Structural equation modeling (SEM) is a statistical technique that allows tests of complex relationships between large numbers of variables. The term “structural equation modeling” is closely related to terms like covariance structure analysis, causal modeling, or path analysis. According to Kline (2005), the main characteristics of SEM are as follows. First of all, there has to be a model that shows how variables are related; it has to be specified prior to statistical analysis. The idea of the model may come from theory, from prior research, or from researchers’ specified knowledge and experience. The procedure is mainly confirmatory, and generally researchers decide which are dependent and independent variables and how they are linked by causal relations.
When specifying a model one must distinguish between observed and latent variables, or at least explicate at which level a variable is located. Most standard procedures do not allow differentiation between observed and latent variables. As an example, if you want to measure how political orientation influences newspaper usage, you have two abstract constructs that cannot be measured directly. These constructs – or latent variables – have to be operationalized through indicators, or observed variables. So political orientation may be measured through statements like “I want to know what is going on in the world” or “I want to know the relevant arguments of current political discourse,” and newspaper usage may be measured through “newspaper reading per day in minutes” or “individual importance of daily newspaper reading.” The strength of SEM in incorporating observed as well as latent variables lies in the validation of constructs that can be achieved through confirmatory factor analysis. Then, the influence of one construct on another can be measured through structural regression analysis. The hypotheses about the modeled relationships can be tested with several indices.
Validation Of Constructs
Constructs in the social sciences may consist of one or more factors, each being represented by one or more indicators. In our example, political orientation can be conceived as a one-factorial construct represented through two indicators: “knowing what is going on in the world” and “knowing the relevant arguments of current political discourse.” As valid constructs are the basis of statistical testing of hypotheses – especially at a higher abstraction level – this part of the modeling is very important for the whole SEM process. First, the dimensionality of given indicators can be tested through exploratory factor analysis. After a reliability test (e.g., with Cronbach’s Alpha), unsuitable indicators can be sorted out in order to strengthen the internal consistency of the construct. Then, construct validity can be tested statistically through confirmatory factor analysis. This particular application of SEM shows to what extent the measurement model matches empirical data. There are several criteria for a good model fit: global criteria for the fitting of the complete measurement model and single criteria for every indicator to explain a construct. The procedure aims at explaining indicator reliability, factor reliability, and the average variance of a factor. If correlation within one factor is high, then one can speak of high convergent validity. If correlation between factors is low, one can speak of high discriminant validity. If the construct is complete (no relevant variables are missing), the indicators relate only to one construct, and if there are no systematic errors (because of bias in either measurement or sampling), one can speak of a fairly validated construct, which can then be used for further modeling.
Structural Regression Analysis
A structural regression model is the most general kind of structural equation model. It contains hypotheses about several aspects:
- whether there is an influence at all;
- how strong the influence is;
- whether effects are direct or mediated by other variables;
- whether there is reciprocity in effects;
- whether the impact is weakened or strengthened by a mediator.
A structural equation model can also be called a hybrid model, as it incorporates a measurement and a structure model. These models are commonly tested with programs like LISREL or AMOS. It allows testing for both direct and indirect causal effects. An example of a direct effect would be the impact of political orientation on newspaper usage. An example of an indirect effect would be the impact of political orientation via interpersonal communication on newspaper usage. The difference from path analysis is that effects can involve latent – not only observed – variables. This setting can test effects from observed to latent variables and vice versa within a single model, and therefore proves more flexibility than do other procedures. The (causal) relationships between variables are conceptualized and illustrated as paths. Those paths can be expressed as mathematical functions. The methodology of SEM thus can show whether there are significant effects in the relationship between two variables. This principle is commonly known from regression analysis; SEM, however, is more powerful, as the impact of a variable on several target variables can be estimated simultaneously.
