Bootstrap confidence intervals for principal covariates regression

Principal covariate regression (PCOVR) is a method for regressing a set of criterion variables with respect to a set of predictor variables when the latter are many in number and/or collinear. This is done by extracting a limited number of components that simultaneously synthesize the predictor variables and predict the criterion ones. So far, no procedure has been offered for estimating statistical uncertainties of the obtained PCOVR parameter estimates. The present paper shows how this goal can be achieved, conditionally on the model specification, by means of the bootstrap approach. Four strategies for estimating bootstrap confidence intervals are derived and their statistical behaviour in terms of coverage is assessed by means of a simulation experiment. Such strategies are distinguished by the use of the varimax and quartimin procedures and by the use of Procrustes rotations of bootstrap solutions towards the sample solution. In general, the four strategies showed appropriate statistical behaviour, with coverage tending to the desired level for increasing sample sizes. The main exception involved strategies based on the quartimin procedure in cases characterized by complex underlying structures of the components. The appropriateness of the statistical behaviour was higher when the proper number of components were extracted

On the white, the black, and the many shades of gray in between:Our reply to Van Ravenzwaaij and Wagenmakers (2021)

In 2019 we wrote an article (Tendeiro & Kiers, 2019) in Psychological Methods over null hypothesis Bayesian testing and its working horse, the Bayes factor. Recently, van Ravenzwaaij and Wagenmakers (2021) offered a response to our piece, also in this journal. Although we do welcome their contribution with thought-provoking remarks on our article, we ended up concluding that there were too many "issues" in van Ravenzwaaij and Wagenmakers (2021) that warrant a rebuttal. In this article we both defend the main premises of our original article and we put the contribution of van Ravenzwaaij and Wagenmakers (2021) under critical appraisal. Our hope is that this exchange between scholars decisively contributes toward a better understanding among psychologists of null hypothesis Bayesian testing in general and of the Bayes factor in particular. (PsycInfo Database Record (c) 2022 APA, all rights reserved)

CAR: A MATLAB Package to Compute Correspondence Analysis with Rotations

Correspondence analysis (CA) is a popular method that can be used to analyse relationships between categorical variables. Like principal component analysis, CA solutions can be rotated both orthogonally and obliquely to simple structure without affecting the total amount of explained inertia. We describe a MATLAB package for computing CA. The package includes orthogonal and oblique rotation of axes. It is designed not only for advanced users of MATLAB but also for beginners. Analysis can be done using a user-friendly interface, or by using command lines. We illustrate the use of CAR with one example.

A New Opinion Polarization Index Developed by Integrating Expert Judgments

Opinion polarization is increasingly becoming an issue in today’s society, producing both unrest at the societal level, and conflict within small scale communications between people of opposite opinion. Often, opinion polarization is conceptualized as the direct opposite of agreement and consequently operationalized as an index of dispersion. However, in doing so, researchers fail to account for the bimodality that is characteristic of a polarized opinion distribution. A valid measurement of opinion polarization would enable us to predict when, and on what issues conflict may arise. The current study is aimed at developing and validating a new index of opinion polarization. The weights of this index were derived from utilizing the knowledge of 58 international experts on polarization through an expert survey. The resulting Opinion Polarization Index predicted expert polarization scores in opinion distributions better than common measures of polarization, such as the standard deviation, Van der Eijk’s polarization measure and Esteban and Ray’s polarization index. We reflect on the use of expert ratings for the development of measurements in this case, and more in general

Are Assumptions of Well-Known Statistical Techniques Checked, and Why (Not)?

