Rescaled Statistic Research Articles

Rescaled Statistics and Wavelet Analysis on Agricultural Drought Disaster Periodic Fluctuations in China from 1950 to 2016

An agricultural drought disaster was analyzed with the new insight of rescaled statistics (R/S) and wavelet analysis in this study. The results showed that: (1) the Hurst index of the agricultural disaster area, the inundated area of agricultural drought disaster, and the grain loss was 0.821, 0.874, and 0.953, respectively, indicating that the process of the agricultural drought disaster had stronger positive continuity during the study period; (2) based on the Morlet analysis of the agricultural disaster area, the inundated area of the agricultural drought disaster, and the grain loss of China from 1950 to 2016, the time series of the agricultural drought had multiple time scale features with the periodic variation on a large scale containing the periodic variation on a small scale; and (3) in the last 67 years, the strong wavelet energy spectrum of the agricultural disaster area, the inundated area of the agricultural drought disaster, and the grain loss was at the time scale of ≈22–32 years, ≈24–32 years, and ≈25–32 years, respectively. In addition, the first major period in the agricultural drought disaster area, the inundated area of agricultural drought disaster, and the grain loss had average periods of approximately 16 years, 16 years, and 18 years, respectively.

Evaluation of Corrections to the Rescaled Statistic for SEM with Big Data

Trustworthy test statistics are of critical importance for evaluating overall model fit in structural equation modeling (SEM). While the likelihood ratio statistic TML derived from the normality as...

The Performance of Ten Modified Rescaled Statistics as the Number of Variables Increases

Among test statistics for assessing overall model fit in structural equation modeling (SEM), the Satorra–Bentler rescaled statistic is most widely used when the normality assumption is violated. However, many researchers have found that tends to overreject correct models when the number of variables (p) is large and/or the sample size (N) is small. Modifications of have been proposed, but few studies have examined their performance against each other, especially when p is large. This article systematically evaluates 10 corrected versions of . Results show that the Bartlett correction and a recently proposed rank correction perform better than others in controlling Type I error rates, according to their deviations from the nominal rate. Nevertheless, the performance of both corrections depends heavily on p in addition to N. As p becomes relatively large, none of the corrected versions can properly control Type I errors even when N is rather large.

Empirically Corrected Rescaled Statistics for SEM with Small N and Large p

ABSTRACTSurvey data often contain many variables. Structural equation modeling (SEM) is commonly used in analyzing such data. With typical nonnormally distributed data in practice, a rescaled statistic Trml proposed by Satorra and Bentler was recommended in the literature of SEM. However, Trml has been shown to be problematic when the sample size N is small and/or the number of variables p is large. There does not exist a reliable test statistic for SEM with small N or large p, especially with nonnormally distributed data. Following the principle of Bartlett correction, this article develops empirical corrections to Trml so that the mean of the empirically corrected statistics approximately equals the degrees of freedom of the nominal chi-square distribution. Results show that empirically corrected statistics control type I errors reasonably well even when N is smaller than 2p, where Trml may reject the correct model 100% even for normally distributed data. The application of the empirically corrected statistics is illustrated via a real data example.

Reliable and More Powerful Methods for Power Analysis in Structural Equation Modeling

The normal-distribution-based likelihood ratio statistic is widely used for power analysis in structural Equation modeling (SEM). In such an analysis, power and sample size are computed by assuming that follows a central chi-square distribution under and a noncentral chi-square distribution under . However, with either violation of normality or not a large enough sample size, both empirical and analytical results indicate that the chi-square distribution assumptions are not realistic and consequently methods of power analysis based on such assumptions are not valid. This article describes a Monte Carlo (MC) method for power analysis. A measure of effect size for characterizing the power property of different rescaled statistics is also provided. Robust methods are proposed to increase the power of and other statistics. Simulation results show that the MC method reliably controls Type I errors and robust estimation methods effectively increase the power, and their combination is thus recommended for conducting power analysis in SEM.

Structural equation modeling with near singular covariance matrices

Conventional structural equation modeling involves fitting a structural model to the sample covariance matrix S . Due to collinearity or small samples with practical data, nonconvergences often occur in the estimation process. For a small constant a , this paper proposes to fit the structural model to the covariance matrix S a = S + a I . When treating S a as the sample covariance matrix in the maximum likelihood (ML) procedure, consistent parameter estimates are still obtained. The asymptotic distributions of the parameter estimates and the corresponding likelihood ratio statistic are studied and compared to those by the conventional ML. Two rescaled statistics for the overall model evaluation with modeling S a are constructed. Empirical results imply that the estimates from modeling S a are more efficient than those of fitting the structural model to S even when data are normally distributed. Simulations and real data examples indicate that modeling S a allows us to evaluate the overall model structure even when S is literally singular. Implications of modeling S a in a broader context are discussed.

On the asymptotic distributions of two statistics for two-level covariance structure models within the class of elliptical distributions

Since data in social and behavioral sciences are often hierarchically organized, special statistical procedures for covariance structure models have been developed to reflect such hierarchical structures. Most of these developments are based on a multivariate normality distribution assumption, which may not be realistic for practical data. It is of interest to know whether normal theory-based inference can still be valid with violations of the distribution condition. Various interesting results have been obtained for conventional covariance structure analysis based on the class of elliptical distributions. This paper shows that similar results still hold for 2-level covariance structure models. Specifically, when both the level-1 (within cluster) and level-2 (between cluster) random components follow the same elliptical distribution, the rescaled statistic recently developed by Yuan and Bentler asymptotically follows a chi-square distribution. When level-1 and level-2 have different elliptical distributions, an additional rescaled statistic can be constructed that also asymptotically follows a chi-square distribution. Our results provide a rationale for applying these rescaled statistics to general non-normal distributions, and also provide insight into issues related to level-1 and level-2 sample sizes.

On normal theory based inference for multilevel models with distributional violations

Data in social and behavioral sciences are often hierarchically organized though seldom normal, yet normal theory based inference procedures are routinely used for analyzing multilevel models. Based on this observation, simple adjustments to normal theory based results are proposed to minimize the consequences of violating normality assumptions. For characterizing the distribution of parameter estimates, sandwich-type covariance matrices are derived. Standard errors based on these covariance matrices remain consistent under distributional violations. Implications of various covariance estimators are also discussed. For evaluating the quality of a multilevel model, a rescaled statistic is given for both the hierarchical linear model and the hierarchical structural equation model. The rescaled statistic, improving the likelihood ratio statistic by estimating one extra parameter, approaches the same mean as its reference distribution. A simulation study with a 2-level factor model implies that the rescaled statistic is preferable.