SUGI 27 Statistics and Data Analysis

Structural Equation Modeling (SEM) is a comprehensive statistical approach to testing hypotheses about relations among observed and latent variables (measured variables and unmeasured constructs) (Hoyle, 1995). SEM takes a confirmatory rather than an exploratory approach, specifies intervariable relations a priori, and estimates measurement errors explicity (Suhr, 1999). The purpose of this paper is to provide an introduction to the SEM statistical approach with examples from health, business, and education fields. SAS code (PROC CALIS), diagrams, and results will be discussed. In the health field, a path analysis investigates the prediction of self-perceived illness with effects of exercise participation, self-perceived fitness, stressful life experiences, and hardiness for promoting stress resistance (Kline, 1998; Roth, Wiebe, Fillingim, & Shay, 1989). Relating to the business field, the example examines the relationship between academic success and career success (e.g., ACT score, cumulative grade point average, salary) (Schumacker & Lomax, 1996). The next example, in the education field, compares results from a baseline latent growth curve model (LGM) of reading achievement to results from an LGM including a categorical variable as a covariate. Examples range from beginner to advanced levels (e.g., path analysis/regression to LGM). PROC CALIS The PROC CALIS procedure (Covariance Analysis of Linear Structural Equations) estimates parameters and tests the appropriateness of structural equation models using covariance structural analysis. Although PROC CALIS was designed to specify linear relations, structural equation modeling (SEM) techniques have the flexibility to test nonlinear trends. Structural Equation Modeling (SEM) SEM is a methodology for representing, estimating, and testing a theoretical network of (mostly) linear relations between variables (Rigdon, 1998). A structural equation model is a hypothesized pattern of directional and nondirectional relationships among a set of observed (measured) and unobserved (latent) variables (MacCallum & Austin, 2000). A measured variable is directly measured whereas a latent variable is not directly or exactly measured. Two main goals in SEM are 1) to understand the patterns of correlation among a set of variables and 2) to explain as much of their variance as possible with the model specified (Kline, 1998). All variables are classified as either dependent or independent. Any variable that has a one-way path (or arrow) pointing at it is classified as a dependent variable. All other variables are classified as independent variables. Relations between variables are of three types: 1) association e.g., correlation, covariance; 2) direct effect, directional relation between two variables, e.g., independent and dependent variables; and 3) indirect effect, effect of an independent variable on a dependent variable through one or more intervening or mediating variables (Hoyle, 1995). Analysis of the relations among the variables to be analyzed takes place after formally stating a model (specification). A path diagram is a pictorial representation of a model. In the model, parameters are specified as fixed or free. Fixed parameters are given a fixed value (e.g., zero or one). Free parameters are estimated from the data. The criterion selected for parameter estimation is know as the discrepancy function. It provides a guideline to minimize the differences between the population covariance matrix, Σ, as estimated by the sample covariance, S, and the covariance matrix derived from the hypothesized model, Σ(0). Identification determines whether a single, unique value for each and every free parameter can be estimated from the observed data. Necessary conditions for model identification are 1) a scale must be established for every latent variable in the model, e.g., fix the variance of each latent variable to 1 or fix the value to 1 of one parameter associated with a latent variable directional influence; 2) the number of free parameters must not exceed the number of nonredundant elements in the sample covariance matrix or q < p* where q is the number of estimated parameters in the model, p* = p(p + 1)/2, and p is the number of measured variables in the model. If the second condition is violated, the researcher has fewer data values than parameters to be estimated (MacCallum, 1995). If a single, unique value cannot be obtained for one or more free parameters, the model is underidentified (q > p*). If a value can be obtained through one and only one manipulation of the observed data, the model is just identified(q = p*). If a value for one or more free parameters can be obtained in multiple ways from the observed data, the model is overidentified (q < p*). In the most common form of SEM, the purpose of the model is to account for variation and covariation of the measured variables (MVs). Path analysis (e.g., regression) tests models and relationships among MVs. Confirmatory factor analysis tests models of relationships between latent variables (LVs or common factors) and MVs which are indicators of common factors. Latent growth curve models (LGM) estimates initial level (intercept), rate of change (slope), structural slopes, and variances. Special cases of SEM are regression, canonical correlation, confirmatory factor analysis, and repeated measures analysis of variance (Kline, 1998). The use of SEM could be impacted by the research hypothesis being testing the requirement of sufficient sample size (e.g., 5-20 cases per parameter estimate) measurement instruments multivariate normality parameter identification outliers missing data interpretation of model fit indices (Schumacker & Lomax, 1996). A suggested approach to SEM analysis proceeds through the following process: review the relevant theory and research literature to support model specification specify a model (e.g., diagram, equations) determine model identification (e.g., if unique values can be found for parameter estimation; the number of degrees of freedom, df, for model testing is positive) select measures for the variables represented in the model and collect data conduct preliminary descriptive statistical analysis (e.g., scaling, missing data, collinearity issues, outlier detection) estimate parameters in the model assess model fit respecify the model if meaningful present and interpret the model. SUGI 27 Statistics and Data Analysis