ABSTRACT

Growth curve modeling is commonly used in psychological, educational, and social science research. The mainstream estimators for growth curve modeling are based on normal theory, but real data are unlikely to be exactly normally distributed. To improve estimation and inference with non-normal data, various estimators have been proposed. Among these estimators, the asymptotically distribution free (ADF) estimator does not need to rely on any distribution assumption but it is not efficient with small and modest sample sizes. We propose a distributionally weighted least squares $DLS$ estimator in the growth curve modeling framework. $DLS$ combines normal theory based and $ADF$ based generalized least squares estimation to balance the information from the data and the normality assumption. Computer simulation results suggest that model-implied covariance-based $DLS$ ($DL{S}_M$) generally provides more accurate and efficient estimates than the examined alternative methods regardless of the distribution. In addition, the relative biases of standard error estimates and the Type I error rates of the Satorra–Bentler test statistic ($T_{SB}$) in $DL{S}_M$ were competitive with the classical methods including maximum likelihood and generalized least squares estimation. We illustrate how to implement $DL{S}_M$ and select the optimal tuning parameter by a bootstrap procedure in a real data example.

摘要

增长曲线模型常用于心理、教育和社会科学研究。增长曲线建模的主流估计量基于正态理论，但实际数据不太可能完全符合正态分布。为了改进对非正态数据的估计和推理，已经提出了各种估计器。在这些估计器中，无渐近分布 ( ADF ) 估计器不需要依赖任何分布假设，但它在样本量小且适中的情况下效率不高。我们提出了一个分布加权最小二乘$D\mathrm{大号}\mathrm{小号}$ 增长曲线建模框架中的估计器。 $D\mathrm{大号}\mathrm{小号}$ 结合了基于正常理论和 $\mathrm{一个}DF$基于广义最小二乘估计来平衡来自数据的信息和正态假设。计算机模拟结果表明，基于模型隐含协方差$D\mathrm{大号}\mathrm{小号}$ ($D\mathrm{大号}{\mathrm{小号}}_{米}$) 通常提供比检查的替代方法更准确和有效的估计，而不管分布如何。此外，标准误差估计的相对偏差和 Satorra–Bentler 检验统计量的 I 类错误率 (${吨}_{\mathrm{小号}乙}$） 在 $D\mathrm{大号}{\mathrm{小号}}_{米}$与包括最大似然和广义最小二乘估计在内的经典方法具有竞争力。我们举例说明如何实现$D\mathrm{大号}{\mathrm{小号}}_{米}$ 并在实际数据示例中通过引导程序选择最佳调整参数。