In order to conduct a structural regression analysis, two approaches are needed: (1) the features of path analysis to test the structural model, and (2) the features of confirmatory factor analysis to test the measurement component (of the factor model). The measurement model consists of the a priori specification of latent factors and the scores on their indicators. The structural model incorporates direct effects between the factors (i.e., latent variables). As in regression analysis, we have preconceptualizations of the direction of effects. According to the underlying theory and model, it can be specified which factor has effects on other factors. So the target factors are called endogenous variables whereas the influencing factors are called exogenous variables. It is important to note that each structural model contains measurement errors and disturbances. The measurement model – i.e., the confirmatory factor analysis – contains measurement errors for each indicator, whereas the error between the effect of one factor and another is called disturbance. A disturbance is not to be understood as a measurement error but rather as an omitted cause. It is something that has not been incorporated into the model but would have contributed to accounting for a large part of the variance. In the SEM procedure, the relationships are freed from measurement error because the error has been estimated and then removed, leaving only common variance. So the options of decomposition of effects and errors exceed the possibilities of conventional techniques like path or regression analysis (Bollen 1987). Thus, when the phenomena of interest are complex and multidimensional, SEM is the only analysis that allows complete and simultaneous tests of all relationships.
Many standard operations in multivariate statistics, such as multiple regression, analysis of variance, or canonical correlation, are special cases of SEM. All the stated procedures actually stem from the principle of the general linear model (GLM). Thus, the generality of SEM is very broad. As in regression analysis, the relationship between two variables can be expressed as an equation, e.g., Y = BX + A. Most statisticians agree that SEM requires large samples. There are several factors that affect sample size, so it is not easy to answer the question of how large a sample needs to be. The analysis of a complex model requires a larger sample than the analysis of a simpler model. Models with greater complexity and more interrelations require the estimation of more statistical effects, and thus larger samples are necessary in order for the results to be reasonably stable. The type of estimation algorithm used in the analysis also affects sample size requirements. There are quite a few different estimation methods in SEM, and some of these may need very large samples. However, some rough guidelines can be offered. Using fewer than 100 cases hardly leads to a reasonable test. A size of 100 to 200 cases is regarded as a minimum, but this is not absolute, because the model’s complexity must also be considered. Furthermore, a small sample size means that the power of statistical tests may be very limited.
There is a great variety of effects that can be tested for significance in SEM. Options range from the variance of single variables up to the evaluation of entire models across several samples. However, tests for statistical significance are less relevant in SEM than to other techniques. Although single effects of related variables can be tested, SEM usually operates on a higher abstraction level. The question is not so much how strong an effect is; the analysis rather concentrates on the whole model, and the questions are whether it is acceptable or whether it had better be rejected – whether it should be modified, and if so, how. The big picture has priority over specific details such as the single path coefficients of two related variables. Furthermore, the significance of effects is not of so much help to the researcher in evaluation, as larger samples are more likely to produce significant results, even if the absolute effect size is minimal. In this case, the significance would only confirm the large sample (see also Tanaka 1987).
The methodology of SEM offers quite a lot of options for evaluating the fit of the estimated model with empirical data. Among the numerous fit indices, the most commonly used include chi-square value, the goodness-of-fit-index (GFI), the adjustedgoodness-of-fit index (AGFI), and the root mean square error of approximation (RMSEA). If indices show a good model fit, the model thus can be accepted. If the test produces implausible values, however, either model or data are insufficient. In any case, checks for logical and theoretical plausibility of the modeled parameters are recommended. Structural equation models are mainly used for confirmatory purposes. As many communication theories are not sufficiently validated, researchers may feel the need to use SEM in an exploratory way (Holbert & Stephenson 2002). Even then, researchers should have some idea about relationships and effects between constructs. To properly evaluate the quality of a model, analyses with SEM need replication. If similar results are achieved through cross-validation with different datasets, the model structure is proved to be reasonable. Finally, it is important to note that SEM does not prove causality in the same way as experiments do. Causality is always a question of design and not of statistical testing.
- Bollen, K. A. (1987). Total, direct, and indirect effects in structural equation models. In C. C. Clogg (ed.), Sociological methodology. Washington, DC: American Sociological Association, pp. 37– 69.
- Holbert, R. L., & Stephenson, M. T. (2002). Structural equation modeling in the communication sciences, 1995 – 2000. Human Communication Research, 28(4), 531–551.
- Kline, R. B. (2005). Principles and practice of structural equation modeling. New York and London: Guilford.
- Peter, J. (1981). Construct validity: A review of psychometric basic issues and marketing practices. Journal of Marketing Research, 18, 133–145.
- Tanaka, J. S. (1987). How big is big enough? Sample size and goodness of fit in structural equation models with latent variables. Child Development, 58, 134–146.
- Ullmann, J. B. (2007). Structural equation modeling. In B. G. Tabachnick & L. S. Fidell (eds.), Using multivariate statistics. Boston and New York: Pearson.