A Latent Variable Mixed-Effects Location Scale Model with an Application to Daily Diary Data

Abstract

A mixed-effects location scale model allows researchers to study within- and between-person variation in repeated measures. Key components of the model include separate variance models to study predictors of the within-person variance, as well as predictors of the between-person variance of a random effect, such as a random intercept. In this paper, a latent variable mixed-effects location scale model is developed that combines a longitudinal common factor model and a mixed-effects location scale model to characterize within- and between-person variation in a common factor. The model is illustrated using daily reports of positive affect and daily stressors for a large sample of adult women.

A latent variable mixed‐effects location scale model that also considers between‐person differences in the autocorrelation

“…We collect the outcome values of person i at a single time point t in the vector y it = (y i1t , … , y iJt ) and all n i values of i are collected in the vector y i = (y i1 , … , y iT i ). We consider the J values at a single time point t and write them as a latent variable model: 15,17…”

“…Thus, the number of random location effects (ie, 𝝂 i ) no longer plays a role in the approximation. 13,15,16 To derive the restriction, we exploit the normality assumption that we made for the random effect vector 𝝓 i and the block diagonal structure of the covariance matrix 𝚽 (see Equation 7). From the properties of the multivariate normal distribution, it follows (see, Reference 32) that the marginal distribution of the scale-related random effects 𝜹 i = (𝜔 i , 𝜅 i ) is a multivariate normal distribution with expectation zero and covariance matrix 𝚽 (𝜔,𝜅) = 𝚽 𝜹 .…”

“…In this paper we will take up this approach by describing an extension of the MELS model in which measurement error of the outcome variable is addressed by using a common factor model to represent the manifest outcome variable(s) and modeling the resulting latent variable as a MELS model. 15,16,24 The latent MELS model, as we call this extension, can be used when multiple indicators of the construct of interest (eg, multiple items reflective of positive affect) have been longitudinally assessed or when only one indicator has been measured. As we show, the single-indicator model is an extension of the general mixed-effects model suggested in Reference 25 (see also References 4,6) that, in contrast to the model of Diggle et al, allows for between-person differences in the occasion-specific residual variance and autoregressive parameters.…”

“…One important limitation to the MELS model, however, is that similar to the standard application of a mixed‐effects model, it assumes that the outcome variable is measured without error. A consequence is that if the variable includes measurement error, the residuals contain true variation of a subject and measurement error, which in turn can bias the parameters used to describe the within‐person variation 15,16 . That is, the average occasion‐specific residual variance and/or the coefficients of the variables used to model this variance may be too high, which in turn may result in invalid substantive conclusions.…”

“…In this paper we will take up this approach by describing an extension of the MELS model in which measurement error of the outcome variable is addressed by using a common factor model to represent the manifest outcome variable(s) and modeling the resulting latent variable as a MELS model 15,16,24 . The latent MELS model, as we call this extension, can be used when multiple indicators of the construct of interest (eg, multiple items reflective of positive affect) have been longitudinally assessed or when only one indicator has been measured.…”