Rank-based test procedures for interaction in the two-way layout with one observation per cell

New aligned-rank test procedures for the composite null hypothesis of no interaction effects (without placing restrictions on the two main effects) against appropriate composite general alternatives are developed for the standard two-way layout with a single observation per cell. Relative power performances of the two new aligned-rank procedures and existing tests due to Tukey (1949) and to de Kroon & van der Laan (1981) are examined via Monte Carlo simulation. Extensive power studies conducted on the 5 × 6 and 5 × 9 two-way layouts with one observation per cell show superior performance of the new procedures for a variety of interaction effects. Simulated critical values for the new procedures are provided in settings where the number of levels for each of the factors is between 3 and 9, inclusive.     

A nonparametric test for interaction in two‐way layouts

The authors present a new nonparametric approach to test for interaction in two‐way layouts. Based on the concept of composite linear rank statistics, they combine the correlated row and column ranking information to construct the test statistic. They determine the limiting distributions of the proposed test statistic under the null hypothesis and Pitman alternatives. They also propose consistent estimators for the limiting covariance matrices associated with the test. They illustrate the application of their test in practical settings using a microarray data set.     

A method for testing interaction in unreplicated two-way tables: using all pairwise interaction contrasts

Several methods exist for testing interaction in unreplicated two-way layouts. Some are based on specifying a functional form for the interaction term and perform well provided that the functional form is appropriate. Other methods do not require such a functional form to be specified but only test for the presence of non-additivity and do not provide a suitable estimate of error variance for a non-additive model. This paper presents a method for testing for interaction in unreplicated two-way tables that is based on testing all pairwise interaction contrasts. This method (i) is easy to implement, (ii) does not assume a functional form for the interaction term, (iii) can find a sub-table of data which may be free from interaction and to base the estimate of unknown error variance, and (iv) can be used for incomplete two-way layouts. The proposed method is illustrated using examples and its power is investigated via simulation studies. Simulation results show that the proposed method is competitive with existing methods for testing for interaction in unreplicated two-way layouts.     

A DISTRIBUTION-FREE TEST FOR TWO-WAY LAYOUT1

Many nonparametric tests have been proposed for the hypothesis of no row (treatment) effect in a one-way layout design. Examples of such tests are Kruskal-Wallis H-test, Bhapkar's (1961) V-test and Deshpande's (1965) L-test. However not many tests are available for testing the same hypothesis in a two-way layout design without interaction. Perhaps the only “established” test is the one due to Friedman (1937). However, it applies to the case of one observation per cell only. In this paper, a new distribution-free test is proposed for the hypothesis of row effect in a two-way layout design. It applies to the case of several observations per cell, not necessarily equal. The asymptotic efficiency of the proposed test relative to other tests is studied.     

A general theory of hypothesis testing based on rankings

In nonparametric statistics, a hypothesis testing problem based on the ranks of the data gives rise to two separate permutation sets corresponding to the null and to the alternative hypothesis, respectively. A modification of Critchlow's unified approach to hypothesis testing is proposed. By defining the distance between permutation sets to be the average distance between pairs of permutations, one from each set, various test statistics are derived for the multi-sample location problem and the two-way layout. The asymptotic distributions of the test statistics are computed under both the null and alternative hypotheses. Some comparisons are made on the basis of the asymptotic relative efficiency.     

Hypothesis Testing Problems in an Unbalanced Longitudinal Ophthalmology Study

ABSTRACT

The analysis of clustered data in a longitudinal ophthalmology study is complicated by correlations between repeatedly measured visual outcomes of paired eyes in a participant and missing observations due to the loss of follow-up. In the present article we consider hypothesis testing problems in an ophthalmology study, where eligible eyes are randomized to two treatments (when two eyes of a participant are eligible, the paired eyes are assigned to different treatments), and vision function outcomes are repeatedly measured over time. A large sample-based nonparametric test statistic and a nonparametric Bootstrap test analog are proposed for testing an interaction effect of two factors and testing an effect of a eye-specific factor within a level of the other person-specific factor on visual function outcomes. Both test statistics allow for missing observations, correlations between repeatedly measured outcomes on individual eyes, and correlations between repeatedly measured outcomes on both eyes of each participant. A simulation study shows that these proposed test statistics maintain nominal significance levels approximately and comparable powers to each other, as well as higher powers than the naive test statistic ignoring correlations between repeated bilateral measurements of both eyes in the same person. For illustration, we apply the proposed test statistics to the changes of visual field defect score in the Advanced Glaucoma Intervention Study.     

A permutation test for multivariate data with grouped components

In this paper, we consider a nonparametric test procedure for multivariate data with grouped components under the two sample problem setting. For the construction of the test statistic, we use linear rank statistics which were derived by applying the likelihood ratio principle for each component. For the null distribution of the test statistic, we apply the permutation principle for small or moderate sample sizes and derive the limiting distribution for the large sample case. Also we illustrate our test procedure with an example and compare with other procedures through simulation study. Finally, we discuss some additional interesting features as concluding remarks.     

