In recent years, the popularity of procedures for collecting intensive longitudinal data, such as the experience-sampling method, has increased greatly. The data collected using such designs allow researchers to study the dynamics of psychological functioning and how these dynamics differ across individuals. To this end, the data are often modeled with multilevel regression models. An important question that arises when researchers design intensive longitudinal studies is how to determine the number of participants needed to test specific hypotheses regarding the parameters of these models with sufficient power. Power calculations for intensive longitudinal studies are challenging because of the hierarchical data structure in which repeated observations are nested within the individuals and because of the serial dependence that is typically present in these data. We therefore present a user-friendly application and step-by-step tutorial for performing simulation-based power analyses for a set of models that are popular in intensive longitudinal research. Because many studies use the same sampling protocol (i.e., a fixed number of at least approximately equidistant observations) within individuals, we assume that this protocol is fixed and focus on the number of participants. All included models explicitly account for the temporal dependencies in the data by assuming serially correlated errors or including autoregressive effects.

We address the question of equivalence between modeling results obtained on intra-individual and inter-individual levels of psychometric analysis. Our focus is on the concept of measurement invariance and the role it may play in this context. We discuss this in general against the background of the latent variable paradigm, complemented by an operational demonstration in terms of a linear state-space model, i.e., a time series model with latent variables. Implemented in a multiple-occasion and multiple-subject setting, the model simultaneously accounts for intra-individual and inter-individual differences. We consider the conditions—in terms of invariance constraints—under which modeling results are generalizable (a) over time within subjects, (b) over subjects within occasions, and (c) over time and subjects simultaneously thus implying an equivalence-relationship between both dimensions. Since we distinguish the measurement model from the structural model governing relations between the latent variables of interest, we decompose the invariance constraints into those that involve structural parameters and those that involve measurement parameters and relate to measurement invariance. Within the resulting taxonomy of models, we show that, under the condition of measurement invariance over time and subjects, there exists a form of structural equivalence between levels of analysis that is distinct from full structural equivalence, i.e., ergodicity. We demonstrate how measurement invariance between and within subjects can be tested in the context of high-frequency repeated measures in personality research. Finally, we relate problems of measurement variance to problems of non-ergodicity as currently discussed and approached in the literature.

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Researchers commonly draw inferences from the group level to the individual and vice versa-that is, across levels. One of the empirical cornerstones of medicine is the clinical trial that tests the efficacy of a drug compared with placebo. If the intervention group outperforms the placebo group, the conclusion is that the drug should be prescribed for individuals with a given disorder. When are such inferences across levels defensible?In their recent paper in PNAS, Fisher et al.(1) state that "statistical findings at the interindividual (group) level only generalize to the intraindividual (person) level if the processes in question are ergodic," meaning that the effects of interest are homogeneous across individuals and stable over time (for formal definitions see, e.g., refs. 2 and 3). Fisher et al. demonstrate that ergodicity does not hold in multiple datasets, concluding that nonergodicity is a "threat to human subjects research."While we commend the authors for the insightful manuscript, we want to stress that ergodicity is sufficient, but not necessary, to draw inferences across levels (3, 4). Accordingly, recent work on ergodicity vs. nonergodicity has shifted away from a binary conceptualization to the idea of a continuum connecting the two (3-6). Fisher et al.(1) briefly acknowledge this perspective, and we want to highlight some important implications here.First, we might encounter different degrees of partial equivalence between levels, depending on where processes are situated on the nonergodicity continuum. In all such cases, ergodicity holds conditional on (i.e., after controlling for) sources of heterogeneity between individuals and/or instability over time. Such "conditional equivalence" (3) allows for conditional inferences across levels. Which and how many sources of heterogeneity will have to be conditioned on, and whether the resulting conditional inferences remain meaningful, will depend on the phenomenon, population, and time span studied (3).Second, statistical approaches such as structural equation or state-space modeling allow one to estimate conditional ergodicity by taking into account (un) observed sources of heterogeneity between individuals and/or instability over time. In addition, as our introductory example alluded to, conditional ergodicity can be accomplished by design; for example, randomization is a powerful way to condition on unobserved heterogeneity between individuals, allowing for treatment effects to be interpreted within (the average) person (conditional on assumptions of temporal stability).Third, research on nonergodicity needs to be comprehensive. Heterogeneity between individuals, as highlighted by Fisher et al. (1), is one important complication for inferences across levels. But instability over time can also produce ambiguous effects and promote interpretational fallacies. For instance, psychological data may involve effects between and within days, and failure to distinguish them will mischaracterize both processes (7). Empirical quests into nonergodicity therefor...

Long-lived simultaneous changes in the autodependency of dynamic system variables characterize crucial events as epileptic seizures and volcanic eruptions and are expected to precede psychiatric conditions. To understand and predict such phenomena, methods are needed that detect such changes in multivariate time series. We put forward two methods: First, we propose KCP-AR, a novel adaptation of the general-purpose KCP (Kernel Change Point) method. Whereas KCP is implemented on the raw data and does not shed light on which parameter changed, KCP-AR is applied to the running autocorrelations, allowing to focus on changes in this parameter. Second, we revisit the regime switching AR(1) approach and propose to fit models wherein only the parameters capturing autodependency differ across the regimes. We perform a simulation study comparing both methods: KCP-AR outperforms regime switching AR(1) when variables are uncorrelated, while the latter is more reliable when multicolinearity is severe. Regime switching AR(1), however, may yield recurrent switches even when the change is long-lived. We discuss an application to psychopathology data where we investigate whether emotional inertia -the autodependency of affective states- changes before a relapse into depression.