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# Structural Equation Modeling

Structural equation modeling is quite similar to path model analysis in that researchers can posit and test complex relations among variables, including mediation (Kline, 2014). However, path model analysis involves only measured variables, which are likely to contain significant amounts of error variance in addition to true score variance. In structural equation modeling, one or more measured variables are replaced with a latent variable constructed of two or more measured variables. The process of estimating the latent variable from the associated measured variables enables researchers to produce an estimate of the error variance in each measured variable, and remove it, thus producing a latent variable that, in theory, is a more accurate and pure representation of the underlying construct (i.e., all true score variance, no error score variance).

For example, many SRL researchers have used the Motivated Strategies for Learning Questionnaire (Pintrich, Smith, Garcia, & McKeachie, 1991) to measure participants’ perceptions of their motivation and strategy use (Vermunt, this volume). This questionnaire includes numerous self-report, Likert-type scale items intended to measure motivation (e.g., “In a class like this, I prefer course material that really challenges me so I can learn new things” and “If I study in appropriate ways, then I will be able to learn the material in this course”) as well as learning strategy use (e.g., “When studying for this course, I often set aside time to discuss course material with a group of students from the class” and “I ask the instructor to clarify concepts I don’t understand well”). In theory, each item is a good but not perfect measure of the underlying latent construct of motivation or learning strategy use. Summing scores across multiple strategy use items would include both true score and error variance in the total. Structural equation modeling allows researchers to estimate and remove the error variance from each individual item, including only true score variance in the estimate of participants’ score on the latent variable. This latent variable is a more accurate and statistically stronger estimate than any of its individual measured variable indicators. Structural equation models can be conceptualized, tested, reconceptualized, and compared similarly to path analysis models. However, structural equation modeling often requires even larger sample sizes than path model analysis.

Baas, Castelijns, Vermeulen, Martens, and Segers (2015) used the self-report Children’s Perceived Use of Self-Regulated Learning Inventory (Vandevelde, Van Keer, & Rosseel, 2013) to measure elementary school students’ surface and deep learning strategies as well as their relations among monitoring, scaffolding, and evaluation. This inventory included two items the authors used to measure perceived use of surface learning strategies, and another eight items used to measure perceived use of deep learning strategies. They used structural equation modeling to determine whether the items appropriately measured each latent perceived strategy use variable, and then they evaluated relations among those latent variables and other phenomena of interest such as scaffolding, monitoring, planning, and product evaluation. They found that scaffolding positively related to perceived use of both surface and deep learning strategies, thus supporting the idea that helping students understand what they need to do next when learning (i.e., scaffolding) is associated with self-reported increases in the perception of strategy use. It is important to note that despite common misconceptions, structural equation modeling does necessarily allow researchers to make causal claims (e.g., scaffolding caused students to perceive greater use of strategies). Causal claims are dependent upon the design of the study (i.e., experimental versus non-experimental designs), rather than the type of analysis technique used (e.g., regression, path model analysis, structural equation modeling; Murnane & Willet, 2010).

Greene, Costa, Robertson, Pan, and Deekens (2010) used a structural equation model to examine relations among college students’ knowledge gains while using a computer-based learning environment, SRL processing including strategy use as captured via think-aloud protocol data (Greene et al., 2018), and implicit theories of intelligence (Dweck & Master, 2008). The authors found that planning, monitoring, and strategy use could be used as measured indicators of a latent SRL variable, and that this variable mediated relations between prior knowledge, implicit theory of intelligence, and posttest performance. This posited model allowed these researchers to test common assumptions about SRL (i.e., SRL mediates relations between individual differences and learning performance; Pintrich, 2000) that would have been impossible to assess with a regression model.

## Growth Curve Modeling

Often, researchers wish to measure changes in strategy use over time, for example in studies of children’s strategic processing as they age through elementary school (Carr & Alexeev, 2011). This change over time can be estimated using growth curve modeling, which is akin to a structural equation modeling approach to GLM repeated measures analysis. In growth curve modeling, the participants’ initial performance ability and their rate of change over time are treated as latent variables that can be more accurately estimated after removing error variance from the repeated measures. Carr and Alexeev captured students’ use of both cognitive and manipulative mathematics strategies in 2nd, 3rd, and 4th grades and then used growth curve modeling to find statistically significant change in the frequency of observed strategy use over that time period. Their analyses extended beyond the scope of this chapter, revealing previously unobserved differences in students that related to 4th grade mathematics competency, as well. The use of growth curve modeling allowed the authors to more accurately discern how strategy use changed over time, and how that change related to covariates such as fluency, accuracy, gender, and subsequent measures of mathematics competency.

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