How To Analyze Social Work Data

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1 Structural Equation Modeling Natasha K. Bowen and Shenyang Guo Print publication date: 2011 Print ISBN-13: Published to Oxford Scholarship Online: Jan-12 DOI: /acprof:oso/ Title Pages Structural Equation Modeling Pocket Guides to Social Work Research Methods Structural Equation Modeling Structural Equation Modeling Determining Sample Size Balancing Power, Precision, and Practicality Patrick Dattalo Preparing Research Articles Bruce A. Thyer Systematic Reviews and Meta-Analysis Julia H. Littell, Jacqueline Corcoran, and Vijayan Pillai Historical Research Elizabeth Ann Danto Confirmatory Factor Analysis Donna Harrington Randomized Controlled Trials Design and Implementation for Community-Based Psychosocial Interventions Page 1 of 5 monograph in OSO for personal use (for details see Subscriber: Otterbein University; date: 21 June 2013 Title Pages

2 Phyllis Solomon, Mary M. Cavanaugh, and Jeffrey Draine Needs Assessment David Royse, Michele Staton-Tindall, Karen Badger, and J. Matthew Webster Multiple Regression with Discrete Dependent Variables John G. Orme and Terri Combs-Orme Developing Cross-Cultural Measurement Thanh V. Tran Intervention Research Developing Social Programs Mark W. Fraser, Jack M. Richman, Maeda J. Galinsky, and Steven H. Day Developing and Validating Rapid Assessment Instruments Neil Abell, David W. Springer, and Akihito Kamata Clinical Data-Mining Integrating Practice and Research Irwin Epstein Strategies to Approximate Random Sampling and Assignment Patrick Dattalo Analyzing Single System Design Data William R. Nugent Survival Analysis Shenyang Guo The Dissertation From Beginning to End Page 2 of 5 monograph in OSO for personal use (for details see Subscriber: Otterbein University; date: 21 June 2013 Title Pages

3 Peter Lyons and Howard J. Doueck Cross-Cultural Research Jorge Delva, Paula Allen-Meares, and Sandra L. Momper Secondary Data Analysis Thomas P. Vartanian Narrative Inquiry Kathleen Wells Policy Creation and Evaluation Understanding Welfare Reform in the United States Richard Hoefer Finding and Evaluating Evidence Systematic Reviews and Evidence-Based Practice Denise E. Bronson and Tamara S. Davis Structural Equation Modeling Natasha K. Bowen and Shenyang Guo (p.iv) Oxford University Press, Inc., publishes works that further Oxford University s objective of excellence in research, scholarship, and education. Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi Page 3 of 5 monograph in OSO for personal use (for details see Subscriber: Otterbein University; date: 21 June 2013 Title Pages

4 New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Copyright 2012 by Oxford University Press, Inc. Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Bowen, Natasha K. Structural equation modeling / Natasha K. Bowen, Shenyang Guo. p. cm. (Pocket guides to social work research methods) Includes bibliographical references and index. ISBN (pbk.: alk. paper) 1. Social sciences Research Data processing. 2. Social service Research. 3. Structural equation modeling. I. Guo, Shenyang. II. Title. III. Series. H61.3.B dc Page 4 of 5 monograph in OSO for personal use (for details see Subscriber: Otterbein University; date: 21 June 2013 Title Pages

5 Printed in the United States of America on acid-free paper Page 5 of 5 monograph in OSO for personal use (for details see Subscriber: Otterbein University; date: 21 June 2013 Title Pages

6 Structural Equation Modeling Natasha K. Bowen and Shenyang Guo Print publication date: 2011 Print ISBN-13: Published to Oxford Scholarship Online: Jan-12 DOI: /acprof:oso/ Acknowledgment DOI: /acprof:oso/ The authors thank Kristina C. Webber for her many wise and helpful contributions to this book, and the University of North Carolina s School of Social Work for giving us the opportunity to teach PhD students about structural equation modeling. Page 1 of 1 monograph in OSO for personal use (for details see Subscriber: Otterbein University; date: 21 June 2013 Acknowledgment

7 Structural Equation Modeling Natasha K. Bowen and Shenyang Guo Print publication date: 2011 Print ISBN-13: Published to Oxford Scholarship Online: Jan-12 DOI: /acprof:oso/ Introduction Natasha K. Bowen, Shenyang Guo DOI: /acprof:oso/ Abstract and Keywords This introductory chapter first sets out the purpose of the book, which is to serve as a concise practical guide for the informed and responsible use of structural equation modeling (SEM). It is designed for social work faculty, researchers, and doctoral students who view themselves more as substantive experts than statistical experts, but who need to use SEM in their research. It is designed for social workers who desire a degree of analytical skill but have neither the time for coursework nor the patience to glean from the immense SEM literature the specifics needed to carry out an SEM analysis. The chapter then discusses what is SEM, the role of theory in SEM, the kinds of data that can or should be analyzed with SEM, and the research questions best answered by SEM. Keywords: structural equation modeling, SEM, social work research, analytical skill Rationale and Highlights of the Book Social work practitioners and researchers commonly measure complex patterns of cognition, affect, and behavior. Attitudes (e.g., racism), cognitions (e.g., self-perceptions), behavior patterns (e.g., aggression), social experiences (e.g., social support), and emotions (e.g., depression) are complex phenomena that can neither be observed directly nor measured accurately with only one questionnaire item. Measuring such phenomena with multiple items is necessary, therefore, in most social work contexts. Often, scores from the multiple items used to measure a construct are Page 1 of 13 Introduction monograph in OSO for personal use (for details see Subscriber: Washington University in St. Louis; date: 21 June 2013

