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Person-centered analyses

Latent Class Analysis (LCA)

As noted by Fryer and Shum (this volume), LCA ‘classes’ can be tested for differences in antecedents or outcomes of strategy use. For example, in unpublished analyses of data reported in Cromley, Snyder-Hogan, and Luciw-Dubas (2010), think-aloud participants were clustered on percentage of inferences, of high-level strategies, and of low-level strategies. The three-cluster solution suggested (1) a low-level strategies cluster who mostly used highlighting and copying definitions, with almost two-thirds of verbalizations being low-level strategies, (2) a high-level strategies cluster, with more than half of verbalizations being high-level strategies such as summarizing and self-questioning and less than one-third of verbalizations being low-level strategies, and (3) a balanced strategies, high inference cluster with one-tenth of verbalizations being inferences, and about one-third each of high- and low-level strategies. Females were significantly over-represented in the high-level strategies cluster, and the high-inference cluster scored significantly higher on free recall than the low-level strategies cluster.

LCA/cluster analysis has the advantage that it is the only approach to analyzing data where subgroups are evident (e.g., when subgroups are evident in scatterplots). An additional advantage is that a class (subgroup) might have a majority from a particular demographic group (e.g., under-represented minority students), but LCA does not force all people from that demographic to be in a particular class; thus, LCA might be a way to lessen effects of stereotyping in research.

As noted by Fryer and Shum, decisions about the number of latent classes require a mix of theory, statistical expertise, and practical knowledge. A statistically preferable solution that results in an uninterpretable class is not the best solution. In addition, LCA sometimes produces classes such as ‘low cognitive strategy-low metacognitive strategy,’ ‘medium-medium,’ and ‘high-high’ which do not add to our understanding (any more than a simple regression would). LCA is not an exploratory technique; as emphasized by Fryer and Shum; a strong theory needs to guide the choice of variables to cluster on. As with exploratory factor analysis, the researcher needs to carefully document and explain the reasons for each choice made at the various decision points in LCA (How many classes? What findings were taken into consideration?). As with SEM, there is a steep learning curve for researchers in learning the nuances of choosing clustering variables, the order of clustering and entering antecedent or outcome variables, fit indices, interpretability, and so on. In addition, each of these analytic choices needs to be carefully described and well justified for the researcher’s audience. A stronger case can be made for LCA if the sample is first split in two, and the analyses are run on both sub-samples to see if they replicate. Thus, by definition, LCA is a large-sample technique.

Growth Mixture Modeling (GMM)

Longitudinal data sometimes show distinct subgroups with quite different shapes of change. For example, some people may show a negative quadratic (‘smile’) pattern of growth on use of a strategy, whereas others in the same dataset show a positive quadratic (‘frown’) pattern of growth on use of the same strategy. Growth mixture modeling is a hybrid of GCM and LCA in that each subset of growth trajectories is a latent class. One advantage of GMM is that statistically significant growth can be found for some subsets of participants, whereas an overall analysis might show no growth. A second advantage is theoretical—GMM can reveal multiple, effective routes to similar, positive outcomes. A third advantage is that, as in LCA, different demographics can be shown to be disproportionate in some classes (think of these as clusters), but not all members of that group are forced to have a certain pattern of growth.

As with LCA, there are a number of disadvantages to GMM. As with LCA and cluster analysis, there are many decision points that must be well justified. As with LCA, it is important to identify the latent growth classes (clusters of people showing a similar pattern of growth) before adding predictors or outcomes of being in a class. The SEM learning curve is somewhat steeper for GMM than for GCM. Finally, GMM is a rapidly evolving area of GCM, and researchers should seek out recent articles to use the best approaches to sample size, number of time points, fit indices, model convergence, and so on.

Latent Transition Analysis

If participants are in different classes over time (or different categorical developmental states over time, as in different Piagetian stages), various antecedent variables can be analyzed as predictors of these changes. For example, Rinne, Ye, and Jordan (2017) categorized students at various stages of fraction problem-solving strategies at three time points between 4th and 6th grades. They identified three latent classes: (1) a large-number bias class showing the well-known whole number bias (e.g., ‘1/4 must be larger than 1/3 because 4 is larger than 3’), (2) an intermediate small-number bias class (e.g., ‘8/9 must be larger than 20/40 because 8 and 9 are smaller than 20 and 40’), and (3) a correct-answer normative group. In the fall of 4th grade, 71% were in class 1, 12% were in class 2, and 17% were in class 3. By spring of 5th grade, only 31% were left in class 1, 26% were in class 2, and 43% had reached class 3. By spring of 6th grade, only 23% were still in class 1, 24% were in class 2, and 53% had reached class 3. Beyond simply describing how many participants were in each class at each time point Rinne et al. (2017) showed that higher number line knowledge at 4th grade significantly predicted making the jump from class 1 to class 3 by 5th grade. Thus, LTA is useful when people are in different categories or classes at each time point (rather than having a score on a continuous variable at each time point as in GCM). All of the advantages of LCA apply to LTA (e.g., avoiding stereotyping). LTA can be thought of as the more statistically powerful SEM (latent variable) version of log-linear analysis.

As with LCA—which is typically used as a first step in LTA—there are a number of decision points, all of which must be well justified. As with GMM, it is important to decide on a number of classes before adding antecedents or outcomes of class membership to the model. Like LCA, LTA is a large-sample technique. As expected, there is a steep learning curve in the SEM (or specialized LCA) software required to conduct LTA.

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