Psychology Path Models
by
Rex B. Kline
  • LAST MODIFIED: 29 May 2015
  • DOI: 10.1093/obo/9780199828340-0173

Introduction

Path analysis belongs to the family of statistical techniques known as structural equation modeling (SEM). Path analysis is the original—and thus the oldest—SEM technique. It can be viewed as an extension of regression techniques to the analysis of directed (causal) effects among a set of variables. Presumed causal effects are specified according to a particular theory, and such hypotheses can be represented graphically in the form of a path model, also known as a structural model of observed variables. But unlike standard regression techniques where the roles of predictor versus criterion are theoretically interchangeable, path models represent directional effects between an outcome variable and its presumed causes, and such effects are not arbitrarily reversible, because they are viewed in a particular theory as properties of nature. The basic logic of path analysis was developed by the geneticist Sewall Wright beginning in the 1920s. For two reasons, Wright’s work on path analysis had relatively little impact in biology at the time. First, the very notion of causality as a scientific construct in biology was usurped by that of statistical association, owing to Karl Pearson’s elaboration on the method of correlation in the late 1890s and early 1900s. Second, Ronald Fisher’s statistical methods from the 1920s, based on the method of analysis of variance applied in the context of experimental designs, appeared to offer a more comprehensive framework for causal inference than Wright’s theory of path coefficients. Fisher also viewed randomization and experimental control as the only real basis for causal inference. The two schools of thought just mentioned no longer predominate, but at the time their proponents resisted Wright’s method.

General Overviews

The basic rationale of path analysis was outlined in several key works by Sewall Wright (Wright 1920, Wright 1921, Wright 1934). An early critical work, Niles 1922, exemplifies the opposition in biology and genetics to path analysis at that time; see also Wright’s response, Wright 1923. Wolfle 1999 is an annotated bibliography that traces the early development of path analysis, and the same history is described in a more narrative form in Shipley 2000.

  • Niles, Henry E. 1922. Correlation, causation and Wright’s theory of “path coefficients.” Genetics 7:258–273.

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    Criticized the philosophical notion of ascribing to causation any basis beyond that implied by statistical association; that is, causation is correlation in this perspective. Also objected to some of Wright’s equations in earlier works, but these formulae are actually correct.

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    • Shipley, Bill. 2000. Cause and correlation in biology: A user’s guide to path analysis, structural equations and causal inference. New York: Cambridge Univ. Press.

      DOI: 10.1017/CBO9780511605949Save Citation »Export Citation »E-mail Citation »

      Reviews the history of path analysis from its development in genetics to its later adoption in the social sciences in the 1960–1970s. Also considers links between Wright’s path analysis method and more modern approaches to causal inference based on graph theory.

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      • Wolfle, Lee M. 1999. Sewall Wright on the method of path coefficients: An annotated bibliography. Structural Equation Modeling 6:280–291.

        DOI: 10.1080/10705519909540134Save Citation »Export Citation »E-mail Citation »

        Traces Wright’s development of path analysis in works that date from 1918 through 1984, at which point path analysis was becoming well known in the social sciences.

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        • Wright, Sewall. 1920. The relative importance of heredity and environment in determining the piebald pattern of guinea-pigs. Proceedings of the National Academy of Sciences 6:320–332.

          DOI: 10.1073/pnas.6.6.320Save Citation »Export Citation »E-mail Citation »

          One of the first path diagrams is presented in this article, in which the basic rationale of the whole method is described.

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          • Wright, Sewall. 1921. Correlation and causation. Journal of Agricultural Research 20:557–585.

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            Demonstrated in this article are two applications of path analysis, one in the area of weight at weaning of guinea pigs, and the other about the rate of transpiration in plants.

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            • Wright, Sewall. 1923. The theory of path coefficients: A reply to Niles’s criticism. Genetics 8:239–255.

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              Wright responded to Nile’s criticisms by emphasizing the importance of a casual model or theory when estimating correlations between variables of interest.

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              • Wright, Sewall. 1934. The method of path coefficients. Annals of Mathematical Statistics 5:161–215.

                DOI: 10.1214/aoms/1177732676Save Citation »Export Citation »E-mail Citation »

                An even more comprehensive statement of the method of path coefficients, with examples of application of the tracing rule, is offered in this work.

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                Path Analysis in the Social Sciences

                The method of path analysis was relatively unknown outside of biology until about the middle of the 20th century. An exception is Burks 1928, a study of the heritability of intelligence among biological children versus foster children, in which Burks used Wright’s path analysis to evaluate the relative contributions of environment and parental intelligence to child intelligence. Otherwise, the modern era of path analysis begins in the 1950–1960s with the study in sociology of occupational mobility. For example, Goodman 1965 described the analysis of statistical models for relating the occupations of fathers to those of their sons while acknowledging that level of education is a factor. Blalock 1964, a work on causal inference with nonexperimental data, probably spurred interest among sociological researchers in the principles of path analysis. Perhaps the first formal use of path analysis in sociology was in Duncan and Hodge 1963, which analyzed occupations as a continuous variable using regression-type techniques. The econometrician Arthur Goldberger described refinements to path analysis, such as the reporting of standard errors, and cast the whole method in the framework of inferential statistics (Goldberger 1970). In 1970, Goldberger and Otis Dudley Duncan organized an influential conference on path analysis in Madison, Wisconsin—see Goldberger and Duncan 1973—where the statistician K. Jöreskog described the mathematical basis of LISREL, later to be the first widely available computer program for SEM analyses. Jöreskog, along with J. Keesling and D. Wiley, integrated measurement concepts from factor analysis with principles of path analysis into what was later called the JKW framework, now known as SEM. Denis and Legerski 2006 and Wolfle 2003 describe the introduction of path analysis to the social sciences in more detail.

                • Blalock, Hubert. M. 1964. Causal inferences in nonexperimental research. Chapel Hill: Univ. of North Carolina Press.

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                  An influential book that introduced causal modeling techniques, including path analysis, to sociology. It offered better descriptions of the logic of causal inference with nonexperimental data than of statistical principles, some of which nowadays would be considered obsolete. For instance, sample estimates were confused with population parameters.

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                  • Burks, Barbara S. 1928. The relative influence of nature and nurture upon mental development: A comparative study of foster parent–foster child resemblance and true parent–true child resemblance. Yearbook of the National Society for the Study of Education 27:219–316.

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                    One of the first applications of path analysis outside of biology. Analyzed a simple, three-variable model of child intelligence, but took account of measurement error and correlated predictors, just like in a modern analysis.

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                    • Denis, Daniel J., and Joanna Legerski. 2006. Causal modeling and the origins of path analysis. Theory and Science 17.1.

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                      Offers a historical review of the development of path analysis, from Sewall Wright in biology in the 1920s to its later “rediscovery” in the social sciences by the middle of the 20th century.