A valid interpretation of most statistical techniques requires that one or more assumptions be met. In published articles, however, little information tends to be reported on whether the data satisfy the assumptions underlying the statistical techniques used. This could be due to self-selection: Only manuscripts with data fulfilling the assumptions are submitted. Another explanation could be that violations of assumptions are rarely checked for in the first place. We studied whether and how 30 researchers checked fictitious data for violations of assumptions in their own working environment. Participants were asked to analyze the data as they would their own data, for which often used and well-known techniques such as the t-procedure, ANOVA and regression (or non-parametric alternatives) were required. It was found that the assumptions of the techniques were rarely checked, and that if they were, it was regularly by means of a statistical test. Interviews afterward revealed a general lack of knowledge about assumptions, the robustness of the techniques with regards to the assumptions, and how (or whether) assumptions should be checked. These data suggest that checking for violations of assumptions is not a well-considered choice, and that the use of statistics can be described as opportunistic

Onset of an outline map to get a hold on the wildwood of clustering methods

The domain of cluster analysis is a meeting point for a very rich multidisciplinary encounter, with cluster-analytic methods being studied and developed in discrete mathematics, numerical analysis, statistics, data analysis and data science, and computer science (including machine learning, data mining, and knowledge discovery), to name but a few. The other side of the coin, however, is that the domain suffers from a major accessibility problem as well as from the fact that it is rife with division across many pretty isolated islands. As a way out, the present paper offers an outline map for the clustering domain as a whole, which takes the form of an overarching conceptual framework and a common language. With this framework we wish to contribute to structuring the domain, to characterizing methods that have often been developed and studied in quite different contexts, to identifying links between them, and to introducing a frame of reference for optimally setting up cluster analyses in data-analytic practice.Comment: 33 pages, 4 figure

Worked-out examples of the adequacy of Bayesian optional stopping

The practice of sequentially testing a null hypothesis as data are collected until the null hypothesis is rejected is known as optional stopping. It is well known that optional stopping is problematic in the context of p value-based null hypothesis significance testing: The false-positive rates quickly overcome the single test's significance level. However, the state of affairs under null hypothesis Bayesian testing, where p values are replaced by Bayes factors, has perhaps surprisingly been much less consensual. Rouder (2014) used simulations to defend the use of optional stopping under null hypothesis Bayesian testing. The idea behind these simulations is closely related to the idea of sampling from prior predictive distributions. Deng et al. (2016) and Hendriksen et al. (2020) have provided mathematical evidence to the effect that optional stopping under null hypothesis Bayesian testing does hold under some conditions. These papers are, however, exceedingly technical for most researchers in the applied social sciences. In this paper, we provide some mathematical derivations concerning Rouder's approximate simulation results for the two Bayesian hypothesis tests that he considered. The key idea is to consider the probability distribution of the Bayes factor, which is regarded as being a random variable across repeated sampling. This paper therefore offers an intuitive perspective to the literature and we believe it is a valid contribution towards understanding the practice of optional stopping in the context of Bayesian hypothesis testing

Simplicity transformations for three-way arrays with symmetric slices, and applications to Tucker-3 models with sparse core arrays

AbstractTucker three-way PCA and Candecomp/Parafac are two well-known methods of generalizing principal component analysis to three way data. Candecomp/Parafac yields component matrices A (e.g., for subjects or objects), B (e.g., for variables) and C (e.g., for occasions) that are typically unique up to jointly permuting and rescaling columns. Tucker-3 analysis, on the other hand, has full transformational freedom. That is, the fit does not change when A,B, and C are postmultiplied by nonsingular transformation matrices, provided that the inverse transformations are applied to the so-called core array G̲. This freedom of transformation can be used to create a simple structure in A,B,C, and/or in G̲. This paper deals with the latter possibility exclusively. It revolves around the question of how a core array, or, in fact, any three-way array can be transformed to have a maximum number of zero elements. Direct applications are in Tucker-3 analysis, where simplicity of the core may facilitate the interpretation of a Tucker-3 solution, and in constrained Tucker-3 analysis, where hypotheses involving sparse cores are taken into account. In the latter cases, it is important to know what degree of sparseness can be attained as a tautology, by using the transformational freedom. In addition, simplicity transformations have proven useful as a mathematical tool to examine rank and generic or typical rank of three-way arrays. So far, a number of simplicity results have been attained, pertaining to arrays sampled randomly from continuous distributions. These results do not apply to three-way arrays with symmetric slices in one direction. The present paper offers a number of simplicity results for arrays with symmetric slices of order 2×2,3×3 and 4×4. Some generalizations to higher orders are also discussed. As a mathematical application, the problem of determining the typical rank of 4×3×3 and 5×3×3 arrays with symmetric slices will be revisited, using a sparse form with only 8 out of 36 elements nonzero for the former case and 10 out of 45 elements nonzero for the latter one, that can be attained almost surely for such arrays. The issue of maximal simplicity of the targets to be presented will be addressed, either by formal proofs or by relying on simulation results

