Ordinal regression

About: Ordinal regression is a research topic. Over the lifetime, 1879 publications have been published within this topic receiving 65431 citations.

Feature extraction and supervised learning on fMRI : from practice to theory

Abstract: Until the advent of non-invasive neuroimaging modalities the knowledge of the human brain came from the study of its lesions, post-mortem analyses and invasive experimentations. Nowadays, modern imaging techniques such as fMRI are revealing several aspects of the human brain with progressively high spatio-temporal resolution. However, in order to answer increasingly complex neuroscientific questions the technical improvements in acquisition must be matched with novel data analysis methods. In this thesis we examine different applications of machine learning to the processing of fMRI data. We propose novel extensions and investigate the theoretical properties of different models. % The goal of an fMRI experiments is to answer a neuroscientific question. However, it is usually not possible to perform hypothesis testing directly on the data output by the fMRI scanner. Instead, fMRI data enters a processing pipeline in which it suffers several transformations before conclusions are drawn. Often the data acquired through the fMRI scanner follows a feature extraction step in which time-independent activation coefficients are extracted from the fMRI signal. The first contribution of this thesis is the introduction a model named Rank-1 GLM (R1-GLM) for the joint estimation of time-independent activation coefficients and the hemodynamic response function (HRF). We quantify the improvement of this approach with respect to existing procedures on different fMRI datasets. The second part of this thesis is devoted to the problem of fMRI-based decoding, i.e., the task of predicting some information about the stimuli from brain activation maps. From a statistical standpoint, this problem is challenging due to the high dimensionality of the data, often thousands of variables, while the number of images available for training is small, typically a few hundreds. We examine the case in which the target variable consist of discretely ordered values. The second contribution of this thesis is to propose the following two metrics to assess the performance of a decoding model: the absolute error and pairwise disagreement. We describe several models that optimize a convex surrogate of these loss functions and examine their performance on different fMRI datasets. Motivated by the success of some ordinal regression models for the task of fMRI-based decoding, we turn to study some theoretical properties of these methods. The property that we investigate is known as consistency or Fisher consistency and relates the minimization of a loss to the minimization of its surrogate. The third, and most theoretical, contribution of this thesis is to examine the consistency properties of a rich family of surrogate loss functions that are used in the context of ordinal regression. We give sufficient conditions for the consistency of the surrogate loss functions considered. This allows us to give theoretical reasons for some empirically observed differences in performance between surrogates.

An approach to the multivariate analysis of ordinal data

Abstract: Techniques of multivariate analysis currently in use are rarely appropriate to an ordinal level of measurement; they are designed either for interval measurement or for nominal categories. It is possible to construct a hypothetical example in which the neglect of the ordered nature of a control variable leads to erroneous conclusions when three ordered variables are jointly cross-classified. This result suggests that, in addition to traditional methods, an additional criterion, based on the logic of partial rank correlation, should be considered in summarizing such data.

Meaningful statistical model formulations for repeated measures

Abstract: When choosing a parametric statistical model two important consider- ations are mathematical soundness and substantive relevance. In this paper, we illustrate and exemplify that a number of issues arise from these considerations, even in relatively simple settings, such as ordinal regression, linear mixed models, models for cross-classified data and generalized linear mixed models. Many of our points are illustrated with data. Choosing a parametric statistical model is a common task in statistical prac- tice. When choosing a model, it is important to reflect on whether the model is sound from a theoretical point of view and whether it is adequate in terms of the scientific research question of interest. While some authors have approached aspects of this problem from a fundamental, theoretical perspective (McCullagh (2002) and references therein) it is fair to say that the problem receives less at- tention in everyday practice than it should. We consider a number of simple but key settings in order to make a number of general and specific points about this topic. Many of these points are illustrated using a few simple settings (Section 2). First, we consider the linear mixed-effects model, that has become a standard tool for analyzing repeated continuous, normally distributed outcomes. While it looks like a relatively straightforward extension of linear regression it is sur- rounded with a number of problems, some of them arising due to the fact that one can adopt either a hierarchical or a marginal point of view which, while having connections, are different. The implications of this fact for variance com- ponent testing ought not to be overlooked (Section 4). Second, when switching from normally distributed to discrete repeated measures (Section 5), one should realize that, even though there are links to the simpler linear mixed model, the situation is dramatically more complicated. In particular, one can choose be- tween a number of relevant but non-equivalent modelling families. Within each