8 combined into one composite score by summing or averaging. The new composite score is then used to guide practice decisions, to evaluate change in social work clients, or in research contexts, is entered as a variable in statistical analyses. Structural equation modeling (SEM) offers a highly desirable alternative to this approach; it is arguably a mandatory tool for researchers developing new measures. In sum, SEM is highly recommended for social work researchers who use or develop multiple-item measures. Using SEM will improve the quality and rigor of research involving such measures, thereby increasing the credibility of results and strengthening the contribution of studies to the social work literature. One barrier to the use of SEM in social work has been the complexity of the literature and the software for the method. SEM software programs vary considerably, the literature is statistically intimidating to many researchers, (p.4) sources disagree on procedures and evaluation criteria, and existing books often provide more statistical information than many social workers want and too little practical information on how to conduct analyses. This book is designed to overcome these barriers. The book will provide the reader with a strong conceptual understanding of SEM, a general understanding of its basic statistical underpinnings, a clear understanding of when it should be used by social work researchers, and step-by-step guidelines for carrying out analyses. After reading the book, committed readers will be able to conduct an SEM analysis with at least one of two common software programs, interpret output, problem-solve undesirable output, and report results with confidence in peer-reviewed journal articles or conference presentations. The book is meant to be a concise practical guide for the informed and responsible use of SEM. It is designed for social work faculty, researchers, and doctoral students who view themselves more as substantive experts than statistical experts, but who need to use SEM in their research. It is designed for social workers who desire a degree of analytical skill but have neither the time for coursework nor the patience to glean from the immense SEM literature the specifics needed to carry out an SEM analysis. Although the book focuses on what the typical social work researcher needs to know to conduct his or her own SEM analyses competently, it also provides numerous references to more in-depth treatments of the topics covered. Because of this feature, readers with multiple levels of skill and statistical fortitude can be accommodated in their search for greater understanding of SEM. At a minimum, however, the book assumes that readers are familiar with basic statistical concepts, such as mean, variance, explained and unexplained Page 2 of 13 Introduction monograph in OSO for personal use (for details see Subscriber: Washington University in St. Louis; date: 21 June 2013

9 variance, basic statistical distributions (e.g., normal distributions), sum of squares, standard deviation, covariance and correlation, linear regression, statistical significance, and standard error. Knowledge of exploratory factor analysis, matrix algebra, and other more advanced topics will be useful to the reader but are not required. Highlights of the book include: (a) a focus on the most common applications of SEM in research by social workers, (b) examples of SEM research from the social work literature, (c) information on best practices in SEM, (d) how to report SEM findings and critique SEM articles, (e) a chronological presentation of SEM steps, (f) strategies for addressing common social work data issues (e.g., ordinal and nonnormal data), (g) information (p.5) on interpreting output and problem solving undesirable output, (h) references to sources of more in-depth statistical information and information on advanced SEM topics, (i) online data and syntax for conducting SEM in Amos and Mplus, and (j) a glossary of terms. In keeping with the goals of the Pocket Guides to Social Work Research Methods series, we synthesize a vast literature into what we believe to be a concise presentation of solid, defensible practices for social work researchers. What is Structural Equation Modeling? SEM may be viewed as a general model of many commonly employed statistical models, such as analysis of variance, analysis of covariance, multiple regression, factor analysis, path analysis, econometric models of simultaneous equation and nonrecursive modeling, multilevel modeling, and latent growth curve modeling. Readers are referred to Tabachnick & Fidell (2007) for an overview of many of these methods. Through appropriate algebraic manipulations, any one of these models can be expressed as a structural equation model. Hence, SEM can be viewed as an umbrella encompassing a set of multivariate statistical approaches to empirical data, both conventional and recently developed approaches. Other names of structural equation modeling include covariance structural analysis, equation system analysis, and analysis of moment structures. Developers of popular software packages for SEM often refer to these terms in the naming of the programs, such as Amos, which stands for analysis of moment structures; LISREL, which stands for linear structural relations; and EQS, which stands for equation systems. A number of software programs can be used for SEM analyses. See Box 1.1 for citations and links for Amos, EQS, LISREL, and Mplus, four SEM programs commonly used by social workers. Page 3 of 13 Introduction monograph in OSO for personal use (for details see Subscriber: Washington University in St. Louis; date: 21 June 2013