A Comparison of Reliability Coefficients for Ordinal Rating Scales

Kappa coefficients are commonly used for quantifying reliability on a categorical scale, whereas correlation coefficients are commonly applied to assess reliability on an interval scale. Both types of coefficients can be used to assess the reliability of ordinal rating scales. In this study, we compare seven reliability coefficients for ordinal rating scales: the kappa coefficients included are Cohen’s kappa, linearly weighted kappa, and quadratically weighted kappa; the correlation coefficients included are intraclass correlation ICC(3,1), Pearson’s correlation, Spearman’s rho, and Kendall’s tau-b. The primary goal is to provide a thorough understanding of these coefficients such that the applied researcher can make a sensible choice for ordinal rating scales. A second aim is to find out whether the choice of the coefficient matters. We studied to what extent we reach the same conclusions about inter-rater reliability with different coefficients, and to what extent the coefficients measure agreement in a similar way, using analytic methods, and simulated and empirical data. Using analytical methods, it is shown that differences between quadratic kappa and the Pearson and intraclass correlations increase if agreement becomes larger. Differences between the three coefficients are generally small if differences between rater means and variances are small. Furthermore, using simulated and empirical data, it is shown that differences between all reliability coefficients tend to increase if agreement between the raters increases. Moreover, for the data in this study, the same conclusion about inter-rater reliability was reached in virtually all cases with the four correlation coefficients. In addition, using quadratically weighted kappa, we reached a similar conclusion as with any correlation coefficient a great number of times. Hence, for the data in this study, it does not really matter which of these five coefficients is used. Moreover, the four correlation coefficients and quadratically weighted kappa tend to measure agreement in a similar way: their values are very highly correlated for the data in this study

PCovR2:A flexible principal covariates regression approach to parsimoniously handle multiple criterion variables

Principal covariates regression (PCovR) allows one to deal with the interpretational and technical problems associated with running ordinary regression using many predictor variables. In PCovR, the predictor variables are reduced to a limited number of components, and simultaneously, criterion variables are regressed on these components. By means of a weighting parameter, users can flexibly choose how much they want to emphasize reconstruction and prediction. However, when datasets contain many criterion variables, PCovR users face new interpretational problems, because many regression weights will be obtained and because some criteria might be unrelated to the predictors. We therefore propose PCovR2, which extends PCovR by also reducing the criteria to a few components. These criterion components are predicted based on the predictor components. The PCovR2 weighting parameter can again be flexibly used to focus on the reconstruction of the predictors and criteria, or on filtering out relevant predictor components and predictable criterion components. We compare PCovR2 to two other approaches, based on partial least squares (PLS) and principal components regression (PCR), that also reduce the criteria and are therefore called PLS2 and PCR2. By means of a simulated example, we show that PCovR2 outperforms PLS2 and PCR2 when one aims to recover all relevant predictor components and predictable criterion components. Moreover, we conduct a simulation study to evaluate how well PCovR2, PLS2 and PCR2 succeed in finding (1) all underlying components and (2) the subset of relevant predictor and predictable criterion components. Finally, we illustrate the use of PCovR2 by means of empirical data