When researchers model multilevel data, often a shared construct of interest is measured by individual-level observations, for example, students’ responses regarding how engaging their instructor’s teaching style is. In such cases, the construct of interest, “engaging teaching,” is shared at the cluster level across individuals, yet rarely are these shared constructs modeled as such. To address this gap, we discuss multilevel confirmatory factor analysis models that have been applied to item-level data obtained from multiple raters within given clusters, focusing particularly on measuring a characteristic at the cluster level. After discussing the parameters in each potential model, we make recommendations as to the appropriate modeling approach and the steps to be taken for model assessment given a set of data and hypothesized construct of interest. In particular, we encourage applied researchers not to use a model without constraints across the within-cluster level and the between-cluster level because such models assume that the average amount of the individual-level construct in a cluster does not differ across clusters. To illustrate this issue, we present simulation results and evaluate a series of models using empirical data from the Trends in International Mathematics and Science Study.

In many applications, the data of interest comprises multiple sequences that evolve over time. Examples include currency exchange rates, network tra c data, and demographic data on multiple variables. We develop a fast method to analyze such co-evolving time sequences jointly to allow (a) estimation/forecasting of missing/delayed/future values, (b) quantitative data mining, discovering correlations (with or without lag) among the given sequences, and (c) outlier detection. Our method, MUSCLES, adapts to changing correlations among time sequences. It can handle inde nitely long sequences e ciently using an incremental algorithm and requires only small amount of storage so that it works well with limited main memory size and does not cause excessive I/O operations. To scale for a large number of sequences, we present a variation, the Selective MUSCLES method and propose an e cient algorithm to reduce the problem size. Experiments on real datasets show that MUSCLES outperforms popular competitors in prediction accuracy up to 10 times, and discovers interesting correlations. Moreover, Selective MUSCLES scales up very well for large numbers of sequences, reducing response time up to 110 times over MUSCLES, and sometimes even improves the prediction quality.

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Latent growth models, a special class of longitudinal models within the broader structural equation modeling (SEM) domain, provide researchers a framework for investigating questions about change over time; yet rarely is time itself modeled as a focal parameter of interest. In the current article, rather than treating time purely as an index of measurement occasions, the proposed Time to Criterion (T2C) model draws from Preacher and Hancock's (2012) latent growth model reparameterization guidelines to model individual variability (i.e., to treat as a random effect) in one's time to achieve a criterion level of a given outcome. As such, the T2C model also allows researchers to model predictors and distal outcomes of time, as well as benefiting more generally from the flexibility afforded by being embedded within the broader SEM framework to accommodate such real-world data issues as missingness, complex error structures, nonnormality, and nested data. In this study we derive T2C from the linear latent growth model and discuss model assumptions and interpretation. By illustrating the model using real data, we demonstrate both its utility for applied research and its implementation in conventional SEM software. We also discuss and illustrate an extension of the model for nonlinear growth. Overall, the T2C model presents a novel and interpretable growth parameterization for further understanding processes of change.

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