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Modern Psychometrics with R

verfasst von: Patrick Mair

Verlag: Springer International Publishing

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Über dieses Buch

This textbook describes the broadening methodology spectrum of psychological measurement in order to meet the statistical needs of a modern psychologist. The way statistics is used, and maybe even perceived, in psychology has drastically changed over the last few years; computationally as well as methodologically. R has taken the field of psychology by storm, to the point that it can now safely be considered the lingua franca for statistical data analysis in psychology. The goal of this book is to give the reader a starting point when analyzing data using a particular method, including advanced versions, and to hopefully motivate him or her to delve deeper into additional literature on the method.

Beginning with one of the oldest psychometric model formulations, the true score model, Mair devotes the early chapters to exploring confirmatory factor analysis, modern test theory, and a sequence of multivariate exploratory method. Subsequent chapters present special techniques useful for modern psychological applications including correlation networks, sophisticated parametric clustering techniques, longitudinal measurements on a single participant, and functional magnetic resonance imaging (fMRI) data. In addition to using real-life data sets to demonstrate each method, the book also reports each method in three parts-- first describing when and why to apply it, then how to compute the method in R, and finally how to present, visualize, and interpret the results. Requiring a basic knowledge of statistical methods and R software, but written in a casual tone, this text is ideal for graduate students in psychology.

Relevant courses include methods of scaling, latent variable modeling, psychometrics for graduate students in Psychology, and multivariate methods in the social sciences.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Classical Test Theory

Abstract

This chapter gives a succinct introduction to classical test theory, an early attempt to formalize a statistical theory of psychological measurement. The main focus is on reliability. After introducing the true score model, the following reliability coefficients are presented: Cronbach’s α, greatest lower bound, and McDonald’s ω’s. In the second part of this chapter, this simple definition of reliability idea is extended to multiple error sources. This leads to generalizability theory which includes concepts like G-studies and D-studies, as well as generalizability and dependability coefficients.

Patrick Mair

Chapter 2. Factor Analysis

Abstract

This chapter introduces exploratory and confirmatory factor analysis. It starts with a section on correlation coefficients since factor analytic techniques are based on covariance/correlation matrices. Special emphasis is on tetrachoric/polychoric correlations for ordinal input data. This is followed by elaborations on exploratory factor analysis including practical aspects such as determining the number of factors and rotation techniques to facilitate factor interpretation. A recent development is Bayesian exploratory factor analysis which, in addition to the loadings, also estimates the number of factors and allows them to be correlated. This approach is explored in a separate section. The second part of this chapter consists of a detailed treatment of confirmatory factor analysis which lays the groundwork for structural equation models presented in the next chapter. In confirmatory factor analysis, the number of factors and the assignment of indicators to factors are determined by substantive considerations. Several extensions in terms of multigroup, longitudinal, and multilevel settings are presented. The chapter concludes with a Bayesian approach to confirmatory factor analysis.

Patrick Mair

Chapter 3. Path Analysis and Structural Equation Models

Abstract

The first part of this chapter introduces simple path analysis structures, not involving any latent variables. Regression-based approaches such as multivariate regression, mediator models, moderator models, and extensions in terms of combined moderator-mediator path models are presented. The second part of this chapter is dedicated to path models with latent variables: structural equation models. This part builds heavily on elaborations on confirmatory factor analysis from the previous chapter. A special focus is on multigroup structural equation models which allow researchers to test hypotheses on group-specific parameters. Within this context non-nested model comparison is illustrated as well. Finally, latent growth models are introduced, an approach for studying changes over time.

Patrick Mair

Chapter 4. Item Response Theory

Abstract

Item response theory (IRT) is a psychometric modeling framework for analyzing categorical data from questionnaires, tests, and other instruments that aim to measure underlying latent traits. Simply speaking, these models estimate a parameter for each item, as well as a parameter for each person. Depending on how many latent traits are involved, a core distinction in IRT is unidimensional vs. multidimensional IRT models. Hence, dimensionality assessment is important before fitting an IRT model, as elaborated in the first section. Subsequently, the focus is on various classical unidimensional models for dichotomous as well as polytomous input data. Afterward, three sections cover various special topics in IRT: item/test information, sample size determination, and differential item functioning, where differences in the item parameters are examined across person subgroups. Some modern IRT flavors are presented in final three sections on multidimensional IRT, longitudinal IRT, and Bayesian IRT.

Patrick Mair

Chapter 5. Preference Modeling

Abstract

This chapter focuses on modeling particular types of input data: ratings, rankings, and paired comparisons. It begins with elaborations on the classical Bradley-Terry model for paired comparisons of objects. Each object gets a parameter on an underlying continuum. Subsequently, the Bradley-Terry model is extended in terms of incorporating predictors. Two modern approaches are considered: recursive partitioning trees and lasso. The second part of the chapter deals with log-linear model formulations for preference data. So called pattern models are introduced, and versions for ratings, rankings, and paired comparisons are presented.

Patrick Mair

Chapter 6. Principal Component Analysis and Extensions

Abstract

This chapter introduces principal component analysis (PCA), a technique for dimension reduction in multivariate datasets. At its core there is a matrix decomposition technique called singular value decomposition, which is introduced at the beginning of this chapter. This is followed by PCA model formulation, computation, and an application. Relationships with exploratory factor analysis are discussed as well. Subsequently, some PCA variants such as robust and sparse PCA are briefly discussed. The final two sections introduce two extensions of PCA. The first one is three-way PCA for three-way input data structures. The second one is called independent component analysis and illustrated using electroencephalography (EEG) data.

