Choosing appropriate covariance matrices in a nonparametric analysis of factorials in block designs

The standard nonparametric, rank-based approach to the analysis of dependent data from factorial designs is based on an estimated unstructured (UN) variance–covariance matrix, but the large number of variance–covariance terms in many designs can seriously affect test performance. In a simulation study for a factorial arranged in blocks, we compared estimates of type-I error probability and power based on the UN structure with the estimates obtained with a more parsimonious heterogeneous-compound-symmetry structure (CSH). Although tests based on the UN structure were anti-conservative with small number of factor levels, especially with four or six blocks, they became conservative at higher number of factor levels. Tests based on the CSH structure were anti-conservative, and results did not depend on the number of factor levels. When both tests were anti-conservative, tests based on the CSH structure were less so. Although use of the CSH structure is concluded to be more suitable than use of the UN structure for the small number of blocks typical in agricultural experiments, results suggest that further improvement of test statistics is needed for such situations.     

Efficiency of tests based on weighted rankings for equality of treatment effects in a complete randomized blocks layout

Quade (1972,1979) proposed a family of nonparametric tests based on a method of weighted within-block rankings, for testing the hypothesis of no treatment effects in a complete randomized blocks layout.

In this paper we obtain an expression for the Pitman asymp-totic relative efficiency of these tests with respect to the Friedman test.     

A monte carlo study of the friedman and conover tests in the single-factor repeated measures design

A Monte Carlo study was used to compare the Type I error rates and power of two nonparametric tests against the F test for the single-factor repeated measures model. The performance of the nonparametric Friedman and Conover tests was investigated for different distributions, numbers of blocks and numbers of repeated measures. The results indicated that the type of the distribution has little effect on the ability of the Friedman and Conover tests to control Type error rates. For power, the Friedman and Conover tests tended to agree in rejecting the same false hyporhesis when the design consisted of three repeated measures. However, the Conover test was more powerful than the Friedman test when the number of repeated measures was 4 or 5. Still, the F test is recommended for the single-factor repeated measures model because of its robustness to non-normality and its good power across a range of conditions.     

Comparing Nonparametric Tests of Equality of Means for Randomized Block Designs

A number of nonparametric tests are compared empirically for a randomized block layout. We assess tests appropriate for when the data are not consistent with normality or when outliers invalidate traditional analysis of variance (ANOVA) tests. The objective is to assess, within this setting, tests that use ranks within blocks, the rank transform procedure that ranks the complete sample and continuous analogs of the Cochran–Mantel–Haenszel tests. The usual linear model is assumed, and our primary foci are tests of equality of means and component tests that assess linear and quadratic trends in the means. These tests include the traditional Page and Friedman tests. We conclude that the rank transform tests have competitive power and warrant greater use than is currently apparent.     

Efficiency of tests based on weighted rankings for randomized blocks when the alternatives have an a prior1 ordering

Salama and Quade (1981) proposed a family of nonparametric tests based on a method of weighted within-block rankings, for testing the hypothesis of no treatment effects against a postulated ranking of the treatments in a complete randomized blocks layout.

These tests and others are compared with respect to asymptotic efficiency.     

An empirical study of the type I error rate and power for some selected normal-theory and nonparametric tests of the independence of two sets of variables

Normal-theory tests of the hypothesis of no relationship among two sets of variables require assumptions of independence, hamoscedasticity, and normality. If, however, the assumption of normality is not tenable, there are few guidelines for properly using these tests. Historically, the lack of a comprehensive hypothesis-testing framework in the nonparametric case has provided few alternatives to normal-theory procedures. Fortunately, this situation has changed with the introduction of nonparametric, general linear model-based tests that can be used with existing computing packages. Multivariate-nonparametric tests due to Puri and Sen (1969, 1971, 1985) and Conover and Iman (1981) are outlined, and the results of a simulation study of the performance of three nonparametric and one normal-theory test of the hypothesis of no relationship among two sets of variables are presented. These results suggest that multivariate-nonparametric tests should be considered for a variety of data conditions. especially heavy-tailed and badly skewed data for small samples and a large number of variates.     

Evaluation of weighted rankings using expected significance level

The use of weighted rankings to analyze complete blocks designs (Quade, 1979) is a practical way of recovering between-block information. A family of old and new test statistics can be generated by this procedure. Selection among these statistics and comparison with parametric and nonparametric competitors are based on expected significance level [ESL] in small designs (3 to 6 blocks, 3 to 5 treatments).     

