First- Versus Second-Order Latent Growth Curve Models: Some Insights From Latent State-Trait Theory

Abstract

First-order latent growth curve models (FGMs) estimate change based on a single observed variable and are widely used in longitudinal research. Despite significant advantages, second-order latent growth curve models (SGMs), which use multiple indicators, are rarely used in practice, and not all aspects of these models are widely understood. In this article, our goal is to contribute to a better understanding of theoretical and practical differences between FGMs and SGMs. We define the latent variables in FGMs and SGMs explicitly on the basis of latent state–trait (LST) theory and discuss insights that arise from this approach. We show that FGMs imply a strict trait-like conception of the construct under study, whereas SGMs allow for both trait and state components. Based on a simulation study and empirical applications to the Center for Epidemiological Studies Depression Scale (Radloff, 1977) we illustrate that, as an important practical consequence, FGMs yield biased reliability estimates whenever constructs contain state components, whereas reliability estimates based on SGMs were found to be accurate. Implications of the state–trait distinction for the measurement of change via latent growth curve models are discussed.

Keywords:

• CES–D
• latent change
• latent growth curve models
• latent state–trait theory
• longitudinal modeling
• reliability estimation

Notes

¹Although Tisak and Tisak (2000) referred to LST models in their article, they did not define their growth models based on the fundamental concepts of LST theory. Mayer et al. (2012) showed how to construct growth components based on latent state variables defined in LST theory. However, in their approach to defining LGMs, they did not explicitly distinguish between state and trait components, as is done in this work.

²For the sake of simplicity, we only consider linear growth models in this article. Note that other types of growth models can also be formulated based on LST theory by postulating a different loading structure on the slope factor [e.g., (t – 1)² for quadratic growth].

³Note that every one of the latent state variables S _it could play the role of the common latent state factor. For example, if we set α_1t = 0 and λ_1t = 1, this would imply that S _1t plays the role of the common latent state factor. Hence, the common latent state factors in this model are uniquely defined only up to positive linear transformations (Steyer, 1988).

⁴Note that Tisak and Tisak (2000) discussed different types of reliability coefficients in the context of LST–LGCM hybrids. Here, we only consider the most inclusive version that is defined as the total portion of systematic variance to observed variance and is referred to as “systematic reliability” in Tisak and Tisak's (2000) work.

⁵This coefficient corresponds closely to Tisak and Tisak's (2000) “dynamic reliability” coefficient.

^aIn line with Eid et al.'s (1999) approach to modeling indicator-specific effects, no indicator-specific factor was included for the first indicator in this model.

^bNo indicator-specific factor was needed in the SGMs for father reports.

⁶It should be noted that item parceling is controversial (e.g., Bandalos, 2002), mainly because the basic assumption underlying the creation of item parcels is that the items are unidimensional, an assumption that might not hold true for many social sciences scales. However, if this assumption is violated, FGMs do not solve the problem, because the creation of a single sum score across multidimensional items would be similarly problematic.