(Re)evaluating the Implications of the Autoregressive Latent Trajectory Model Through Likelihood Ratio Tests of Its Initial Conditions

ABSTRACT

The autoregressive latent trajectory (ALT) model synthesizes the autoregressive model and the latent growth curve model. The ALT model is flexible enough to produce a variety of discrepant model-implied change trajectories. While some researchers consider this a virtue, others have cautioned that this may confound interpretations of the model's parameters. In this article, we show that some—but not all—of these interpretational difficulties may be clarified mathematically and tested explicitly via likelihood ratio tests (LRTs) imposed on the initial conditions of the model. We show analytically the nested relations among three variants of the ALT model and the constraints needed to establish equivalences. A Monte Carlo simulation study indicated that LRTs, particularly when used in combination with information criterion measures, can allow researchers to test targeted hypotheses about the functional forms of the change process under study. We further demonstrate when and how such tests may justifiably be used to facilitate our understanding of the underlying process of change using a subsample (N = 3,995) of longitudinal family income data from the National Longitudinal Survey of Youth.

KEYWORDS:

• Autoregressive latent trajectory model
• initial condition
• latent growth curve models

Notes

1 It is assumed in the mathematical derivation of the ICS. However, the model is used as a local description of a process and does not imply that the actual process has been following Equation (1(1) ) from minus infinity.

2 Strictly speaking, the stationarity beyond the second timepoint is not needed to establish the equivalence between the ongoing EALT and the predetermined ALT models as long as the AR parameter linking the first to the second measurement occasion is assumed to also characterize the stationary AR process prior to the first timepoint. If ρ is freed to be varying from t = 2 onward, then the mean and individual-specific latent change trajectories would no longer be linear. However, as stationarity is assumed to hold prior and up to the second available timepoint, it is arguably more reasonable to assume stationarity across all timepoints for consistency's sake.

3 From Equations (10b(10) ) and (10c(10) ), for t ≥ 2, , when μ₁ = μ_α; or , when μ₁ ≠ μ_α.

4 Specifically, d(t) is obtained as (11) when μ_α and ρ are held constant.

5 When all four latent variables are included in the model, the μ and variance-covariance parameters only have to be estimated once (e.g., only for α_i and β_i but not A_i and B_i, and vice versa) because the other set of parameters can be obtained through transformations.

6 Lower nonconvergence rates were observed when we fit the linear LGC model with AR disturbances (Equation [5(5) ]) as opposed to the ongoing EALT model with nonlinear constraints to the same sets of simulated data. Respectively for N = 100, 500, 1,000 sample-size conditions, the proportions of nonconvergent trials across 500 Monte Carlo runs were 10.20%, 0.20%, 0.00% if the true model was the predetermined ALT model (compared to 41.40%, 29.00%, and 21.40% in Table 2); 3.20%, 0.00%, 0.00% when the true model was the ongoing EALT model (compared to 7.00%, 0.60%, and 0.40% in Table 3); and 60.20%, 73.40%, 79.20% when the true model was the HI EALT model (compared to 91.00%, 99.80%, and 100.00% in Table 4). This finding supports our postulate that misspecification is a major reason for nonconvergence, but the nonlinear constraints in the ongoing EALT model further aggravated the convergence issue.

7 This decision was driven primarily by the extremely low convergence rates of the ongoing EALT model when it was a misspecified model. We also considered the option of dropping all nonconvergent trials, but that entailed an excessively low number of retained trials in some conditions.