Testing for the Equality of Two Nonparametric Regression Curves with Long Memory Errors

This article discusses the problem of testing the equality of two nonparametric regression functions against two-sided alternatives for uniform design on [0,1] with long memory moving average errors. The standard deviations and the long memory parameters are possibly different for the two errors. The article adapts the partial sum process idea used in the independent observations settings to construct the tests and derives their asymptotic null distributions. The article also shows that these tests are consistent for general alternatives and obtains their limiting distributions under a sequence of local alternatives. Since the limiting null distributions of these tests are unknown, we first conducted a Monte Carlo simulation study to obtain a few selected critical values of the proposed tests. Then based on these critical values, another Monte Carlo simulation is conducted to study the finite sample level and power behavior of these tests at some alternatives. The article also contains a simulation study that assesses the effect of estimating the nonparametric regression function on an estimate of the long memory parameter of the errors. It is observed that the estimate based on direct observations is generally preferable over the one based on the estimated nonparametric residuals.     

Power comparison of data depth-based nonparametric tests for testing equality of locations

In the recent years, the notion of data depth has been used in nonparametric multivariate data analysis since it gives natural ‘centre-outward’ ordering of multivariate data points with respect to the given data cloud. In the literature, various nonparametric tests are developed for testing equality of location of two multivariate distributions based on data depth. Here, we define two nonparametric tests based on two different test statistic for testing equality of locations of two multivariate distributions. In the present work, we compare the performance of these tests with the tests developed by Li and Liu [New nonparametric tests of multivariate locations and scales using data depth. Statist Sci. 2004;(1):686–696] for testing equality of locations of two multivariate distributions. Comparison in terms of power is done for multivariate symmetric and skewed distributions using simulation for three popular depth functions. Application of tests to real life data is provided. Conclusion and recommendations are also provided.     

Performance of nonparametric multiple comparison tests under heteroscedasticity,dependency, and skewed error distribution

In this article, an extensive Monte Carlo simulation study is conducted to evaluate and compare nonparametric multiple comparison tests under violations of classical analysis of variance assumptions. Simulation space of the Monte Carlo study is composed of 288 different combinations of balanced and unbalanced sample sizes, number of groups, treatment effects, various levels of heterogeneity of variances, dependence between subgroup levels, and skewed error distributions under the single factor experimental design. By this large simulation space, we present a detailed analysis of effects of the violations of assumptions on the performance of nonparametric multiple comparison tests in terms of three error and four power measures. Observations of this study are beneficial to decide the optimal nonparametric test according to requirements and conditions of undertaken experiments. When some of the assumptions of analysis of variance are violated and number of groups is small, use of stepwise Steel-Dwass procedure with Holm's approach is appropriate to control type I error at a desired level. Dunn's method should be employed for greater number of groups. When subgroups are unbalanced and number of groups is small, Nemenyi's procedure with Duncan's approach produces high power values. Conover's procedure successfully provides high power values with a small number of unbalanced groups or with a greater number of balanced or unbalanced groups. At the same time, Conover's procedure is unable to control type I error rates.     

Multivariate nonparametric methods in two-way balanced designs: performances and limitations in small samples

Investigations of multivariate population are pretty common in applied researches, and the two-way crossed factorial design is a common design used at the exploratory phase in industrial applications. When assumptions such as multivariate normality and covariance homogeneity are violated, the conventional wisdom is to resort to nonparametric tests for hypotheses testing. In this paper we compare the performances, and in particular the power, of some nonparametric and semi-parametric methods that have been developed in recent years. Specifically, we examined resampling methods and robust versions of classical multivariate analysis of variance (MANOVA) tests. In a simulation study, we generate data sets with different configurations of factor''s effect, number of replicates, number of response variables under null hypothesis, and number of response variables under alternative hypothesis. The objective is to elicit practical advice and guides to practitioners regarding the sensitivity of the tests in the various configurations, the tradeoff between power and type I error, the strategic impact of increasing number of response variables, and the favourable performance of one test when the alternative is sparse. A real case study from an industrial engineering experiment in thermoformed packaging production is used to compare and illustrate the application of the various methods.     

NONPARAMETRIC ESTIMATION OF GENEWISE VARIANCE FOR MICROARRAY DATA

Estimation of genewise variance arises from two important applications in microarray data analysis: selecting significantly differentially expressed genes and validation tests for normalization of microarray data. We approach the problem by introducing a two-way nonparametric model, which is an extension of the famous Neyman-Scott model and is applicable beyond microarray data. The problem itself poses interesting challenges because the number of nuisance parameters is proportional to the sample size and it is not obvious how the variance function can be estimated when measurements are correlated. In such a high-dimensional nonparametric problem, we proposed two novel nonparametric estimators for genewise variance function and semiparametric estimators for measurement correlation, via solving a system of nonlinear equations. Their asymptotic normality is established. The finite sample property is demonstrated by simulation studies. The estimators also improve the power of the tests for detecting statistically differentially expressed genes. The methodology is illustrated by the data from MicroArray Quality Control (MAQC) project.     

Nonparametric tests for multivariate multi-sample locations based on data depth

Several nonparametric tests for multivariate multi-sample location problem are proposed in this paper. These tests are based on the notion of data depth, which is used to measure the centrality/outlyingness of a given point with respect to a given distribution or a data cloud. Proposed tests are completely nonparametric and implemented through the idea of permutation tests. Performance of the proposed tests is compared with existing parametric test and nonparametric test based on data depth. An extensive simulation study reveals that proposed tests are superior to the existing tests based on data depth with regard to power. Illustrations with real data are provided.     

