Structural Equation Model

Structural Equation Model

A structural equation model (SEM) represents relationships among observed and potentially unobserved variables through a system of equations. It can combine a measurement model, which connects latent constructs to measured indicators, with a structural model, which specifies associations or hypothesised pathways among constructs. An observed-variable path analysis is also a form of SEM. The arrows express the researcher's assumptions; fitting the model does not by itself establish that they are causal.

Measurement and structural components

Suppose an unobserved construct $H$ represents health burden and observed questionnaire items $Z_1,Z_2,Z_3$ are proposed indicators. A simple reflective measurement specification is

$$Z_j=\nu_j+\lambda_jH+\epsilon_j,\qquad j=1,2,3,$$

where $\lambda_j$ is a factor loading, $\nu_j$ an intercept and $\epsilon_j$ an indicator-specific residual. The choice of reflective direction means the latent construct is hypothesised to influence responses; it is not appropriate for every index of health or deprivation. An alternative formative construction has different meaning and identification requirements. Specify indicator scales, reference periods, residual covariances and whether the same construct is measured comparably across groups and time.

A structural component might specify $M=aX+\zeta_M$ and $Y=c'X+bM+\zeta_Y$, where $X$ is an exposure, $M$ a mediator and $Y$ an outcome. Covariates, nonlinear links, repeated measures and correlated disturbances can be incorporated when substantively justified and identified. An SEM usually implies restrictions on the means, variances and covariances of observed variables; estimation compares those implications with data under a selected estimator and distributional assumptions.

Latent variables lack a natural scale until one loading or the factor variance is fixed, so report the scaling constraint. Ensure the number and arrangement of indicators and paths give an identified model: a numerical optimiser cannot recover unique parameters from data if several parameter sets produce the same observable distribution. Three indicators are an illustrative starting point, not a universal guarantee of identification; additional paths, correlated errors, categories or multiple factors change the accounting.

Worked mediation paths

For a transparent observed-variable path example, assume all three variables are standardised, $X$ has variance one, residuals are mean-zero and uncorrelated with the prior variables as specified, and the fitted path equations are

$$M=0.50X+\zeta_M,\qquad Y=0.20X+0.40M+\zeta_Y.$$

The path from $X$ to $M$ is $a=0.50$, from $M$ to $Y$ conditional on $X$ is $b=0.40$, and the direct path from $X$ to $Y$ conditional on $M$ is $c'=0.20$. In this linear additive specification, the algebraic indirect path is $ab=0.50(0.40)=0.20$ and the total association implied by substitution is $c=c'+ab=0.40$. If $X$ rises by one standard-deviation unit, the model predicts a 0.50-standard-deviation change in $M$ and a 0.40-standard-deviation total change in $Y$, under the stated linear equations. These coefficients are invented for teaching.

The standardisation and residual assumptions can be checked for internal consistency. If $\operatorname{Var}(M)=1$, then $\operatorname{Var}(\zeta_M)=1-0.50^2=0.75$. The explained variance of $Y$ from the two correlated predictors is $0.20^2+0.40^2+2(0.20)(0.40)(0.50)=0.28$ because $\operatorname{Cov}(X,M)=0.50$. Setting $\operatorname{Var}(\zeta_Y)=0.72$ gives $\operatorname{Var}(Y)=1$ under the stated independent residual structure. The example therefore does not quietly assume that $X$ and $M$ are uncorrelated.

The indirect product is an association decomposition until causal identification assumptions are defended. Even if $X$ were randomised, $M$ usually is not; unmeasured common causes of $M$ and $Y$, post-treatment confounding, reverse temporal order or measurement error can invalidate a causal mediation claim. Nonlinear, interaction or time-to-event models need estimand-specific definitions; $c=c'+ab$ is not a universal identity on every effect scale. Use appropriate intervals for an indirect effect, commonly resampling or suitable model-based methods, rather than treating a product as normally distributed without checking.

Estimation and model fit

Maximum likelihood, robust likelihood, weighted least squares or Bayesian estimation may be appropriate depending on variable types, missingness, sample size and distributional assumptions. Ordered questionnaire responses should not automatically be treated as continuous normal variables. State the estimator, handling of missing data and clustering, parameter constraints and any prior assumptions. Report unstandardised estimates with units and uncertainty, and standardised coefficients only as a supplement when meaningful.

Fit statistics such as a model chi-square comparison, comparative fit index, root mean square error of approximation and standardised root mean square residual describe aspects of agreement between model-implied and observed data under a chosen estimator. They do not prove that the specified arrows, latent construct or causal story are correct. A model with few or no degrees of freedom can fit covariance data by construction while remaining scientifically weak. Do not rely on a single universal cutoff without considering model size, sample, estimator and substantive misfit.

Inspect residuals, implausible coefficients, negative variance estimates, factor loadings, influential observations and identification warnings. Modification indices can suggest alternative specifications, but repeatedly adding paths after observing fit can overfit the sample; document revisions and validate in independent data. Compare plausible theory-led alternatives, not only the model that yields the best fit index. For longitudinal or group comparisons, test measurement invariance before interpreting differences in a latent mean as differences in the same construct.

Applications in health economics

An SEM can investigate how access, symptoms, treatment burden and health-related quality of life are associated, while accounting for error in measured indicators. It can analyse pathways between an intervention, intermediate health states and outcomes when the design and assumptions support them. A latent health-burden score is not automatically a preference-based utility for QALYs; mapping it to utility requires appropriate evidence and validation. Nor does a path coefficient directly give an incremental cost-effectiveness ratio.

SEM may help distinguish correlations among observed patient-reported measures from a proposed latent construct, or explore heterogeneity in how socioeconomic factors relate to service use. Specify the target population, time ordering and causal question. If the purpose is prediction, evaluate out-of-sample calibration and utility; if the purpose is causal explanation, address confounding, selection and intervention definitions. Good in-sample covariance fit alone does neither job.

Reporting and source guidance

Provide a path diagram or an unambiguous equation list, construct definitions, indicator items and scaling, hypothesised directions, omitted paths and residual covariances, data timing, sample and missingness, identification constraints, estimator, fit and diagnostic results, parameter uncertainty and sensitivity to alternatives. Separate a measured association, model-implied indirect effect and causal effect in the conclusions.

The official lavaan SEM tutorial distinguishes latent-variable, regression and covariance specifications. Gunzler and colleagues' methodological mediation introduction describes SEM pathways in applied health research. Valente and colleagues' discussion of mediation confounding clarifies why randomising an exposure alone does not identify every mediation effect. The path numbers above are illustrative and were checked algebraically, not fitted to patients.