Abstract
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.
Highlights
Population heterogeneity exists when multiple distinct statistical models are required to adequately describe a population (Muthén, 1989)
We show that measurement invariance (MI) holding simultaneously over time and subject can be interpreted as constituting a mode of structural equivalence between the intra- and the interindividual level of analysis that is distinct from full structural equivalence
Tying into the ergodicity debate (e.g., Molenaar, 2004), we clarified the relationship between the concepts of MI and ergodicity in the context of general latent variable modeling as well as in
Summary
Population heterogeneity exists when multiple distinct statistical models are required to adequately describe a population (Muthén, 1989). The converse argument would be that, if MI across persons selected on basis of X holds, the interpretation of the latent variable is the same across these persons (e.g., Mellenbergh, 1989; Horn and McArdle, 1992; Lubke et al, 2003a; Dolan et al, 2004; Borsboom and Dolan, 2007; Nesselroade et al, 2007; Wicherts and Dolan, 2010; Raykov et al, 2012) This notion of MI as theoretical invariance, as compared to the above notion of unbiasedness, can mainly be found for operationalizations of MI in the linear factor model. Any uni- or multivariate autoregressive moving average model can be accommodated (i.e., reformulated in terms of a first order vector autoregressive process) by extending the state vector by the relevant process components (e.g., Harvey, 1989; Hamaker and Dolan, 2009; Shumway and Stoffer, 2011)
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