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                      • Duncan, Otis D., and Robert W. Hodge. 1963. Education and occupational mobility: A regression analysis. American Journal of Sociology 68:629–644.

                        DOI: 10.1086/223461Save Citation »Export Citation »E-mail Citation »

                        Perhaps the first substantive application of path analysis in sociology. Dealt with the estimation of the direct effects of a father’s occupation on a son’s occupation, and with the indirect effects of the former on the latter through a son’s education.

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                        • Goldberger, Arthur S. 1970. On Boudon’s method of linear causal analysis. American Sociological Review 35:97–101.

                          DOI: 10.2307/2093856Save Citation »Export Citation »E-mail Citation »

                          Offered suggestions about ways to improve the statistical rigor of results from path analysis, including the distinction between sample estimators and population parameters and the reporting of standard errors, which quantify the amount of sampling error.

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                          • Goldberger, Arthur S., and Otis Dudley Duncan. 1973. Structural equation models in the social sciences. New York: Seminar Press.

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                            This book consists of papers presented at the 1970 conference in Madison, Wisconsin, on statistical causal models organized by Goldberger and Duncan. The mathematical basis of LISREL was first described by Jöreskog at this conference.

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                            • Goodman, Leo A. 1965. On the statistical analysis of mobility tables. American Journal of Sociology 70:564–585.

                              DOI: 10.1086/223932Save Citation »Export Citation »E-mail Citation »

                              Described the analysis of contingency table data relating the occupations of fathers to those of their sons. Motivated the search for ways to include other variables in the analysis, and for alternative representations of occupational mobility, such as a continuum.

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                              • Wolfle, Lee M. 2003. The introduction of path analysis to the social sciences, and some emergent themes: An annotated bibliography. Structural Equation Modeling 10:1–34.

                                DOI: 10.1207/S15328007SEM1001_1Save Citation »Export Citation »E-mail Citation »

                                This annotated bibliography follows the introduction of path analysis in the social sciences, beginning with sociology in the 1960s and then later in other disciplines such as psychology and education. Many early examples of the application of path analysis are described.

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                                Innovations of Path Analysis

                                Wright’s theory of path coefficients consisted of four essential theoretical and statistical advances. In hindsight, these innovations are remarkable in that they are nearly a century old, but they continue to be refined to this day. Basic insights from path analysis are summarized next: (1) Path models offer a diagrammatic way to represent both causal and noncausal hypotheses. The former correspond to direct or indirect causal effects between variables. Causal effects can be either recursive (or unidirectional, such that no two variables are specified as causing each other) or nonrecursive. The latter refers to feedback loops, or a series of variables connected by causal effects that point back to earlier variables. Noncausal hypotheses concern spurious associations, or the specification that two variables have a common cause, which could give rise to a substantial observed covariation, but one with nothing to do with causality. (2) Path models as graphical representations of hypotheses can be analyzed in their own right. Examples include the generation of all testable implications embodied in a path model and the determination of whether two different models for the same variables generate different sets of testable implications, as explained in Pearl 2009. (3) Wright demonstrated how to express coefficients for direct causal effects in terms of the data, or sample covariances among observed variables, as a series of equations. Solving these equations by hand using Wrights’ rule of tracing—the only option in Wright’s time—or nowadays using a computer leads to unique estimates of each direct causal effect in the model, if all such equations can be mathematically resolved. A path model with this property is referred to as “identified.” Kenny and Milan 2012 describe identification requirements for path models. (4) Indirect causal effects are the basis for the concept of mediation, defined by Little 2013 as the causal hypothesis that changes in one variable causes changes in another variable, which in turn leads to changes in the outcome variable. In other words, mediation refers to an indirect causal pathway through which effects are conducted from a source through a conduit—the mediator—to the final outcome. MacKinnon 2008 describes a large theoretical and empirical literature that is devoted to mediation, a point elaborated in a later section.

                                • Kenny, David A., and Stephanie Milan. 2012. Identification: A nontechnical discussion of a technical issue. In Handbook of structural equation modeling. Edited by R. H. Hoyle, 145–163. New York: Guilford.

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                                  Outlines requirements for identifying path models and other kinds of structural equation models, including latent variable models.

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                                  • Little, Todd D. 2013. Longitudinal structural equation modeling. New York: Guilford.

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                                    Mediation is clearly defined in chapter 9 and distinguished from moderation, with which mediation is often confused. Estimation of mediation in longitudinal designs with time precedence between measurement of the cause, mediator, and outcome is also described.

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                                    • MacKinnon, David P. 2008. Introduction to statistical mediation analysis. Mahwah, NJ: Lawrence Erlbaum.

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                                      A widely cited book about statistical mediation analysis, which has its origins in path analysis.

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                                      • Pearl, Judea. 2009. Causal inference in statistics: An overview. Statistics Surveys 3:96–146.

                                        DOI: 10.1214/09-SS057Save Citation »Export Citation »E-mail Citation »

                                        Describes the role of graphical causal models, which include path models, in causal inference. Emphasizes how casual graphs encode both assumptions that are testable with data and also assumptions that cannot be directly tested.

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                                        Mediation