Rank-based analysis of repeated measures block designs

A rank-based inference is developed for repeated measures balanced incomplete block and randomized complete block designs using a suitable dispersion function. Asymptotic distributions of rank estimators are developed after establishing approximate linearity of the gradient vector of the dispersion function. Unlike available nonparametric procedures for those designs, estimation and testing are tied together. Three different test statistics are developed for testing the linear hypotheses. Friedman's (1937) statistic and Durbin's (1951) statistic are particular cases of one of the three proposed statistics. An estimate of a scale parameter which appears in the ARE expression as well as as in the variences and covariances of the rank estimators is discussed.     

Small sample power of multivariate nonparametric tests for monotone trend

Bhattacharyya and Kioiz (1966) propose two multivariate nonparametric tests for monotone trend, one involving coordinate-wise Mann statistics and the other, coordinate-wise Spearman statistics. Dietz and Killeen (1981) propose a different test statistic based on coordinate-wise Mann statistics. The Pitman asymptotic relative efficiency of all three tests with respect to a normal theory competitor equals the cube root of the efficiency of a multivariate signed rank test with respect to Hotelling's T2. In this article, the small sample power of the nonparametric tests, the normal theory test, and a Bonferroni approach involving coordinate-wise univariate Mann or Spearman tests is examined in a simulation study. The Mann statistic of Dietz and Killeen and the Spearman statistic of Bhattacharyya and Klotz are found to perform well under both null and alternative hypotheses     

Using -splines to test the linearity of partially linear models

The nonparametric component in a partially linear model is estimated by a linear combination of fixed-knot cubic B-splines with a second-order difference penalty on the adjacent B-spline coefficients. The resulting penalized least-squares estimator is used to construct two Wald-type spline-based test statistics for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, the first test statistic asymptotically has the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom, under the null hypothesis. The smoothing parameter is determined by specifying a value for the asymptotically expected value of the test statistic under the null hypothesis. When the number of knots is fixed and under the null hypothesis, the second test statistic asymptotically has a chi-squared distribution with K=q+2 degrees of freedom, where q is the number of knots used for estimation. The power performances of the two proposed tests are investigated via simulation experiments, and the practicality of the proposed methodology is illustrated using a real-life data set.     

Tests of hypotheses based on ranks in the general linear model

A unified approach is developed for testing hypotheses in the general linear model based on the ranks of the residuals. It complements the nonparametric estimation procedures recently reported in the literature. The testing and estimation procedures together provide a robust alternative to least squares. The methods are similar in spirit to least squares so that results are simple to interpret. Hypotheses concerning a subset of specified parameters can be tested, while the remaining parameters are treated as nuisance parameters. Asymptotically, the test statistic is shown to have a chi-square distribution under the null hypothesis. This result is then extended to cover a sequence of contiguous alternatives from which the Pitman efficacy is derived. The general application of the test requires the consistent estimation of a functional of the underlying distribution and one such estimate is furnished.     

A Jonckheere-Terpstra-type test for perfect ranking in balanced ranked set sampling

Many methods based on ranked set sampling (RSS) assume perfect ranking of the samples. Here, by using the data measured by a balanced RSS scheme, we propose a nonparametric test for the assumption of perfect ranking. The test statistic that we use formally corresponds to the Jonckheere-Terpstra-type test statistic. We show formal relations of the proposed test for perfect ranking to other methods proposed recently in the literature. Through an empirical power study, we demonstrate that the proposed method performs favorably compared to many of its competitors.     

Goodness-of-fit tests for one-shot device accelerated life testing data

In this article, we develop a formal goodness-of-fit testing procedure for one-shot device testing data, in which each observation in the sample is either left censored or right censored. Such data are also called current status data. We provide an algorithm for calculating the nonparametric maximum likelihood estimate (NPMLE) of the unknown lifetime distribution based on such data. Then, we consider four different test statistics that can be used for testing the goodness-of-fit of accelerated failure time (AFT) model by the use of samples of residuals: a chi-square-type statistic based on the difference between the empirical and expected numbers of failures at each inspection time; two other statistics based on the difference between the NPMLE of the lifetime distribution obtained from one-shot device testing data and the distribution specified under the null hypothesis; as a final statistic, we use White's idea of comparing two estimators of the Fisher Information (FI) to propose a test statistic. We then compare these tests in terms of power, and draw some conclusions. Finally, we present an example to illustrate the proposed tests.     

A non-parametric statistic for testing conditional heteroscedasticity for unobserved component models

When prediction intervals are constructed using unobserved component models (UCM), problems can arise due to the possible existence of components that may or may not be conditionally heteroscedastic. Accurate coverage depends on correctly identifying the source of the heteroscedasticity. Different proposals for testing heteroscedasticity have been applied to UCM; however, in most cases, these procedures are unable to identify the heteroscedastic component correctly. The main issue is that test statistics are affected by the presence of serial correlation, causing the distribution of the statistic under conditional homoscedasticity to remain unknown. We propose a nonparametric statistic for testing heteroscedasticity based on the well-known Wilcoxon''s rank statistic. We study the asymptotic validation of the statistic and examine bootstrap procedures for approximating its finite sample distribution. Simulation results show an improvement in the size of the homoscedasticity tests and a power that is clearly comparable with the best alternative in the literature. We also apply the test on real inflation data. Looking for the presence of a conditionally heteroscedastic effect on the error terms, we arrive at conclusions that almost all cases are different than those given by the alternative test statistics presented in the literature.     

