Structural equation modeling: Difference between revisions

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Traces of the historical convergence of the factor analytic and path analytic traditions persist as the distinction between the measurement and structural portions of models; and as continuing disagreements over model testing, and whether measurement should precede or accompany structural estimates.<ref name="HG00a">Hayduk, L.; Glaser, D.N. (2000) "Jiving the Four-Step, Waltzing Around Factor Analysis, and Other Serious Fun". Structural Equation Modeling. 7 (1): 1-35.</ref><ref name="HG00b">Hayduk, L.; Glaser, D.N. (2000) "Doing the Four-Step, Right-2-3, Wrong-2-3: A Brief Reply to Mulaik and Millsap; Bollen; Bentler; and Herting and Costner". Structural Equation Modeling. 7 (1): 111-123.</ref> Viewing factor analysis as a data-reduction technique deemphasizes testing, which contrasts with path analytic appreciation for testing postulated causal connections – where the test result might signal model misspecification. The friction between factor analytic and path analytic traditions continue to surface in the literature.

Wright's path analysis influenced Hermann Wold, Wold's student Karl Jöreskog, and Jöreskog's student Claes Fornell, but SEM never gained a large following among U.S. econometricians, possibly due to fundamental differences in modeling objectives and typical data structures. The prolonged separation of SEM's economic branch led to procedural and terminological differences, though deep mathematical and statistical connections remain.<ref name="Westland15">Westland, J.C. (2015). Structural Equation Modeling: From Paths to Networks. New York, Springer.</ref><ref>{{Cite journal|last=Christ|first=Carl F.|date=1994|title=The Cowles Commission's Contributions to Econometrics at Chicago, 1939-1955|url=https://www.jstor.org/stable/2728422|journal=Journal of Economic Literature|volume=32|issue=1|pages=30–59|jstor=2728422|issn=0022-0515}}</ref> The economic version of SEM can be seen in SEMNET discussions of endogeneity, and in the heat produced as Judea Pearl's approach to causality via directed acyclic graphs (DAG's) rubs against economic approaches to modeling.<ref name="Pearl09"/> Discussions comparing and contrasting various SEM approaches are available<ref name="Imbens20">Imbens, G.W. (2020). "Potential outcome and directed acyclic graph approaches to causality: Relevance for empirical practice in economics". Journal of Economic Literature. 58 (4): 11-20-1179.</ref><ref name="BP13">Bollen, K.A.; Pearl, J. (2013) "Eight myths about causality and structural equation models." In S.L. Morgan (ed.) Handbook of Causal Analysis for Social Research, Chapter 15, 301-328301–328, Springer. doi:10.1007/978-94-007-6094-3_15</ref> but disciplinary differences in data structures and the concerns motivating economic models make reunion unlikely. Pearl<ref name=Pearl09 /> extended SEM from linear to nonparametric models, and proposed causal and counterfactual interpretations of the equations. Nonparametric SEMs permit estimating total, direct and indirect effects without making any commitment to linearity of effects or assumptions about the distributions of the error terms.<ref name=BP13 />

SEM analyses are popular in the social sciences because computer programs make it possible to estimate complicated causal structures, but the complexity of the models introduces substantial variability in the quality of the results. Some, but not all, results are obtained without the "inconvenience" of understanding experimental design, statistical control, the consequences of sample size, and other features contributing to good research design.{{Citation needed|date=July 2023}}

