Reconsidering the Use of Autoregressive Latent Trajectory (ALT) Models

Abstract

The simultaneous estimation of autoregressive (simplex) structures and latent trajectories, so called ALT (autoregressive latent trajectory) models, is becoming an increasingly popular approach to the analysis of change. Although historically autoregressive (AR) and latent growth curve (LGC) models have been developed quite independently from each other, the underlying pattern of change is often highly similar. In this article it is shown that their integration rests on the strong assumption that neither the AR part nor the LGC part contains any misspecification. In practice, however, this assumption is often violated due to nonlinearity in the LGC part. As a consequence, the autoregressive (simplex) process incorrectly accounts for part of this nonlinearity, thus rendering any substantive interpretation of parameter estimates virtually impossible. Accordingly, researchers are advised to exercise extreme caution when using ALT models in practice. All arguments are illustrated by empirical data on skill acquisition, and a simulation study is provided to investigate the conditions and consequences of mistaking nonlinear growth curve patterns as autoregressive processes.

Notes

¹ Strictly speaking, by linear I refer to the first order polynomial equation, which could still correspond to a nonlinear growth curve, given a different parameterization of time or a transformation of the repeated measures.

² Rogosa and Willett (1985) speak of a “constant rate of change model.”

³ Kenny (1974; see also Campbell & Erlebacher, 1970) coined the term “fan-spread effect” for this pattern of increasing variance.

*p < .05.

**p < .01, ns. p ≥ .05.

⁴ Introducing a single error covariance between ε₂ and ε₃ would result in an acceptable model fit with χ² = 15.6, df = 11, p > .05; CFI = 1.00; RMSEA = 0.07; SRMR = 0.01. Given there exists a plausible post hoc explanation for this covariance, this would probably be an appropriate strategy in the context of discovery but not in the (stricter) context of justification.

⁵ Although the model quickly converges, it results in an improper solution by estimating two negative error variances (ε₁ and ε₆), sometimes also referred to as Heywood cases. Because neither of the variances is significant, following the procedure proposed by Chen, Bollen, Paxton, Curran, and Kirby (2001) and constraining the two error variances to a small positive value (VAR(ε₁) = VAR(ε₆) = 0.01) results in an excellent model fit (see text) and appropriate parameter estimates (Mean(α) = 8.13**; Mean(β₁) = 3.85**; Mean(β₂) = −0.52**; Mean(β₃) = 0.04 ns; VAR(α) = 22.15**; VAR(β₁) = 13.69**; VAR(β₂) = 2.14**; VAR(β₃) = 0.03**; COR(α,β₁) = 0.06 ns; COR(β,β₂) = 0.01 ns; COR(α,β₃) = −0.06 ns; COR(β₁,β₂) = −0.92**; COR(β₁,β₃) = 0.805**; COR(β₂,β₃) = −0.97**). Again, all parameter estimates are in the expected direction and of expected size and correspond to the underlying theory of skill acquisition. Finally, with df = 9, the third order polynomial growth curve model is more parsimonious than the AR model (df = 8), and the model fit of the two models is almost identical.

⁶ To enhance the clarity of presentation, the results of only eight instead of nine conditions are reported in this article. This results in a total of 24,000 structural equation models to be estimated.

⁷ For the quadratic growth factor it is not reasonable to compute the average estimate across conditions because it was varied from a positive to a negative value. Thus, the mean is not reported in Table 2.

⁸ This applies also to the second example employed by Bollen and Curran (2004, p. 370) to illustrate the ALT model. In this example, data from the National Longitudinal Study of Youth (NLSY) were used to analyze reported total net family income of young people transitioning into the labor force across five 2-year intervals. It seems quite likely that this is not a linear process.