Patrick Mair

Chapter 7. Correspondence Analysis

Abstract

Correspondence analysis (CA) aims to scale the row and column categories of a contingency table in a low-dimensional space. In this space basic structures and associations among the row categories and among the column categories are represented. The basic distinction we make in CA is simple CA vs. multiple CA. Simple CA involves two categorical variables only, whereas multiple CA is used for higher-dimensional frequency tables. In this chapter we present the French approach to CA which tackles the CA problem analytically, whereas in the next chapter, we use a numerical approach. At the end of this chapter, we present a method called configural frequency analysis which can be used to investigate CA outputs in more detail.

Patrick Mair

Chapter 8. Gifi Methods

Abstract

The Gifi system is a powerful and flexible framework for exploratory multivariate data analysis. It is especially attractive for categorical input data or, more general, input variables with mixed scale levels. At the core of Gifi is the idea of optimal scaling, introduced in the first part of this chapter. Subsequently, two of the most prominent Gifi models are presented. The first model is called Princals. In its basic form, it is a principal component analysis variant for ordinal input data. The second model is called Homals which performs multiple correspondence analyses. Both models can be extended in various directions. In this chapter we focus on a combined Homals-Princals strategy for input data with mixed scale levels. In the last part, another optimal scaling approach called Lineals is introduced which can be used as a preprocessing tool for factor analysis and structural equation models with categorical indicators.

Patrick Mair

Chapter 9. Multidimensional Scaling

Abstract

Multidimensional scaling (MDS) a multivariate method, applicable to a variety of data scenarios. It aims to represent input proximities among objects, such as variables or persons, by means of fitted distances in a low-dimensional space. The chapter starts with general elaborations on proximities, followed by exploratory MDS using the SMACOF framework. Within this context, goodness-of-fit assessment in MDS is discussed in detail. Another section covers confirmatory MDS where it is distinguished between internal and external constraints on the configuration. What follows is a section on unfolding, a technique for dual scaling based on preference data. In the last part of this chapter, basic MDS is extended to multiple input dissimilarity matrices (individual differences scaling). In addition, Procrustes is introduced for matching multiple MDS configurations.

Patrick Mair

Chapter 10. Biplots

Abstract

Biplots are a multivariate scatterplot concept to visualize row and column structures in complex data. It can be applied to any of the exploratory methods presented in previous chapters. We introduce biplots by means of a multivariate regression with two predictors. Subsequently, this concept is applied to principal component analysis, where biplots are one of the classical output visualization techniques. The same plotting principle is adopted to Princals. In the following section, biplots for multidimensional scaling are introduced where external covariates are mapped onto the configuration. Finally, biplots within a correspondence analysis context are discussed where they are simply asymmetric maps.

Patrick Mair

Chapter 11. Networks

Abstract

This chapter introduces network approaches to analyze associations in multivariate datasets. The first part of this chapter deals with classical (social) network analysis with relational input data. In psychological applications, however, it is rather uncommon to directly observe relational data. Therefore, a main focus in the remainder of this chapter is on networks based on a correlation input matrix. After introducing basic correlation networks, they are extended to partial correlation networks including the graphical lasso, which removes edges using regularization. The following section illustrates how scaling approaches can be integrated into networks. This leads to eigenmodels, which in a subsequent step are extended toward the incorporation of clustering (latent class networks). The final section is about Bayesian networks, a method based on the graph-theoretic concept of directed acyclic graphs. Bayesian networks do not involve correlations and allow researchers to study directed relationships among variables.

Patrick Mair

Chapter 12. Parametric Cluster Analysis and Mixture Regression

Abstract

This chapter is about advanced parametric clustering techniques based on the concept of mixture distributions. The first section introduces mixture distributions from a general perspective, followed by two popular applications in clustering: normal mixture models (latent profile analysis) for metric input variables and multinomial mixture models (latent class analysis) for categorical variables. Subsequently, these ideas are extended to mixed input scale levels. In the following section, the mixture distribution concept is embedded into a regression framework. In mixture regression models, clustering and estimation of regression parameters are performed simultaneously. By means of Dirichlet process regression, we add another complexity layer to the modeling framework by letting an algorithm determine the optimal number of clusters. Finally, the focus is on latent Dirichlet allocations: topic models for clustering text data.

Patrick Mair

Chapter 13. Modeling Trajectories and Time Series

Abstract

This chapter presents various modeling options for trajectories and time series. The first part covers hidden Markov models where the aim is to find latent states between which a participant can switch back and forth during an experimental task. Extended modeling options in terms of including covariates are presented as well. The second part introduces time series analysis. The main focus is on a parametric model class called ARIMA, representing a flexible regression framework for time series able to handle autocorrelated residuals. As a special ARIMA flavor, intervention analysis is presented which allows researchers to study whether a critical event had an impact on the series or not. The third part covers functional data analysis, applicable to data settings where each individual produces its own trajectory, subject to smoothing. In addition to functional regression modeling, a functional version of principal component analysis is presented.

Patrick Mair

Chapter 14. Analysis of fMRI Data

Abstract

fMRI stands for “functional magnetic resonance imaging” and represents a noninvasive, indirect method for measuring neural activity over time. Such brain scans result in large, complex, and noisy data, which makes data analysis challenging. The first part of the chapter focuses on data preparation and visualization techniques. This is followed by standard univariate linear modeling approaches, for which some effort needs to go into the computation of the expected BOLD signal and the design matrix specification. After fitting the regression models, a huge multiple testing problem arises. A corresponding section focuses on the false discovery rate, Gaussian random fields, and permutation tests including cluster-based thresholding. The last few sections describe specific multivariate methods popular in fMRI: independent component analysis, representational similarity analysis, and connectivity analysis.

Patrick Mair

Backmatter

Titel: Modern Psychometrics with R
verfasst von: Patrick Mair
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-93177-7
Print ISBN: 978-3-319-93175-3
DOI: https://doi.org/10.1007/978-3-319-93177-7