Nonparametric Phase-II monitoring for detecting monotone trend based on inverse sampling

Recently, Mukherjee and Bandyopadhyay (J Stat Plan Inference, 2011, doi:10.1016/j.jspi.2011.02.017) introduced some partially sequential tests for detecting liner trend among the incoming series of observations when a training sample is available a-priori. Their work is very useful in econometric or environmental monitoring under certain situations. The present work is intended for generalization of their tests for any monotone trend. We develop two nonparametric tests for the identity of some unknown univariate continuous distribution functions against monotone or unidirectional trend in location. One of these two tests is based on usual ranks and the other is based on sequential ranks. These are typical nonparametric tests for monitoring structural changes. Performance of the two tests are compared using asymptotic studies as well as through some numerical results based on Monte-Carlo simulations. An illustration is offered using a real data on monthly production of certain beverage.     

Semi-Sequential One-Shot Monitoring of Small Disorders With Controlled Type I Error Rate

We propose a simple two-stage monitoring rule for detecting small disorders in a two-sample location problem. The proposed rule is based on ranks and hence is nonparametric in nature. In the first stage, we use a sequential monitoring scheme to decide the necessity of employing a location test at some point of time. If there is urgency, we simply use a two-sample Wilcoxon rank sum test in the second stage. This leads to a semi sequential one-shot monitoring procedure. We study some asymptotic performance of the proposed rule. We also present some numerical findings obtained through Monte Carlo studies. The proposed rule meets the challenge of controlling type I error rate in sequential monitoring of an incoming series of observations.     

Generalized sequential ranks and tests of randomness

Generalized sequential ranks are defined and are proved to be independent and uniformly distributed under the hypothesis of randomness. By comparison with a Spear-MAsr-type statistic it is shown that in certain cases the test based on the sum of all sequential ranks is an asymptotically optimum test for trend against logistic regression. The test is equivalent to tests proposed by Jonckheere and Terpstra and has a high efficiency when compared with the optimal parametric test for normal regression alternatives.     

Rank-based multiple change-point detection

Abstract

A nonparametric procedure is proposed to estimate multiple change-points of location changes in a univariate data sequence by using ranks instead of the raw data. While existing rank-based multiple change-point detection methods are mostly based on sequential tests, we treat it as a model selection problem. We derive the corresponding Schwarz’s information criterion for rank-statistics, theoretically prove the consistency of the change-point estimator and use a pruned dynamic programing algorithm to achieve the change-point estimator. Simulation studies show our method’s robustness, effectiveness and efficiency in detecting mean-changes. We also apply the method to a gene dataset as an illustration.     

Simultaneous tests for patterned recognition using nonparametric partially sequential procedure

In the present paper we introduce a partially sequential sampling procedure to develop a nonparametric method for simultaneous testing. Our work, as in [U. Bandyopadhyay, A. Mukherjee, B. Purkait, Nonparametric partial sequential tests for patterned alternatives in multi-sample problems, Sequential Analysis 26 (4) (2007) 443–466], is motivated by an interesting investigation related to arsenic contamination in ground water. Here we incorporate the idea of multiple hypotheses testing as in [Y. Benjamini, T. Hochberg, Controlling the false discovery rate: A practical and powerful approach to multiple testing, Journal of Royal Statistical Society B 85 (1995) 289–300] in a typical way. We present some Monte Carlo studies related to the proposed procedure. We observe that the proposed sampling design minimizes the expected sample sizes in different situations. The procedure as a whole effectively describes the testing under dual pattern alternatives. We indicate in brief some large sample situations. We also present detailed analysis of a geological field survey data.     

Controlling Type-I Error Rate in Monitoring Structural Changes Using Partially Sequential Procedures

In the present article, we develop some asymptotically power on partially sequential nonparametric tests for monitoring structural changes. Our test procedures are based on Wilcoxon score. We use the idea of curved stopping boundaries. We derive some exact results and perform simulation studies to provide various properties of the tests. We see that one of the proposed procedures significantly controls the Type I error rate. This procedure may be very effective for fluctuation monitoring. We illustrate the procedures by using real life data from the stock market.     

On group sequential tests for data in unequally sized groups and with unknown variance

We present a method to generalise the scope of application of group sequential tests designed for equally sized groups of normal observations with known variance. Preserving the significance levels against which standardised statistics are compared leads to tests for unequally grouped data which maintain Type I error probabilities to a high degree of accuracy. The same approach can be followed when observations have unknown variance by setting critical values for Studentised statistics at percentiles of the appropriate t-distributions. This significance level approach is equally applicable to group sequential one-sided tests and two-sided tests, possibly with early stopping permitted to accept the null hypothesis. In applications to equivalence testing, tests are required to maintain a specified power, rather than Type I error rate: such tests can be constructed by defining the standardised test statistics used in the significance level approach with respect to appropriately chosen hypotheses.     