10 This book provides instructions and online resources for using Amos and Mplus, each of which has distinct advantages for the social work researcher. The general principles covered, however, apply to all SEM software. For social work researchers, SEM may most often be used as an approach to data analysis that combines simultaneous regression equations and factor analysis (Ecob & Cuttance, 1987). Factor analysis models test hypotheses about how well sets of observed variables in an existing dataset measure latent constructs (i.e., factors). Latent constructs represent (p.6) theoretical, abstract concepts or phenomena such as attitudes, behavior patterns, cognitions, social experiences, and emotions that cannot be observed or measured directly or with single items. Factor models are also called measurement models because they focus on how one or more latent constructs are measured, or represented, by a set of observed variables. Confirmatory factor analysis (CFA) in the SEM framework permits sophisticated tests of the factor structure and quality of social work measures. (Shortly we will provide examples and much more detail about the terms being introduced here.) Latent variables with adequate statistical properties can then be used in cross-sectional and longitudinal regression analyses. Box 1-1 Examples of SEM Software Programs Used by Social Work Researchers The following four programs are widely used for SEM analyses: Amos (Arbuckle, , ). Website: EQS (Bentler & Wu, 1995; Bentler & Wu, 2001). Website: LISREL (Jöreskog & Sörbom, 1999; Sörbom & Jöreskog, 2006). Website: Mplus (Muthén & Muthén, ; Muthén & Muthén, 2010). Website: Page 4 of 13 Introduction monograph in OSO for personal use (for details see Subscriber: Washington University in St. Louis; date: 21 June 2013

11 Regression models test hypotheses about the strength and direction of relationships between predictor variables and an outcome variable. Unlike standard regression models, SEM accommodates regression relationships among latent variables and between observed and latent variables. Unlike conventional regression models, SEM can estimate in a single analysis procedure models in which one or more variables are simultaneously predicted and predictor variables. Structural equation models with directional relationships among latent variables are often called general structural equation models (general SEMs). In sum, SEM is a general statistical approach with many applications. Over the past two decades, statistical theories and computing software packages for SEM have developed at an accelerated pace. Newer SEM approaches include methods for analyzing latent classes cross-sectionally and over time (mixture modeling), and latent growth curve modeling (Bollen & Curran, 2006). Consistent with the goals of the pocket guides, (p.7) this book focuses on a manageable subset of SEM topics that are relevant to social work research. Specifically, we focus on SEM s most common social work applications confirmatory factor analysis and cross-sectional structural models with latent variables. In addition, we focus on proper methods for addressing common data concerns in social work research, ordinal-level data, nonnormal data, and missing data. The Role of Theory in Structural Equation Modeling The primary goal of an SEM analysis is to confirm research hypotheses about the observed means, variances, and covariances of a set of variables. The hypotheses are represented by a number of structural parameters (e.g., factor loadings, regression paths) that is smaller than the number of observed parameters. As a confirmatory approach, it is crucial for researchers using SEM to test models that have strong theoretical or empirical foundations. Nugent and Glisson (1999), for example, operationalized two ways children s service systems might respond to children: either as responsive or reactive systems. Responsive systems, the ideal, were defined as [quick] to respond appropriately or sympathetically to each child s specific mental health needs (p. 43). Reactive systems were operationalized as those that refuse to provide services, provide disruptive services, or otherwise fail to provide children with needed mental health treatments. With well-defined hypotheses based on previous research, the authors tested the nature of services provided in 28 counties in one state and the relationship between reactivity and Page 5 of 13 Introduction monograph in OSO for personal use (for details see Subscriber: Washington University in St. Louis; date: 21 June 2013

12 responsiveness of the systems. Similarly, confirmatory factor analyses should be based on theory and/or the results of exploratory factor analyses and other psychometric tests. SEM models are commonly presented in path diagrams. The path diagram is a summary of theoretically suggested relationships among latent variables and indicator variables, and directional (regression) and nondirectional (i.e., correlational) relationships among latent variables. Importantly, correlated errors of measurement and prediction can also be modeled in SEM analyses. We emphasize throughout the book that having a theoretical model and/or theory-derived constructs prior to any empirical modeling is mandated for both CFA and structural modeling with latent variables. (p.8) Path diagrams are graphics with geometric figures and arrows suggesting causal influences. SEM, however, has no better ability to identify causal relationships than any other regression or factor analytic procedure. Cross-sectional SEMs reveal associations among variables (one criterion for causality), and repeated measures in SEM can model time order of variables (another criterion for causality), but SEM in and of itself cannot definitively rule out other potential explanations for relationships among variables (the third criterion for establishing causality). The arrows in SEM illustrations reflect hypothesized relationships based on theory and previous research. SEM results may or may not provide support for the theory being tested, but they cannot prove or disprove theory or causality. Reversing the direction of arrows in any SEM may yield equally significant parameter estimates and statistics on model quality. For another brief treatment of this subject, see Fabrigar, Porter, and Norris (2010). These authors point out that although SEM cannot compensate for a nonexperimental design, it can be a useful analysis technique for experimental data and can be superior to other techniques with quasi-experimental data for ruling out competing causes of intervention outcomes. Because models proposing opposite effects can yield similar statistics, it is a common and desirable practice to test alternative models in SEM. Good model statistics for an SEM model support its validity; model statistics that are superior to those obtained for a competing model provide valuable additional credibility. But neither establishes causality nor proves theory. Using experimental or quasi-experimental designs or statistical models specially developed for observational data in research studies remains the best way to identify causal effects. Page 6 of 13 Introduction monograph in OSO for personal use (for details see Subscriber: Washington University in St. Louis; date: 21 June 2013

