• Introduction
• General Overviews
• Multivariate Assumptions
• Multivariate Effect Sizes
• Multivariate Analysis of Variance
• Discriminant Analysis
• Canonical Correlation Analysis
• Exploratory Factor Analysis
• Principal Components Analysis and Common Factor Analysis
• Factor Retention
• Rotation Criteria
• Confirmatory Factor Analysis
• Structural Equation Modeling
• Multidimensional Scaling
• Nonparametric Multivariate Analysis

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# Multivariate Research MethodologybyRobin K. Henson, Krystal HinermanLAST REVIEWED: 28 April 2016LAST MODIFIED: 28 April 2016DOI: 10.1093/obo/9780199756810-0145

## Introduction

Broadly defined, multivariate research methods involve the inclusion of more than one outcome in a singular analysis. Instead of conducting a series of univariate analysis, one for each outcome, multivariate analyses consider all the outcomes of interest at the same time. When compared to multiple univariate analysis, the multivariate approach allows for the reduction in Type I error, for an increase in power, and for the relationship between the outcome variables to be expressed. Perhaps most importantly from an applied perspective, the multivariate approach allows the researcher to analyze the data in a way that is most reflective of the actual research context and environment. In education, research scenarios involving multiple outcomes could include examples such as how a character education program impacts student self-efficacy, attitudes, and behavior, or how mentoring influences reading, math, and science performance. Beyond more applied uses of multivariate designs, multivariate analysis can extend to include models such as those specific to testing structural validity of instruments designed to measure latent constructs such as intelligence and personality or to predicting group placement. In other words, multivariate research methods are warranted in most situations where multiple dependent variables are being considered in the same research scenario. In order to appropriately analyze data gathered under a multivariate design, researchers must shift away from univariate analysis into the multivariate analysis framework. The sections in this article are intended as an introduction to the applications, considerations, and mechanics of conducting multivariate research. Although the sections are presented as discrete categories, the topics often overlap and inform each other. The sections are General Overviews of multivariate topics, Multivariate Assumptions, Multivariate Effect Sizes, Multivariate Analysis of Variance (MANOVA), Discriminant Analysis (DA), Canonical Correlation Analysis (CCA), Exploratory Factor Analysis (EFA), Principal Components Analysis and Common Factor Analysis, Factor Retention, Rotation Criteria, Confirmatory Factor Analysis (CFA), Structural Equation Modeling (SEM), Multidimensional Scaling (MDS), and Nonparametric Multivariate Analysis.

## General Overviews

The text and articles in this section provide overviews and introductions to major concepts for consideration while conducting multivariate analysis. Huberty and Morris 1989 and Huberty 1994 emphasize the reason for conducting a multivariate analysis instead of a series of univariate analyses. Tabachnick and Fidell 2012 and Stevens 2009 provide comprehensive introductions to understanding and conducting multivariate analysis. Grimm and Yarnold 1995 and Grimm and Yarnold 2000 include introductory chapters on a broad range of multivariate topics written by several experts in the field. Written in a conversational style, Harris 2001 introduces multivariate analysis to the novice researcher, while Johnson and Wichern 2007 provides in-depth chapters for those with stronger statistical backgrounds. Meyers, et al. 2013 presents introductions and step-by-step analysis examples using SPSS (Statistical Package for the Social Sciences). Raykov and Marcoulides 2008 provides several applied examples that include syntax for SPSS, SAS, and MPlus. Loehlin 2004 introduces latent variable models, including factor, path, and structural equation analysis.

• Grimm, L. G., and P. R. Yarnold, eds. 1995. Reading and understanding multivariate statistics. Washington, DC: American Psychological Association.

Written for those with little to no exposure to multivariate analyses but a basic understanding of univariate statistics. Includes chapter topics such as path analysis, multidimensional scaling, analysis of cross-classified data, logistic regression, DA, MANOVA, and meta-analysis. Each chapter includes suggestions for further reading and a glossary of terms and symbols.

