# Whether to use structural equation modelling to analyse observational studies in psychology

I've noticed this issue coming up a lot in statistical consulting settings and i was keen to get your thoughts.

### Context

I often speak to research students that have conducted a study approximately as follows:

• Observational study
• Sample size might be 100, 200, 300, etc.
• Multiple psychological scales have been measured (e.g., perhaps anxiety, depression, personality, attitudes, other clinical scales, perhaps intelligence, etc.)

The researchers have read the relevant literature and have some thoughts about possible causal processes. Often there will be some general conceptualisation of variables into antecedents, process variables, and outcome variables. They have also often heard that structural equation modelling is more appropriate for testing overall models of the relationships between the set of variables that they are studying.

### Question

• Under what conditions do you think structural equation modelling is an appropriate technique for analysing such studies?
• If you would not recommend structural equation modelling, what alternative techniques would you recommend?
• What advice would you give to researchers considering using structural equation modelling in such cases?

My disclaimer: I realize this question has laid dormant for some time, but it seems to be an important one, and one that you intended to elicit multiple responses. I am a Social Psychologist, and from the sounds of it, probably a bit more comfortable with such designs than Henrik (though his concerns about causal interpretations are totally legitimate).

Under What Conditions Is SEM An Appropriate Data Analysis Technique?

To me, this question actually actually gets at two distinct sub-questions:

1. Why use SEM in the first place?
2. If a researcher has decided to use SEM, what are the data-related requirements for using SEM?

Why use SEM in the first place?

SEM is a more nuanced and complicated--and therefore less accessible--approach to data analysis than other, more typical, general linear modelling approaches (e.g., ANOVAs, correlations, regression, and their extensions, etc.,). Anything you can think of doing with those approaches, you can do with SEM.

As such, I think would-be users should first strongly evaluate why they are compelled to use SEM in the first place. To be sure, SEM offers some powerful benefits to its users, but I have reviewed papers in which none of these benefits are utilized, and the end-product is a data analysis section in a paper that is needlessly more difficult for typical readers to understand. It's just simply not worth the trouble--for the researcher, or the reader--if the benefits of SEM vs. other data analysis approaches are not being reaped.

So what do I see as the primary benefits of an SEM approach? The big ones, in my opinion are:

(1) Modeling latent variables: SEM allows users to examine structural relations (variances, covariances/correlations, regressions, group mean differences) among unobserved latent variables, which are essentially the shared covariance between a group of variables (e.g., items from an anxiety measure your students might use).

The big selling point for analyzing latent variables (e.g., latent anxiety) vs. an observed score of the construct (e.g., an average of the anxiety items) is that latent variables are error-free--latent variables are formed of shared covariance, and error is theorized to covary with nothing. This translates to increased statistical power, as users no longer have to worry about measurement unreliability attenuating the effects they are trying to model.

Another, more understated, reason to consider using SEM is in some cases it is a more construct-valid way of testing our theories about constructs. If your students, for example, were using three different measures of anxiety, wouldn't it be better to understand the causes/consequences of what those three measures have in common--presumably anxiety-- in an SEM framework, instead of privileging any particular one measure as the measure of anxiety?

(2) Modeling multiple dependent variables: Even if someone isn't going to use SEM to model latent variables, it can still be quite useful as a framework for simultaneously analyzing multiple outcome variables in one model. For example, perhaps your students are interested in exploring how the same predictors are associated with a number of different clinically relevant outcomes (e.g., anxiety, depression, loneliness, self-esteem, etc.,). Why run four separate models (increasing Type I error rate), when you can just run one model for all four outcomes that you are interested in? This is also a reason to use SEM when dealing with certain types of dependent data, where multiple, dependent respondents might both yield predictor and outcome responses (e.g., dyadic data; see Kenny, Kashy, and Cook, 2006, for a description of the SEM approach to using the Actor-Partner Interdependence Model [APIM]).

(3) Modeling assumptions, instead of making them: With many other approaches to data analysis (e.g., ANOVA, correlation, regression), we make a ton of assumptions about the properties of the data we are dealing with--such as homogeneity of variance/homoskedasticity. SEM (usually combined with a latent variable approach) enables users to actually model variance parameters simultaneously alongside means and/or correlations/regressive pathways. This means that users can begin theorizing about and testing hypothesis about variability, in addition to mean differences/covariability, instead of just treating variability as an annoying assumption-related afterthought.

