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1 Structural Equation Modeling (SEM) Essentials by Jim Grace Purpose of this module is to provide a very brief presentation of the things one needs to know about SEM before learning how apply SEM.

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2 Where You can Learn More about SEM Grace (2006) Structural Equation Modeling and Natural Systems. Cambridge Univ. Press. Shipley (2000) Cause and Correlation in Biology. Cambridge Univ. Press. Kline (2005) Principles and Practice of Structural Equation Modeling. (2 nd Edition) Guilford Press. Bollen (1989) Structural Equations with Latent Variables. John Wiley and Sons. Lee (2007) Structural Equation Modeling: A Bayesian Approach. John Wiley and Sons.

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3 I. Essential Points about SEM Outline II. Structural Equation Models: Form and Function

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4 I. SEM Essentials: 1. SEM is a form of graphical modeling, and therefore, a system in which relationships can be represented in either graphical or equational form. x1x1 y1y1 1 11 graphical form y 1 = γ 11 x 1 + ζ 1 equational form 2. An equation is said to be structural if there exists sufficient evidence from all available sources to support the interpretation that x 1 has a causal effect on y 1.

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5 y2y2 y1y1 x1x1 y3y3 ζ1ζ1 ζ2 ζ2 ζ3ζ3 Complex Hypothesis e.g. y 1 = γ 11 x 1 + ζ 1 y 2 = β 21 y 1 + γ 21 x 1 + ζ 2 y 3 = β 32 y 2 + γ 31 x 1 + ζ 3 Corresponding Equations 3. Structural equation modeling can be defined as the use of two or more structural equations to represent complex hypotheses.

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6 a. manipulations of x can repeatably be demonstrated to be followed by responses in y, and/or b. we can assume that the values of x that we have can serve as indicators for the values of x that existed when effects on y were being generated, and/or c. if it can be assumed that a manipulation of x would result in a subsequent change in the values of y Relevant References: Pearl (2000) Causality. Cambridge University Press. Shipley (2000) Cause and Correlation in Biology. Cambridge 4. Some practical criteria for supporting an assumption of causal relationships in structural equations:

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7 5. A Grossly Oversimplified History of SEM Wright (1918) Pearson (1890s) Fisher (1922) Joreskog (1973) Lee (2007) Neyman & E. Pearson (1934) Spearman (1904) Bayes & LaPlace (1773/1774) MCMC (1948-) testing alt. models likelihood r, chi-square factor analysis path analysis SEM Contemporary Conven- tional Statistics Bayesian Analysis Raftery (1993) note that SEM is a framework and incorporates new statistical techniques as they become available (if appropriate to its purpose)

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8 6. SEM is a framework for building and evaluating multivariate hypotheses about multiple processes. It is not dependent on a particular estimation method. 7. When it comes to statistical methodology, it is important to distinguish between the priorities of the methodology versus those of the scientific enterprise. Regarding the diagram below, in SEM we use statistics for the purposes of the scientific enterprise. Statistics and other Methodological Tools, Procedures, and Principles. The Scientific Enterprise

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9 The Methodological Side of SEM

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10 The Relationship of SEM to the Scientific Enterprise modified from Starfield and Bleloch (1991) Understanding of Processes univariate descriptive statistics exploration, methodology and theory development realistic predictive models simplistic models multivariate descriptive statistics detailed process models univariate data modeling Data structural equation modeling

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11 8. SEM seeks to progress knowledge through cumulative learning. Current work is striving to increase the capacity for model memory and model generality. exploratory/ model-building applications structural equation modeling confirmatory/ hypothesis-testing applications one aim of SEM

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12 9. It is not widely understood that the univariate model, and especially ANOVA, is not well suited for studying systems, but rather, is designed for studying individual processes, net effects, or for identifying predictors. 10. The dominance of the univariate statistical model in the natural sciences has, in my personal view, retarded the progress of science.

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13 11. An interest in systems under multivariate control motivates us to explicitly consider the relative importances of multiple processes and how they interact. We seek to consider simultaneously the main factors that determine how system responses behave. 12. SEM is one of the few applications of statistical inference where the results of estimation are frequently you have the wrong model!. This feedback comes from the unique feature that in SEM we compare patterns in the data to those implied by the model. This is an extremely important form of learning about systems.

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14 13. Illustrations of fixed-structure protocol models: Univariate Models x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 Multivariate Models x1x1 x2x2 x3x3 x4x4 x5x5 F y1y1 y2y2 y3y3 y4y4 y5y5 Do these model structures match the causal forces that influenced the data? If not, what can they tell you about the processes operating?

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15 14. Structural equation modeling and its associated scientific goals represent an ambitious undertaking. We should be both humbled by the limits of our successes and inspired by the learning that takes place during the journey.