                                        The specification and estimation of indirect causal effects originated in the technique of path analysis, and mediation always involves indirect effects such as Exercise → Fitness → Health, which represents the hypothesis that exercise changes health by first changing fitness. If no direct effect of exercise on health is expected, then the causal effect of the former on the latter is purely indirect through the variable of fitness. Baron and Kenny 1986 describes the product method for estimating indirect causal effects as the product of the path coefficients for the direct effects that make up the indirect causal pathway. It involves using regression techniques to estimate the direct effect of a cause on a mediator, and also the direct effect of the mediator on the outcome. For continuous variables in a linear model where no interactions are assumed, the product of the two path coefficients just described estimates the impact of the cause on the outcome through the mediator controlling for the direct effect of the mediator on the outcome; it also controls for the direct effect of the cause on the outcome, if any. This product estimator also assumes that there are no unmeasured common causes of any pair of variables among the cause, mediator, and outcome. Baron and Kenny 1986 did not describe the use of statistical significance as a decision criterion in mediational analysis, but many researchers use significance testing in this way. A drawback is that small indirect effects could be significant in large samples, but large indirect effect may not be significant in smaller samples. There is also the possibility that results of significance tests, or p values, are typically wrong for indirect effects. This is because the distributional assumptions of such tests are implausible, as noted in Kline 2013. For example, significance tests of product estimators of indirect effects assume normal distributions, but distributions of products are not generally normal. Bootstrapped significance tests of product estimators for indirect effects are an alternative, but results of bootstrapping in samples that are not large can be very inaccurate. Statistical significance tests of any kind assume random sampling, but the use of true random sampling in human studies is rare. Instead, most samples in human studies are ad hoc or convenience samples made up of cases that happen to be locally available, and what error terms of significance tests measure in convenience samples is generally unknown. A more modern view distinguishes between indirect effects and mediation; that is, mediation always involves indirect effects, but not all indirect effects should be described as mediation without a proper research design. Such designs feature time precedence, or appropriate lags between the measurement of the cause, mediator, and outcome. One example, described in Bullock, et al. 2010, is an experimental mediation design where the cause is randomized (e.g., treatment vs. control), the mediator—typically an individual difference variable—is measured at a later time, and the outcome is measured at a still later, third time. Stone-Romero and Rosopa 2011 describes experimental mediation designs where the mediator is manipulated in addition to the causal variable. Longitudinal designs also allow time precedence in measurement. For example, Cole and Maxwell 2003 describes the half-longitudinal mediation design, where the mediator and outcome variables are each measured at times 1 and 2, but the cause is measured only at time 1. In the path model for such a design, the coefficient for the direct effect of the cause at time 1 on the mediator at time 2 is estimated, controlling for the mediator at time 1, while the coefficient for the direct effect of the mediator at time 1 on the outcome at time 2 controls for the outcome at time 1. The product of the two coefficients just mentioned estimates mediation in a linear model, assuming no interactions. Selig and Preacher 2009 describes the full-longitudinal mediation design, where the cause, mediator, and outcome are each measured at three different times. Such a design contains within it two half-longitudinal designs, and there are multiple estimators of mediation effects. But without a proper research design, an indirect effect in the corresponding path model would not be described as mediation. For example, nonexperimental designs where all variables are concurrently measured (i.e., a cross-sectional design) do not feature time precedence. In such designs, indirect effects may be estimated, but such effects should not be referred to as mediation.

                                        • Baron, Reuben M., and David A. Kenny. 1986. The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology 51:1173–1182.

                                          DOI: 10.1037/0022-3514.51.6.1173Save Citation »Export Citation »E-mail Citation »

                                          Describes the product method for estimating mediation in linear models for continuous variables where no interaction is assumed. Other works described in a later section expand the basic Baron-Kenny product method to allow for interactions between the cause and the mediator and also allow for mediators or outcomes that are not continuous.

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                                          • Bullock, John G., Donald P. Green, and Shang E. Ha. 2010. Yes, but what’s the mechanism? (Don’t expect an easy answer). Journal of Personality and Social Psychology 98:550–558.

                                            DOI: 10.1037/a0018933Save Citation »Export Citation »E-mail Citation »

                                            Describes the experimental analysis of mediation and points out potential sources of bias when the mediator is an individual difference variable that is not manipulated. Concludes that inference about mediation is actually much more difficult than many researchers appreciate, even in experimental designs where at least some forms of bias may be controlled.

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                                            • Cole, David A., and Scott E. Maxwell. 2003. Testing mediational models with longitudinal data: Questions and tips in the use of structural equation modeling. Journal of Abnormal Psychology 112:558–577.

                                              DOI: 10.1037/0021-843X.112.4.558Save Citation »Export Citation »E-mail Citation »

                                              Argues that estimating mediation in longitudinal designs instead of in cross-sectional designs allows for more rigorous inferences about the causal effects implied by a mediational model. A five-step method for using SEM (including path analysis) to test hypotheses about mediation is outlined.

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                                              • Kline, Rex B. 2013. Beyond significance testing: Statistics reform in the behavioral sciences. 2d ed. Washington, DC: American Psychological Association.

                                                DOI: 10.1037/14136-000Save Citation »Export Citation »E-mail Citation »

                                                Reminds researchers of the limitations of significance testing, including the possibility that p values are routinely wrong and widely misinterpreted. In SEM, using statistical significance as a decision criterion for model respecification will do little more than fit the model to sample-specific variation, or sampling error.

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                                                • Selig, James P., and Kristopher J. Preacher. 2009. Mediation models for longitudinal data in developmental research. Research in Human Development 6:144–164.

                                                  DOI: 10.1080/15427600902911247Save Citation »Export Citation »E-mail Citation »

                                                  Describes longitudinal mediation designs for developmental research and also the roles of the theory of change for variables in the model, time precedence in measurement, and types of indirect effects in the model.

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                                                  • Stone-Romero, Eugene F., and Patrick J. Rosopa. 2011. Experimental tests of mediation models: Prospects, problems, and some solutions. Organizational Research Methods 14:631–646.

                                                    DOI: 10.1177/1094428110372673Save Citation »Export Citation »E-mail Citation »

                                                    Reviews experimental designs for estimating mediation, including ones where both the cause and mediator are manipulated by the researcher. Challenges in manipulating mediators are considered in addition to characteristics of alternative estimators of mediation.

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                                                    Moderation and Conditional Process Modeling

                                                    Another extension of Wright’s method of path analysis is conditional process modeling, which is defined by Hayes 2013 and Hayes and Preacher 2013 as dealing with the analysis of contingent causal processes where both mediation and moderation are analyzed together. Moderation refers to the hypothesis that the causal effect of one variable, or the focal variable, on an outcome changes across the levels of third variable, or the moderator. Because moderation is symmetrical, it is also true that (1) the causal effect of the moderator changes across the levels of the focal variable, and (2) the designation of one cause as the focal variable and the other cause as the moderator is arbitrary. A key concept in conditional process modeling is that of mediated moderation, or the specification that moderation is transmitted at least in part through one or more intervening variables, or mediators. For example, Lance 1988 studied the relation of memory demand and complexity of social perception, and their interaction on recall accuracy for the script of a lecture. The model also included a mediator, recollection of specific behaviors mentioned in the script, through which the interaction effect was specified to influence recall accuracy. The results suggested that the size of the mediated moderation effect just described was greater than that of the direct effects of the individual components on recall accuracy. A second essential idea in conditional process modeling is that of moderated mediation, also known as a conditional indirect effect. It is indicated when the strength of some component of an indirect effect varies across the levels of another variable. As described in Edwards and Lambert 2007, there is more than one kind of moderated mediation, of which only three are mentioned here. In first-stage moderation for the indirect causal pathway XMY, the magnitude of the direct effect for the first path, or XM, depends on moderator variable W, but the size of the direct effect for the second path, or MY, is not conditional (i.e., there is no moderator for this direct effect). In second-stage moderation, the magnitude of the direct effect for the second path, or MY, changes across the levels of a moderator variable, but not the size of the direct effect for the first path, or XM. But in first- and-second stage moderation, both direct effects of the indirect causal pathway are conditional; that is, each varies in strength over the levels of at least one moderator variable. Preacher, et al. 2007 gives additional examples of moderated mediation and mediated moderation, and also describes analysis strategies. Two examples of conditional process modeling are then outlined. Curran, et al. 2013 evaluated a conditional process model of children’s engagement and disaffection with soccer. The authors found that structure from coaches related positively to engagement and negatively to disaffection, and that these relations are indirect through children’s psychological need satisfaction. These indirect effects were appreciable only among children who reported higher levels of autonomy support from their coaches. Desrosiers, et al. 2014 found in a sample of adults with anxiety and depressive disorders that nonreactivity, or the ability to observe internal or external experience without becoming preoccupied, moderated both the direct effect of observing on depression symptoms and the indirect effect of observing on symptoms through emotion regulation strategies.