Another testable assumption, when comparing group mean levels on some variable, is whether that variable actually means the same thing to each group--referred to as measurement invariance in the SEM literature (see Vandenberg & Lance, 2000, for a review of this process). If so, then comparisons on mean levels of that variable are valid, but if groups have a significantly different understanding of what something is, comparing mean levels between groups is questionable. We make this particular assumption implicitly all the time in research using group-comparisons.

And then there is the assumption, that when you average or sum item scores (e.g., on an anxiety measure) to create an aggregate index, that each item is an equally good measure of the underlying construct (because each item is weighted equally in the averaging/summing). SEM eliminates this assumption when latent variables are used, by estimating different factor loading values (the association between the item and the latent variable) for each item.

Lastly, other assumptions about the data (e.g, normality), though still important for SEM, can be managed (e..g, through the use of "robust" estimators, see Finney & DiStefano, 2008) when the data fail to meet certain criteria (low levels of skewness and kurtosis).

(4) Specifying model constraints: The last big reason, in my opinion, to consider using SEM, is because it makes it very easy to test particular hypotheses you might have about your model of data, by forcing ("constraining" in SEM terms) certain paths in your model to take on particular values, and examining how that impacts the fit of your model to your data. Some examples include: (A) constraining a regression pathway to zero, to test whether it's necessary in the model; (B) containing multiple regression pathways to be equal in magnitude (e.g., is the associative strength for some predictor roughly equal for anxiety and depression?); (C) constraining the measurement parameters necessary to evaluate measurement invariance (described above); (D) constraining a regression pathway to be equal in strength between two different groups, in order to test moderation by group.

What are the data-related requirements for SEM?

The data-related requirements for SEM are pretty modest; you need an adequate sample size, and for your data to meet the assumptions of the model estimator you have selected (Maximum-Liklihood is typical).

It is difficult to give a one-size-fits-all recommendation for sample size. Based on some straightforward simulations, Little (2013) suggests that for very simple models, 100-150 observations might be enough, but sample size needs will increase as models become more complex, and/or as the reliability/validity of the variables used in the model decreases. If model complexity is a concern, you could consider parcelling the indicators of your latent variables, but not all are onboard with this approach (Little, Cunningham, Shahar, & Widaman, 2002). But generally speaking, all else being equal, bigger samples (I strive for 200 minimum in my own research) are better.

As for meeting the assumptions of a selected estimator, usually this is pretty easy to assess (e.g., look at skewness and kurtosis values for a maximum likelihood estimator). And even if data depart from assumed properties, a research could consider the use of a "robust" estimator (Finney & DiStefano, 2008), or an estimator that assumes a different kind of data (e.g., a categorical estimator, like diagonally weighted least squares).

Alternatives To SEM for Data Analysis?

If a researcher isn't going to take advantage of the benefits provided by an SEM approach that I've highlighted above, I'd recommend sticking to the more straight-forward and accessible version of that particular analysis (e..g, t-tests, ANOVAs, correlation analysis, regression models [including mediation, moderation, and conditional process models]). Readers are more familiar with them, and will therefore more easily understand them. It's just not worth confusing readers with the minutiae of SEM if you're essentially using SEM to the same effect as a simpler analytic approach.

Advice to Researchers Considering The Use Of SEM?

For those brand new to SEM:

1. Get a comprehensive, accessibly-written foundation SEM text. I like Beaujean (2014), Brown (2015; the earlier edition is solid too), and Little (2013; good overall introduction, even though it later focuses specifically on longitudinal models).
2. Learn how to use the lavaan package for R (Rosseel, 2012). It's syntax is as easy as SEM syntax can get, it's functionality is broad enough for many folks' SEM needs (definitely for beginners), and it's free. The Beaujean book gives a great simultaneous introduction to SEM and the lavaan package.
3. Consult/use CrossValidated and StacksOverflow regularly. Unexpected things can happen when fitting SEM models, and chances are, many of the weird things you might experience have already been described and troubleshot on Stacks.
4. As Herik points out, note that just because you are specifying a model that implies causal associations, it does not mean that SEM helps to establish causality in a cross-sectional/non-experimental study. Also, it's totally worth considering the use of SEM to analyze data from longitudinal and/or experimental designs.

