Distinguishing state variability from trait change in longitudinal data: The role of measurement (non)invariance in latent state-trait analyses

Abstract

Researchers analyzing longitudinal data often want to find out whether the process they study is characterized by (1) short-term state variability, (2) long-term trait change, or (3) a combination of state variability and trait change. Classical latent state-trait (LST) models are designed to measure reversible state variability around a fixed set-point or trait, whereas latent growth curve (LGC) models focus on long-lasting and often irreversible trait changes. In the present article, we contrast LST and LGC models from the perspective of measurement invariance testing. We show that establishing a pure state-variability process requires (1) the inclusion of a mean structure and (2) establishing strong factorial invariance in LST analyses. Analytical derivations and simulations demonstrate that LST models with noninvariant parameters can mask the fact that a trait-change or hybrid process has generated the data. Furthermore, the inappropriate application of LST models to trait change or hybrid data can lead to bias in the estimates of consistency and occasion specificity, which are typically of key interest in LST analyses. Four tips for the proper application of LST models are provided.

Appendices

Appendix 1 Formal definition of the multiple-indicator LGC model used in the simulation study

The multiple-indicator linear LGC model used in the simulation study (see Fig. 3) was first presented by Eid et al. (2012). Here, we show how this model can be formally defined on the basis of concepts of LST theory to illustrate its mathematical relationship to the STMS model. For more details, see Bishop et al. (2013). In the first step, we assume that all latent trait variables measured at the same time point t are congeneric:

$$ {\upxi}_{it}={\alpha}_{it}+{\lambda}_{it}{\upxi}_t. $$

(20)

This is different from the assumption stated in Equation 3 for the STMS model in that we no longer postulate homogeneity of all latent trait variables ξ _it but only of those that are measured on the same time point t. In the second step, we define an intercept factor to be equal to the common latent trait factor measured at time 1 (ξ₁) and a slope factor to be equal to the latent difference variable (ξ₂ − ξ₁):

$$ Intercept\equiv {\upxi}_1 $$

(21)

$$ Slope\equiv \left({\upxi}_2-{\upxi}_1\right). $$

(22)

In the third step, we make the assumption that change over time is linear by postulating the following relation:

$$ \begin{array}{c}\kern-2em {\upxi}_t={\upxi}_1+\left(t-1\right)\left({\upxi}_2-{\upxi}_1\right)\\ {}= Intercept+\left(t-1\right) Slope.\end{array} $$

(23)

In the fourth step, we assume, as in the STMS model, that occasion-specific effects are homogeneous for all indicators measured at the same time point, so that they can be represented by common state residual factors (compare Equation 4):

$$ {\zeta}_{it}={\delta}_{it}{\zeta}_t. $$

(24)

Inserting Equation 23 into Equation 20, we obtain

$$ \begin{array}{c}\kern-9.7em {\upxi}_{it}={\alpha}_{it}+{\lambda}_{it}{\upxi}_t\\ {}\kern-1em ={\alpha}_{it}+{\lambda}_{it}\left[{\upxi}_1+\left(t-1\right)\left({\upxi}_2-{\upxi}_1\right)\right]\\ {}\kern-1em ={\alpha}_{it}+{\lambda}_{it}{\upxi}_1+{\lambda}_{it}\left(t-1\right)\left({\upxi}_2-{\upxi}_1\right)\\ {}={\alpha}_{it}+{\lambda}_{it} Intercept+{\lambda}_{it}\left(t-1\right) Slope.\end{array} $$

(25)

Inserting Equations 24 and 25 into the basic LST measurement equation (Equation 5) yields

$$ \begin{array}{c}\kern-13.5em {Y}_{it}={\upxi}_{it}+{\upzeta}_{it}+{\upvarepsilon}_{it}\\ {}\kern-.7em ={\alpha}_{it}+{\lambda}_{it}{\upxi}_1+{\lambda}_{it}\left(t-1\right)\left({\upxi}_2-{\upxi}_1\right)+{\delta}_{it}{\zeta}_t+{\upvarepsilon}_{it}\\ {}\kern1em ={\alpha}_{it}+{\lambda}_{it} Intercept+{\lambda}_{it}\left(t-1\right) Slope+{\delta}_{it}{\zeta}_t+{\upvarepsilon}_{it}.\end{array} $$

(26)

The STMS model is a special case of the model in Equation 26 that results if Var(ξ₂ − ξ₁) = E(ξ₂ − ξ₁) = 0.

Appendix 2 Parameter specification in the simulation study

In this appendix, we describe the parameter specification of the population model used in the simulation study.

Parameter	Value
Indicator intercepts, α it	Fixed to 0 for all indicators
Loadings on the latent intercept factor, λξ1	Fixed to 1 for all indicators
Loadings on the linear slope factor, λ(ξ2 – ξ1)	Fixed to 0 (time 1), 1 (time 2), 2 (time 3), 3 (time 4)
Loadings on the latent state residual factors, δ it	Fixed to 1 for all indicators
Indicator residual variances, Var(ε it )	Varied depending on the slope factor variance so that reliability would equal .8 for all indicators at all time points in all conditions
Variance of the intercept factor, Var(ξ1)	0.5
Mean of the intercept factor, E(ξ1)	0
Variance of the linear slope factor, Var(ξ2 – ξ1)	Three conditions: 0.005 (1 % of the intercept factor variance), 0.025 (5 % of the intercept factor variance), and 0.05 (10 % of the intercept factor variance)
Mean of the linear slope factor, E(ξ2 – ξ1)	Varied depending on the slope factor variance condition to obtain small (0.2), medium (0.5), and large (0.8) mean differences in terms of Cohen’s d measure, respectively; Cohen’s d was defined as E(ξ2 – ξ1)/[Var(ξ2 – ξ1)0.5]
Covariance between intercept and linear slope factor, Cov[ξ1,(ξ2 – ξ1)]	Fixed to 0
Latent state residual factor variances, Var(ζ t )	Always 0.3 at time 1. Remaining values were varied depending on the slope factor variance condition, so that occasion specificity would equal .3 for all indicators at all time points in all conditions
Covariances among latent state residual factors and between latent state residual factors and all other factors	Fixed to 0

Appendix 3 Mplus scripts for the proper specification of the STMS and linear growth models with time-invariant factor loadings and intercepts

STMS model with time-invariant parameters

Linear LGC model with time-invariant parameters

MTMS model with time-invariant parameters

Multiple-indicator linear LGC model with indicator-specific intercept and slope factors

Appendix 4 Simulation results for individual fit statistics

Table 1 Mean χ ² and SD for each cell of the simulation design

In Figs. 8, 9, and 10, we summarize the results for RMSEA, CFI, and SRMR.

Fig. 8

Mean root mean square error of approximation (RMSEA; middle of the box) and 90 % confidence limits (upper and lower limits of the box) across the simulation conditions. STMS, singletrait–multistate

Fig. 9

Mean comparative fit index (CFI; middle of the box) and SD (upper and lower limits of the box) across the simulation conditions. STMS, singletrait–multistate

Fig. 10

Mean standardized root mean square residual (SRMR; middle of the box) and SD (upper and lower limits of the box) across the simulation conditions