Inter-individual differences in multivariate time-series: Latent class vector-autoregressive modelling

Theories of emotion regulation posit the existence of individual differences in emotion dynamics. Current multi-subject time-series models account for differences in dynamics across individuals only to a very limited extent. This results in an aggregation that may poorly apply at the individual level. We present the exploratory method of latent class vector-autoregressive modelling (LCVAR), which extends the time-series models to include clustering of individuals with similar dynamic processes. LCVAR can identify individuals with similar emotion dynamics in intensive time-series, which may be of unequal length. The method performs excellently under a range of simulated conditions. The value of identifying clusters in time-series is illustrated using affect measures of 410 individuals, assessed at over 70 time points per individual. LCVAR discerned six clusters of distinct emotion dynamics with regard to diurnal patterns and augmentation and blunting processes between eight emotions. Introduction In recent years, research has expanded into the dynamic processes of psychological phenomena. Smartphones and tablets are providing an infrastructure that allows for the collection of time-intensive, multivariate data using ambulatory assessment (AA). Intensive longitudinal AA data can provide insight into emotion dynamics. These dynamics reflect how individuals react to their environment and regulate their emotions. For instance, emotional inertia, the propensity of affective states to persist over time and resist to change, is often quantified as the autoregression of an emotion (Kuppens & Verduyn, 2017). High LATENT CLASS VECTOR-AUTOREGRESSIVE MODELLING 2 emotional inertia may be indicative of an inability to recover from negative emotions as a result of impaired emotion regulation. High inertia may also be a sign of increased preoccupation, resulting in emotional insensitivity and decreased involvement with the environment (Brose, Schmiedek, Koval, & Kuppens, 2015). Inertia is negatively associated with psychological well-being (Houben, Van Den Noortgate, & Kuppens, 2015; Koval & Kuppens, 2012) and has been linked to mental disorders. For instance, manic and depressive episodes are characterised by high levels of inertia of positive or negative affect. This expresses general under-reactivity and inflexibility (American Psychiatric Association, 2013). While single-subject time-series models can provide insight into emotion dynamics, these models are unwieldy in a large sample and prevent a proper characterisation of the population of individuals. To arrive at a proper generalisation, one needs to account for the presence of inter-individual differences in emotion dynamics in multi-subject AA data. Theories surrounding emotion regulation suggest that individuals perceive and predict situations differently and construct qualitatively different emotional processes (Barrett, 2018) that differ in their persistence (Verduyn & Lavrijsen, 2015) and therefore display qualitatively distinct dynamic affect patterns (Houben et al., 2015). Good emotion regulation may manifest itself, for instance, by rarely moving out of positive affect states and by moving out of negative affect states quickly (Hay & Diehl, 2011). Such individual differences can be captured with multi-subject time-series models that identify distinct subgroups of people with similar emotion dynamics. Before we introduce our novel latent class vector-autoregressive modelling (LCVAR) strategy to identify groups of people with similar emotion dynamics we first outline the benefits of such time-series clustering models to popular alternatives. When using time-series models the inter-individual differences that are allowed follow a continuum from single-subject models to narrow panel models that estimate dynamic LATENT CLASS VECTOR-AUTOREGRESSIVE MODELLING 3 patterns with one set of fixed-coefficients.1 Such fixed-coefficients can become substantially biased in the presence of inter-individual differences (Hamaker, Kuiper, & Grasman, 2015). Inter-individual differences arise as observed or unobserved heterogeneity (Lubke & Muthén, 2005). Observed time-series heterogeneity can be explained by an observed variable, for instance gender. Unobserved heterogeneity is caused by variables that remain unknown, which occurs rather often. Unobserved heterogeneity may emerge from a distribution of quantitative deviations around a uniform effect or reflect discrete and qualitatively different processes. Such quantitative inter-individual differences can be modelled through random-coefficients. Hierarchical Bayesian models use random-coefficients to allow for time-series models where all coefficients may vary across individuals and Bayesian estimation offers the distribution of the coefficients great flexibility (e.g., Driver & Voelkle, 2018). Thereby, the multilevel vector-autoregressive (VAR) model captures within-individual