A Nonparametric Test Statistic for the General Linear Model

Puri and Sen (1969Puri and Sen (1985) presented a nonparametric test statistic based on a general linear model approach that is appropriate for testing a wide class of hypotheses. The two forms of this statistic, pure- and mixed-rank, differ according to whether the original predictor values or their ranks are used. Both forms permit the use of standard statistical packages to perform the analyses. The applicability of these statistics in testing a number of hypotheses is highlighted, and an example of their use is given. A simulation study for the multivariate-multiple-regression case is used to examine the distributional behavior of the pure- and mixed-rank statistics and an important competitor, the rank transformation of Conover and Iman (1981). The results suggest that the pure- and mixed-rank statistics are superior with respect to minimizing liberal Type I error rates, whereas the Conover and Iman statistic produces larger power values.

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A Nonparametric Test Statistic for the General Linear Model

Journal of Educational Statistics ◽

10.2307/1164944 ◽

1989 ◽

Vol 14 (4) ◽

pp. 351 ◽

Cited By ~ 16

Author(s):

Michael R. Harwell ◽

Ronald C. Serlin

Keyword(s):

Linear Model ◽

Nonparametric Test ◽

General Linear Model ◽

General Linear ◽

Test Statistic ◽

Nonparametric Test Statistic

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A fast and accurate algorithm to test for binary phenotypes and its application to PheWAS

10.1101/109876 ◽

2017 ◽

Cited By ~ 1

Author(s):

Rounak Dey ◽

Ellen M. Schmidt ◽

Goncalo R. Abecasis ◽

Seunggeun Lee

Keyword(s):

Type I Error ◽

Score Test ◽

Saddlepoint Approximation ◽

Case Control ◽

Error Rates ◽

Type I ◽

Test Statistic ◽

Association Analyses ◽

Type I Error Rates ◽

And Control

AbstractThe availability of electronic health record (EHR)-based phenotypes allows for genome-wide association analyses in thousands of traits, and has great potential to identify novel genetic variants associated with clinical phenotypes. We can interpret the phenome-wide association study (PheWAS) result for a single genetic variant by observing its association across a landscape of phenotypes. Since PheWAS can test 1000s of binary phenotypes, and most of them have unbalanced (case:control = 1:10) or often extremely unbalanced (case:control = 1:600) case-control ratios, existing methods cannot provide an accurate and scalable way to test for associations. Here we propose a computationally fast score test-based method that estimates the distribution of the test statistic using the saddlepoint approximation. Our method is much faster than the state of the art Firth’s test (∼ 100 times). It can also adjust for covariates and control type I error rates even when the case-control ratio is extremely unbalanced. Through application to PheWAS data from the Michigan Genomics Initiative, we show that the proposed method can control type I error rates while replicating previously known association signals even for traits with a very small number of cases and a large number of controls.

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Comparison of Test Statistics of Nonnormal and Unbalanced Samples for Multivariate Analysis of Variance in terms of Type-I Error Rates

Computational and Mathematical Methods in Medicine ◽

10.1155/2019/2173638 ◽

2019 ◽

Vol 2019 ◽

pp. 1-8 ◽

Cited By ~ 8

Author(s):

Can Ateş ◽

Özlem Kaymaz ◽

H. Emre Kale ◽

Mustafa Agah Tekindal

Keyword(s):

Normal Distribution ◽

Type I Error ◽

Error Rates ◽

Type I ◽

Test Statistics ◽

Test Statistic ◽

Heterogeneous Variance ◽

Heterogeneous Variances ◽

Trace Test ◽

Wilks Lambda

In this study, we investigate how Wilks’ lambda, Pillai’s trace, Hotelling’s trace, and Roy’s largest root test statistics can be affected when the normal and homogeneous variance assumptions of the MANOVA method are violated. In other words, in these cases, the robustness of the tests is examined. For this purpose, a simulation study is conducted in different scenarios. In different variable numbers and different sample sizes, considering the group variances are homogeneous σ12=σ22=⋯=σg2 and heterogeneous (increasing) σ12<σ22<⋯<σg2, random numbers are generated from Gamma(4-4-4; 0.5), Gamma(4-9-36; 0.5), Student’s t(2), and Normal(0; 1) distributions. Furthermore, the number of observations in the groups being balanced and unbalanced is also taken into account. After 10000 repetitions, type-I error values are calculated for each test for α = 0.05. In the Gamma distribution, Pillai’s trace test statistic gives more robust results in the case of homogeneous and heterogeneous variances for 2 variables, and in the case of 3 variables, Roy’s largest root test statistic gives more robust results in balanced samples and Pillai’s trace test statistic in unbalanced samples. In Student’s t distribution, Pillai’s trace test statistic gives more robust results in the case of homogeneous variance and Wilks’ lambda test statistic in the case of heterogeneous variance. In the normal distribution, in the case of homogeneous variance for 2 variables, Roy’s largest root test statistic gives relatively more robust results and Wilks’ lambda test statistic for 3 variables. Also in the case of heterogeneous variance for 2 and 3 variables, Roy’s largest root test statistic gives robust results in the normal distribution. The test statistics used with MANOVA are affected by the violation of homogeneity of covariance matrices and normality assumptions particularly from unbalanced number of observations.

