In jeksterslabds/jeksterslabRnotes: R Statistics Notes

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Structural Equation Modeling

\begin{equation} \mathbf{\Sigma} = \mathbf{\Sigma} \left( \mathbf{\theta} \right) \end{equation}

where

• $\mathbf{\Sigma}$ (sigma) is the population covariance matrix of observed variables
• $\mathbf{\theta}$ (theta) is a vector that contains the model parameters
• $\mathbf{\Sigma} \left( \mathbf{\theta} \right)$ is the covariance matrix written as a function of $\mathbf{\theta}$

Regression

\begin{equation} y = \gamma x + \zeta \end{equation}

where

• $\gamma$ (gamma) is regression coefficient,

• $\zeta$ (zeta) is the disturbance variable

• uncorrelated with $x$, and

• the expected value of $\zeta$, $\mathbb{E} = 0$

\begin{equation} \begin{split} \mathbf{\Sigma} &= \mathbf{\Sigma} \left( \mathbf{\theta} \right) \ \begin{bmatrix} \mathrm{Var} \left( y \right) & \ \mathrm{Cov} \left( x, y \right) & \mathrm{Var} \left( x \right) \end{bmatrix} &= \begin{bmatrix} \gamma^2 \sigma_{x}^{2} + \sigma_{\zeta}^{2} & \ \gamma \sigma_{x}^{2} & \sigma_{x}^{2} \end{bmatrix} \end{split} \end{equation}

Factor Model

\begin{equation} \begin{split} x_1 &= \xi + \delta_1 \ x_2 &= \xi + \delta_2 \end{split} \end{equation}

where

• $x_1$ and $x_2$ are indicators of a factor or latent random variable called $\xi$ (xi)

• $\delta_1$ and $\delta_2$ are random disturbance terms

• uncorrelated with $\xi$ and with each other, and

• $\mathbb{E} \left( \delta_1 \right) = \mathbb{E} \left( \delta_2 \right) = 0$

• the latent variable $\xi$ has a variance $\phi$ (phi)

\begin{equation} \begin{split} \mathbf{\Sigma} &= \mathbf{\Sigma} \left( \mathbf{\theta} \right) \ \begin{bmatrix} \mathrm{Var} \left( x_1 \right) & \ \mathrm{Cov} \left( x_1, x_2 \right) & \mathrm{Var} \left( x_2 \right) \end{bmatrix} &= \begin{bmatrix} \phi + \sigma^{2}{\delta_1} & \ \phi & \phi + \sigma^{2}{\delta_2} \end{bmatrix} \end{split} \end{equation}

Regression and Factor Model

\begin{equation} \begin{split} y &= \gamma \xi + \zeta \ x_1 &= \xi + \delta_1 \ x_2 &= \xi + \delta_2 \end{split} \end{equation}

where

• $y$ is associated with an unobserved variable $\xi$

• the factor model is identical to the previous model

• $\zeta$, $\delta_1$, and $\delta_2$ are uncorrelted with $\xi$ and with each other

\begin{equation} \begin{split} \mathbf{\Sigma} &= \mathbf{\Sigma} \left( \mathbf{\theta} \right) \ \begin{bmatrix} \mathrm{Var} \left( y \right) & & \ \mathrm{Cov} \left( x_1, y \right) & \mathrm{Var} \left( x_1 \right) & \ \mathrm{Cov} \left( x_2, y \right) & \mathrm{Cov} \left( x_2, x_1 \right) & \mathrm{Var} \left( x_2 \right) \end{bmatrix} &= \begin{bmatrix} \gamma^2 \phi + \sigma^{2}{\zeta} & & \ \gamma \phi & \phi + \sigma^{2}{\delta_1} & \ \gamma \phi & \phi & \phi + \sigma^{2}_{\delta_2}
\end{bmatrix} \end{split} \end{equation}

Historical Background

Sociology

• Bielby and Hauser (1977)

Psychology

• Bentler (1980, 1986)

Economics

• Goldberger (1972)
• Aigner et al. (1984)

Multidisciplinary

• Goldberger and Duncan (1973)
• Blalock (1971, 1985)