dynamics (level 1) and continuous inter-individual differences (level 2) (Schuurman, Ferrer, de Boer-Sonnenschein, & Hamaker, 2016). Random-coefficients allow for quantitative differences (i.e., differences in coefficient values) but are unable to capture qualitative differences (i.e., differences in time-series models) and cannot explain why certain coefficients are positive for some and negative for others. In contrast, discrete unobserved heterogeneity is accounted for by inferring latent subgroups from the data using exploratory methods. This can uncover subgroups with distinct patterns in affect dynamics. In the context of unobserved heterogeneity, subgroups are denoted latent classes (Lubke & Muthén, 2005). The latent classes themselves are unknown, and which latent class an individual belongs to needs to be inferred from the data. 1To circumvent the ambiguity surrounding the term “random effects” we refer to a coefficient as fixed when it is identical across individuals and random when it may vary across individuals. LATENT CLASS VECTOR-AUTOREGRESSIVE MODELLING 4 Time-series models for discrete unobserved heterogeneity Latent Markov models (LMMs) have been proposed to discover latent classes in AA data. These models estimate latent class membership at time point t based on on the latent class membership at t − 1 and the current observed variable(s). Latent classes represent different mood states and the stability of mood states is expressed by transition probabilities between latent classes. To allow for inter-individual differences in mood fluctuation LMMs may nest latent classes within latent classes (Crayen, Eid, Lischetzke, & Vermunt, 2017) or model transition probabilities through random-coefficients (Asparouhov, Hamaker, & Muthén, 2017). However, the transition probabilities in LMMs give no insight into the dynamics between all observed emotions. In this article we propose a time-series clustering model that will identify latent classes (henceforth called clusters) of individuals based on similarity in emotion dynamics between all observed emotions. Individuals belong to exactly one cluster throughout. These models allow researchers to assess various theories on emotion regulation and pathology that imply groups of individuals who display different emotion dynamic patterns. Time-series clustering. We focus here exclusively on time-series clustering methods which cluster on the temporal persistence (inertia) of emotions and their dynamic interplay. Of the methods introduced so far, some are non-probabilistic (i.e., they operate by minimisation of a criterion, e.g., Bulteel, Tuerlinckx, Brose, & Ceulemans, 2016). However, model-based clustering methods (i.e., based on a mixture model, McLachlan & Peel, 2004) are generally preferred because they offer a framework for formal inference and have theoretically favourable properties; For example, if the model is correct and the sample is large enough, the clustering of each individual is optimal (McLachlan, 2011). In line with Jacques and Preda (2014) we distinguish between filtering and adaptive time-series clustering (sometimes called multi-step and simultaneous clustering, respecLATENT CLASS VECTOR-AUTOREGRESSIVE MODELLING 5 tively), see Ernst, Timmerman, Jeronimus, and Albers (2019) for an overview of these methods. Filtering methods firstly summarise the dynamics of every individual (e.g., by summarising the time-series of every individual into ideographic VAR coefficients), and subsequently form clusters based on the individual summaries. Conversely, adaptive methods combine summarising and clustering individuals’ dynamics (i.e., the summary into VAR coefficients depends on the clusters and vice versa). Because filtering methods summarize individual time-series before they cluster, the data summary may provide a suboptimal representation of the time-series for the clustering approach. This occurs if the data resulting from the compression is not discriminant for the different clusters (Bouveyron & Brunet-Saumard, 2014), or when the individual time-series are overfitted. Adaptive methods avoid this problem, because the description of dynamic processes adapts during the clustering, which prevents premature compression of the data. Adaptive methods thus enable a cluster-specific description of the time-series. Consequently, different clusters can be modelled by qualitatively different time-series models. Existing adaptive time-series clustering methods (e.g, Anderlucci & Viroli, 2015; Vermunt, 2007) are suboptimal for time-series with more than 10 measurement occasions and/or varying lengths, which are both common in AA data. Our adaptive, model-based LCVAR model combines: (1) VAR time-series models, describing intra-individual dynamic processes and (2) a latent class model, clustering individuals with similar dynamic characteristics via fixed-coefficients. This combination provides some crucial advantages: (a) characteristics of AA emotion data are accounted for, including multiple variables, a hi