                                                    • Curran, Thomas, Andrew P. Hill, and Christopher P. Niemiec. 2013. A conditional process model of children’s behavioral engagement and behavioral disaffection in sport based on self-determination theory. Journal of Sport & Exercise Psychology 35:30–43.

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                                                      Found that indirect effects of coaching structure on children’s soccer engagement through their psychological need satisfaction depended on the level of autonomy support from coaches.

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                                                      • Desrosiers, Althea, Vera Vine, Joshua Curtiss, and David H. Klemanski. 2014. Observing nonreactively: A conditional process model linking mindfulness facets, cognitive emotion regulation strategies, and depression and anxiety symptoms. Journal of Affective Disorders 165:31–37.

                                                        DOI: 10.1016/j.jad.2014.04.024Save Citation »Export Citation »E-mail Citation »

                                                        Reports that the capability to engage in some experience without becoming preoccupied moderates the direct and indirect effects of observing on depression through emotional regulation strategies.

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                                                        • Edwards, Jeffrey R., and Lisa Schurer Lambert. 2007. Methods for integrating moderation and mediation: A general analytical framework using moderated path analysis. Psychological Methods 12:1–22.

                                                          DOI: 10.1037/1082-989X.12.1.1Save Citation »Export Citation »E-mail Citation »

                                                          Outlines a general analytical approach for combining and estimating mediation and moderation as part of the same model. Defines types of moderated mediation, including first-stage, second-stage, and first-and-second-stage moderation, among other kinds of conditional indirect effects.

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                                                          • Hayes, Andrew F. 2013. Introduction to mediation, moderation, and conditional process analysis: A regression-based approach. New York: Guilford.

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                                                            This book deals with the analysis of mediation and moderation in both multiple regression and path analysis. It also addresses the theoretical background of conditional process modeling, and offers several examples.

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                                                            • Hayes, Andrew F., and Kristopher J. Preacher. 2013. Conditional process modeling: Using structural equation modeling to examine contingent causal processes. In Structural equation modeling: A second course. 2d ed. Edited by G. R. Hancock and R. O. Mueller, 219–266. Greenwich, CT: Information Age.

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                                                              This chapter-length work introduces conditional process modeling in the context of path analysis and SEM.

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                                                              • Lance, Charles E. 1988. Residual centering, exploratory and confirmatory moderator analysis, and decomposition of effects in path models containing interaction effects. Applied Psychological Measurement 12:163–175.

                                                                DOI: 10.1177/014662168801200205Save Citation »Export Citation »E-mail Citation »

                                                                Early example of the estimation of mediated moderation in a path model of recall accuracy for the script of a lecture as a function of memory demand, social perception complexity, and recall for specific behaviors.

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                                                                • Preacher, Kristopher J., Derek D. Rucker, and Andrew F. Hayes. 2007. Addressing moderated mediation hypotheses: Theory, methods, and prescriptions. Multivariate Behavioral Research 42:185–227.

                                                                  DOI: 10.1080/00273170701341316Save Citation »Export Citation »E-mail Citation »

                                                                  Reviews statistical and conceptual definitions of moderated mediation, suggests strategies for hypothesis testing, and describes methods to probe appreciable conditional effects, including graphical summaries with confidence intervals.

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                                                                  Graph Theory and the Structural Causal Model

                                                                  Pearl 2009 and Pearl 2012 describe an approach to causal inference known as the structural causal model (SCM) that is well known in epidemiology but less so in traditional social science disciplines such as psychology and education. Graph theory for causal modeling originated in Pearl’s work in the 1980s on Bayesian networks in artificial intelligence. Such a network can be represented as a graph-type structure in computer memory. It is then virtually navigated by the computer in order to update conditional probabilities of particular events, given information about other events. While initially applied to discrete variables with multinomial distributions, Koski and Noble 2012 notes that graphical models can also represent dependence relations among other types of variables, including continuous ones with joint multivariate distributions. In this way, models in graph theory can be seen as nonparametric generalizations of Wright’s theory of path coefficients and parametric path models. Basic features of the SCM are summarized as follows: (1) Causal hypotheses are represented both graphically and in expressions that are a kind of mathematical language subject to theorems, lemmas, and proofs. (2) The SCM provide a precise language for communicating the assumptions behind causal questions to be answered. (3) The SCM also explicitly distinguishes between questions that can be empirically tested and those that are unanswerable, given the model. It also provides ways to determine what new measurements would be needed in order to address an “unanswerable” question. (4) The SCM subsumes other useful theories or methods for causal inference, including Rubin’s potential outcomes model and parametric SEM (see Rubin 2005). Causal hypotheses in the SCM are represented in either a directed acyclic graph with no causal loops or in a directed cyclic graph with causal loops. Elwert 2013 describes ways to analyze directed graphs in order to determine whether any individual causal effect, direct or total, is identified or not identified. Causal effects are identified through the specification of covariates that sever or block biasing paths—also known as back-door paths—that contribute to noncausal statistical associations. This capability does not require data, and analyzing a causal graph in the planning stages of a study can help to determine whether additional variables should be measured, ones that would identify causal effects of interest—Fleischer and Diez Roux 2008 gives several examples. After the data are collected, the graph can be analyzed in order to locate conditional independences, or pairs of variables that should be rendered independent, controlling for certain other variables, or covariates, in the model. As outlined by Hayduk, et al. 2003, these covariates distance-separate (d-separate) the focal pair, given the hypotheses represented in the causal graph. If all variables are continuous, then conditional independences are estimated with sample data as partial correlations, which should “vanish” (be about zero in value), if the model is correct.

                                                                  • Elwert, Felix. 2013. Graphical causal models. In Handbook of causal analysis for social research. Edited by S. L. Morgan, 245–273. New York: Springer.

                                                                    DOI: 10.1007/978-94-007-6094-3Save Citation »Export Citation »E-mail Citation »

                                                                    Offers a clear introduction to graphical identification criteria and statistical control of bias from the perspective of graphical causal models. Also gives helpful advice for selecting covariates in regression analysis when there is a casual model, and considers potential effects of sampling on the introduction of bias in the results.