13 What Kinds of Data Can or Should Be Analyzed with SEM? Ideally, SEM is conducted with large sample sizes and continuous variables with multivariate normality. The number of cases needed varies substantially based on the strength of the measurement and structural relationships being modeled, and the complexity of the model being tested. CFA models and general SEM with strong relationships among variables (e.g., standardized values of 0.80), for example, with all else (p.9) being equal, can be tested with smaller samples than models with weak relationships (e.g., standardized values of 0.20) among variables. Sample size and statistical power are discussed further in Chapters 3 and 7. Social workers often work with variables that are ordinal and/or nonnormally distributed, and datasets containing missing values. SEM software provides a number of satisfactory options for handling data with these statistically undesirable characteristics. In addition to its advantages over traditional regression approaches, therefore, SEM software provides solutions to common social work methodological issues that, if ignored, reduce the quality of social work studies, and consequently, the literature used to guide social work practice. What Research Questions Are Best Answered with SEM? Examples from Social Work Studies Measurement Questions Answered with SEM Measurement questions relate to the reliability and validity of data collected with questionnaires, checklists, rating sheets, interview schedules, and so on. SEM s ability to model sets of questions as indicators of hypothesized latent constructs (such as depression, social support, attitudes toward health care, organizational climate) provides a number of major statistical advantages, which will become evident later. Questions about the quality of multiple items as indicators of one or more dimensions of a construct are factor analysis questions. The questions answered by CFA differ from those answered by exploratory factor analysis (EFA) procedures. As implied in the title, confirmatory factor analysis is used to test the adequacy of a well-defined model. The specified model is predetermined by theory or past research. The questions asked are closed ended: Do these indicators measure the phenomenon well? Do the data support the existence of multiple dimensions of the phenomenon, Page 7 of 13 Introduction monograph in OSO for personal use (for details see Subscriber: Washington University in St. Louis; date: 21 June 2013

14 each measured by prespecified items? EFA is used earlier in the scale development process to answer more open-ended questions for example, how many dimensions of the phenomenon are represented by these items? Which items are associated with each dimension? More about the distinction between EFA and CFA and their roles in the scale development process will be presented in Chapter 4. (p.10) CFA provides answers to questions about the structure of latent phenomena (e.g., the nature and number of dimensions), and the individual and collective performance of indicators. For example, researchers in one study (Bride, Robinson, Yegidis, & Figley, 2004) used data from 287 social workers who completed the Secondary Traumatic Stress Scale (STSS) to validate the scale as a measure of indirect trauma. Items on the STSS assess dimensions of traumatic stress as defined in the diagnostic criteria for posttraumatic stress disorder in the Diagnostic and Statistical Manual of Mental Disorders (American Psychiatric Association, 1994). Therefore, the hypothesized factor structure was derived from a strong foundation in theory and previous research. The results of the researchers CFA provided answers to the following measurement questions: 1. Did the items measure the three hypothesized dimensions of trauma symptomatology? Yes, each of the 17 items on the scale was associated with the one dimension of trauma it was hypothesized to measure and not strongly associated with the other two dimensions it was not hypothesized to measure. 2. How well did each indicator perform? Factor loadings were moderate to high (0.58 to 0.79) and statistically significant. The size of the factor loadings indicates which items are most strongly related to each dimension. 3. How good was the model overall? The model explained 33% to 63% of the variance of each indicator, which is reasonable according to Bride et al. (2004). Other measures of the quality of the model met or exceeded standard criteria. 4. How highly correlated were the three dimensions of trauma symptomatology? Intercorrelations of the three dimensions ranged from 0.74 to 0.83 and were statistically significant. These correlations are consistent with theory and previous research about the components of trauma, according to the authors. Bride et al. (2004) did not report the variances of the latent variables associated with the three dimensions of trauma symptoms in their model, Page 8 of 13 Introduction monograph in OSO for personal use (for details see Subscriber: Washington University in St. Louis; date: 21 June 2013

