Questions: Structural Equation Modeling with Latent Variables

Question 1

A researcher builds an SEM with CFI = .97 and RMSEA = .04 and concludes: 'These fit indices confirm that my theory of how socioeconomic status influences civic participation through institutional trust is correct.' What is wrong with this conclusion?

• A. The CFI threshold for confirmation is .99, not .95
• B. Good fit only shows the model is consistent with the data — many alternative models could fit equally well
• C. RMSEA must be below .03 for any theoretical conclusion to be valid
• D. SEM cannot test mediation models, so the conclusion is procedurally invalid

Answer: (B)

Good fit means your model's implied covariance matrix closely matches the observed covariance matrix — it means your model is *consistent* with the data, not that it is correct. Multiple theoretically different models can fit the same data equally well. A model can achieve excellent fit while encoding wrong theoretical assumptions; equivalently, a correct theory might fit poorly if it is mis-specified. This is the central epistemological limitation of SEM: fit is necessary but not sufficient evidence for a theory.

Question 2

What is SEM's primary advantage over ordinary multiple regression when predictors are psychological constructs measured by survey items?

• A. SEM produces larger R-squared values by including more variables simultaneously
• B. SEM explicitly models measurement error in latent variables, yielding unbiased structural coefficients
• C. SEM does not require multivariate normality, making it more robust
• D. SEM generates larger sample sizes through data imputation

Answer: (B)

When you regress one observed variable on others, any measurement error in the predictors biases the regression coefficients toward zero (attenuation bias). SEM solves this by distinguishing the latent variable (the true construct) from its observed indicators, explicitly modeling the measurement error in each indicator. The structural paths then connect the latent variables — purged of measurement error — so the coefficients are unbiased estimates of the true relationships. This is the main advantage over ordinary regression for psychological and social science constructs.

Question 3

When SEM modification indices indicate that freeing a fixed parameter (adding a path) would improve model fit, this tells the researcher which paths are theoretically meaningful and should be added.

• T. True
• F. False

Answer: False

Modification indices are purely statistical: they report how much the chi-square fit statistic would decrease if a currently-fixed parameter were freed. They say nothing about whether that parameter makes theoretical sense. Chasing modification indices produces models that are over-fit to a specific sample rather than to a theory — they capitalize on chance variation in the data and will not replicate. Adding paths based on modification indices alone is a form of post-hoc data dredging that undermines SEM's confirmatory purpose.

Question 4

In SEM, estimating the measurement model and structural model simultaneously allows the structural paths to account for measurement error in the latent constructs.

• T. True
• F. False

Answer: True

This simultaneous estimation is what distinguishes SEM from a two-step approach (first factor analysis, then regression on factor scores). When both models are estimated together, the structural paths connect latent variables whose measurement error is explicitly modeled — giving unbiased estimates of the relationships among the true underlying constructs. Estimating separately would introduce bias because factor scores themselves carry residual measurement error.

Question 5

Why is good model fit in SEM not sufficient evidence that a theoretical model is correct, even when fit indices like CFI and RMSEA meet recommended thresholds?

Model answer: Good fit means the model's implied covariance matrix is consistent with the observed data — but there are always multiple models that can produce the same covariance structure. Two models with opposite causal directions, or with entirely different mediating variables, can be statistically equivalent — they fit identically and cannot be distinguished by fit indices alone. Additionally, a model can fit because it is flexible enough to accommodate the data even if its theoretical claims are wrong. Fit is a necessary but not sufficient condition for theoretical validity; substantive interpretation, experimental replication, and alternative model comparison are also required.

This connects to the broader principle that statistical consistency is not causal identification. SEM is a confirmatory technique, but 'confirmatory' means you've committed to a structure before looking at data — it doesn't mean the structure is uniquely supported. The remedy is triangulation: reporting alternative equivalent models, using longitudinal data to test temporal order, replicating in independent samples, and treating good fit as 'not yet falsified' rather than 'confirmed.'