Department of Psychology, University of Notre Dame, Notre Dame, IN, USA.
Department of Statistics, Nanjing University of Posts and Telecommunications, Nanjing, China.
Multivariate Behav Res. 2022 Mar-May;57(2-3):223-242. doi: 10.1080/00273171.2020.1858018. Epub 2021 Jan 5.
Chi-square type test statistics are widely used in assessing the goodness-of-fit of a theoretical model. The exact distributions of such statistics can be quite different from the nominal chi-square distribution due to violation of conditions encountered with real data. In such instances, the bootstrap or Monte Carlo methodology might be used to approximate the distribution of the statistic. However, the sample quantile may be a poor estimate of the population counterpart when either the sample size is small or the number of different values of the replicated statistic is limited. Using statistical learning, this article develops a method that yields more accurate quantiles for chi-square type test statistics. Formulas for smoothing the quantiles of chi-square type statistics are obtained. Combined with the bootstrap methodology, the smoothed quantiles are further used to conduct equivalence testing in mean and covariance structure analysis. Two real data examples illustrate the applications of the developed formulas in quantifying the size of model misspecification under equivalence testing. The idea developed in the article can also be used to develop formulas for smoothing the quantiles of other types of test statistics or parameter estimates.
卡方类型检验统计量广泛用于评估理论模型的拟合优度。由于实际数据中遇到的条件违反，此类统计量的精确分布可能与名义卡方分布有很大不同。在这种情况下，可以使用自举或蒙特卡罗方法来近似统计量的分布。但是，当样本量较小时或重复统计量的不同值数量有限时，样本分位数可能是总体对应物的较差估计。本文利用统计学习方法，提出了一种为卡方类型检验统计量生成更准确分位数的方法。得到了对卡方类型统计量的分位数进行平滑的公式。结合自举方法，平滑分位数进一步用于在均值和协方差结构分析中进行等效性检验。两个实际数据示例说明了所开发公式在等效性检验下量化模型误定大小的应用。本文中提出的思想也可用于开发其他类型的检验统计量或参数估计的分位数平滑公式。
