Power of Latent Growth Curve Models to Detect Piecewise Linear Trajectories

Abstract

Latent curve models (LCMs) have been used extensively to analyze longitudinal data. However, little is known about the power of LCMs to detect nonlinear trends when they are present in the data. This simulation study was designed to investigate the Type I error rates, rates of nonconvergence, and the power of LCMs to detect piecewise linear growth and mean differences in the slopes of the 2 joined longitudinal processes represented by the piecewise model. The impact of 7 design factors was examined: number of time points, growth magnitude (slope mean), interindividual variability, sample size, position of the turning point, and the correlation of the intercept and the second slope as well between the 2 slopes. The results show that previous results based on linear LCMs cannot be fully generalized to a nonlinear model defined by 2 linear slopes. Interestingly, design factors specific to the piecewise context (position of the turning point and correlation between the 2 growth factors) had some effects on the results, but these effects remained minimal and much lower than the effects of other design factors. Similarly, observed rates of inadmissible solutions are comparable to those previously reported for linear LCMs. The major finding of this study is that a moderate sample size (N = 200) is needed to detect piecewise linear trajectories, but that much larger samples (N = 1,500) are required to achieve adequate statistical power to detect slope mean difference of small magnitude.

Keywords:

• convergence
• latent curve models
• mean difference
• Monte Carlo
• piecewise trajectory
• power analysis
• structural equation models
• Type I error rates

Notes

¹ Power is known to be influenced by the magnitude of the effect to be detected so that the magnitude of ${\mu }_{S_2}$ and ${\mu }_{S_1}-{\mu }_{S_2}$ reflect one of the critical elements to consider in this study. However, another indicator of the magnitude of the effect associated with the full LCM (less relevant here to the detection of a specific parameter) is the R² of the repeated measures. R² values are a function of the time score, the variances and covariances of the growth factors, and the variances of the residuals. More precisely, R² values of the PWL model can be calculated using the following formula:

${\mathrm{R}}^2({\mathrm{y}}_{\mathrm{t}})=\frac{{\varphi }_I+{{\lambda }^2}_{1t}{{\varphi }_S}_{{\hspace{0.17em}}_1}+{\lambda }^2{{\hspace{0.17em}}_2}_t{\varphi }_{S_2}+2{\lambda }_{1t}{{\varphi }_{IS}}_{{\hspace{0.17em}}_1}+2{\lambda }_{2t}{\varphi }_{I{S}_2}+2{{\lambda }_1}_t{\lambda }_{2t}{\varphi }_{S_1{S}_2}}{{\varphi }_I+{{\lambda }^2}_{1t}{{\varphi }_S}_{{\hspace{0.17em}}_1}+{\lambda }^2{{\hspace{0.17em}}_2}_t{\varphi }_{S_2}+2{\lambda }_{1t}{{\varphi }_{IS}}_{{\hspace{0.17em}}_1}+2{\lambda }_{2t}{\varphi }_{I{S}_2}+2{{\lambda }_1}_t{\lambda }_{2t}{\varphi }_{S_1{S}_2}+{\theta }_t},$

where y_t is the outcome at time t, λ_1t is the time score for S₁, λ_2t is the time score for S₂, ${\varphi }_I$ is the interindividual variance of I, ${\varphi }_{S_1}$ is the interindividual variance of S₁, ${\varphi }_{S_2}$ is the interindividual variance of S₂, ${\varphi }_{I{S}_1}$ is the covariance between I and S₁, ${\varphi }_{I{S}_2}$ is the covariance between I and S₂, ${\varphi }_{S_1{S}_2}$ is the covariance between S₁ and S₂, and ${\theta }_t$ is the time-specific residual variance. Thus, R² is essentially composed by two terms: ${\varphi }_I+{{\lambda }^2}_{1t}{{\varphi }_S}_{{\hspace{0.17em}}_1}+{\lambda }^2{\hspace{0.17em}}_{2}t{\varphi }_{S_2}+2{\lambda }_{1t}{{\varphi }_{IS}}_{{\hspace{0.17em}}_1}+2{\lambda }_{2t}{\varphi }_{I{S}_2}+2{{\lambda }_1}_t{\lambda }_{2t}{\varphi }_{S_1{S}_2}$ and ${\theta }_t$. The first term is strictly increasing over time given that all growth factor variances and covariances are greater than zero in this study. The second term is constant due to the homoscedastic assumption (${\theta }_t=\theta$) of this study. As a result the R² strictly increases over time. Based on this formula, it is clear that the specific R² values change across design conditions under the influence of multiple design conditions, making it complex to report specific values associated with each of the time points within each of the 7,524 design conditions. Overall, R² values range from .5 for the first time point to .99 for the 10th time point. This increase over time is reasonable, as more time points usually provide more precision in the estimation of the underlying trajectories. However, the higher values observed at later time points might have resulted in slightly inflated power estimates.

² To test the statistical significance of ${\mu }_{S_1}-{\mu }_{S_2}$, we relied on a Wald test of statistical significance. Alternatively, likelihood ratio tests (LRTs), which are well suited for situations where the normality assumption is met and multiple parameters are estimated, could also have been used. The Wald test is routinely provided by most software, making it simpler to use than the comparison of models based on LRTs. For this reason, the Wald test is more frequently used in practice. However, the squared version of the Wald test and the LRT are asymptotically equivalent and follow a chi-square distribution of 1 df (Bollen, 1989; DasGupta, 2008). These two tests should thus give similar results in most situations. In fact, small differences could potentially be expected when the asymptotic equivalence between the Wald test and the LRT no longer holds, such as in small sample sizes. The specific conditions where this equivalence breaks down should clearly be more systematically investigated in the context of future studies.