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                                                                    • Fleischer, Nancy L., and Ana Victoria Diez Roux. 2008. Using directed acyclic graphs to guide analyses of neighbourhood health effects: An introduction. Journal of Epidemiology and Community Health 62:842–846.

                                                                      DOI: 10.1136/jech.2007.067371Save Citation »Export Citation »E-mail Citation »

                                                                      Demonstrates the analysis of directed acyclic graphs for a hypothetical study on neighborhood health effects for the sake of identifying covariates that would control for biasing paths between presumed causes and effects.

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                                                                      • Hayduk, Leslie, Greta Cummings, Rainer Stratkotter, et al. 2003. Pearl’s d-separation: One more step into causal thinking. Structural Equation Modeling 10:289–311.

                                                                        DOI: 10.1207/S15328007SEM1002_8Save Citation »Export Citation »E-mail Citation »

                                                                        Explains the connections between basic concepts in graph theory, such as d-separation and graphical identification criteria, and the representation of causal effects and their estimation in standard path analysis.

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                                                                        • Koski, Timo J. T., and John M. Noble. 2012. A review of Bayesian networks and structure learning. Mathematica Applicanda 40:53–103.

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                                                                          Reviews basic features of Bayesian networks as represented by directed acyclic graphs in the context of machine learning. Also deals with assumptions of such graphs.

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                                                                          • Pearl, Judea. 2009. Causality: Models, reasoning, and inference. 2d ed. New York: Cambridge Univ. Press.

                                                                            DOI: 10.1017/CBO9780511803161Save Citation »Export Citation »E-mail Citation »

                                                                            The book offers a rigorous and comprehensive framework for causal inference in graph theory, but it may challenge readers without strong mathematical or quantitative backgrounds.

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                                                                            • Pearl, Judea. 2012. The causal foundations of structural equation modeling. In Handbook of structural equation modeling. Edited by R. H. Hoyle, 68–91. New York: Guilford.

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                                                                              This more accessible work elaborates on how SEM and the potential outcomes model are both subsumed and extended by the SCM.

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                                                                              • Rubin, Donald. B. 2005. Causal inference using potential outcomes: Design, modeling, decisions. Journal of the American Statistical Association 100:322–331.

                                                                                DOI: 10.1198/016214504000001880Save Citation »Export Citation »E-mail Citation »

                                                                                Describes the potential outcomes model, which emphasizes the estimation of causal effects as counterfactuals in data sets where not all potential outcomes are measured (e.g., treated cases are not observed in the control condition, and vice versa).

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                                                                                Causal Mediation Analysis

                                                                                The analysis of mediation in graph theory is referred to as causal mediation analysis. It is based on the mediation formula in Pearl 2014, which, as explained in Valeri and VanderWeele 2013, defines direct, indirect, and total causal effects in a consistent way regardless of the statistical model, such as linear versus nonlinear, and regardless of possible interactions between causes and mediators. Causal mediation analysis can also be extended to linear, logistic, log-linear, or Poisson-type regression analyses (e.g., odds ratios or risk ratios can be analyzed for dichotomous outcomes), and mediators can be continuous or dichotomous. Interaction effects between the causal variable and the mediator are routinely estimated in causal mediation analysis. In contrast, the classical Baron-Kenny product method, in which indirect effects of continuous variables in linear models are estimated as products of coefficients from the constituent paths, assumes no interaction. As noted in Valeri and VanderWeele 2013, if there is truly no causal variable-mediator interaction in the population, the results of a causal mediation analysis and the Baron-Kenny product method yield the same results in linear models for continuous mediators and outcomes; otherwise, the two approaches can generate quite different estimates for the same data. This is why causal mediation analysis extends the Baron-Kenny approach for the types of models just described to allow for such interactions. Direct and indirect effects are defined from a counterfactual perspective in causal mediation analysis. This perspective comes from Rubin’s potential outcomes model, which refers to counterfactual conditional statements about what would be the case, given a prior event. In randomized experiments, for example, there are two basic counterfactuals: (1) what would be the outcomes of control cases, if they were treated; and (2) what would be the outcomes of treated cases, if they were not treated? If each case was either treated or not treated, these potential outcomes are not observed. This means that the observed data—outcomes for treated versus control cases—is a subset of all possible combinations, so the potential outcomes model is concerned with causal inference based on incomplete data. In causal mediation analysis for a randomized cause (e.g., treatment vs. control) and continuous mediator and outcome variables, the controlled direct effect is defined as the average difference between the treated and untreated cases if the mediator were controlled at the same level for all cases in the population. There is a different value of the controlled direct effect of the causal variable for each level of the mediator, so for a continuous mediator there are actually an infinite number of controlled direct effects. The natural direct effect is the average difference in outcome if the causal variable were allowed to change from control to treatment, but the mediator is kept to the level that it would have taken in the control condition. Unlike the case for the controlled direct effect, the level of the mediator is not fixed to the same constant for all cases. Instead, the mediator is allowed to vary, but only over values that would be naturally observed in the control condition. The natural indirect effect estimates the amount of change among treated cases as the mediator changes from values that would be observed in the control group to the values it would obtain in the treatment group; that is, the outcome is influenced by the cause due solely to its influence on the mediator. The total causal effect is the sum of the natural direct effect and the natural indirect effect. The controlled direct effect does not have a simple additive relation with either the natural direct effect or natural indirect effect, so it is not part of an effect decomposition in causal mediation analysis. Petersen, et al. 2006 defines the direct and indirect effects just described in a hypothetical study of antiviral treatment for HIV infection where the mediator is the level of viral load, among other examples. Imai, et al. 2010 describes causal mediation analysis in the social sciences.

                                                                                • Imai, Kosuke, Luke Keele, and Teppei Tingley. 2010. A general approach to causal mediation analysis. Psychological Methods 15:309–334.

                                                                                  DOI: 10.1037/a0020761Save Citation »Export Citation »E-mail Citation »

                                                                                  Describes the estimation of mediation from a causal perspective, and outlines assumptions in mediational analyses. Also deals with sensitivity analysis, which formally assesses the robustness of empirical estimates of direct, indirect, and total causal effects to violations of assumptions about omitted common causes.

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                                                                                  • Pearl, Judea. 2014. Interpretation and identification of causal mediation. Psychological Methods 19:459–481.

                                                                                    DOI: 10.1037/a0036434Save Citation »Export Citation »E-mail Citation »

                                                                                    Describes identification requirements for estimating controlled and natural direct effects and also natural indirect effects. These requirements can be validated algorithmically from the diagrammatic representation of hypotheses in a directed graph.

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                                                                                    • Petersen, Maya L., Sandra E. Sinisi, and Mark J. van der Laan. 2006. Estimation of direct causal effects. Epidemiology 17:276–284.

