Quantitative psychology is the study of psychological phenomena using mathematical or statistical methods. Among fields within psychology, quantitative psychology is not recognized by the general public as part of psychology, despite its relatively long history within psychology. This lack of identity is in part due to the dual character of quantitative psychology. The field includes both the application of quantitative methods in studying psychological phenomena and the development of new quantitative methods as tools for research. Quantitative psychologists divide their time between each of these two general activities, or may pursue one activity exclusively. For example, some quantitative psychologists are associated with a particular research area (e.g., cognitive psychology) and apply quantitative tools with the goal of acquiring new knowledge about that area of inquiry. Other quantitative psychologists focus on the development of particular quantitative tools, which could later be applied across multiple research areas. From the viewpoint of psychology in general, the most valuable contributions made by quantitative psychologists are those that have enduring applicability to real psychological research or practice. These contributions usually rest on a foundation of prior theoretical work however, and this theoretical work may have had no obvious application initially.

The precise historical origins of quantitative psychology are difficult to pinpoint. Two broad influences were the experimental psychological tradition from the nineteenth century, and the rise of mental testing in the late nineteenth and early twentieth centuries. The psychophysical research of nineteenth century psychologists such as Gustav Fechner and William Wundt introduced the application of mathematical methods to the study of sensation and perception. The idea that psychological phenomena might be accurately described using mathematical functions was new, and it inspired later generalizations to other domains of psychology, such as the measurement of attitudes (Thurstone, 1928). With regard to measurement, the development of measures of intelligence by Alfred Binet (Binet and Simon, 1905) and by James McKeen Cattell (Cattell, 1890) are early examples of influential psychometric research. Cattell first used the term ‘mental test’ to describe a set of common tasks used to assess the examinee's intelligence. In truth, the idea of tests as indicators of psychological traits was much older, going back to the use of written examinations in early China (Thorndike and Lohman, 1990).

Motivations for new developments in quantitative psychology have often come from applications of quantitative methods to practical problems. For example, the application of testing to problems in educational measurement and in employment selection has led to new developments and extensions. The entire field of item response theory arose in part as a response to the practical inadequacies in classical models of measurement, for example (van der Linden and Hambleton, 1997). In educational measurement, the need for multiple forms of the same test and the resulting problems of equivalence in such tests led to new developments in test equating methods (Kolen and Brennan, 1995). In the area of research design, the inability to apply randomization principles in some domains of psychological research has [Page xi]led to new developments in causal modeling and quasi-experimental design (Shadish, Cook, and Campbell, 2002). However, some quantitative developments are slow to be adopted by practicing researchers, even when doing so might enhance research quality. Borsboom (2007) noted that important quantitative developments go unnoticed by psychologists in some cases. One obvious barrier to dissemination is that new quantitative developments often appear first in technical form, which discourages immediate adoption by the wider psychological community. The need then arises for work that will translate the technical details into a form that can be understood by a wider audience.

The present book is an attempt to meet this need. Here we survey the field of quantitative psychology as it exists today, providing an overview with some depth while still making the contents accessible to readers who are not experts in statistics. We do assume that the reader has been exposed to fundamental concepts in statistics. Nearly all psychologists acquire some familiarity with statistical reasoning as part of their training. Some of the quantitative topics discussed in the book are necessarily more technical than others. We have tried to achieve relative uniformity in the level of discourse across the different contributed chapters, and where possible, to limit the technical level without sacrificing information. In choosing the topics to be covered, we have admittedly been influenced by our own perceptions of the important trends in the field. We do not claim to be exhaustive in our coverage. Arguably, there are additional quantitative topics that could have been covered as part of quantitative psychology. To our knowledge, this book is the first attempt to bring together the many different topics within quantitative psychology in a single volume.

