for fitting models and (iv) algorithms to optimise the criteria. Model formulations may be based either on data that are in the form of proximities or on configurational matrices. In its configurational version, individual difference scaling may be formulated as a form of generalized Procrustes analysis. Algorithms are introduced for fitting the new models. An application from sensory evaluation illustrates the performance of the methods and their solutions

Recent advances in methodology for clinical trials in small populations : the InSPiRe project

Where there are a limited number of patients, such as in a rare disease, clinical trials in these small populations present several challenges, including statistical issues. This led to an EU FP7 call for proposals in 2013. One of the three projects funded was the Innovative Methodology for Small Populations Research (InSPiRe) project. This paper summarizes the main results of the project, which was completed in 2017. The InSPiRe project has led to development of novel statistical methodology for clinical trials in small populations in four areas. We have explored new decision-making methods for small population clinical trials using a Bayesian decision-theoretic framework to compare costs with potential benefits, developed approaches for targeted treatment trials, enabling simultaneous identification of subgroups and confirmation of treatment effect for these patients, worked on early phase clinical trial design and on extrapolation from adult to pediatric studies, developing methods to enable use of pharmacokinetics and pharmacodynamics data, and also developed improved robust meta-analysis methods for a small number of trials to support the planning, analysis and interpretation of a trial as well as enabling extrapolation between patient groups. In addition to scientific publications, we have contributed to regulatory guidance and produced free software in order to facilitate implementation of the novel methods

Zig-zag exploratory factor analysis with more variables than observations

In this paper, the problem of fitting the exploratory factor analysis (EFA) model to data matrices with more variables than observations is reconsidered. A new algorithm named ‘zig-zag EFA’ is introduced for the simultaneous least squares estimation of all EFA model unknowns. As in principal component analysis, zig-zag EFA is based on the singular value decomposition of data matrices. Another advantage of the proposed computational routine is that it facilitates the estimation of both common and unique factor scores. Applications to both real and artificial data illustrate the algorithm and the EFA solutions

Exploratory factor and principal component analyses: some new aspects

Exploratory Factor Analysis (EFA) and Principal Component Analysis (PCA) are popular techniques for simplifying the presentation of, and investigating the structureof, an (n × p) data matrix. However, these fundamentally different techniques are frequently confused, and the differences between them are obscured, because they give similar results in some practical cases. We therefore investigate conditions under which they are expected to be close to each other, by considering EFA as a matrix decomposition so that it can be directly compared with the data matrix decomposition underlying PCA. Correspondingly, we propose an extended version of PCA, called the EFA-like PCA, which mimics the EFA matrix decomposition in the sense that they contain the same unknowns. We provide iterative algorithms for estimating the EFA-like PCA parameters, and derive conditions that have to be satisfied for the two techniques to give similar results. Throughout, we consider separately the cases n > p and p ≥ n. All derived algorithms and matrix conditions are illustrated on two data sets, one for each of these two cases

The relative frailty variance and shared frailty models

The relative frailty variance among survivors provides a readily interpretable measure of how the heterogeneity of a population, as represented by a frailty model, evolves over time. We discuss the properties of the relative frailty variance, show that it characterizes frailty distributions, and that, suitably rescaled, it may be used to compare patterns of dependence across models and data sets. In shared frailty models, the relative frailty variance is closely related to the cross-ratio function, which is estimable from bivariate survival data. We investigate the possible shapes of the relative frailty variance function for the purpose of model selection, and review available frailty distribution families in this context. We introduce several new families with contrasting properties, including simple but flexible time-varying frailty models. The benefits of the approach we propose are illustrated with two applications to bivariate current status data obtained from serological surveys

Correlated infections: quantifying individual heterogeneity in the spread of infectious diseases

In this paper, we propose new methods for investigating the extent of heterogeneity in effective contact rates relevant to the transmission of infections. These methods exploit the correlations between ages at infection for different infections within individuals. The methods are developed for serological surveys, which provide accessible individual data on several infections, and are applied to a wide range of infections. We find that childhood infections are often highly correlated within individuals in early childhood, with the correlations persisting into adulthood only for infections sharing a transmission route. We discuss 2 applications of the methods: 1) to making inferences about routes of transmission when these are unknown or uncertain and 2) to estimating epidemiologic parameters such as the basic reproduction number and the critical immunization threshold. Two examples of such applications are presented: elucidating the transmission route of polyomaviruses BK and JC and estimating the basic reproduction number and critical immunization coverage of varicella-zoster infection in Belgium, Italy, Poland, and England and Wales. We speculate that childhood correlations stem from confounding of different transmission routes and represent heterogeneity in childhood circumstances, notably nursery-school attendance. In contrast, it is suggested that correlations in adulthood are route-specific

Estimation of basic reproduction numbers: individual heterogeneity and robustness to perturbation of the contact function

The basic reproduction number of an infection in a given population, R0, is inflated by individual heterogeneity in contact rates. Recently, new methods for estimating R0 using social contact data and serological survey data have been proposed. These methods, like most of their predecessors, ignore individual heterogeneity, and are sensitive to perturbation of the contact function. Using a frailty framework, we derive expressions for R0 in the presence of age-varying heterogeneity. In this case, R0 is the spectral radius of a population version of the next generation operator, which involves the variance function of the age-dependent frailty. This variance can be estimated within a shared frailty framework from paired data on two infections transmitted by the same route. We propose two estimators of R0 for infections in endemic equilibrium. We investigate their performance by simulation, and find that one is generally less efficient but more robust than the other to perturbation of the effective contact function. These methods are applied to data on varicella zoster virus infection from two European countries