Structural Equation Modeling (SEM)

What would you need to measure to find out whether urban parks help residents feel mentally restored? You could survey together how easy it is to get from home to a park, how satisfied people were when they used it, and how psychologically restored they feel after a visit.

The problem is that these concepts are hard to measure accurately with a single number. Park accessibility, satisfaction with use and mental restoration are concepts measured indirectly through several survey items. On top of that, the researcher may want to know whether park accessibility is directly related to restoration, or related indirectly through satisfaction with use.

Structural equation modeling (SEM) is a method that measures latent variables with several observed items like these and estimates the relationships among the latent variables in a single model. In this article, we use a concrete example to look at the components of SEM and the analysis process, path coefficients and indirect effects, model fit, identification and estimation methods, modification indices, and how to report results.

Whereas ordinary regression is mainly used to analyze conditional relationships among observed variables, SEM can explicitly handle latent variables that are hard to observe directly, and measurement error.

SEM is broadly divided into a measurement model and a structural model. The measurement model expresses how the survey items measure each concept, and the structural model shows what directional relationships are assumed among those concepts. A key feature is that the two parts can be estimated together.

It is also used in studies that handle several regression equations at once or that want to estimate direct and indirect effects together.

Exploratory factor analysis (EFA) focuses on discovering factor structure, and confirmatory factor analysis (CFA) on examining a measurement structure specified in advance. In SEM, you add paths between latent variables to this measurement structure so the research hypotheses can be evaluated together.

The researcher wants to see whether park accessibility (X) is related to satisfaction with use (M), and whether satisfaction is in turn related to mental restoration (Y). A path by which accessibility relates directly to restoration is also specified.

Each of the three concepts is measured with four items. The items were set up to be answered from 1, 'Strongly disagree,' to 5, 'Strongly agree.'

This case is an example for explaining the principles of the analysis, and the figures below are not results from real resident data.

For example, connect A1–A4 to the accessibility factor, S1–S4 to satisfaction and R1–R4 to restoration. In the basic model, we assume each item loads only on its designated factor and that the covariances between measurement errors are 0.

Here x_i is an observed item, ξ is the latent factor, λ_i is the factor loading and δ_i is the item's unique variance and measurement error. To set the scale of a factor, an identification constraint is needed, such as fixing each factor's variance to 1 or fixing the unstandardized loading of a marker item to 1.

If each item's loading is high and the estimates are stable, that is evidence the latent factor is closely related to those items. High loadings alone, however, do not automatically prove the validity of the item content or unidimensionality.

As equations, this can be written M = aX + ζ_M, Y = c′X + bM + ζ_Y.

Here a is the path between accessibility and satisfaction, b is the path between satisfaction and restoration accounting for accessibility, and c′ is the direct path between accessibility and restoration accounting for satisfaction. ζ_M and ζ_Y are the residuals of each endogenous latent variable.

In a real model, you also need to specify the necessary covariates, identification conditions and the residual structure.

If the items can be treated as close to continuous and the relevant distributional assumptions are appropriate, you can use maximum likelihood (ML). To deal with non-normality you can consider robust maximum likelihood (MLR), and if you analyze ordinal Likert items as categorical, you can consider methods such as WLSMV.

Before the analysis, check missing values and response distributions, excessively high correlations between variables, outliers, sample size and whether the model is identified. Handling of missing values also depends on the estimator. For example, FIML can handle missing data in continuous variables under appropriate assumptions, but it is not used the same way with every estimator and data type.

The sample size needed for SEM is hard to set with a single rule such as "a certain multiple of the number of items." It depends on the factor loadings, number of factors, number of free parameters, missingness and distributions, the estimation method and power, so you need to consider the complexity of the model.

For example, A2's standardized loading of .82 means there is a relatively strong relationship between the accessibility factor and responses to A2. In a standard single-factor measurement model, the proportion of A2's variance explained by that factor is roughly .82² = .6724, or 67.2%.

When interpreting loadings, also check the confidence intervals, standard errors and item content. It is not appropriate to automatically delete an item because its loading is low, or to treat the content of different items as the same because their coefficients are high.

So you need to look at model fit as well as the structural paths.

Values such as .90 or .95 for CFI and TLI, .06 or .08 for RMSEA and .08 for SRMR are commonly used reference points, not pass marks for every model. Interpretation depends on sample size, model degrees of freedom, the estimator and the characteristics of the data.

In particular, the χ² test is sensitive to sample size, so it is best to look at several fit indices, the residuals and theoretical plausibility as a whole.

In this simple linear model, the indirect effect is calculated as a × b = .50 × .60 = .30, and the total effect is the sum of the direct and indirect effects, .15 + .30 = .45.

So you can describe accessibility as showing a standardized relationship of .30 through the indirect path via satisfaction and .15 through the direct path controlling for satisfaction. Because the distribution of a product can be asymmetric, the confidence interval of the indirect effect is generally estimated with methods such as the bootstrap.

For example, if satisfaction's R² is .25, accessibility explains about 25% of the variance in satisfaction in this model.

Restoration is influenced by both accessibility and satisfaction, so its explained variance can be higher. For example, if the standardized coefficients of the two predictors are .15 and .60 and the correlation between accessibility and satisfaction is .50, the explained variance of restoration in this simple structure is .15² + .60² + 2(.15)(.60)(.50) = .4725, or about 47.3%.

High explained variance alone, however, does not prove the model's causal validity or generalizability.

Convergent validity checks whether items designed to measure the same concept are sufficiently related, and discriminant validity checks whether different latent factors are distinguishable.

For convergent validity, you can refer to average variance extracted (AVE), construct reliability, McDonald's omega and similar measures. For discriminant validity, you can look at factor correlations, HTMT and so on. For example, if the correlation between two factors is close to 1, you should further examine whether they are really distinct concepts.

However, AVE above .50 or a particular reliability index above .70 does not by itself fully prove the validity of the measurement. It is important to also evaluate the item content and the theoretical basis from previous research.

If a Heywood case appears, where a standardized loading is abnormally large or a residual variance is estimated as negative, there may be a problem with the model or the data.

A modification index shows how much fit might improve if a constrained parameter were freely estimated. But you should not add correlations between measurement errors or repeatedly connect new paths just because an index is high. Consider it only when there is a theoretically convincing reason, such as shared variation from identical wording or the same measurement method.

If you modified the model to fit the data, report the reasons and the changes, and if possible re-test the model on independent data. Improving model fit and establishing the validity of the research hypotheses are different matters.

When reporting a mediation path, clearly state the assumptions required for a causal interpretation.

In real research, you should also report standard errors, the estimator, the confidence interval of each path and detailed results of the measurement model.

Whereas CFA mainly evaluates the measurement relationships between items and factors, SEM can estimate structural paths between latent variables along with the measurement model.

But good fit or significant path coefficients do not automatically mean a correct causal model. You need to examine the validity of the measurement model, the identification and estimation assumptions, the data-generating process and the possibility of alternative models together.