Bayesian Inference and Application of Robust Growth Curve Models Using Student's t Distribution

Abstract

Despite the widespread popularity of growth curve analysis, few studies have investigated robust growth curve models. In this article, the t distribution is applied to model heavy-tailed data and contaminated normal data with outliers for growth curve analysis. The derived robust growth curve models are estimated through Bayesian methods utilizing data augmentation and Gibbs sampling algorithms. The analysis of mathematical development data shows that the robust latent basis growth curve model better describes the mathematical growth trajectory than the corresponding normal growth curve model and can reveal the individual differences in mathematical development. Simulation studies further confirm that the robust growth curve models significantly outperform the normal growth curve models for both heavy-tailed t data and normal data with outliers but lose only slight efficiency for normal data. It appears convincing to replace the normal distribution with the t distribution for growth curve analysis. Three information criteria are evaluated for model selection. Online software is also provided for conducting robust analysis discussed in this study.

Keywords:

• Bayesian inference
• mathematical development
• model comparison
• robust growth curve models
• t distribution

Notes

¹The posterior standard deviation is analogical to the frequentist standard error and the posterior credible interval, also called credible interval or Bayesian confidence interval, is analogical to the frequentist confidence interval. For the credible interval, [ _r ,û _r ] = [ _r +Φ⁻¹(0.025) × ŝ _r , _r +Φ⁻¹(0.975) × ŝ _r ] where Φ is the normal distribution function.

²For the sake of space, only results for T = 5 and N = 100, 200, 300, and 400 are reported here. Results for other conditions are similar and available on request.

³In all simulations, the robust no growth curve model has never been selected and thus is excluded in the table for the sake of space.