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"Accepting" failing models as "close enough" is also not a reasonable alternative. A cautionary instance was provided by Browne, MacCallum, Kim, Anderson, and Glaser who addressed the mathematics behind why the {{math|χ²}} test can have (though it does not always have) considerable power to detect model misspecification.<ref name="BMKAG02">Browne, M.W.; MacCallum, R.C.; Kim, C.T.; Andersen, B.L.; Glaser, R. (2002) "When fit indices and residuals are incompatible." Psychological Methods. 7: 403-421.</ref> The probability accompanying a {{math|χ²}} test is the probability that the data could arise by random sampling variations if the current model, with its optimal estimates, constituted the real underlying population forces. A small {{math|χ²}} probability reports it would be unlikely for the current data to have arisen if the current model structure constituted the real population causal forces – with the remaining differences attributed to random sampling variations. Browne, McCallum, Kim, Andersen, and Glaser presented a factor model they viewed as acceptable despite the model being significantly inconsistent with their data according to {{math|χ²}}. The fallaciousness of their claim that close-fit should be treated as good enough was demonstrated by Hayduk, Pazkerka-Robinson, Cummings, Levers and Beres<ref name="HP-RCLB05">Hayduk, L. A.; Pazderka-Robinson, H.; Cummings, G.G.; Levers, M-J. D.; Beres, M. A. (2005) "Structural equation model testing and the quality of natural killer cell activity measurements." BMC Medical Research Methodology. 5 (1): 1-9. doi: 10.1186/1471-2288-5-1. Note the correction of .922 to .992, and the correction of .944 to .994 in the Hayduk, et al. Table 1.</ref> who demonstrated a fitting model for Browne, et al.'s own data by incorporating an experimental feature Browne, et al. overlooked. The fault was not in the math of the indices or in the over-sensitivity of {{math|χ²}} testing. The fault was in Browne, MacCallum, and the other authors forgetting, neglecting, or overlooking, that the amount of ill fit cannot be trusted to correspond to the nature, ___location, or seriousness of problems in a model's specification.<ref name="Hayduk14a">Hayduk, L.A. (2014a) "Seeing perfectly-fitting factor models that are causally misspecified: Understanding that close-fitting models can be worse." Educational and Psychological Measurement. 74 (6): 905-926. doi: 10.1177/0013164414527449</ref>

Many researchers tried to justify switching to fit-indices, rather than testing their models, by claiming that {{math|χ²}} increases (and hence {{math|χ²}} probability decreases) with increasing sample size (N). There are two mistakes in discounting {{math|χ²}} on this basis. First, for proper models, {{math|χ²}} does not increase with increasing N,<ref name="Hayduk14b"/> so if {{math|χ²}} increases with N that itself is a sign that something is detectably problematic. And second, for models that are detectably misspecified, {{math|χ²}} increase with N provides the good-news of increasing statistical power to detect model misspecification (namely power to detect Type II error). Some kinds of important misspecifications cannot be detected by {{math|χ²}},<ref name="Hayduk14a"/> so any amount of ill fit beyond what might be reasonably produced by random variations warrants report and consideration.<ref name="Barrett07"/><ref name="Hayduk14b"/> The {{math|χ²}} model test, possibly adjusted,<ref name="SB94">Satorra, A.; and Bentler, P. M. (1994) “Corrections to test statistics and standard errors in covariance structure analysis”. In A. von Eye and C. C. Clogg (Eds.), Latent variables analysis: Applications for developmental research (pp. 399-419399–419). Thousand Oaks, CA: Sage.</ref> is the strongest available structural equation model test.

Numerous fit indices quantify how closely a model fits the data but all fit indices suffer from the logical difficulty that the size or amount of ill fit is not trustably coordinated with the severity or nature of the issues producing the data inconsistency.<ref name="Hayduk14a"/> Models with different causal structures which fit the data identically well, have been called equivalent models.<ref name="Kline16"/> Such models are data-fit-equivalent though not causally equivalent, so at least one of the so-called equivalent models must be inconsistent with the world's structure. If there is a perfect 1.0 correlation between X and Y and we model this as X causes Y, there will be perfect fit and zero residual error. But the model may not match the world because Y may actually cause X, or both X and Y may be responding to a common cause Z, or the world may contain a mixture of these effects (e.g. like a common cause plus an effect of Y on X), or other causal structures. The perfect fit does not tell us the model's structure corresponds to the world's structure, and this in turn implies that getting closer to perfect fit does not necessarily correspond to getting closer to the world's structure – maybe it does, maybe it doesn't. This makes it incorrect for a researcher to claim that even perfect model fit implies the model is correctly causally specified. For even moderately complex models, precisely equivalently-fitting models are rare. Models almost-fitting the data, according to any index, unavoidably introduce additional potentially-important yet unknown model misspecifications. These models constitute a greater research impediment.