15 but CFA results do indicate the magnitude of variances and whether they are statistically significantly different from zero. Subscales with little variance are not useful in practice, so it is important to examine these variance estimates in SEM output. (p.11) Like Bride et al. (2004), social workers may use CFA as a final test in a process of developing a new scale. Another important measurement question for social workers that can be answered with CFA is whether measures have the same meaning for different groups and over time (Maitland, Dixon, Hultsch, & Hertzog, 2001, p. 74). If scores on a measure are compared for individuals from different populations (e.g., of different ages, gender, cultural backgrounds) or for the same individuals over time, it is critical to establish that the scores obtained from different groups or at different times have the same meaning. Maitland et al. (2001) used CFA to study the measurement equivalence or invariance of the Bradburn Affect Balance Scale (Bradburn ABS) across gender and age groups and over time. The researchers found that a small number of items from the two-dimension scale performed differently across groups and time, leading them to conclude that comparisons of scores across groups and time from past and future studies needed to be interpreted cautiously. Observed group and longitudinal differences in positive and negative affect could be partly attributed to variations in item performance rather than differences in the true scores for affect. Structural Questions Answered with SEM Relationships among latent variables (or factors) and other variables in an SEM model are structural relationships. Structural questions relate to the regression and correlational relationships among latent variables and among latent and observed variables. SEM structural models can include any combination of latent variables and observed variables. Observed demographic variables can be included as covariate or control variables, for example, in a model with latent independent and dependent variables. As with CFA models, all variables and relationships in structural models should be justifiable with theory and/or previous research. SEM permits simultaneous regression equations, that is, equations in which one variable can serve as both an independent and a dependent variable. It is therefore a valuable tool for testing mediation models, that is, models in which the relationship between an independent variable and a dependent variable is hypothesized to be partially or completely explained by a third, Page 9 of 13 Introduction monograph in OSO for personal use (for details see Subscriber: Washington University in St. Louis; date: 21 June 2013

16 intervening variable. It also permits tests of models in which there are multiple dependent variables. In Nugent and Glisson s (1999) model of predictors of child service system characteristics, for example, system reactivity and system responsivity were simultaneously (p.12) predicted by all other variables in the model (either directly, indirectly, or both). SEM is also a useful framework for testing moderation (interaction) models, or models in which the effects of one variable on another vary by the values or levels of a third variable. It provides more detailed output about moderation effects than typical regression procedures. In multiple regression, for example, moderation effects are obtained by creating product terms of the variables that are expected to interact (e.g., gender stress). The results indicate the magnitude, direction, and statistical significance of interaction terms. In an SEM analysis, in contrast, the estimate and statistical significance of each parameter for each group (e.g., boys and girls) can be obtained, and differences across groups can be tested for statistical significance. Every parameter or any subset of parameters can be allowed to vary across groups, while others are constrained to be equal. The quality of models with and without equality constraints can be compared to determine which is best. Such information is useful for determining the validity of measures across demographic or developmental groups. A study by Bowen, Bowen, and Ware (2002) provides examples of the flexibility of SEM to answer structural questions. The study examined the direct and indirect effects of neighborhood social disorganization on educational behavior using self-report data from 1,757 adolescents. Supportive parenting and parent educational support were hypothesized mediators of the relationship between neighborhood characteristics and educational behavior. Race/ethnicity and family poverty were observed control variables in the model. The rest of the variables in the structural model were latent. The authors hypothesized that the magnitude of the direct and indirect effects in the model would be different for middle and high school students a moderation hypothesis based on past research. Results of the analysis answered the following structural questions: 1. Did neighborhood disorganization have a direct effect on educational behavior? Yes, negative neighborhood characteristics had a statistically significant moderate and negative direct effect on adolescents educational behavior. 2. Was the effect of neighborhood disorganization on educational behavior mediated by parental behaviors (supportive parenting and parent educational behavior)? Page 10 of 13 Introduction monograph in OSO for personal use (for details see Subscriber: Washington University in St. Louis; date: 21 June 2013

17 Yes, the effect was partially (p.13) mediated by a three-part path with statistically significant coefficients between neighborhood disorganization and supportive parenting (negative), between supportive parenting and parent educational support (positive), and between parent educational support and educational behavior (positive). 3. Were race/ethnicity and family poverty predictive of educational behavior? No. Race/ethnicity and family poverty were significantly correlated with each other and with neighborhood disorganization, but the regression path between each observed variable and the dependent variable was not statistically significant. 4. Did the structural paths differ for middle and high school students as hypothesized? No. The moderation hypothesis was not supported. The relationships among the constructs were statistically equivalent for adolescents at both school levels. 5. How good was the model overall? Multiple measures of the quality of the final model met or exceeded standard criteria. As with traditional regression analyses, SEM results indicate the percent of variance of dependent variables explained by predictor variables. In this study: 14% to 33% of the variance of the mediators was explained, and 34% to 44% of educational behavior was explained. It bears repeating that even when SEM models are grounded in theory and previous research, support for models in the form of statistically significant regression paths, factor loadings, and correlations, and good overall model fit does not prove that the model or the theory from which it is derived is correct. Nor does such support indicate causality. Such support, as we will discuss in more detail later, can only be interpreted as consistency with the observed data used to test the model. SEM as a Useful and Efficient Tool in Social Work Research Many challenging questions confronted by social work researchers can be answered efficiently, effectively, and succinctly by SEM. SEM is often the best choice for social work analyses given the nature of their measures (p.14) and data. The topics and characteristics of SEM articles in a sampling of social work journals were examined by Guo and Lee (2007). The authors reviewed all articles published during the period of January 1, 1999 to December 31, 2004 in the following eight social work or social-workrelated journals: Child Abuse & Neglect, Journal of Gerontology Series B: Psychological Sciences and Social Sciences, Journal of Social Service Page 11 of 13 Introduction monograph in OSO for personal use (for details see Subscriber: Washington University in St. Louis; date: 21 June 2013