Two non-parametric, analysis-of-means-type tests for homogeneity of variances

After a brief review of the literature, two non-parametric tests for homogeneity of variances are presented. The first test is based on the analysis of means for ranks, which is a non-parametric version of the analysis of means (ANOM) that uses ranks as input for an ANOM test. The second test uses inverse normal scores of the ranks of scale transformations of the observations as input to the ANOM. Both homogeneity of variances tests can be presented in a graphical form, which makes it easy for practitioners to assess the practical and the statistical significance. A Monte Carlo study is used to show that these tests have power comparable with that of well-known robust tests for homogeneity of variances.     

Nonparametric multiple testing procedures

Assume that we have n_i independent observations from each of k independent populations. Each population has the same distribution except for a translation parameter. We are interested in specific pairwise differences of the parameters in various settings, such as treatment vs. control, change point or all pairwise differences. We propose new multiple testing procedures for the pairwise differences. The new procedures are based on ranks and they have desirable practical properties not shared by existing procedures. These include tests that satisfy the interval property. Furthermore, the test method provides an interval that serves as an estimate of the difference in the parameters of interest.     

Concordance of Complete or Right-Censored Rankings Based on Spearman's Footrule

A right-censored ranking is what results when a judge ranks only the “top K” of M objects. Complete uncensored rankings constitute a special case. We present two measures of concordance among the rankings of N ≥ 2 such judges, both based on Spearman's footrule. One measure is unweighted, while the other gives greatest weight to the first rank, less to the second, and so on. We consider methods for calculating or estimating the P-values of the corresponding tests of the hypothesis of random ranking.     

Nonparametric tests of hypotheses for umbrella alternatives

The author proposes a general method for constructing nonparametric tests of hypotheses for umbrella alternatives. Such alternatives are relevant when the treatment effect changes in direction after reaching a peak. The author's class of tests is based on the ranks of the observations. His general approach consists of defining two sets of rankings: the first is induced by the alternative and the other by the data itself. His test statistic measures the distance between the two sets. The author determines the asymptotic distribution for some special cases of distances under both the null and the alternative hypothesis when the location of the peak is known or unknown. He shows the good power of his tests through a limited simulation study     

A non-parametric maximum test for the Behrens–Fisher problem

Non-normality and heteroscedasticity are common in applications. For the comparison of two samples in the non-parametric Behrens–Fisher problem, different tests have been proposed, but no single test can be recommended for all situations. Here, we propose combining two tests, the Welch t test based on ranks and the Brunner–Munzel test, within a maximum test. Simulation studies indicate that this maximum test, performed as a permutation test, controls the type I error rate and stabilizes the power. That is, it has good power characteristics for a variety of distributions, and also for unbalanced sample sizes. Compared to the single tests, the maximum test shows acceptable type I error control.     

Schätzungen in Stochastischen Differenzen-und Differentialgleiehungssystemen 2

Some ways of Studentizing the parametric c-sample tests (c≧2) for location are examined and their asymptotic properties established. A way of Studentizing the c-sample Puni test based on the ranks of the observations is proposed. The resulting test is shown to be asymptotically valid and consistent for a reasonable class of alternatives     

Closed testing procedures for all pairwise comparisons in a randomized block design

We consider multiple comparison test procedures among treatment effects in a randomized block design. We propose closed testing procedures based on maximum values of some two-sample t test statistics and based on F test statistics. It is shown that the proposed procedures are more powerful than single-step procedures and the REGW (Ryan/Einot–Gabriel/Welsch)-type tests. Next, we consider the randomized block design under simple ordered restrictions of treatment effects. We propose closed testing procedures based on maximum values of two-sample one-sided t test statistics and based on Batholomew’s statistics for all pairwise comparisons of treatment effects. Although single-step multiple comparison procedures are utilized in general, the power of these procedures is low for a large number of groups. The closed testing procedures stated in the present article are more powerful than the single-step procedures. Simulation studies are performed under the null hypothesis and some alternative hypotheses. In this studies, the proposed procedures show a good performance.     

