On the Mathematical Relationship Between Latent Change Score and Autoregressive Cross-Lagged Factor Approaches: Cautions for Inferring Causal Relationship Between Variables

Abstract

The present paper focuses on the relationship between latent change score (LCS) and autoregressive cross-lagged (ARCL) factor models in longitudinal designs. These models originated from different theoretical traditions for different analytic purposes, yet they share similar mathematical forms. In this paper, we elucidate the mathematical relationship between these models and show that the LCS model is reduced to the ARCL model when fixed effects are assumed in the slope factor scores. Additionally, we provide an applied example using height and weight data from a gerontological study. Throughout the example, we emphasize caution in choosing which model (ARCL or LCS) to apply due to the risk of obtaining misleading results concerning the presence and direction of causal precedence between two variables. We suggest approaching model specification not only by comparing estimates and fit indices between the LCS and ARCL models (as well as other models) but also by giving appropriate weight to substantive and theoretical considerations, such as assessing the justifiability of the assumption of random effects in the slope factor scores.

KEYWORDS:

• latent change score model
• autoregressive cross-lagged model
• causality
• longitudinal data

Notes

Other models still, such as the autoregressive latent trajectory (ALT) model (Bollen & Curran, 2004, 2006; Curran & Bollen, 2001) and state-trait model (Kenny & Zautra, 2001), also aim to infer causal relationship between variables. However, here we restrict our attention to the LCS and ARCL models due to their recent popularity in applied research settings.

However, on this point, Hamaker et al. (2015) support applying the concept of Granger causality when using the ARCL and related models, although they acknowledge that the omitted variable problem implies that we cannot make strong causal statements based on correlational data.

In many cases, as a simple start, the model parameters of the change coefficients (β_y, β_x, γ_y, γ_x), which will be explained below, may be considered invariant over time (e.g., ergodic; McArdle, 2009). This assumption can be relaxed so that different patterns of (causal) influence can be assumed at different times, although the resulting (causal) relationships become more complex and more difficult to interpret. Additionally, with such model specifications one certainly cannot deal with all possible threats to validity from time-dependent causal agents (Shadish et al., 2002, cf. McArdle). The same is equally true of the ARCL model (and other alternative models).

Although these residual variances and covariance are typically fixed to zero (e.g., McArdle, 2009; McArdle & Hamagami, 2001), to enhance the generalizability of the discussion we consider the present model specification here. When residual variances and covariances are fixed to zero, error covariance may be assumed in the model instead of residual covariance.

As will be shown below, an intercept can be included in the ARCL model to capture mean-structure. However in some treatments, such as Ferrer and McArdle (2003), the intercept has been omitted from the model due to the focus in SEM on covariance-structure rather than mean-structure analysis.

In the ARCL model, it is sometimes assumed that residual (and error) variances can vary among different timepoints. But to clarify the discussion, time-invariant residual (and error) variances (i.e., measurement invariance; Meredith, 1964, 1993) are assumed here. Regardless of this assumption, the mathematical relationship derived in the present research is satisfied.

Although Bollen and Curran's (2004) work is undoubtedly innovative in its methodological development, there are many other reasons that this work is limited in its explication of the mathematical relationship between the LCS and ARCL (factor) models. First, Bollen and Curran's (2004) discussion generally assumes that repeated measures are observed rather than latent. As a result, these authors do not explain the ARCL factor model in their paper, leading them to omit further details of the mathematical relationship between the LCS and ARCL models. Second, even if we imagine the case where repeated measures are latent variables, these authors do not explain that latent variables for (predetermined) measures in the first timepoint in the ALT model actually correspond to the intercept factors in the LCS model, and that the original intercept factor in the ALT model corresponds to the slope factor score in the LCS model. This impedes applied (and theoretically oriented) researchers’ understanding of the mathematical relationship between the LCS model and the ALT (factor) model and results in difficulty understanding the relationship between the LCS and ARCL (factor) models. Third, Bollen and Curran did not show how to combine the equations of the LCS model, as shown in Equation (5(5) ) of this paper. So the relationship regarding the autoregressive parameters between the LCS and ARCL models (as in Equations [4(4) ] and [5(5) ]) was unclear.

Here we did not use data of older adults other than those in the 63–80 age range because models with the full sample did not fit the data. This could be due to a violation of the strong assumption of equal cross-lagged and coupling effects over time, but it could also be attributed to the limited sample of individuals aged 80 or older. In this illustration we simply intended to show the mathematical equivalence of the LCS and ARCL models, so we do not discuss this point further.

In this study, we analyzed the data with analysis of moment structure (AMOS). Supplemental materials, which include path diagrams and estimation results, are available on our website http://www.satoshiusami.com/supplementalmaterialsUsamietal2015/ or upon request.

For both forms of causal precedence, changes in X (weight) precedes Y (height) and Y (height) precedes X (weight), it might be more reasonable to assume that some third variable (e.g., osteoporosis) might directly cause decreases in weight and height simultaneously. However, the body of research on longitudinal changes in height and weight in elderly samples is unfortunately very limited, lending little help in interpreting these patterns. Because our analyses were performed primarily for the purpose of illustration we do not pursue this issue further.

In the systems literature from physics (e.g., Oppenheim, Willsky, & Nawab, 1998) an alternative interpretation of random effects on slope factor scores is the perturbation of error variances. As one reviewer commented, one taking a systems approach assumes perturbation and is unlikely to be interested in making a causal argument in the first place, which is different from standard behavioral research for causal inference. However, it is worthwhile to say that we may take a similar approach to the systems literature inasmuch as we may not have to stick to making a causal argument when using the LCS model in the initial stages of research.