18 Research, Journal of Studies on Alcohol, Research on Social Work Practice, Social Work Research, Social Work, and Social Service Review. During the 6- year period, Social Work and Social Service Review published no studies that employed SEM. A total of 139 articles using SEM were published by the seven remaining journals that were examined. Table 1.1 summarizes the 139 SEM publications by substantive areas and types of SEM. As the table shows, the majority of SEM applications in the targeted social work journals were general structural models (54.7%). The finding is not surprising because many social work research questions concern theoretically derived relationships among concepts that are best measured with latent variables. The second most common type of SEM was CFA (33.1%). Again, this finding is reasonable because developing measures of unobservable constructs is a primary task of Table 1.1 SEM Applications by Social Work Research Area and SEM Type Substantive area CFA General structural models Path analysis Total Aging Child welfare Health/Mental health School social work Substance abuse % 64.4% 15.6% 100% % 78.6% 7.1% 100% % 18.5% 7.4% 100% % 33.3% 33.3% 100% % 61.7% 10.6% 100% Total Total % 33.1% 54.7% 12.2% 100% (p.15) social work research. The remaining SEM articles reported on studies using path analysis (12.2%). Path analysis is useful for examining simultaneous regression equations among observed variables but does not exploit fully the advantages of SEM. In addition, it is possible (albeit more difficult) to obtain many of the results of a path analysis with more Page 12 of 13 Introduction monograph in OSO for personal use (for details see Subscriber: Washington University in St. Louis; date: 21 June 2013

19 conventional analyses and software. Therefore, it makes sense that fewer social work articles used path analysis than the two SEM procedures with latent variables. Across substantive areas, the proportion of studies using different types of SEM varied, with general structural models more common in the fields of child welfare, aging, and substance abuse. CFA was the most common type of analysis used in SEM studies of health and mental health. The Guo and Lee (2007) study indicated that SEM was being used by researchers in many major topical areas of social work research. It is hoped that by the end of this book, readers will agree that SEM is the most appropriate analysis tool for much of the research done by social researchers. Page 13 of 13 Introduction monograph in OSO for personal use (for details see Subscriber: Washington University in St. Louis; date: 21 June 2013

20 Structural Equation Modeling Natasha K. Bowen and Shenyang Guo Print publication date: 2011 Print ISBN-13: Published to Oxford Scholarship Online: Jan-12 DOI: /acprof:oso/ Structural Equation Modeling Concepts Natasha K. Bowen, Shenyang Guo DOI: /acprof:oso/ Abstract and Keywords This chapter discusses a number of theoretical and statistical concepts and principles that are central to SEM. It introduces SEM notation and equations in the context of more familiar graphics and terminology. It explains the role of matrices in SEM analyses. Keywords: structural equation modeling, SEM, social work research, SEM notation, equations In this chapter we discuss in detail a number of theoretical and statistical concepts and principles that are central to SEM. SEM notation and equations are introduced in the context of more familiar graphics and terminology. The role of matrices in SEM analyses is explained. The material in this chapter is essential to understanding the more detailed treatment of topics in later chapters, but later chapters also reinforce and help illustrate concepts introduced here. Iacobucci (2009) also provides a complementary and instructive summary of SEM notation and its relationship to the matrices. For more in-depth information on basic statistical concepts, refer to a social science statistics text (e.g., Cohen & Cohen, 1983; Pagano, 1994; Rosenthal, 2001). More advanced treatment of the statistical foundations of SEM can be found in Bollen (1989), Long (1983), and Kaplan (2009), and among other SEM texts in the reference list. Page 1 of 33 monograph in OSO for personal use (for details see Subscriber: University of Toronto Libraries; date: 21 June 2013 Structural Equation Modeling Concepts