Turning now to the contents, Part I of the book addresses issues in research design and causal inference. Chapter 1, by Michael Sobel, describes the current thinking on the conditions needed for inferences of causality from empirical studies: What evidence is needed to conclude that A causes B? As noted in the chapter, the interest among researchers in developing formal principles of causal inference from real data has greatly increased in recent decades. Chapter 2, by Roger Kirk, on experimental design describes the basic principles of design for true experiments in which randomization ispossible for at least one independent variable. While most psychologists are exposed at some point in their training to statistical methods for the analysis of experimental data, research design principles are now less frequently taught in graduate schools of psychology (Aiken, West, and Millsap, 2008). Chapter 3, by Charles Reichardt, continues the design discussion by describing issues in the design of quasi-experiments, or studies in which interventions are made without full randomization. This topic is of deep interest to the many researchers who, for various practical reasons, cannot conduct true experiments. Chapter 4, by Paul D. Allison, addresses the problem of missing data, a topic that is now considered essential in the education of nearly any researcher in psychology, and is particularly important for longitudinal researchers. We know now that some informal methods of handling missing data can distort conclusions, and that better methods are available. We have placed this chapter in Part I because effective handling of missing data usually requires careful research design, especially with regard to the choice of which variables are measured.

Part II of the book considers topics in psychological measurement, which has been an essential part of quantitative psychology from its early beginnings. Chapter 5, by James Algina and Randall Penfield, describes classical test theory (CTT). CTT has roots in work done in the late nineteenth century, yet it still guides much thinking about measurement in psychology today. An understanding of CTT is essential for psychologists who must critically evaluate tests and measures used in psychological research or in applied settings. Chapter 6, by Robert MacCallum, discusses the traditional common factor analysis model, which bears a close relationship to models used in CTT. The linear factor model is the most commonly used latent variable model in psychology. It is used primarily to explore or confirm the number of latent dimensions that underlie a set of measures. Chapter 7, by David Thissen and Lynne Steinberg, concerns item [Page xii]response theory (IRT), which is a set of models for how people respond to test items. IRT models are latent variable models that make stronger assumptions than do models in CTT, but these stronger assumptions also permit useful applications such as computerized adaptive testing. This chapter gives an overview of IRT models and assumptions. Chapter 8, by Michael Edwards and Maria Orlando Edelen, addresses three special topics within IRT: computerized adaptive testing, the detection of differential item functioning, and multidimensional IRT. Computerized adaptive testing represents an important innovation in actual testing practice. Differential item functioning refers to group or temporal differences in the probabilities of various responses to test items, given scores on the latent variables. Multidimensional IRT is a collection of models for items in which more than one latent variable affects the response probabilities. The last chapter in this section, Chapter 9, by David Rindskopf, addresses latent class analysis, which is a latent variable model that applies when the latent variable is categorical rather than continuous in scale. These models have important applications within areas of psychology that posit multiple subpopulations defined by psychological status. For example, latent class models are the focus of recent debates about taxons versus continuous dimensions as models for personality measurements (e.g., Waller and Meehl, 1998).

Part III of the book addresses psychological scaling methods. Whereas Part II focused on measurement models for the psychological attributes of people, Part III focuses on the scaling of psychological stimuli. To illustrate, we may want to understand how people evaluate political figures on a set of attributes. Psychological scaling methods can be used to help understand: (1) the dimensions along which people evaluate the political figures; and (2) the estimated location for each political figure on the dimensions. To begin, Chapter 10, by Yoshio Takane, Sunho Jung, and Yuriko Oshima-Takane, describes procedures for metric and non-metric scaling of stimuli. The distinction between these two forms of scaling lies in the initial measures that form the input for the scaling procedure, and whether those measures can be viewed as metric or simply ordinal in scale. Chapter 11, by Heungsun Hwang, Marc A. Tomiuk, and Yoshio Takane, addresses correspondence analysis and multiple correspondence analysis. Correspondence analysis provides exploratory representations of data in two-way cross-tabulation tables in terms of several latent dimensions. Under this representation, it is possible to calculate distances between rows and columns in the original table on the latent dimensions. This group of methods is not yet widely known in North America. Finally, Chapter 12, by Alberto Maydeu-Olivares and Ulf Böckenholt, describes scaling methods for preference data. Preference data are the data gathered by asking participants to indicate their choices or preferences among a set of stimuli. Thurstone (1927) presented models for such data that provided a basis for scaling the stimuli on psychological dimensions using the relative frequencies with which one stimuli is preferred to the other stimuli. Recent developments have greatly expanded the number of models that can be applied to preference data, as illustrated in the chapter.