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Investigating the Behaviors of M2 and RMSEA2 in Fitting a Unidimensional Model to Multidimensional Data

Applied Psychological Measurement ◽

10.1177/0146621617710464 ◽

2017 ◽

Vol 41 (8) ◽

pp. 632-644

Author(s):

Jie Xu ◽

Insu Paek ◽

Yan Xia

Keyword(s):

Goodness Of Fit ◽

Type I Error ◽

Error Rates ◽

Multidimensional Data ◽

Ratio Test ◽

Type I ◽

Limited Information ◽

Test Statistic ◽

Goodness Of Fit Test ◽

Goodness Of Fit Tests

It has been widely known that the Type I error rates of goodness-of-fit tests using full information test statistics, such as Pearson’s test statistic χ2 and the likelihood ratio test statistic G2, are problematic when data are sparse. Under such conditions, the limited information goodness-of-fit test statistic M2 is recommended in model fit assessment for models with binary response data. A simulation study was conducted to investigate the power and Type I error rate of M2 in fitting unidimensional models to many different types of multidimensional data. As an additional interest, the behavior of RMSEA2 was also examined, which is the root mean square error approximation (RMSEA) based on M2. Findings from the current study showed that M2 and RMSEA2 are sensitive in detecting the misfits due to varying slope parameters, the bifactor structure, and the partially (or completely) simple structure for multidimensional data, but not the misfits due to the within-item multidimensional structures.

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Challenges and Opportunities in the Estimation of Dynamic Models

Organizational Research Methods ◽

10.1177/1094428119842638 ◽

2019 ◽

Vol 23 (4) ◽

pp. 595-619 ◽

Cited By ~ 1

Author(s):

Ran Xu ◽

Richard P. DeShon ◽

Christopher R. Dishop

Keyword(s):

Linear Model ◽

Multilevel Models ◽

Type I Error ◽

Generalized Method Of Moments ◽

Hierarchical Linear Model ◽

Error Rates ◽

Equation Modeling ◽

Type I ◽

Type I Error Rates ◽

Organizational Science

Interest in modeling longitudinal processes is increasing rapidly in organizational science. Organizational scholars often employ multilevel or hierarchical linear models (HLMs) to study such processes given that longitudinal data in organizational science typically consist of observations over a relatively small number of time intervals ( T) nested within a relatively large number of units ( N; e.g., people, teams, organizations). In this paper, we first distinguish change and dynamics as common research foci when modeling longitudinal processes and then demonstrate that a unique set of inferential hazards exists when investigating change or dynamics using multilevel models. Specifically, multilevel models that include one or more time-lagged values of the dependent variable as predictors often result in substantially biased estimates of the model parameters, inflated Type I error rates, and ultimately inaccurate inference. Using Monte Carlo simulations, we investigate the bias and Type I error rates for the standard centered/uncentered hierarchical linear model (HLM) and compare them with two alternative estimation methods: the Bollen and Brand structural equation modeling (SEM) approach and the Arrelano and Bond generalized method of moments using instrumental variables (GMM-IV) approach. We find that the commonly applied hierarchical linear model performs poorly, whereas the SEM and GMM-IV approaches generally perform well, with the SEM approach yielding slightly better performance in small samples with large autoregressive effects. We recommend the Bollen and Brand SEM approach for general use when studying change or dynamics in organizational science.

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One-Sided Nonparametric Tests for Ordinal Data

Perceptual and Motor Skills ◽

10.2466/pms.101.2.510-514 ◽

2005 ◽

Vol 101 (2) ◽

pp. 510-514 ◽

Cited By ~ 1

Author(s):

Markus Neuhäuser

Keyword(s):

Ordinal Data ◽

Type I Error ◽

Nonparametric Test ◽

Psychological Research ◽

Nonparametric Tests ◽

Wilcoxon Test ◽

Type I ◽

The Novel ◽

Test Statistic ◽

Wilcoxon Rank Sum Test

Baumgartner, Weiß, and Schindler (1998) introduced a novel nonparametric test for the two-sample comparison that is superior to commonly used tests such as the Wilcoxon rank-sum test. A modification of the novel test statistic can be used for one-sided comparisons based on ordinal data. Such comparisons frequently occur in psychological research, and the Wilcoxon test is often recommended for their analysis. Here, the two tests were compared in a simulation study. According to this study the tests have a similar type I error rate, but the modified Baumgartner-Weiß-Schindler test is more powerful than the Wilcoxon test.

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A Maximum Test for Scale: Type I Error Rates and Power

Journal of Educational and Behavioral Statistics ◽

10.3102/10769986020001027 ◽

1995 ◽

Vol 20 (1) ◽

pp. 27-39 ◽

Cited By ~ 2

Author(s):

James Algina ◽

R. Clifford Blair ◽

William T. Coombs

Keyword(s):

Type I Error ◽

Error Rates ◽

Type I ◽

Test Statistics ◽

Test Statistic ◽

Nominal Level ◽

Scale Type ◽

Type I Error Rates ◽

Maximum Test ◽

Study Type

A maximum test in which the test statistic is the more extreme of the Brown-Forsythe and O’Brien’s test statistics is developed. Estimated Type I error rates and power are presented for the Brown-Forsythe test, O’Brien’s test, and the maximum test. For the conditions included in the study, Type I error rates for the maximum test are near the nominal level. In all conditions, the power of the maximum test tended to be equal to or greater than that of the test—O’Brien or Brown-Forsythe—that had the larger power.

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