A nonparametric procedure for the analysis of balanced crossover designs

The authors propose nonparametric tests for the hypothesis of no direct treatment effects, as well as for the hypothesis of no carryover effects, for balanced crossover designs in which the number of treatments equals the number of periods p, where p ≥ 3. They suppose that the design consists of n replications of balanced crossover designs, each formed by m Latin squares of order p. Their tests are permutation tests which are based on the n vectors of least squares estimators of the parameters of interest obtained from the n replications of the experiment. They obtain both the exact and limiting distribution of the test statistics, and they show that the tests have, asymptotically, the same power as the F‐ratio test.     

Balanced incomplete Latin square designs

Latin squares have been widely used to design an experiment where the blocking factors and treatment factors have the same number of levels. For some experiments, the size of blocks may be less than the number of treatments. Since not all the treatments can be compared within each block, a new class of designs called balanced incomplete Latin squares (BILS) is proposed. A general method for constructing BILS is proposed by an intelligent selection of certain cells from a complete Latin square via orthogonal Latin squares. The optimality of the proposed BILS designs is investigated. It is shown that the proposed transversal BILS designs are asymptotically optimal for all the row, column and treatment effects. The relative efficiencies of a delete-one-transversal BILS design with respect to the optimal designs for both cases are also derived; it is shown to be close to 100%, as the order becomes large.     

Conditional Score Tests for Heteroscedasticity in the Two-Way Error Components Model

This paper extends the one-way heteroskedasticity score test of Holly and Gardiol (2000, In: Krishnakumar, J, Ronchetti, E (Eds.), Panel Data Econometrics: Future Directions, North-Holland, Amsterdam, pp. 199–211) to two conditional Lagrange Multiplier (LM) tests of heteroskedasticity under contiguous alternatives within the two-way error components model framework. In each case, the derivation of Rao's efficient score statistics for testing heteroskedasticity is first obtained. Then, based on a specific set of assumptions, the asymptotic distribution of the score under contiguous alternatives is established. Finally, the expression for the score test statistic in the presence of heteroskedasticity and related asymptotic local powers of these score test statistics are derived and discussed.     

An adjusted friedman test for the nested design

As far as is known, no distribution-free test for treatment effects in the presence of nested column effects, is available, Therefore anadjusted Friedman-type test is derived, which is asymptotically the same as the Friedman-type test of Mack and Skillings (1980). If the assumption of commensurability holds, the adjustment is made by means of the renumbering of the cells within rows.     

Rank tests for patterned alternatives in two-way non-parametric analysis of variance

We consider the problem of testing against trend and umbrella alternatives, with known and unknown peak, in two-way layouts with fixed effects. We consider the non-parametric two-way layout ANOVA model of Akritas and Arnold (J. Amer. Statist. Assoc. 89 (1994) 336), and use the non-parametric formulation of patterned alternatives introduced by Akritas and Brunner (Research Developments in Probability and Statistics: Festschrift in honor of Madan L. Puri, VSP, Zeist, The Netherlands, 1996, pp. 277–288). The hypotheses of no main effects and of no simple effects are both considered. New rank test statistics are developed to specifically detect these types of alternatives. For main effects, we consider two types of statistics, one using weights similar to Hettmansperger and Norton (J. Amer. Statist. Assoc. 82 (1987) 292) and one with weights which maximize the asymptotic efficacy. For simple effects, we consider in detail only statistics to detect trend or umbrella patterns with known peaks, and refer to Callegari (Ph.D. Thesis, University of Padova, Italy) for a discussion about possible statistics for umbrella alternatives with unknown peaks. The null asymptotic distributions of the new statistics are derived. A number of simulation studies investigate their finite sample behaviors and compare the achieved alpha levels and power with some alternative procedures. An application to data used in a clinical study is presented to illustrate how to utilize some of the proposed tests for main effects.     

Testing for pairwise serial independence via the empirical distribution function

Built on Skaug and Tjøstheim's approach, this paper proposes a new test for serial independence by comparing the pairwise empirical distribution functions of a time series with the products of its marginals for various lags, where the number of lags increases with the sample size and different lags are assigned different weights. Typically, the more recent information receives a larger weight. The test has some appealing attributes. It is consistent against all pairwise dependences and is powerful against alternatives whose dependence decays to zero as the lag increases. Although the test statistic is a weighted sum of degenerate Cramér–von Mises statistics, it has a null asymptotic N (0, 1) distribution. The test statistic and its limit distribution are invariant to any order preserving transformation. The test applies to time series whose distributions can be discrete or continuous, with possibly infinite moments. Finally, the test statistic only involves ranking the observations and is computationally simple. It has the advantage of avoiding smoothed nonparametric estimation. A simulation experiment is conducted to study the finite sample performance of the proposed test in comparison with some related tests.