Part IV of the book presents chapters on topics within the general subject of data analysis and statistics. The section begins with Chapter 13, by Razia Azen and David Budescu, on the use of multiple regression in psychological research. Multiple regression is the most widely used multivariate statistical method in psychology, and it serves as a kind of ‘gateway’ to more elaborate forms of multivariate statistics. A clear understanding of regression methods is thus essential for any researcher who wishes to use multivariate statistics. Chapter 14, by Carolyn Anderson, addresses the analysis of categorical data. Given that most measurement in psychology is done with items that have discrete response scales, categorical data analysis is vital to the study of psychological measures, as noted in the chapter. Also, substantial advances in categorical analysis methods have been made in the last 30 years. Chapter 15, by Jee-Seon Kim, concerns the analysis of multilevel data, or data in which some hierarchical structure is present among the individuals who provide the data. Multilevel data analysis is now regarded as a standard tool for psychologists who study data containing pre-existing groups, such as families, [Page xiii]siblings, couples, work groups, schools, or organizations. In Chapter 16, William H. Beasley and Joseph L. Rodgers describe methods of analysis that employ resampling in various forms, such as the bootstrap or the jackknife. Resampling methods are highly useful for the analysis of psychological data because standard distributional assumptions are often violated in such data. In these cases, resampling methods offer one approach to obtaining accurate standard errors and confidence intervals. Chapter 17, by Rand Wilcox, addresses robust data analysis, or methods of data analysis that permit accurate estimation and statistical inference when distributional assumptions are violated. Psychologists who are unfamiliar with recent developments in this area will be surprised to learn of what is known about the negative impact of distributional problems on standard inference procedures, and what new alternatives are available. Chapter 18, by Andy Field, examines methods of meta-analysis, or the statistical integration of the results of many independent empirical studies. Meta-analytic methods have undergone rapid growth in the last 25 years, and have led to important advances in some areas of psychology, such as industrial-organizational psychology. Chapter 19, by Herbert Hoijtink, describes developments in data analysis that are motivated from a Bayesian perspective, in contrast to the frequentist perspective that has dominated much statistical practice in psychology. Psychologists as a whole are unaware of the impact of Bayesian statistical methods in the field of statistics generally. The chapter provides an introduction to many ideas that are now standard in the field of statistics, and that will become more widely used in psychology. The last chapter in this section is Chapter 20, by Lawrence J. Hubert, Hans-Friedrich Köhn, and Douglas Steinley, on cluster analysis. Cluster analysis is a collection of methods for grouping objects using some measures of distance or similarity between the objects. The chapter focuseson two broad clustering methods, hierarchical clustering and K-means partitioning. Software routines written in MATLAB are used throughout to illustrate the methods.

Part V of the book is devoted to structural equation modeling. Structural equation models (SEMs) are found in nearly every area of psychology at present, moving in 30 years from a topic largely confined to technical journals to a part of the standard statistical training in many graduate schools of psychology. At present, the topic is too large to be covered in a single chapter, and so we have included several chapters in this section. Chapter 21, by Robert Cudeck and Stephen H.C. du Toit, gives an overview of general SEM theory and practice. This chapter covers the specification and identification of structural models, parameter estimation methods, and the evaluation of fit, using regression theory as a basic building block leading to the full SEM. In fact, many of the statistical models that are already familiar to psychologists, such as the analysis of variance model and regression models, can be represented as SEMs. Chapter 22, by Melanie M. Wall, addresses nonlinear structural equation models, in contrast to the linear models that form the basis for many applications of structural modeling. The need for such nonlinear models becomes apparent, for example, when theory suggests that two latent variables might interact in their causal effects on a third variable. Interactions at the latent level provide one application for these nonlinear models, and the chapter mentions other potential applications while also describing several broad approaches to estimation and fit evaluation in nonlinear SEM. Chapter 23, by Conor Dolan, addresses the topic of mixture modeling in the context of SEM. Mixture models arise from the combination of several distinct statistical models corresponding to distinct subpopulations within a general population. Mixture SEM models exist when the component models are SEM models in the various subpopulations. For example, in clinical applications, a mixture model might posit several distinct subpopulations corresponding to different levels of psychopathology in the general population. The last chapter in this section, Chapter 24, by David Kaplan, Jee-Seon Kim, and Su-Young Kim, describes developments in multilevel latent variable modeling. This chapter shares a multilevel perspective with the earlier Chapter 15, but focuses here on latent variable modeling in multilevel data. Given the frequent use of latent variable models in psychology to describe psychological measures, [Page xiv]the extension to multilevel data structures is important in expanding the scope of these latent variable models.

