The Curious Case of the Cross-Sectional Correlation.

The cross-sectional correlation is frequently used to summarize psychological data, and can be considered the basis for many statistical techniques. However, the work of Peter Molenaar on ergodicity has raised concerns about the meaning and utility of this measure, especially when the interest is in discovering general laws that apply to (all) individuals. Through using Cattell's databox and adopting a multilevel perspective, this paper provides a closer look at the cross-sectional correlation, with the goal to better understand its meaning when ergodicity is absent. An analytical expression is presented that shows the cross-sectional correlation is a function of the between-person correlation (based on person-specific means), and the within-person correlation (based on individuals' temporal deviations from their person-specific means). Two curiosities related to this expression of the cross-sectional correlation are elaborated on, that is: a) the difference between the within-person correlation and the (average) person-specific correlation; and b) the unexpected scenarios that can arise because the cross-sectional correlation is a weighted sum rather than a weighted average of the between-person and within-person correlations. Seven specific examples are presented to illustrate various ways in which these two curiosities may combine; R code is provided, which allows researchers to investigate additional scenarios.

At the Frontiers of Modeling Intensive Longitudinal Data: Dynamic Structural Equation Models for the Affective Measurements from the COGITO Study.

With the growing popularity of intensive longitudinal research, the modeling techniques and software options for such data are also expanding rapidly. Here we use dynamic multilevel modeling, as it is incorporated in the new dynamic structural equation modeling (DSEM) toolbox in Mplus, to analyze the affective data from the COGITO study. These data consist of two samples of over 100 individuals each who were measured for about 100 days. We use composite scores of positive and negative affect and apply a multilevel vector autoregressive model to allow for individual differences in means, autoregressions, and cross-lagged effects. Then we extend the model to include random residual variances and covariance, and finally we investigate whether prior depression affects later depression scores through the random effects of the daily diary measures. We end with discussing several urgent-but mostly unresolved-issues in the area of dynamic multilevel modeling.

On the Use of Mixed Markov Models for Intensive Longitudinal Data.

Markov modeling presents an attractive analytical framework for researchers who are interested in state-switching processes occurring within a person, dyad, family, group, or other system over time. Markov modeling is flexible and can be used with various types of data to study observed or latent state-switching processes, and can include subject-specific random effects to account for heterogeneity. We focus on the application of mixed Markov models to intensive longitudinal data sets in psychology, which are becoming ever more common and provide a rich description of each subject's process. We examine how specifications of a Markov model change when continuous random effect distributions are included, and how mixed Markov models can be used in the intensive longitudinal research context. Advantages of Bayesian estimation are discussed and the approach is illustrated by two empirical applications.

Using a Few Snapshots to Distinguish Mountains from Waves: Weak Factorial Invariance in the Context of Trait-State Research.

In this article, we show that the underlying dimensions obtained when factor analyzing cross-sectional data actually form a mix of within-person state dimensions and between-person trait dimensions. We propose a factor analytical model that distinguishes between four independent sources of variance: common trait, unique trait, common state, and unique state. We show that by testing whether there is weak factorial invariance across the trait and state factor structures, we can tackle the fundamental question first raised by Cattell; that is, are within-person state dimensions qualitatively the same as between-person trait dimensions? Furthermore, we discuss how this model is related to other trait-state factor models, and we illustrate its use with two empirical data sets. We end by discussing the implications for cross-sectional factor analysis and suggest potential future developments.

A Comparison of Inverse-Wishart Prior Specifications for Covariance Matrices in Multilevel Autoregressive Models.

Multilevel autoregressive models are especially suited for modeling between-person differences in within-person processes. Fitting these models with Bayesian techniques requires the specification of prior distributions for all parameters. Often it is desirable to specify prior distributions that have negligible effects on the resulting parameter estimates. However, the conjugate prior distribution for covariance matrices-the Inverse-Wishart distribution-tends to be informative when variances are close to zero. This is problematic for multilevel autoregressive models, because autoregressive parameters are usually small for each individual, so that the variance of these parameters will be small. We performed a simulation study to compare the performance of three Inverse-Wishart prior specifications suggested in the literature, when one or more variances for the random effects in the multilevel autoregressive model are small. Our results show that the prior specification that uses plug-in ML estimates of the variances performs best. We advise to always include a sensitivity analysis for the prior specification for covariance matrices of random parameters, especially in autoregressive models, and to include a data-based prior specification in this analysis. We illustrate such an analysis by means of an empirical application on repeated measures data on worrying and positive affect.