                                                                                      DOI: 10.1097/01.ede.0000208475.99429.2dSave Citation »Export Citation »E-mail Citation »

                                                                                      Illustrates the difference between controlled and natural direct effects in two examples of hypothetical studies. Also explains how to estimate such effects in linear models with continuous mediators and outcomes using standard statistical software.

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                                                                                      • Valeri, Linda, and Tyler J. VanderWeele. 2013. Mediation analysis allowing for exposure-mediator interactions and causal interpretation: Theoretical assumptions and implementation with SAS and SPSS macros. Psychological Methods 2:137–150.

                                                                                        DOI: 10.1037/a0031034Save Citation »Export Citation »E-mail Citation »

                                                                                        Describes macros (scripts) for SPSS and SAS/STAT for causal mediation analysis that are freely available over the Internet. Also considers types of inferences about mediation that can be evaluated with these macros.

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                                                                                        Latent Variable Extensions and SEM Textbooks

                                                                                        All of the types of analyses just described for path models can be extended to latent variable models, where some constructs are measured with multiple observed variables, or indicators. The main advantage of multiple-indicator measurement is that the reliability of construct measurement is increased compared with single-indicator measurement, assuming that multiple indicators have good psychometric characteristics. As explained in Cole and Preacher 2014, the failure to take account of measurement error in analyses of observed variables, such as in multiple regression or in traditional path analysis, can potentially bias the results. In SEM, it is no special problem within the limits of identification to analyze models with any combination of observed or latent variables, and in this sense the technique of path analysis is just a special case of SEM for models with observed variables only. The family of SEM techniques, which also includes confirmatory factor analysis, can also be extended to include the estimation of means on latent variables, and also the simultaneous analysis of models using data from two or more samples. The latter refers to invariance testing. This is because a key question in such analyses is whether the groups differ appreciably on parameters of interest, such as direct effects, error variances, and so on, that make it impossible to conclude that the same set of indicators measures the same constructs over different populations, such as women versus men. All of these extensions are described in textbooks for SEM, which at the introductory level include Kline 2010, Loehlin 2004, and Schumacker and Lomax 2010. Books at a more advanced level include Kaplan 2009 and Mulaik 2009, while the edited volumes Hoyle 2012 and Hancock and Mueller 2013 describe both basic and more advanced applications.

                                                                                        • Cole, David A., and Kristopher J. Preacher. 2014. Manifest variable path analysis: Potentially serious and misleading consequences due to uncorrected measurement error. Psychological Methods 19:300–315.

                                                                                          DOI: 10.1037/a0033805Save Citation »Export Citation »E-mail Citation »

                                                                                          Describes possible consequences of analyzing unreliable scores in observed-variable statistical methods, including multiple regression and path analysis, if measurement error is not taken into account. These problems tend to become worse as the complexity of the model increases.

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                                                                                          • Hancock, Gregory R., and Ralph O. Mueller, eds. 2013. Structural equation modeling: A second course. 2d ed. Greenwich, CT: Information Age.

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                                                                                            Covers foundations (basic concepts), extensions (advanced applications), and assumptions in SEM.

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                                                                                            • Hoyle, Rick H., ed. 2012. Handbook of structural equation modeling. New York: Guilford.

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                                                                                              Covers SEM from an introductory level up through more advanced applications in particular research areas, including genetic and spatial mapping.

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                                                                                              • Kaplan, David. 2009. Structural equation modeling: Foundations and extensions. 2d ed. Thousand Oaks, CA: SAGE.

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                                                                                                Advanced level. Concise coverage of more advanced applications, including multilevel SEM.

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                                                                                                • Kline, Rex B. 2010. Principles and practice of structural equation modeling. 3d ed. New York: Guilford.

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                                                                                                  Introductory level. Covers all phases of SEM, from model specification to data screening to the analysis to the reporting of results.

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                                                                                                  • Loehlin, John C. 2004. Latent variable models. 4th ed. Mahwah, NJ: Lawrence Erlbaum.

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                                                                                                    Introductory level. Extensive coverage of factor analysis methods, both exploratory and confirmatory.

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                                                                                                    • Mulaik, Stanley A. 2009. Linear causal modeling with structural equations. Boca Raton, FL: CRC Press.

                                                                                                      DOI: 10.1201/9781439800393Save Citation »Export Citation »E-mail Citation »

                                                                                                      Advanced level. Covers many advanced methods and also gives good historical reviews.

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                                                                                                      • Schumacker, Randall E., and Richard G. Lomax. 2010. A beginner’s guide to structural equation modeling. 3d ed. New York and London: Routledge.

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                                                                                                        Introductory level. Covers both basic and more advanced analyses in LISREL. Also deals in detail with Monte Carlo (computer simulation) methods, such as in power analysis in SEM.

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                                                                                                        Computer Tools for SEM

                                                                                                        Computer tools for SEM (which also analyze path models) can be broken down into two major categories: (1) stand-alone programs that do not require a larger computing environment; and (2) procedures, packages, routines, or commands that are part of a larger environment. Stand-alone SEM computer tools include Amos (Amos Development Corporation 1983–2013), EQS (Bentler 2006), LISREL (Scientific Software International 2013), Mplus (Müthen and Müthen 1998–2014), and Ωnyx (pronounced “onix”) (von Oertzen, et al. 2015). The latter is a freely available graphical environment for analyzing structural equation models, but the other computer programs are all commercial products. All five of these software programs allow the user to specify the model by drawing it on screen. Another alternative supported by most SEM computer programs is batch mode, where the user specifies the model, data, and analysis in syntax using the programming language native to that particular computer tool. There is a freely available student version of LISREL that is quite capable and makes a good learning tool. Examples of procedures or commands for SEM within larger computing environments include the CALIS routine in SAS/STAT (SAS Institute 2013) and the sem and gsem commands in Stata (StataCorp 2013). There are more and more freely available SEM packages for R, which is an open-source computing environment for statistical analyses and graphical production. Fox 2012 (a sem package) and Rosseel 2012 (a lavaan package) each have extensive analytical capabilities. Boker, et al. 2011 (an OpenMx package and library) is a rewrite of the matrix algebra processor Mx with extensive SEM capabilities. There are additional SEM packages for R, and more are being developed all the time. It is expected that R-based analyses will play an ever larger role in SEM studies in the near future.

                                                                                                        • Amos Development Corporation. 1983–2013. IBM SPSS Amos (Version 22.0). Meadville, PA: Amos Development Corporation.

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                                                                                                          Commercial program. Features a graphical editor that allows the user to specify the model by drawing in on screen with drawing shape tools.

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                                                                                                          • Bentler, Peter M. 2006. EQS 6 structural equations program manual. Encino, CA: Multivariate Software.

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                                                                                                            Commercial program. Features a graphical editor and offers several estimation methods for analyzing outcomes that are either continuous or categorical variables.