21 Latent Versus Observed Variables Latent variable is a central concept in SEM. Latent variables are measures of hidden or unobserved phenomena and theoretical constructs. In social (p.17) work, latent variables represent complex social and psychological phenomena, such as attitudes, social relationships, or emotions, which are best measured with multiple observed items. Many terms for latent variables are encountered in the SEM literature, for example, factors, constructs, measures, or dimensions. In contrast, observed variables are variables that exist in a database or spreadsheet. They are variables whose raw scores for sample members can be seen, or observed, in a dataset. Observed variables may comprise scores from survey items or interview questions, or they may have been computed from other variables (e.g., a dichotomous income variable obtained by categorizing a continuous measure of income). Individual observed variables may be called items, indicators, manifest items, variables, questionnaire items, measures, or other terms in different sources. The observed items that measure latent variables may collectively be called a scale, subscale, instrument, measure, questionnaire, etc. The use of terms is not always consistent. The main point, however, is that observed variables come from raw data in data files. We ll see later that the actual input data for SEM is usually the covariance matrix derived from a set of indicators. We follow Bollen (1989) in making a critical distinction between the terms scale and index. Note that this distinction is not made consistently in the literature! The latent variable modeling that is the subject of this book specifically involves scales, which in our conceptualization, are used to measure unobserved phenomena that cause scores on multiple, correlated indicators (Bollen). An underlying workplace climate will cause employees to respond in a generally negative or positive way to a set of indicators on a workplace support scale. In contrast, indicators of indices cause scores on the index and are not necessarily highly correlated. Checking off items on a list (index or inventory) of life stressors, for example, might lead to an individual s high score on the index, but experiencing the death of a close family member, trouble with boss, or pregnancy are not necessarily or on average correlated or caused by some underlying phenomenon (Holmes & Rahe, 1967). Scores on indices are not driven by latent phenomena so are not of interest here. The distinction made between latent and observed variables represents a fundamental difference between SEM and conventional regression modeling. Page 2 of 33 monograph in OSO for personal use (for details see Subscriber: University of Toronto Libraries; date: 21 June 2013 Structural Equation Modeling Concepts

22 In the SEM framework, latent variables are of interest but cannot be directly measured. Observed variables are modeled as functions of model-specific latent constructs and latent measurement errors. (p.18) In this framework, researchers are able to isolate true causes of scores and variations in scores due to irrelevant causes. Tests of relationships among the resulting latent variables are therefore superior to tests among variables containing irrelevant variance (i.e., error variance). As we have described, latent variables are measured indirectly through multiple observed variables. Researchers (Glisson, Hemmelgarn, & Post, 2002), for example, examined the quality of a 48-item instrument called the Shortform Assessment for Children (SAC) as a measure of overall mental health and psychosocial functioning (p. 82). The instrument includes 48 items, 24 of which are hypothesized to represent an internalizing dimension or factor, and 24 of which represent an externalizing dimension of mental health and psychosocial functioning. The internalizing items relate to affect, psychosomatic complaints, and social engagement. In this example, internalizing behavior is a latent (hidden, unobservable) phenomenon with a continuum of values. Each person is believed to have a true but unknowable score on a continuum of internalizing behavior. This internal personal truth is believed to largely determine each person s scores on the set of direct questions about emotion, psychosomatic complaints, and social engagement. Observed scores derived from responses to the instrument s questions are expected to be correlated with each other because they are all caused by each respondent s true, unobservable internalizing status. Similarly, in the study by Bride et al. (2004), social workers differing experiences with the latent phenomenon indirect trauma were expected to influence their responses to the 17 items on the STSS. Scores on the items are expected to be correlated with each other and with the latent variable because they are caused by the same experience. If a worker s exposure to indirect trauma has been low, responses to all 17 items are expected to reflect that level of exposure. Overall and in general, if a worker s exposure to indirect trauma is high, his or her scores on all items should reflect that reality. Latent constructs also apply to characteristics of organizations. In a study of turnover among employees of child welfare agencies, for example, researchers (McGowan, Auerbach, & Strolin-Goltzman, 2009) describe constructs such as clarity and coherence of practice, technology, training, and record keeping, and job supports and relationships. In another study using SEM, Jang (2009) also used measures of workplace characteristics, Page 3 of 33 monograph in OSO for personal use (for details see Subscriber: University of Toronto Libraries; date: 21 June 2013 Structural Equation Modeling Concepts

23 for example, perceived supervisory support, and perceived workplace support. The assumption behind such measures (p.19) is that some true but unobservable characteristic of an organization will systematically affect the responses of individuals within the organization to questions related to those characteristics. In the SEM framework, the presence and nature of a latent variable such as indirect trauma exposure or perceived workplace support is inferred from relationships (correlations or covariances) among the scores for observed variables chosen to measure it. Specifically, one starts with known information e.g., a covariance between two observed variables and applies statistical principles to estimate the relationship of each indicator to the hypothesized latent variable. If we hypothesize the existence of the latent variable ability, shown in Figure 2.1, for example, and we know from the questionnaire responses of 200 subjects that the correlation between items Q1 and Q2 is 0.64, we know (from measurement theory) that the product of the standardized paths from ability to Q1 and Q2 equals 0.64 (DeVellis, 2003). If we assumed that the two observed variables are influenced equally by the latent variable ability, we would know that the path coefficients were both 0.80 (because = 0.64). Squaring the path coefficients also indicates the Figure 2.1 Calculating the Relationships of Observed Variables to a Latent Variable. (p.20) amount of variance of each indicator explained by the latent variable 64% in example in Figure 2.1. Because the explained and unexplained variance of a variable must equal 100%, we also know how much of the variance of each indicator is error (unexplained variance) (d1 or d2; 36% in the example). The variance of the error term is the difference between Page 4 of 33 monograph in OSO for personal use (for details see Subscriber: University of Toronto Libraries; date: 21 June 2013 Structural Equation Modeling Concepts