Part VI of the book examines statistical models for longitudinal data. The analysis of longitudinal data has undergone many new developments in the last 30 years, resulting in new approaches that are unfamiliar to most psychologists. Chapter 25, by Suzanne E. Graham, Judith D. Singer, and John B. Willett, provides an overview of longitudinal methods by focusing on models for individual change over time. The shift in recent years from modeling averages across time to developing random effects models for individual change trajectories is an important theme here. The problem of modeling change is a long-standing one in psychology (e.g., Harris, 1963) and will certainly continue to be of interest. Chapter 26, by Emilio Ferrer and Guangjian Zhang, examines the use of time series models in psychological research. Time series analysis is traditionally an important topic in longitudinal data analysis, but panel studies in psychology often include too few measurements to enable the use of such analyses. This situation has changed however with newer methods of data collection that seek many repeated measurements (Walls and Schafer, 2006), such as the use of electronic devices to record repeated self-reports of mood or stress levels. These methods have provided new scope for the application of times series analysis. The final chapter of this section is Chapter 27, by Jeroen K. Vermunt, that describes methods for event history analysis. Event history analysis includes methods for modeling the occurrence and timing of discrete events in a longitudinal sequence. For example, we may want to study causal influences on the elapsed time between the initial hiring of an employee and that employee's departure from a job. As another example, we may study the time spent in recovery following a major psychological trauma. Event history analysis methods are less familiar to psychologists than are other longitudinal methods based on regression models.

The last section of the book examines some specialized quantitative methods that are important, but are not easily classified in any of the preceding categories. Chapters 28 and 29 are related by a common emphasis on quantitative methods for the analysis of neuroimaging data. Chapter 28, by Josep Marco-Pallerés, Estela Camara, Thomas F. Münte, and Antoni Rodríguez-Fornells, describes methods for handling data provided by electroencephalography (EEG). EEG measurements typically provide a wealth of time-related data from multiple channels, and are often recorded in response to various stimuli to study variation in people's responses. Some form of data reduction is often necessary (e.g., principal components analysis). Methods for looking at cross-series relations in multiple time series are also important. The chapter describes the statistical methods that are most often used for these data. Chapter 29, by Estela Camara, Josep Marco-Pallerés, Thomas F. Münte, and Antoni Rodríguez-Fornells, describes methods for handling magnetic resonance imaging (MRI) data. Like EEG data, the data provided by MRI is extensive and complex, requiring careful multivariate analyses that both simplifies and brings important trends into relief. MRI is an extraordinary tool for the analysis of brain processes, but methods for the analysis of these data are still under development. Please note that color versions of the plates in Chapters 28 and 29 are available at the end of the Handbook. The final chapter in this section is Chapter 30, by James O. Ramsay, on functional data analysis. Functional data analysis is a collection of methods for working with functions of data as the basic object of analysis. These functions ordinarily operate on individual-level data, as in a collection of individual growth curves over time. Once the functions to be used are specified, it is possible to also model selected features of these functions, such as differentials or acceleration. Functional data analysis makes it possible to model data in ways that would be difficult with more conventional approaches.

While it is difficult to span the entire field of quantitative psychology in 30 chapters, we feel that the chapters in this volume represent a fair sampling of the many contributions made by quantitative psychologists to design, measurement, and analysis. We hope that people who [Page xv]are interested in learning more about quantitative psychology will find these chapters to be informative, and that psychologists who seek to use quantitative methods will find the book to be a useful resource.