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                                                                                                            • Boker, Steven, Michael Neale, Hermine Maes, et al. 2011. OpenMx: An open source extended structural equation modeling framework. Psychometrika 76:306–317.

                                                                                                              DOI: 10.1007/s11336-010-9200-6Save Citation »Export Citation »E-mail Citation »

                                                                                                              Freely available. This package implements the capabilities of the Mx matrix algebra processor in the R environment. Has many capabilities for SEM analyses. Available online.

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                                                                                                              • Fox, John. 2012. Structural equation modeling in R with the sem package.

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                                                                                                                Freely available. Fox’s sem was one of the first SEM packages for R. It has been recently updated. Available online.

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                                                                                                                • Müthen, Linda K., and Bengt O. Müthen. 1998–2014. Mplus (Version 7.3). Los Angeles: Authors.

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                                                                                                                  Commercial program. Extensive capabilities for latent variable analyses, including SEM, multilevel modeling, and mixture modeling.

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                                                                                                                  • Rosseel, Yves. 2012. lavaan: An R package for structural equation modeling. Journal of Statistical Software 48.2.

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                                                                                                                    Freely available. This capable SEM package for R has been updated many times. Available online.

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                                                                                                                    • SAS Institute. 2013. SAS/STAT (Version 13.1). Cary, NC: SAS Institute.

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                                                                                                                      Commercial program. The CALIS procedure in SAS/STAT is for SEM analyses.

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                                                                                                                      • Scientific Software International. 2013. LISREL (Version 9.1). Skokie, IL: Scientific Software International.

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                                                                                                                        Commercial program. LISREL is the original computer tool for SEM, and its PRELIS program has extensive capabilities for data preparation and simulation. Free student version available online.

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                                                                                                                        • StataCorp. 2013. Stata statistical software: Release 13. College Station, TX: StataCorp.

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                                                                                                                          Commercial program. The Stata sem and gsem commands for SEM are syntax based. There is also Builder, which allows the specification of a model by drawing it onscreen.

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                                                                                                                          • von Oertzen, Timo, Andreas M. Brandmaier, and Siny Tsang. 2015. Structural equation modeling with Ωnyx. Structural Equation Modeling 22:148–161.

                                                                                                                            DOI: 10.1080/10705511.2014.935842Save Citation »Export Citation »E-mail Citation »

                                                                                                                            Freely available. A graphical environment for analyzing structural equation models with raw data. There is no native batch mode, although Ωnyx can read syntax written in the lavaan, sem, and OpenMx packages and then draw the model onscreen. Available online.

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                                                                                                                            Books about SEM Computer Tools

                                                                                                                            There are also several books that show readers how to use a particular SEM computer tool. They do not generally deal with conceptual issues in SEM; that is, you should already know something about SEM before starting to use a computer tool. Examples of books for particular computer programs include Acock 2013 for Stata; Blunch 2013 and Byrne 2010 for Amos; Vieira 2011 for LISREL; Beaujean 2014 for laavan; and Byrne 2012, Geiser 2013, and Wang and Wang 2012 for Mplus.

                                                                                                                            • Acock, Alan C. 2013. Discovering structural equation modeling using Stata 13. College Station, TX: Stata Press.

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                                                                                                                              Deals with both syntax-based SEM analyses in Stata and graphical-based analyses conducted using the Builder command.

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                                                                                                                              • Beaujean, A. Alexander. 2014. Latent variable modeling using R: A step-by-step guide. New York: Routledge.

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                                                                                                                                Describes many examples of lavaan syntax for specifying the data, model, and analysis output. Covers some advanced applications, too, such as multiple-samples analysis.

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                                                                                                                                • Blunch, Niels. 2013. Introduction to structural equation modeling using IBM SPSS Statistics and Amos. 2d ed. Thousand Oaks, CA: SAGE.

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                                                                                                                                  Covers both basic and some more advanced analyses using the Amos graphical editor to control the analysis.

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                                                                                                                                  • Byrne, Barbara M. 2010. Structural equation modeling with Amos: Basic concepts, applications, and programming. 2d ed. New York: Routledge.

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                                                                                                                                    Demonstrates many applications of Amos to analyze a wide range of data. Good coverage of conceptual issues, too.

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                                                                                                                                    • Byrne, Barbara M. 2012. Structural equation modeling with Mplus: Basic concepts, applications, and programming. New York: Routledge.

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                                                                                                                                      Describes basic and more advanced analyses in Mplus. The latter include multitrait-multimethod models, multilevel models, and latent growth models.

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                                                                                                                                      • Geiser, Christian. 2013. Data analysis with Mplus. New York: Guilford.

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                                                                                                                                        Both basic and advanced analyses with Mplus are described, including multilevel regression analysis and latent class analysis.

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                                                                                                                                        • Vieira, Armando Luis. 2011. Interactive LISREL in practice: Getting started with a SIMPLIS approach. New York: Springer.

                                                                                                                                          DOI: 10.1007/978-3-642-18044-6Save Citation »Export Citation »E-mail Citation »

                                                                                                                                          Covers SEM analyses using the SIMPLIS programming language in LISREL, which is an easier-to-learn alternative to LISREL’s original programming language based on matrix algebra.

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                                                                                                                                          • Wang, Jichuan, and Xiaoqian Wang. 2012. Structural equation modeling: Applications using Mplus. Chichester, UK: Wiley.

                                                                                                                                            DOI: 10.1002/9781118356258Save Citation »Export Citation »E-mail Citation »

                                                                                                                                            Offers many examples of Mplus analyses, including latent growth modeling for longitudinal data, mixture modeling, and power analysis in SEM.

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                                                                                                                                            Computer Tools for Analyzing Directed Acyclic Graphs

                                                                                                                                            Textor, et al. 2011 describes DAGitty, which is a freely accessible, Internet browser–based graphical environment for drawing, editing, and analyzing directed acyclic graphs. It features a visual editor that permits the user to draw the graph onscreen. Variables can be defined as exposures versus outcomes in epidemiological studies. As the model is drawn, DAGitty automatically lists, in a series of text windows, testable conditional independences and sufficient sets of covariates that minimize bias in estimating causal effects. The program also automatically writes text-based syntax that describes the diagram, and this syntax can be saved for later analyses. Porter, et al. 1999–2009, a Belief and Decision Network Tool, is a freely available Java applet for learning about directed acyclic graphs. After drawing a DAG onscreen, this program can then be optionally run in quiz mode, where it poses true-false questions about whether pairs of variables are conditionally independent, controlling for certain other variables in the graph. The correct answer is shown after the user enters his or her response. In “ask the applet” mode, the user clicks on two focal variables and a set of covariates, and the program automatically indicates whether the focal variables are independent, given those covariates. These capabilities provide tutorials for learning about concepts in graphical causal models. In the freely available DAG Program Knüppel and Stang 2010, the user specifies the graph by entering syntax that describes the variables and presumed causal effects among them in a series of templates that make up the user interface. Model specifications are summarized in text fields that enumerate covariates and also exposure, outcome, and unmeasured (latent) variables, but the DAG Program does not draw the corresponding diagram. Minimal sets of covariates that minimize bias in estimating causal effects are also automatically listed. Breitling 2010 (a dagR package for R) provides a set of functions for drawing, manipulating, and analyzing directed acyclic graphs, and also for simulating data that would be consistent with the corresponding diagram. It is intended for epidemiological studies, but can be used by researchers in other disciplines, too. The dagR package also allows researchers to evaluate the effects of analyzing different subsets of covariates when analyzing presumed causal effects of exposure variables on outcome variables, and also for finding spurious associations. Models are specified in syntax, but the corresponding graph of the model can be manipulated in the R environment. A model specified in dagR can be saved as an R programming object that can be transferred to another researcher for analysis in R.

