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.
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Notes
Extensions of LST models that also account for trait changes have been presented in the literature (e.g., Eid, Courvoisier, & Lischetzke, 2012; Eid & Hoffmann, 1998; Geiser, Keller, & Lockhart, 2013; Steyer, Krambeer, & Hannöver, 2004; Tisak & Tisak, 2000). However, our focus in this article is on classical LST models as state-variability models that do not allow for trait changes, since these models are the most frequently used LST models in the applied literature.
For simplicity, we assume in this article that all indicators are homogeneous in the sense that they share the same trait within scaling differences. Geiser and Lockhart (2012) discuss LST models that allow for indicator heterogeneity (unique trait components) and/or method effects. The MI issues discussed in the present article are general in nature and apply to both LST models for homogeneous and heterogeneous indicators.
Note that some of these restrictions follow by definition of the theoretical concepts in LST theory, whereas others require additional assumptions. For the issues discussed in the present article, a distinction between restrictions that follow by definition and restrictions that require additional assumptions is not essential. We refer readers interested in the specific details to Steyer et al. (1992) or Steyer et al. (2012).
It should be noted that the STMS model is often specified as a higher order factor model, in which the observed variables load onto common latent state factors τ t , which themselves load onto a second-order latent trait factor ξ (see, e.g., Steyer et al., 1992). In this type of specification, not only the first-order factor loadings and intercepts should be tested for time invariance, but also the second-order factor loadings and intercepts that relate the latent state factors to the latent trait factor.
Note that this model could be extended to include additional latent change variables (ξ3 − ξ1) and (ξ4 − ξ1) to measure specific components of change. For simplicity, such more complex latent change score models are not considered in the present article.
We realize that cutoff criteria for approximate fit indices are to some extent arbitrary and that they should not be taken as “golden rules,” as has been pointed out in the literature (Chen, Curran, Bollen, Kirby, & Paxton, 2008; Marsh, Hau, & Wen, 2004). In the present article, we are using a summary fit criterion mostly to simplify the presentation.
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Author Note
The authors would like to thank Jacob Bishop for creating the path diagrams for this article.
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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:
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):
In the third step, we make the assumption that change over time is linear by postulating the following relation:
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):
Inserting Equation 23 into Equation 20, we obtain
Inserting Equations 24 and 25 into the basic LST measurement equation (Equation 5) yields
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
In Figs. 8, 9, and 10, we summarize the results for RMSEA, CFI, and SRMR.
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Geiser, C., Keller, B.T., Lockhart, G. et al. Distinguishing state variability from trait change in longitudinal data: The role of measurement (non)invariance in latent state-trait analyses. Behav Res 47, 172–203 (2015). https://doi.org/10.3758/s13428-014-0457-z
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DOI: https://doi.org/10.3758/s13428-014-0457-z