24 100% and the amount of variance explained by ability (Long, 1983). In other words, 36% of the variance of Q1 is error variance, or variance that is unrelated to the construct of interest, ability. Given the correlation between Q1 and Q2 and the magnitude of the relationship between the unobserved construct ability and observed scores on Q1 and Q2, it is possible to estimate scores for subjects on the new latent variable ability and the variance of those scores. This illustration is simplified, but the process of working backward from known relationships (usually covariances among observed variables) to estimates of unknown parameters is a central notion in SEM. In this discussion, we have illustrated an important property of SEM, that is, the product of the standardized path coefficients (i.e., 0.80 and 0.80) from one latent variable to two observed variables equals the correlation (i.e., 0.64) of the observed variables. In Box 2.1, we provide a proof of the property, which was developed by Spearman in 1904, marking the birth of SEM. In any SEM, researchers have observed data, such as a known correlation of The known (or observed) data are used to estimate path coefficients, such as the two coefficients reflecting the net influence of ability on Q1 and Q2. Of course, the estimation becomes more complicated when there are multiple correlations or covariances as input data, latent variable effects are not assumed to be the same on all indicators, there are more than two indicators of a latent variable, and so on. In more complicated models, in fact, more than one solution is possible more than one set of parameters might satisfy the multiple equations defining the model. An important component of the analysis therefore becomes determining which solution is the best. We will examine that issue more thoroughly shortly. Parts of a Measurement Model We will now look more closely at the statistical and conceptual foundations of a measurement model building on the terms introduced in the (p.21) (p.22) previous section. In this section, and throughout the rest of the book, we will employ the common practice of using Greek notation to refer to specific elements in the models presented. For example, using Greek notation, error terms are indicated by δ (delta), rather than the d used in Figure 2.1. Readers are encouraged to refer to the guide to Greek notation provided in the Appendix 1 for an explanation of all symbols used. The notation for SEM equations, illustrations, and matrices varies across sources. We present one set of notations that we believe minimizes confusion across Page 5 of 33 monograph in OSO for personal use (for details see Subscriber: University of Toronto Libraries; date: 21 June 2013 Structural Equation Modeling Concepts

25 measurement and structural examples, but readers should be aware that they will encounter other notation protocols in other sources. Box 2-1 Proof of an SEM Property and a First Peek at SEM Notation In Spearman s original work, he claimed that observed intercorrelations among scores on tests of different types of mental ability could be accounted for by a general underlying ability factor. Using our current example, we can imagine that the general ability factor affecting all test scores is the latent variable ability. Scores on Q1 and Q2 in this example represent observed scores on two mental ability subtests. Variance in Q1 and Q2 that is not explained by ability is captured in d1 and d2, respectively. Denoting the two path coefficients (now called factor loadings) as λ 1 and λ 2 (lambda 1 and lambda 2), Spearman proved that the observed correlation between Q1 and Q2 (i.e., ρ 12 ) equals the product of the two factor loadings λ 1 and λ 2, or ρ 12 = λ 1 λ 2, or 0.64 = 0.80 * To prove this, we first express our model of Figure 2.1 in the following equations: Assuming we work with standardized scores for all variables, then the correlation ρ 12 is simply the covariance of Q1 and Q2, or ρ 12 = Cov(Q1, Q2). Using the algebra of expectations, we can further write Because E(Abilityd2) = 0 and E(Abilityd1) = 0 (because there is no correlation between the common factor Ability and each error), and E(d1d2 = 0) (because the two measurement errors are not correlated), then the equation becomes ρ 12 = λ 1 λ 2 E(Ability 2 ). Because E(Ability 2 ) is Variance(Ability) and equals 1 (because Ability is a standardized score), then ρ 12 = λ 1 λ 2. That is, the observed correlation between two variables is a product of two path coefficients. Figure 2.2 presents a simple CFA model using common symbols. The model has three latent variables: Risk1, Risk2, and Behavior. Latent variables are indicated by circles or ovals. Because they are latent, by definition the three variables do not exist in a dataset. They are hidden, unobservable, theoretical variables. In the model, each is hypothesized to have three Page 6 of 33 monograph in OSO for personal use (for details see Subscriber: University of Toronto Libraries; date: 21 June 2013 Structural Equation Modeling Concepts

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