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16 II. Structural Equation Models: Form and Function A. Anatomy of Observed Variable Models

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17 x1x1 y1y1 y2y2 1 2 Some Terminology exogenous variable endogenous variables 21 11 21 path coefficients direct effect of x 1 on y 2 indirect effect of x 1 on y 2 is 11 times 21

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18 nonrecursive y1y1 x2x2 x1x1 y2y2 ζ1ζ1 ζ2ζ2 C y1y1 x2x2 x1x1 y2y2 ζ1ζ1 ζ2ζ2 D x1x1 y1y1 ζ1ζ1 y2y2 A ζ2ζ2 x1x1 y1y1 y2y2 B ζ1ζ1 ζ2ζ2 model B, which has paths between all variables is saturated (vs A, which is unsaturated)

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19 First Rule of Path Coefficients: the path coefficients for unanalyzed relationships (curved arrows) between exogenous variables are simply the correlations (standardized form) or covariances (unstandardized form). x1x1 x2x2 y1y1.40 x 1 x 2 y 1 ----------------------------- x 1 1.0 x 2 0.401.0 y 1 0.500.601.0

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20 x1x1 y1y1 y2y2 11 =.50 21 =.60 (gamma) used to represent effect of exogenous on endogenous. (beta) used to represent effect of endogenous on endogenous. Second Rule of Path Coefficients: when variables are connected by a single causal path, the path coefficient is simply the standardized or unstandardized regression coefficient (note that a standardized regression coefficient = a simple correlation.) x 1 y 1 y 2 ------------------------------------------------- x 1 1.0 y 1 0.501.0 y 2 0.300.601.0

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21 Third Rule of Path Coefficients: strength of a compound path is the product of the coefficients along the path. x1x1 y1y1 y2y2.50.60 Thus, in this example the effect of x 1 on y 2 = 0.5 x 0.6 = 0.30 Since the strength of the indirect path from x 1 to y 2 equals the correlation between x 1 and y 2, we say x 1 and y 2 are conditionally independent.

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22 What does it mean when two separated variables are not conditionally independent? x 1 y 1 y 2 ------------------------------------------------- x 1 1.0 y 1 0.551.0 y 2 0.500.601.0 x1x1 y1y1 y2y2 r =.55r =.60 0.55 x 0.60 = 0.33, which is not equal to 0.50

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23 The inequality implies that the true model is x1x1 y1y1 y2y2 Fourth Rule of Path Coefficients: when variables are connected by more than one causal pathway, the path coefficients are "partial" regression coefficients. additional process Which pairs of variables are connected by two causal paths? answer: x 1 and y 2 (obvious one), but also y 1 and y 2, which are connected by the joint influence of x 1 on both of them.

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24 And for another case: x1x1 x2x2 y1y1 A case of shared causal influence: the unanalyzed relation between x 1 and x 2 represents the effects of an unspecified joint causal process. Therefore, x 1 and y 1 connected by two causal paths. x 2 and y 1 likewise.

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25 x1x1 y1y1 y2y2.40.31.48 How to Interpret Partial Path Coefficients: - The Concept of Statistical Control The effect of y 1 on y 2 is controlled for the joint effects of x 1. I have an article on this subject that is brief and to the point. Grace, J.B. and K.A. Bollen 2005. Interpreting the results from multiple regression and structural equation models. Bull. Ecological Soc. Amer. 86:283-295.

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26 Interpretation of Partial Coefficients Analogy to an electronic equalizer from Sourceforge.net With all other variables in model held to their means, how much does a response variable change when a predictor is varied?

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27 x1x1 y1y1 y2y2 Fifth Rule of Path Coefficients: paths from error variables are correlations or covariances. R 2 = 0.16.92 R 2 = 0.44.73 2 1.31.40.48 equation for path from error variable.56 alternative is to show values for zetas, which = 1-R 2.84

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28 x1x1 y1y1 y2y2 R 2 = 0.16 R 2 = 0.25 2 1.50.40 x 1 y 1 y 2 ------------------------------- x 1 1.0 y 1 0.401.0 y 2 0.500.601.0 Now, imagine y 1 and y 2 are joint responses Sixth Rule of Path Coefficients: unanalyzed residual correlations between endogenous variables are partial correlations or covariances.