Prediction bands for the EDF of a future sample

We develop both nonparametric and parametric methods for obtaining prediction bands for the empirical distribution function (EDF) of a future sample. These methods yield simultaneous prediction intervals for all order statistics of the future sample, and they also correspond to tests for the two-sample problem. The nonparametric prediction bands correspond to the two-sample Kolmogorov-Smirnov test and related nonparametric tests, but the parametric prediction bands correspond to entirely new parametric two-sample tests. The parametric prediction bands tend to outperform the nonparametric bands when the parametric assumptions hold, but they may have true coverage probabilities well below their nominal levels when the parametric assumptions fail. A new computational algorithm is used to obtain critical values in the nonparametric case.     

Paradoxes in nonparametric tests

This paper is a continuation of one (1992) in which the author studied the paradoxes that can arise when a nonparametric statistical test is used to give an ordering of k samples and the subsets of those samples. This article characterizes the projection paradoxes that can occur when using contingency tables, complete block designs, and tests of dichotomous behaviour of several samples. This is done by examining the “dictionaries” of possible orderings of each of these procedures. Specifically, it is shown that contingency tables and complete block designs, like the Kruskal-Wallis nonparametric test on k samples, minimize the number and kinds of projection paradoxes that can occur; however, using a test of dichotomous behaviour of several samples does not. An analysis is given of two procedures used to determine the ordering of a pair of samples from a set of k samples. It is shown that these two procedures may not have anything in common.     

Tests for Differences in Dispersion Based on Quantiles

It is commonly known that the validity of the F test for testing differences in variability is highly sensitive to the assumption that the population distributions are normal. Hence there is a need for nonparametric tests that do not rely on the assumption of normal population distributions. Several nonparametric tests for testing differences in dispersion have been developed in the past 40 years. These include Mood's test, Klotz's test, and the Siegel-Tukey test. Unfortunately, many of these tests do not have a natural or easily calculated measure of dispersion associated with them. This article introduces a test for differences in dispersion based on quantiles that is easy to compute and readily comprehended by the casual user of statistics.     

Comparison of Nonparametric and Parametric Methods in Repeated Measures Designs - A Simulation Study

This study investigates the performance of parametric and nonparametric tests to analyze repeated measures designs. Both multivariate normal and exponential distributions were simulated for varying values of the correlation and ten or twenty subjects within each cell. For multivariate normal distributions, the type I error rates were lower than the usual 0.05 level for nonparametric tests, whereas the parametric tests without the Greenhouse-Geisser or the Huynh-Feldt adjustment produced slightly higher type I error rates. Type I error rates for nonparametric tests, for multivariate exponential distributions, were more stable than parametric, Greenhouse-Geisser or Huynh-Feldt adjusted tests. For ten subjects within each cell, the parametric tests were more powerful than nonparametric tests. For twenty subjects per cell, the power of the nonparametric and parametric tests was comparable.     

Multivariate goodness-of-fit tests based on statistically equivalent blocks

Various nonparametric procedures are known for the goodness-of-fit test in the univariate case. The distribution-free nature of these procedures does not extend to the multivariate case. In this paper, we consider an application of the theory of statistically equivalent blocks(SEB)to obtain distribution-free procedures for the multivariate case. The sample values are transformed to random variables which are distributed as sample spacings from a uniform distribution on [0, 1], under the null hypothesis. Various test statistics are known, based on the spacings, which are used for testing uniformity in the univariate case. Any of these statistics can be used in the multivariate situation, based on the spacings generated from the SEB. This paper gives an expository development of the theory of SEB and a review of tests for goodness-of-fit, based on sample spacings. To show an application of the SEB, we consider a test of bivariate normality.     

Nonparametric methods in multivariate factorial designs for large number of factor levels

We propose different multivariate nonparametric tests for factorial designs and derive their asymptotic distribution for the situation where the number of replications is limited, whereas the number of treatments goes to infinity (large a, small n case). The tests are based on separate rankings for the different variables, and they are therefore invariant under separate monotone transformations of the individual variables.     

Testing independence based on Bernstein empirical copula and copula density

In this paper we provide three nonparametric tests of independence between continuous random variables based on the Bernstein copula distribution function and the Bernstein copula density function. The first test is constructed based on a Cramér-von Mises divergence-type functional based on the empirical Bernstein copula process. The two other tests are based on the Bernstein copula density and use Cramér-von Mises and Kullback–Leibler divergence-type functionals, respectively. Furthermore, we study the asymptotic null distribution of each of these test statistics. Finally, we consider a Monte Carlo experiment to investigate the performance of our tests. In particular we examine their size and power which we compare with those of the classical nonparametric tests that are based on the empirical distribution function.