[Paper Review] Estimating Knots and Their Association in Parallel Bilinear Spline Growth Curve Models in the Framework of Individual Measurement Occasions

Jin Liu, Robert A. Perera|arXiv (Cornell University)|Apr 7, 2020

Meta-analysis and systematic reviewsDecision Sciences44 references14 citations

TL;DR

This paper proposes a parallel bilinear spline growth curve model (PBLSGM) to estimate unknown knot locations and their variability in multiple correlated developmental trajectories using individual measurement occasions. The method extends univariate bilinear spline models to jointly model nonlinear change patterns, enabling estimation of knot-knot associations, with simulation and real-data validation showing unbiased parameter estimates and appropriate confidence interval coverage.

ABSTRACT

Latent growth curve models with spline functions are flexible and accessible statistical tools for investigating nonlinear change patterns that exhibit distinct phases of development in manifested variables. Among such models, the bilinear spline growth model (BLSGM) is the most straightforward and intuitive but useful. An existing study has demonstrated that the BLSGM allows the knot (or change-point), at which two linear segments join together, to be an additional growth factor other than the intercept and slopes so that researchers can estimate the knot and its variability in the framework of individual measurement occasions. However, developmental processes usually unfold in a joint development where two or more outcomes and their change patterns are correlated over time. As an extension of the existing BLSGM with an unknown knot, this study considers a parallel BLSGM (PBLSGM) for investigating multiple nonlinear growth processes and estimating the knot with its variability of each process as well as the knot-knot association in the framework of individual measurement occasions. We present the proposed model by simulation studies and a real-world data analysis. Our simulation studies demonstrate that the proposed PBLSGM generally estimate the parameters of interest unbiasedly, precisely and exhibit appropriate confidence interval coverage. An empirical example using longitudinal reading scores, mathematics scores, and science scores shows that the model can estimate the knot with its variance for each growth curve and the covariance between two knots. We also provide the corresponding code for the proposed model.

Motivation & Objective

• To develop a parallel bilinear spline growth curve model (PBLSGM) that estimates unknown knot locations and their variability in multiple correlated developmental processes.
• To extend existing univariate bilinear spline models to a multivariate framework capable of modeling joint development with correlated change patterns.
• To enable estimation of knot-knot associations, in addition to intercept and slope associations, in parallel growth curves.
• To provide transformation methods for interpreting model estimates in the original parameter space, enhancing interpretability.
• To address practical challenges in longitudinal data with heterogeneous measurement times and potential convergence issues.

Proposed method

• Proposes a multivariate extension of the bilinear spline growth model (BLSGM) to simultaneously model two or more nonlinear growth trajectories with unknown knot locations.
• Models each outcome's growth curve using a piecewise linear function with a knot as a random effect, allowing individual variability in the timing of change points.
• Employs reparameterization techniques to express growth factors in a form that allows direct interpretation of knot means, variances, and covariances.
• Applies inverse transformation functions to convert estimates from the reparameterized space back to the original parameter space for meaningful interpretation.
• Uses full information maximum likelihood (FIML) for handling individual measurement occasions and missing data under missing at at random (MAR) assumptions.
• Provides OpenMx and Mplus 8 syntax in the online appendix for model implementation and replication.

Experimental results

Research questions

• RQ1 Can a parallel bilinear spline growth model estimate unknown knot locations and their variability in multiple correlated developmental trajectories?
• RQ2 How well do the PBLSGM estimates perform in terms of bias, precision, and coverage under various sample sizes and measurement conditions?
• RQ3 What is the impact of model complexity on convergence and proper solution rates, especially with small samples or moderate knot variance?
• RQ4 How do different time metrics (e.g., age vs. grade-in-school) affect knot estimates and model interpretation in empirical applications?
• RQ5 What is the best practice for model selection when comparing PBLSGMs with other functional forms (e.g., quadratic) in joint developmental modeling?

Key findings

• The PBLSGM estimates the means of growth factors (intercept, slopes, knots) with low bias, high precision, and appropriate 95% confidence interval coverage in simulation studies.
• Estimates of variances and covariances of growth factors were generally accurate, though slight biases (>10%) occurred under small sample sizes or low measurement precision.
• The full PBLSGM model sometimes produced improper solutions (e.g., negative knot variances or out-of-range correlations), especially when knot variance was set at 0.3.
• The reduced PBLSGM model, which excludes knot-knot covariance, showed comparable standard errors and only slightly increased bias, making it a viable alternative when convergence fails.
• In the empirical application using ECLS-K: 2011 data, the PBLSGM outperformed quadratic models in capturing early developmental changes in reading and mathematics scores.
• Sensitivity analysis showed that knot estimates were consistent across different time metrics (age vs. grade-in-school), though grade-based knots were more heterogeneous, underscoring the importance of research-driven time metric selection.