A Monte Carlo study of rank tests for block designs

Tests based on ranks and the F-test are compared for block designs with n observations per block-treatment combination. Com-parisons are made on level of significance and on power. Rank tests examined include the Friedman as well as those using aligned ranks, weighted ranks, and the rank transformation. It is seen that the performance of these tests in relationship to each other depends on sample size, distribution of the random error term, and the severity of the block effects.     

A Comparison of the Cost-efficiencies of the Sequential, Group-sequential, and Variable-sample-size-sequential Probability Ratio Tests

Wald and Wolfowitz (1948) have shown that the Sequential Probability Ratio Test (SPRT) for deciding between two simple hypotheses is, under very restrictive conditions, optimal in three attractive senses. First, it can be a Bayes-optimal rule. Second, of all level α tests having the same power, the test with the smallest joint-expected number of observations is the SPRT, where this expectation is taken jointly with respect to both data and prior over the two hypotheses. Third, the level α test needing the fewest conditional-expected number of observat ions is the SPRT, where this expectation is now taken with respect to the data conditional on either hypothesis being true. Principal among the strong restrictions is that sampling can proceed only in a one-at-a-time manner. In this paper, we relax some of the conditions and show that there are sequential procedures that strictly dominate the SPRT in all three senses. We conclude that the third type of optimality occurs rarely and that decision-makers are better served by looking for sequential procedures that possess the first two types of optimality. By relaxing the one-at-a-time sampling restriction, we obtain optimal (in the first two senses) variable-s ample-size- sequential probability ratio tests.     

On a class of partially sequential two-sample test procedures for multivariate continuous data

In a two-sample testing problem, sometimes one of the sample observations are difficult and/or costlier to collect compared to the other one. Also, it may be the situation that sample observations from one of the populations have been previously collected and for operational advantages we do not wish to collect any more observations from the second population that are necessary for reaching a decision. Partially sequential technique is found to be very useful in such situations. The technique gained its popularity in statistics literature due to its very nature of capitalizing the best aspects of both fixed and sequential procedures. The literature is enriched with various types of partially sequential techniques useable under different types of data set-up. Nonetheless, there is no mention of multivariate data framework in this context, although very common in practice. The present paper aims at developing a class of partially sequential nonparametric test procedures for two-sample multivariate continuous data. For this we suggest a suitable stopping rule adopting inverse sampling technique and propose a class of test statistics based on the samples drawn using the suggested sampling scheme. Various asymptotic properties of the proposed tests are explored. An extensive simulation study is also performed to study the asymptotic performance of the tests. Finally the benefit of the proposed test procedure is demonstrated with an application to a real-life data on liver disease.     

Effect size for comparing two or more normal distributions based on maximal contrasts in outcomes

Effect size is a concept that can be especially useful in bioequivalence and studies designed to find important and not just statistically significant differences among responses to treatments based on independent random samples. We develop and explore a new effect size related to a maximal superiority ordering for assessing the separation among two or more normal distributions, possibly having different means and different variances. Confidence intervals and tests of hypothesis for this effect size are developed using a p value obtained by averaging over a distribution on variances. Since there is almost always some difference among treatments, instead of the usual hypothesis test of exactly no effect, researchers should consider testing that an appropriate effect size has at least, or at most, some meaningful magnitude, when one is available, possibly established using the framework developed here. A simulation study of type I error rate, power and interval length is presented. R-code for constructing the confidence intervals and carrying out the tests here can be downloaded from Author’s website.     

Permutation tests for between-unit fixed effects in multivariate generalized linear mixed models

A permutation testing approach in multivariate mixed models is presented. The solutions proposed allow for testing between-unit effect; they are exact under some assumptions, while approximated in the more general case. The classes of models comprised by this approach include generalized linear models, vector generalized additive models and other nonparametric models based on smoothing. Moreover it does not assume observations of different units to have the same distribution. The extensions to a multivariate framework are presented and discussed. The proposed multivariate tests exploit the dependence among variables, hence increasing the power with respect to other standard solutions (e.g. Bonferroni correction) which combine many univariate tests in an overall one. Examples are given of two applications to real data from psychological and ecological studies; a simulation study provides some insight into the unbiasedness of the tests and their power. The methods were implemented in the R package flip, freely available on CRAN.     

Šidák-type tests for the two-sample problem based on precedence and exceedance statistics

This paper deals with a class of nonparametric two-sample tests for ordered alternatives. The test statistics proposed are based on the number of observations from one sample that precede or exceed a threshold specified by the other sample, and they are extensions of ?idák's test. We derive their exact null distributions and also discuss a large-sample approximation. We then study their power properties exactly against the Lehmann alternative and make some comparative comments. Finally, we present an example to illustrate the proposed tests.