Structural equation model

Chong-ho Yu, Ph.Ds.

Structural Equation Model

n this course it was mentioned twice that researchers should not design a complex experiment with too many variables when the knowledge of the subject matter is not well-understood. Nevertheless, a good architect should build a house rather than just making bricks. As the knowledge of the field accumulates to a degree that a synthesis is needed, researchers should put many variables together to build a coherent model, which specifies causal-effect relationships among those variables.
The importance of model building can be best illustrated by the Keynesian model. What? An economic model? Yes, you are not in the wrong class. Let me explain it. The Keynesian model was introduced by John Maynard Keynes, the most well-known British economist in the 20^th century. This model hypothesizes how government spending can trigger more spending in other economic sectors. This idea, "multiplier," originated from another economist instead of John Keynes. But that economist is virtually unknown while John Keynes has gained world-wide reputation. It is because instead of introducing a piece of concept, John Keynes built a comprehensive model for explaining the relationships among government expenditures, employment, money supply, inflation, interest rate, investment, and gross domestic product.

While dealing with many sets of variables and relationships, inexperienced researchers may perform several separate ANOVA and regression analyses. Indeed, a structural equation model (SEM) should be developed to avoid "missing links" among variables. SEM is built upon a measurement model and a structural model.

Measurement model
A measurement model, as its name implies, is about measurement and data collection. In the section concerning reliability and Cronbach Alpha, we discussed how researchers develop instrument with high reliability and low measurement error. In factor analysis, researchers extract latent variables from observed variables. The relationships are indicated in the following figure. The ellipse represents a latent variable such as a mental construct, the rectangles represent observed variables, which are items in a scale. The circles denote measurement errors.

Structural model
A structural model specifies how well some variables could predict some other variables. Because prediction involves relationships, it could be viewed as a regression model. Also, since the relationships form a "chain" or a "path," it is also known as path model.
When two models are combined, they form a structural equation model. The following is an example given by Lomax (1992). Assume that based upon literature research, a researcher hypothesizes that "home background" could be a predictor to "school achievement," and "school achievement" could predict "career success", he should define such vague concepts as home background, school achievement, and career success. The next logical step he should go is to develop instrument and data collection schemes to measure those latent constructs.

This example is simplified. A real-life structural equation model is more complicated than that. In addition, this is an example of a formative model (Roberts & Thatcher 2009), in which several indicators are combined to form a socially generated construct i.e. "Household background" or "socio-economic status" (SES) does not exist independently. Thus, in the diagram the arrows point to the construct, not the other way around. However, in psychology usually it is opposite: the arrows point from the construct to the indicators. For example, the latent construct "anxiety" exists in our cognitive structure and this mental state drives our behaviors, as measured by the indicators. This type of model is known as the reflective model (Coltman, Devinney, Midgley, & Venaik, 2008).

While applying an input-output model to research on Web-based instruction, researchers may find only one dependent variable and independent variable, namely, test performance and the treatment. However, while applying the input-process-output structural framework, between the input and output are involved many other observed and latent variables.

AMOS

There are many software packages for SEM. LISREL, marketed by SPSS Inc., is used to be the most recommended program, but SPSS has replaced LISREL with AMOS 2020). Many SEM programs require a strong background in matrix algebra and programming skills. On the contrary, AMOS is GUI-based. You can literally draw a model on the program canvas, enter the data, and then test the model. Please use the navigation buttons below the figure to walk through the simulated modeling building process.

Know exactly what you're doing

The model shown in the above example contains sets of relationships. How could that researcher formulate those relationships? Not by taking drugs, of course. In structural equation modeling, you have to know exactly what you are doing by drawing prior knowledge from past research. In the measurement model for SEM, the factor analysis employed should be confirmatory factor analysis rather than exploratory factor analysis. American Psychological Association (1996) pointed out the danger of treating exploratory research as confirmatory research:

The premature formulation of theoretical models has often led to the worst problems seen in the use of null hypothesis testing, such as misrepresentation of exploratory results as confirmatory studies, or poor design of confirmatory studies in the absence of necessary exploratory results. We propose that the field become more open to well formulated and well conducted exploratory studies with the appropriate quantitative treatment of their results, thereby enhancing the quality and utility of future theory generation and assessment.

Reference

American Psychological Association. (1996). Task force on statistical inference initial report. http://www.apa.org/science/tfsi.html
Coltman, T., Devinney, T. M., Midgley, D. F., & Venaik, S. (2008). Formative versus reflective measurement models: Two applications of formative measurement. Journal of Business Research, 61(12), 1250-1262. https://doi.org/10.1016/j.jbusres.2008.01.013
Hoyle, R. H. (Ed.) (1995). Structural equation modeling: Concepts, issues, and applications. Thousand Oaks: Sage Publications.
Roberts, N., & Thatcher, J. (2009). Conceptualizing and testing formative constructs: Tutorial and annotated example. ACM SIGMIS Database: the DATABASE for Advances in Information Systems, 40(3), 9-39. https://doi.org/10.1145/1592401.1592405
Schumacker, R. E. & Lomax, R. G. (1996). A beginner's guide to structural equation modeling. Mahwah, NJ: Lawrence Erlbaum Associates.
SPSS, Inc. (2020). AMOS http://www.spss.com/software/spss/base/Amos/overview.htm