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29 x1x1 y1y1 y2y2 R 2 = 0.16 R 2 = 0.25 2 1.50.40 This implies that some other factor is influencing y 1 and y 2 the partial correlation between y1 and y2 is typically represented as a correlated error term

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30 Seventh Rule of Path Coefficients: total effect one variable has on another equals the sum of its direct and indirect effects. y1y1 x2x2 x1x1 y2y2 ζ1ζ1 ζ2ζ2.80.15.64 -.11.27 x 1 x 2 y 1 ------------------------------- y 1 0.64-0.11--- y 2 0.32-0.030.27 Total Effects: Eighth Rule of Path Coefficients: sum of all pathways between two variables (causal and noncausal) equals the correlation/covariance. note: correlation between x 1 and y 1 = 0.55, which equals 0.64 - 0.80*0.11

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31 Suppression Effect - when presence of another variable causes path coefficient to strongly differ from bivariate correlation. x 1 x 2 y 1 y 2 ----------------------------------------------- x 1 1.0 x 2 0.801.0 y 1 0.550.401.0 y 2 0.300.230.351.0 y1y1 x2x2 x1x1 y2y2 ζ1ζ1 ζ2ζ2.80.15.64 -.11.27 path coefficient for x 2 to y 1 very different from correlation, (results from overwhelming influence from x 1.)

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32 II. Structural Equation Models: Form and Function B. Anatomy of Latent Variable Models

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33 Latent Variables Latent variables are those whose presence we suspect or theorize, but for which we have no direct measures. Intelligence IQ score *note that we must specify some parameter, either error, loading, or variance of latent variable. ζ latent variableobserved indicatorerror variable 1.0 fixed loading* 1.0

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34 Latent Variables (cont.) Purposes Served by Latent Variables: (2) Allow us to estimate and correct for measurement error. (3) Represent certain kinds of hypotheses. (1) Specification of difference between observed data and processes of interest.

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35 Range of Examples single-indicator Elevation estimate from map multi-method Soil Organic soil C loss on ignition Territory Size singing range, t1 singing range, t2 singing range, t3 repeated measures Caribou Counts observer 1 observer 2 repeatability

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36 The Concept of Measurement Error the argument for universal use of latent variables 1. Observed variable models, path or other, assume all independent variables are measured without error. 2. Reliability - the degree to which a measurement is repeatable (i.e., a measure of precision). error in measuring x is ascribed to error in predicting/explaining y xy 0.60 R 2 = 0.30 x y illustration

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37 Example Imagine that some of the observed variance in x is due to error of measurement. calibration data set based on repeated measurement trials plotx-trial1x-trial2x-trial3 11.2721.2061.281 21.6041.5771.671 32.1772.1922.104 41.9832.0801.999........................ n2.4602.2662.418 average correlation between trials = 0.90 therefore, average R-square = 0.81 reliability = square root of R 2 measurement error variance = (1 - R 2 ) times VARx imagine in this case VARx = 3.14, so error variance = 0.19 x 3.14 = 0.60 LV1 x LV2 y.90.65 1.0.60 R2 =.42

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38 II. Structural Equation Models: Form and Function C. Estimation and Evaluation

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39 1. The Multiequational Framework (a) the observed variable model We can model the interdependences among a set of predictors and responses using an extension of the general linear model that accommodates the dependences of response variables on other response variables. y = p x 1 vector of responses α = p x 1 vector of intercepts Β = p x p coefficient matrix of y s on y s Γ = p x q coefficient matrix of y s on x s x = q x 1 vector of exogenous predictors ζ = p x 1 vector of errors for the elements of y Φ = cov ( x ) = q x q matrix of covariances among x s Ψ = cov ( ζ ) = q x q matrix of covariances among errors y = α + Βy + Γx + ζ

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40 The LISREL Equations Jöreskög 1973 (b) the latent variable model η = α + Β η + Γξ + ζ x = Λ x ξ + δ y = Λ y η + ε where: η is a vector of latent responses, ξ is a vector of latent predictors, Β and Γ are matrices of coefficients, ζ is a vector of errors for η, and α is a vector of intercepts for η (c) the measurement model where: Λ x is a vector of loadings that link observed x variables to latent predictors, Λ y is a vector of loadings that link observed y variables to latent responses, and δ and ε are vectors are errors

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41 2. Estimation Methods (a) decomposition of correlations (original path analysis) (b) least-squares procedures (historic or in special cases) (c) maximum likelihood (standard method) (d) Markov chain Monte Carlo (MCMC) methods (including Bayesian applications)

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42 Bayesian References: Bayesian Networks: Neopolitan, R.E. (2004). Learning Bayesian Networks. Upper Saddle River, NJ, Prentice Hall Publs. Bayesian SEM: Lee, SY (2007) Structural Equation Modeling: A Bayesian Approach. Wiley & Sons.

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43 SEM is Based on the Analysis of Covariances! Why?Analysis of correlations represents loss of information. A B r = 0.86r = 0.50 illustration with regressions having same slope and intercept Analysis of covariances allows for estimation of both standardized and unstandardized parameters.