• Grimm, L. G., and P. R. Yarnold, eds. 2000. Reading and understanding MORE multivariate statistics. Washington, DC: American Psychological Association.

An extension of the previous edition focusing on conceptual understanding. Includes chapters on generalizability theory, item response theory, assessing validity of measurement, cluster analysis, SEM, CCA, repeated measures analyses, and survival analysis. Each chapter includes suggestions for further reading and a glossary of terms and symbols.

• Harris, R. J. 2001. A primer of multivariate statistics. 3d ed. New York: Psychology Press.

Presented in a conversational style, includes discussion of the theoretical reasoning behind multivariate analysis as well has how the analyses are conducted. Emphasizes the importance of testing the interpretations of latent variables produced through multivariate analyses.

• Huberty, C. J. 1994. Why multivariate analyses? Educational and Psychological Measurement 54:620–627.

Overviews the definition and purpose of multivariate analyses. Emphasizes the importance of choosing the appropriate response variables. Refutes the idea of an obligatory multivariate analysis followed by a series of univariate analysis as protection against Type I error.

• Huberty, C. J., and J. D. Morris. 1989. Multivariate analysis versus multiple univariate analyses. Psychological Bulletin 105:302–308.

Discusses the methodological argument surrounding multiple ANOVAs versus a MANOVA as it relates to Type I error. Emphasizes that analysis decisions should be based on the established research questions. Multivariate analysis is more appropriate for research questions involving outcome variable selection, variable ordering, and variable system structure.

• Johnson, R. A., and D. W. Wichern. 2007. Applied multivariate statistical analysis. 6th ed. Upper Saddle River, NJ: Pearson.

Written from more of a business perspective for those with a stronger statistical background. Presents in-depth chapters on matrix algebra, sampling, and multivariate normal distribution in addition to commonly applied multivariate analyses. Includes examples with SAS syntax and output for two-way MANOVA, PCA, factor analysis, and DA.

• Loehlin, J. C. 2004. Latent variable models: An introduction to factor, path, and structural equation analysis. 4th ed. Mahwah, NJ: Lawrence Erlbaum.

Provides a comprehensive overview of latent variable modeling as it pertains to factor analysis, path analysis, and structural equation modeling. Includes both introductory and advanced topics, accessibly presented using path diagrams, applied examples from published research, and illustrative exercises. Both the novice and more advanced user will find the explanations and extensive references useful.

• Meyers, L. S., G. Gamst, and A. J. Guarino. 2013. Applied multivariate research: Design and interpretation. Los Angeles: SAGE.

Presented using conversational narrative, provides an introduction of multivariate research to the novice to intermediate user. Includes numerous graphical illustrations including path diagrams and segments of output. Readers will find the step-by-step approach with screenshots of SPSS menus and output helpful to conducting and interpreting analysis.

• Raykov, T., and G. A. Marcoulides. 2008. An introduction to applied multivariate analysis. New York: Routledge.

Written as an introductory applied text, covers commonly used multivariate analyses while focusing on necessary methodological considerations such as data screening, meeting assumptions, and missing data. Includes explanations through extensions of univariate concepts. Examples of data analysis in SPSS, SAS, and MPlus included.

• Stevens, J. 2009. Applied multivariate statistics for the social sciences. 5th ed. New York: Routledge.

Comprehensive text covering many background topics such as matrix algebra and multiple regression as well as the major multivariate analyses such as MANOVA, DA, factor analysis, CCA, and multivariate categorical data analysis. Each chapter begins with a brief introduction to the topic and includes example data sets with instructions for conducting the analysis and interpreting output in SPSS and SAS.

• Tabachnick, B. G., and L. S. Fidell. 2012. Using multivariate statistics. 6th ed. Boston: Pearson Education.

A comprehensive applied review including conceptual background and discussion of analytical decisions, as well as a demonstration of how to conduct multivariate analysis. Demonstrations include DA, logistic regression, survival analysis, CCA, PCA, factor analysis, SEM, and multilevel linear modeling. Includes SPSS and SAS syntax and sample output.