Measurement invariance within and between individuals: a distinct problem in testing the equivalence of intra- and inter-individual model structures

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.

Keywords: ergodicity; intra-individual level of analysis; latent variables; measurement invariance; state-space modeling.

Figure 1

Model taxonomy in terms of model equations and verbalized form.

Figure 2

Relatively best fitting models for subjects 7, 13, and 22. Paths fixed to zero are not drawn. Note that these include the regression parameters of the vector eta on the constant, i.e., vector alpha, which are fixed to zero for scaling purposes. Paths fixed to one are dashed. These include the latent residual variances in order to provide a latent metric. Freely estimated paths are drawn in black and parameter point estimates are provided. Items denoted with e are extraversion marker items, whereas items denoted with a are agreeableness marker items. The numerical ordering of the items employed here corresponds to the ordering of the items as given in the data description section. Index i is dropped as the models describe single individuals.

Figure 3

Individual model for subject 7 including the neuroticism marker item “bad tempered” as a potentially biasing (fixed) variable. According to this representation, the neuroticism item possibly affects the agreeableness marker items above the potential effect it has through the agreeableness factor.