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44 Σ = { σ 11 σ 12 σ 22 σ 13 σ 23 σ 33 } Model-Implied CorrelationsObserved Correlations* { 1.0.24 1.0.01.70 1.0 } S = * typically the unstandardized correlations, or covariances 2. Estimation (cont.) – analysis of covariance structure The most commonly used method of estimation over the past 3 decades has been through the analysis of covariance structure (think – analysis of patterns of correlations among variables). compare

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45 x1 y1 y2 Hypothesized Model Σ = { σ 11 σ 12 σ 22 σ 13 σ 23 σ 33 } Implied Covariance Matrix Observed Covariance Matrix { 1.3.24.41.01 9.7 12.3 } S = compare Model Fit Evaluations + Parameter Estimates estimation (e.g., maximum likelihood) 3. Evaluation

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46 Model Identification - Summary 2. Several factors can prevent identification, including: a. too many paths specified in model b. certain kinds of model specifications can make parameters unidentified c. multicollinearity d. combination of a complex model and a small sample 1. For the model parameters to be estimated with unique values, they must be identified. As in linear algebra, we have a requirement that we need as many known pieces of information as we do unknown parameters. 3. Good news is that most software checks for identification (in something called the information matrix) and lets you know which parameters are not identified.

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47 The most commonly used fitting function in maximum likelihood estimation of structural equation models is based on the log likelihood ratio, which compares the likelihood for a given model to the likelihood of a model with perfect fit. Fitting Functions Note that when sample matrix and implied matrix are equal, terms 1 and 3 = 0 and terms 2 and 4 = 0. Thus, perfect model fit yields a value of F ML of 0.

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48 Maximum likelihood estimators, such as F ML, possess several important properties: (1) asymptotically unbiased, (2) scale invariant, and (3) best estimators. Assumptions: (1) and S matrices are positive definite (i.e., that they do not have a singular determinant such as might arise from a negative variance estimate, an implied correlation greater than 1.0, or from one row of a matrix being a linear function of another), and (2) data follow a multinormal distribution. Fitting Functions (cont.)

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49 One of the most commonly used approaches to performing such tests (the model Χ 2 test) utilizes the fact that the maximum likelihood fitting function F ML follows a X 2 (chi-square) distribution. The Χ 2 Test X 2 = n-1(F ML ) Here, n refers to the sample size, thus X 2 is a direct function of sample size. Assessment of Fit between Sample Covariance and Model- Implied Covariance Matrix

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50 Illustration of the use of Χ 2 X 2 = 3.64 with 1 df and 100 samples P = 0.056 X 2 = 7.27 with 1 df and 200 samples P = 0.007 x y1y1 y2y2 1.0 0.41.0 0.350.51.0 r xy2 expected to be 0.2 (0.40 x 0.50) X 2 = 1.82 with 1 df and 50 samples P = 0.18 correlation matrix issue: should there be a path from x to y 2 ? 0.400.50 Essentially, our ability to detect significant differences from our base model, depends as usual on sample size.

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51 Additional Points about Model Fit Indices: 1.The chi-square test appears to be reasonably effective at sample sizes less than 200. 2. There is no perfect answer to the model selection problem. 4. A lot of attention is being paid to Bayesian model selection methods at the present time. 3. No topic in SEM has had more attention than the development of indices that can be used as guides for model selection. 5. In SEM practice, much of the weight of evidence falls on the investigator to show that the results are repeatable (predictive of the next sample).

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52 Alternatives when data extremely nonnormal Robust Methods: Satorra, A., & Bentler, P. M. (1988). Scaling corrections for chi- square statistics in covariance structure analysis. 1988 Proceedings of the Business and Economics Statistics Section of the American Statistical Association, 308-313. Bootstrap Methods: Bollen, K. A., & Stine, R. A. (1993). Bootstrapping goodness- of-fit measures in structural equation models. In K. A. Bollen and J. S. Long (Eds.) Testing structural equation models. Newbury Park, CA: Sage Publications. Alternative Distribution Specification: - Bayesian and other:

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53 Residuals: Most fit indices represent average of residuals between observed and predicted covariances. Therefore, individual residuals should be inspected. Correlation Matrix to be Analyzed y1 y2 x -------- -------- -------- y1 1.00 y2 0.50 1.00 x 0.40 0.35 1.00 Fitted Correlation Matrix y1 y2 x -------- -------- -------- y1 1.00 y2 0.50 1.00 x 0.40 0.20 1.00 residual = 0.15 Diagnosing Causes of Lack of Fit (misspecification) Modification Indices: Predicted effects of model modification on model chi-square.

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54 The topic of model selection, which focuses on how you choose among competing models, is very important. Please refer to additional tutorials for considerations of this topic.

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55 While we have glossed over as many details as we could, these fundamentals will hopefully help you get started with SEM. Another gentle introduction to SEM oriented to the community ecologist is Chapter 30 in McCune, B. and J.B. Grace 2004. Analysis of Ecological Communities. MJM. (sold at cost with no profit)

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