                                                                                                                                            Computer Tools for Causal Mediation Analysis

                                                                                                                                            Valeri and VanderWeele 2013 describes macros for SPSS and SAS/STAT for causal mediation analysis that also allow covariates, such as variables that control for treatment history. Imai, et al. 2010 describes the mediation package for R that also conducts sensitivity analyses about the effects of violated assumptions, such as the requirement for no omitted common causes of the mediator and the outcome variable. Capabilities for causal mediation analysis were implemented in version 7.2 of Mplus, as described in Muthén and Asparouhov 2015. Hicks and Tingley 2011 describes the MEDIATION package for casual mediation analysis in Stata.

                                                                                                                                            • Hicks, Raymond, and Dustin Tingley. 2011. Causal mediation analysis. Stata Journal 11:605–619.

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                                                                                                                                              This freely macro for Stata is for causal mediation analysis. Also calculates sensitivity analyses that take account of nonrandom assignment for the mediating variable. Available online.

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                                                                                                                                              • Imai, Kosuke, Luke Keele, and Teppei Yamamoto. 2010. Identification, inference, and sensitivity analysis for causal mediation effects. Statistical Science 25:51–71.

                                                                                                                                                DOI: 10.1214/10-STS321Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                The freely available mediation package for R is for causal mediation analysis. It also estimates the effects of violated assumptions in the analysis. Available online.

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                                                                                                                                                • Muthén, Bengt, and Tihomir Asparouhov. 2015. Causal effects in mediation modeling: An introduction with applications to latent variables. Structural Equation Modeling 22:12–23.

                                                                                                                                                  DOI: 10.1080/10705511.2014.935843Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                  Describes special syntax in Mplus versions 7.2 and later that supports causal mediation analysis. Mediators or outcomes can be binary or continuous variables. Either variable can be observed or latent.

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                                                                                                                                                  • Valeri, Linda, and Tyler J. VanderWeele. 2013. Mediation analysis allowing for exposure-mediator interactions and causal interpretation: Theoretical assumptions and implementation with SAS and SPSS macros. Psychological Methods 2:137–150.

                                                                                                                                                    DOI: 10.1037/a0031034Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                    Describes macros (scripts) for SPSS and SAS/STAT for causal mediation analysis that are freely available over the Internet. Available online.

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                                                                                                                                                    Best Practices

                                                                                                                                                    There are many problems with the typical published path analytic study in particular or SEM study in general. These include the failure to report the complete solution (e.g., unstandardized estimates with standard errors); state the method of estimation used; give a full account of directionality specifications (i.e., explain why X causes Y instead of the reverse); provide information about the residuals, or the details of model-data correspondence; and address the issue of equivalent models, which are just as complex as the researcher’s model and have the same fit to the data, but feature a different configuration of paths among the same variables. See MacCallum and Austin 2000 and Shah and Goldstein 2006 for descriptions of additional kinds of problems. Fortunately, there are several resources, including a chapter in Mueller and Hancock 2008 about best practice recommendations for SEM, a chapter in Boomsma, et al. 2012 about what to include in written reports of SEM analyses for various types of models, and recommendations in Schreiber 2008 and Hoyle and Isherwood 2011 about reporting practices.

                                                                                                                                                    • Boomsma, Anne, Rick H. Hoyle, and Abigail T. Panter. 2012. The structural equation modeling research report. In Handbook of structural equation modeling. Edited by R. H. Hoyle, 341–358. New York: Guilford.

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                                                                                                                                                      Offers guidance about reporting the results of SEM analyses for five types of models, including factor analysis, measurement invariance, latent variable interactions, latent growth curve, and Monte Carlo (computer simulation).

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                                                                                                                                                      • Hoyle, Rick H., and Jennifer C. Isherwood. 2011. Reporting results from structural equation modeling analyses in Archives of Scientific Psychology. Archives of Scientific Psychology 1:14–22.

                                                                                                                                                        DOI: 10.1037/arc0000004Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                        Extended the Journal Article Reporting Standards of the American Psychological Association for SEM studies. A helpful checklist for complete reporting is provided.

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                                                                                                                                                        • MacCallum, Robert C., and James T. Austin. 2000. Applications of structural equation modeling in psychological research. Annual Review of Psychology 51:201–236.

                                                                                                                                                          DOI: 10.1146/annurev.psych.51.1.201Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                          Reviewed five hundred published SEM studies in sixteen psychology research journals. Many problems were found, including the use of samples that are too small, failure to report the complete solution, and neglect of the mention of equivalent models, which is a form of confirmation bias.

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                                                                                                                                                          • Mueller, Ralph O., and Gregory R. Hancock. 2008. Best practices in structural equation modeling. In Best practices in quantitative methods. Edited by J. W. Osborne, 488–508. Thousand Oaks, CA: SAGE.

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                                                                                                                                                            Offers helpful suggestions for constructing clear and complete model diagrams, describes analysis strategies for latent variable models with both a measurement component and a structural component, and gives examples of reporting the complete set of parameter estimates.

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                                                                                                                                                            • Schreiber, James B. 2008. Core reporting practices in structural equation modeling. Research in Social and Administrative Pharmacy 4:83–97.

                                                                                                                                                              DOI: 10.1016/j.sapharm.2007.04.003Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                              Presents two examples of reporting results of SEM analyses: the first example for a confirmatory factor analysis model, and the second example for a latent variable (structural regression) model.

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                                                                                                                                                              • Shah, Rachna, and Susan M. Goldstein. 2006. Use of structural equation modeling in operations management research: Looking back and forward. Journal of Operations Management 24:148–169.

                                                                                                                                                                DOI: 10.1016/j.jom.2005.05.001Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                Reviews ninety-three published SEM studies in four operations management journals. Common shortcomings include the failure to justify directionality specifications, inadequate statistical power, poor data screening, and mismatches between tables, figures, and text.

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