Other Collections

• Aigner and Goldberg (1977)
• Joreskog and Wold (1982)
• the November 1982 issue of the Journal of Marketing Research, and
• the May-June 1983 issue of the Journal of Econometrics

Origins

Wright and Path Analysis

• Wright (1918, 1921, 1934, 1960)

• the path diagram,

• the equations relating correlations or covariances to parameters (tracing rules), and

• the decomposition of effects (total effects, direct effects, and indirect effects)

• Wright's development of equations for covariances of variables in terms of model parameters is the same as that of $\mathbf{\Sigma} = \mathbf{\Sigma} \left( \mathbf{\theta} \right)$, except that he developed these equations from path diagrams rather than the matrix methods employed today.

Latent Variable and Measurement Models

• The factor analysis tradition spawned by Spearman (1904) emphasized the relation of latent factors to observed variables.
• The central concern was on what we now call the measurement model.
• The structural relations between latent variables other than their correlation (or lack of correlation) were not examined.
• In econometrics the focus was the structural relation between observed variables with an occasional reference to error-in-the-variable situations.

Path Analysis in the Social Sciences

• Social scientists and statisticians did not pay more attention to his work until the 1960s
• Following Duncan's (1966) expository account, the late 1960s and early 1970s saw many applications of path analysis in the sociological journals.
• The rediscovery of path analysis in sociology diffused to political science and several other social science disciplines.
• Stimulated by work in sociology, Werts and Linn (1970) wrote an expository treatment of path analysis, but it was slow to catch on in psychology.

Synthesis

• Wright's path analysis examples demonstrated that econometric-type models with variables measured with error could be identified and estimated.

• The conceptual synthesis of models containing structurally related latent variables and more elaborate measurement models was developed extensively in sociology during the 1960s and early 1970s.

• Blalock (1963) argued that sociologists should use causal models containing both indicators and underlying variables to make inferences about the latent variables based on the covariances of the observed indicators.

• He suggested that observed variables can be causes or effects of latent variables or observed variables can directly affect each other.
• He contrasted this with the restrictive implicit assumptions of factor analysis where all indicators are viewed as effects of the latent variable.

The LISREL Model

• Joreskog (1973), Keesing (1972), and Wiley (1973) developed very general structural equation models, incorporated path diagrams and other features of path analysis into their presentations. Researchers know these techniques by the abbreviation of the JKW model (Bentler 1980), or more commonly as the LISREL model.

• Its present form has some elaboration in the symbols employed in path diagrams,

• has equations relating covariances to parameters that are derived with matrix operations rather than from "reading" the path diagram, and
• has a more refined and clearly defined decomposition of direct, indirect, and total effects (see, e.g., Duncan 1971; Alwin and Hauser 1975; Fox 1980; Graff and Schmidt 1982).

Representation

• LISREL Joreskog et al. (1967, 1970, 1973, 1977, 1978)
• Bentler and Weeks (1980)
• McArdle and McDonald (1984)

Estimators

• In econometrics the properties of estimators for structural equations with observed variables were well established (see, e.g., Goldberger 1964)
• In psychometrics the work of Lawley (1940), Anderson and Rubin (1956), and Joreskog (1969) helped lay the foundations for hypothesis testing in factor analysis.
• Bock and Bargmann (1966) proposed an analysis of covariance structures to estimate the components of variance due to latent variables in multinormal observed variables.
• Joreskog (1973) proposed a maximum likelihood estimator (based on the multinormality of the observed variables) for general structural equation models which is today the most widely used estimator.

• Joreskog and Goldberger (1972) and Browne (1974, 1982, 1984) suggested generalized least squares (GLS) estimators that offer additional flexibility in the assumptions under which they apply.

• Browne (1982, 1984), for example, proposed estimators that assume arbitrary distributions or elliptical distributions for the observed variables.

• Bentler (1983) suggested estimators that treat higher-order product moments of the observed variables. He demonstrated that these moments can help identify model parameters that are not identified by the covariances and the gains in efficiency that may result.

• Muthen (1984, 1987), among others, has generalized these models to ordinal or limited observed variables.

Software

• LISREL (Joreskog and Sorbom)
• EQS (Bentler)
• Mplus (Muthen and Muthen)
• OpenMX
• sem (Fox)

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