And for those who are beginning to actually use SEM:

1. You will, at some point, be tempted to specify correlated residuals willy-nilly, in an effort to improve the fit of your model. Don't. At least not without a good a priori reason. More often than not, a larger sample, or a simpler model is the cure.
2. Avoid the use of the marker-variable method of identification for latent variables (i.e., fixing the first factor loading to 1). It privileges that indicator as the "gold-standard" indicator of your latent variable, when in most cases, there is no reason to assume this is the case. Be aware that this is the default identification setting in most programs.

References

Beaujean, A. A. (2014). Latent variable modeling using R: A step-by-step guide. New York, NY: Routledge.

Brown, T. A. (2015). Confirmatory factor analysis for applied researchers (2nd edition). New York, NY: Guilford Press.

Finney, S. J., & DiStefano, C. (2008). Non-normal and categorical data in structural equation modeling. In G. R. Hancock & R. D. Mueller (Eds.), Structural equation modeling: A second course (pp. 269-314). Information Age Publishing.

Kenny, D. A., Kashy, D. A., & Cook, W. L. (2006). Dyadic data analysis. New York, NY: Guilford Press.

Little, T. D. (2013). Longitudinal structural equation modeling. New York, NY: Guilford Press.

Little, T. D., Cunningham, W. A., Shahar, G., & Widaman, K. F. (2002). To parcel or not to parcel: Exploring the question, weighing the merits. Structural Equation Modeling, 9, 151-173.

Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48(2), 1-36.

Vandenberg, R. J., & Lance, C. E. (2000). A review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational researchers. Organizational Research Methods, 3, 4-70.

• +1 good answer. I'm looking forward to your other contributions! – Momo Dec 18 '15 at 17:12
• +1 Great answer. I agree with most of what you say. One follow up point: I don't think SEM has greater statistical power because it is estimating relationship between latent variables. In general, I would think that if anything SEM introduces an additional source of error. Thus, there is the usual error in estimating the observed variable, and additional error in estimating error variance or other elements of the latent structure. – Jeromy Anglim Dec 19 '15 at 5:01
• I would respectively maintain that SEM, by modelling error in unique factors, does, in fact, increase power This can be most easily demonstrated, I think, by comparing an SEM model of group differences in latent means to a model of observed mean differences of the same data. If you run the SEM using an effects-coding approach (see Little, Slegers, & Card, 2006), your latent means for each group will take on the same value as your observed means. But the variances for each group in the SEM model will be smaller than in the observed, rendering a larger effect size that is easier to detect. – jsakaluk Dec 20 '15 at 6:24
• I realize the above comment might not be the best way to respond to your concern; if you ask a separate question about SEM and power on CV, I'd be happy to post a more elaborate answer, with some example output that could be helpful. – jsakaluk Dec 20 '15 at 16:05

Disclaimer: I consider myself an experiemtal psychologist with an emphasis on experimental. Hence, I have a natural unease with designs like this.

To answer your first and second question: I think for a design like this a SEM or, depending on the number of variables involved, mediation or moderation analyses is the natural way of dealing with the data. I have no good idea what else to recommend.

For your third question: I think the main advantage with a design like this is it's main disadvantage. Namely that you (given enough variables) will find significant results. The question is, how you interpret these results.

That is, you can look at so many hypotheses (some more some less inspired by the relevant literature) that you will probably find something significant (not in the literal sense of rejecting a SEM) that will be interpratable in a psychological sense. Therefore, my advice to anyone doing this would be twofold:

1. Stress the problem with causal interpretation of these designs. I am not an expert in this but know, that a fully cross-sectional design can hardly be interpreted causal, independent of how intuitively plausible that may sound. More advanced designs like cross-lagged pnael designs or stuff like this is needed for causal interpetations. I think the work by Shadish, Cook & Campbell (or at least some of them) are a good ressource for further discussion of these topics.
2. Stress the individual responsibility and scientific ethics. If you see that your initial idea is not supported by the data, it is the natural next step to inspect the data further. However, you shall never rely on HARKing (Hypothesizing After the Results are Known; Kerr, 1998, see also Maxwell, 2004). That is, you should stress that there is a thin line between a reasonable adaption of your hypotheses given the data and cherry picking of significant results.
• and Bernd: Ditto! Great acronym and I hope it catches on. – rolando2 May 27 '11 at 21:50