VARIANCE TESTING WITH SIMPLICIAL DATA DEPTH

A method is developed and studied for testing equality of variances based on simplicial data depth and Mood\u27s nonparametric test in the case of two samples. A method for calculating univariate simplicial data depth using a rank transformation is introduced. Type I error rates and power curves are compared for three existing tests for equality of variances and the data depth test using data simulated from the nonnal distribution and 5 nonnormal distributions. In addition, a new method of aligning two samples with unequal location parameters is proposed. This method shows significant improvement over aligning by either the median or mean in controlling Type I error rates of skewed distributions

A Power Comparison of Robust Test Statistics Based On Adaptive Estimators

Seven test statistics known to be robust to the combined effects of nonnormality and variance heterogeneity were compared for their sensitivity to detect treatment effects in a one-way completely randomized design containing four groups. The six Welch-James-type heteroscedastic tests adopted either symmetric or asymmetric trimmed means, were transformed for skewness, and used a bootstrap method to assess statistical significance. The remaining test, due to Wilcox and Keselman (2003), used a modification of the well-known one-step M-estimator of central tendency rather than trimmed means. The Welch-James-type test is recommended because for nonnormal data likely to be encountered in applied research settings it should be more powerful than the test presented by Wilcox and Keselman. However, the reverse is true for data that are extremely nonnormal

A Power Comparison Of Robust Test Statistics Based On Adaptive Estimators.

Seven test statistics known to be robust to the combined effects of nonnormality and variance heterogeneity were compared for their sensitivity to detect treatment effects in a one-way completely randomized design containing four groups

Trimming, Transforming Statistics, And Bootstrapping: Circumventing the Biasing Effects Of Heterescedasticity And Nonnormality

Researchers can adopt different measures of central tendency and test statistics to examine the effect of a treatment variable across groups (e.g., means, trimmed means, M-estimators, & medians. Recently developed statistics are compared with respect to their ability to control Type I errors when data were nonnormal, heterogeneous, and the design was unbalanced: (1) a preliminary test for symmetry which determines whether data should be trimmed symmetrically or asymmetrically, (2) two different transformations to eliminate skewness, (3) the accuracy of assessing statistical significance with a bootstrap methodology was examined, and (4) statistics that use a robust measure of the typical score that empirically determined whether data should be trimmed, and, if so, in which direction, and by what amount were examined. The 56 procedures considered were remarkably robust to extreme forms of heterogeneity and nonnormality. However, we recommend a number of Welch-James heteroscedastic statistics which are preceded by the Babu, Padmanaban, and Puri (1999) test for symmetry that either symmetrically trimmed 10% of the data per group, or asymmetrically trimmed 20% of the data per group, after which either Johnson\u27s (1978) or Hall\u27s (1992) transformation was applied to the statistic and where significance was assessed through bootstrapping. Close competitors to the best methods were found that did not involve a transformation

Trimming, Transforming Statistics, And Bootstrapping: Circumventing the Biasing Effects Of Heterescedasticity And Nonnormality.

Researchers can adopt different measures of central tendency and test statistics to examine the effect of a treatment variable across groups

The Impact Of Multiple Imputation On The Type Ii Error Rate Of The T Test

ABSTRACT THE IMPACT OF MULTIPLE IMPUTATION ON THE TYPE II ERROR RATE OF THE T TEST by TAMMY A. GRACE August 2016 Advisor: Shlomo Sawilowsky, PhD Major: Evaluation and Research Degree: Doctor of Philosophy The National Academy of Science identified numerous high priority areas for missing data research. This study addresses several of those areas by systematically investigating the impact of multiple imputation on the rejection rate of the independent samples t test under varying conditions of sample size, effect size, fraction of missing data, distribution shape, and alpha. In addition to addressing gaps in the missing data literature, this study also provides an overview of the multiple imputation procedure, as implemented in SPSS, with a focus on the practical aspects and challenges of using this method

A Monte Carlo Comparison of Robust MANOVA Test Statistics

Multivariate Analysis of Variance (MANOVA) is a popular statistical tool in the social sciences, allowing for the comparison of mean vectors across groups. MANOVA rests on three primary assumptions regarding the population: (a) multivariate normality, (b) equality of group population covariance matrices and (c) independence of errors. When these assumptions are violated, MANOVA does not perform well with respect to Type I error and power. There are several alternative test statistics that can be considered including robust statistics and the use of the structural equation modeling (SEM) framework. This simulation study focused on comparing the performance of the P test statistics with fifteen other test statistics across seven manipulated factors. These statistics were evaluated across 12,076 different conditions in terms of Type I error and power. Results suggest that when assumptions were met, the standard MANOVA test functioned well. However, when assumptions were violated, it performed poorly, whereas several of the alternatives performed better. Discussion focuses on advice for selecting alternatives in practice. This study’s focus on all these in one simulation and the 3 group case should be helpful to the practitioner making methodological sections

Comparing linear discriminant analysis and supervised learning algorithms for binary classification - a method comparison study

In psychology, linear discriminant analysis (LDA) is the method of choice for two-group classification tasks based on questionnaire data. In this study, we present a comparison of LDA with several supervised learning algorithms. In particular, we examine to what extent the predictive performance of LDA relies on the multivariate normality assumption. As nonparametric alternatives, the linear support vector machine (SVM), classification and regression tree (CART), random forest (RF), probabilistic neural network (PNN), and the ensemble k conditional nearest neighbor (EkCNN) algorithms are applied. Predictive performance is determined using measures of overall performance, discrimination, and calibration, and is compared in two reference data sets as well as in a simulation study. The reference data are Likert-type data, and comprise 5 and 10 predictor variables, respectively. Simulations are based on the reference data and are done for a balanced and an unbalanced scenario in each case. In order to compare the algorithms' performance, data are simulated from multivariate distributions with differing degrees of nonnormality. Results differ depending on the specific performance measure. The main finding is that LDA is always outperformed by RF in the bimodal data with respect to overall performance. Discriminative ability of the RF algorithm is often higher compared to LDA, but its model calibration is usually worse. Still LDA mostly ranges second in cases it is outperformed by another algorithm, or the differences are only marginal. In consequence, we still recommend LDA for this type of application

UU-tests for variance components in one-way random effects models

We consider a test for the hypothesis that the within-treatment variance component in a one-way random effects model is null. This test is based on a decomposition of a $U$-statistic. Its asymptotic null distribution is derived under the mild regularity condition that the second moment of the random effects and the fourth moment of the within-treatment errors are finite. Under the additional assumption that the fourth moment of the random effect is finite, we also derive the distribution of the proposed $U$-test statistic under a sequence of local alternative hypotheses. We report the results of a simulation study conducted to compare the performance of the $U$-test with that of the usual $F$-test. The main conclusions of the simulation study are that (i) under normality or under moderate degrees of imbalance in the design, the $F$-test behaves well when compared to the $U$-test, and (ii) when the distribution of the random effects and within-treatment errors are nonnormal, the $U$-test is preferable even when the number of treatments is small.Comment: Published in at http://dx.doi.org/10.1214/193940307000000149 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org