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Exponential models are essential for describing rapid, multiplicative changes in natural systems, such as population growth. When a population doubles at regular intervals, the process can be modeled using a suitable base. For instance, a bacterial culture that doubles every three hours follows the model n(t)=n0⋅2t/3, where n(t) is the population at the time t.A more general model uses the natural base e, especially for continuous growth. This takes the form n(t)=n0⋅ert, where r is...
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Area of Science:

  • Quantitative Psychology
  • Statistical Modeling
  • Longitudinal Data Analysis

Background:

  • Multilevel and latent growth modeling analysis (GMA) is crucial for comparing independent groups in longitudinal studies, especially randomized controlled trials.
  • Linear GMA allows direct calculation of effect sizes from coefficients, but quadratic GMA presents challenges due to non-linear effect variations.
  • Estimating effect sizes at specific time points, including intermediate or extrapolated points, is difficult with standard quadratic GMA approaches.

Purpose of the Study:

  • To formulate equations for calculating time-varying effect sizes in quadratic growth modeling analysis (GMA).
  • To provide associated Mplus input commands for implementing these time-varying effect size calculations.
  • To address the limitations of previous GMA methods in estimating effect sizes for quadratic growth trajectories.

Main Methods:

  • Development of novel equations to derive time-varying effect sizes specifically for quadratic GMA.
  • Implementation of these equations using Mplus software, including specific input command guidance.
  • Validation through illustrative analyses and a Monte Carlo simulation study to assess bias and confidence intervals.

Main Results:

  • The proposed equations successfully formulate time-varying effect sizes for quadratic GMA.
  • Mplus commands are provided for practical application of the new methodology.
  • Monte Carlo simulations indicated that the bias in the estimated effect sizes and their confidence intervals was negligible.

Conclusions:

  • The developed methods offer a robust approach to quantifying time-varying effect sizes in quadratic growth models.
  • This facilitates more accurate and nuanced comparisons of group differences across the entire study duration.
  • The findings support the reliable application of these techniques in longitudinal research, enhancing the interpretation of complex growth patterns.