Robust Bayesian Approaches in Growth Curve Modeling: Using Student’s t Distributions versus a Semiparametric Method

Abstract

Despite broad applications of growth curve models, few studies have dealt with a practical issue – nonnormality of data. Previous studies have used Student’s t distributions to remedy the nonnormal problems. In this study, robust distributional growth curve models are proposed from a semiparametric Bayesian perspective, in which intraindividual measurement errors follow unknown random distributions with Dirichlet process mixture priors. Based on Monte Carlo simulations, we evaluate the performance of the robust semiparametric Bayesian method and compare it to the robust method using Student’s t distributions as well as the traditional normal-based method. We conclude that the semiparametric Bayesian method is more robust against nonnormal data. An example about the development of mathematical abilities is provided to illustrate the application of robust growth curve modeling, using school children’s Peabody Individual Achievement Test mathematical test scores from the National Longitudinal Survey of Youth 1997 Cohort.

Keywords:

• Semiparametric Bayesian
• robust method
• Dirichlet process mixture
• growth curve modeling

Notes

¹ Notice that for semiparametric Bayesian method, the maximum of the potential number of mixture components $C$ is set at 20 in the simulation. This is reasonable because the estimated number of clusters is always below 20 based on Tong (2014). So setting C at 20 or a larger value such as 50 does not affect the model estimation. But a very small value of $C$ (e.g., 5) may reduce the accuracy of the estimation. In addition, the hyperparameter $\alpha$ is specified to follow $Gamma(2,2)$, as suggested by Ishwaran (2000).

² $relativebias=\left(\begin{array}{cc}\hat{\theta }\times 100\mathrm{\%}& \theta =0\\ \frac{\hat{\theta }-\theta }{\theta }\times 100\mathrm{\%}& \theta \ne 0\end{array}\right.$, where $\theta$ is the true parameter value and $\hat{\theta }$ is the estimate of $\theta$.

³ Posterior credible interval, also called credible interval or Bayesian confidence interval, is analogical to the frequentist confidence interval. The 95% HPD credible interval $[l,u]$ satisfies the following: 1. $Prob(l\le \theta \le u|data)=0.95$; 2. for ${\theta }_1\in [l,u]$ and ${\theta }_2\notin [l,u],$ $Prob({\theta }_1|data)>Prob({\theta }_2|data)$. In general, HPD intervals have the smallest volume in the parameter space of $\theta$, and numerical methods have to be used to find HPD intervals.

⁴ Note that all the detailed parameter estimation results are provided on our webpage in the same structure as Table 3..

⁵ In addition to the illustrative example in this article, we also provide another example to help substantive researchers understand the application of the proposed robust method in growth curve modeling, using data from the Virginia Cognitive Aging Project. The detailed description of this example as well as the programming code are available on our webpage: https://www.dropbox.com/sh/njhhnm4tdi4ahp9/AADXXBbB79qvp0x37YZSNeK7a?dl=0..

⁶ Infinite-dimensional can be interpreted as of finite but unbounded dimension.