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.     

Analysis of Means Using Ranks

A nonparametric procedure, called the analysis of means using ranks (ANOMR), is proposed for testing the equality of several population means. The ANOMR procedure may be used graphically in the form of a Shewhart control chart and so has the advantage of pinpointing which population mean, if any, is significantly different from the others. Exact and asymptotic critical values are given for the implementation of ANOMR. Results from a Monte Carlo power study are presented which indicate that for light-tailed distributions such as the uniform and the normal, ANOMR is only slightly less powerful than the parametric competitive procedures based on analysis of variance and analysis of means. For heavy-tailed distributions such as the Cauchy, ANOMR is shown to provide greater power than the parametric procedures. The results also indicate that for both light and heavy-tailed distributions the use of the ANOMR test instead of the Kruskal-Wallis test leads to only a small loss of power for a range of alternatives.     

Analysis of means using ranks for the randomized complete block design

A nonparametric test procedure is proposed for the analysis of randomized complete block designs. Such a procedure may be carried out graphically in the form of a Shewhart control chart. Exact and asymptotic critical values are given for the implementation of the proposed procedure. A Monte Carlo study is made to compare the powers of the proposed procedure to those of analysis of variance, the analysis of means, and the Friedman procedures. Results of the study indicate that the proposed procedure has superior power performance when testing against slippage alternative hypotheses under heavy-tailed distributions such as the Cauchy distribution. However, when testing against symmetric alternatives under light-tailed distributions, the proposed procedure does not perform well     

Nonparametric multiple comparison procedures for unbalanced one-way factorial designs

In this paper, we present several nonparametric multiple comparison (MC) procedures for unbalanced one-way factorial designs. The nonparametric hypotheses are formulated by using normalized distribution functions and the comparisons are carried out on the basis of the relative treatment effects. The proposed test statistics take the form of linear pseudo rank statistics and the asymptotic joint distribution of the pseudo rank statistics for testing treatments versus control satisfies the multivariate totally positive of order two condition irrespective of the correlations among the rank statistics. Therefore, in the context of MCs of treatments versus control, the nonparametric Simes test is validated for the global testing of the intersection hypothesis. For simultaneous testing of individual hypotheses, the nonparametric Hochberg stepup procedure strongly controls the familywise type I error rate asymptotically. With regard to all pairwise comparisons, we generalize various single-step and stagewise procedures to perform comparisons on the relative treatment effects. To further compare with normal theory counterparts, the asymptotic relative efficiencies of the nonparametric MC procedures with respect to the parametric MC procedures are derived under a sequence of Pitman alternatives in a nonparametric location shift model for unbalanced one-way layouts. Monte Carlo simulations are conducted to demonstrate the validity and power of the proposed nonparametric MC procedures.     

Semiparametric statistical inferences for longitudinal data with nonparametric covariance modelling

In this paper, semiparametric modelling for longitudinal data with an unstructured error process is considered. We propose a partially linear additive regression model for longitudinal data in which within-subject variances and covariances of the error process are described by unknown univariate and bivariate functions, respectively. We provide an estimating approach in which polynomial splines are used to approximate the additive nonparametric components and the within-subject variance and covariance functions are estimated nonparametrically. Both the asymptotic normality of the resulting parametric component estimators and optimal convergence rate of the resulting nonparametric component estimators are established. In addition, we develop a variable selection procedure to identify significant parametric and nonparametric components simultaneously. We show that the proposed SCAD penalty-based estimators of non-zero components have an oracle property. Some simulation studies are conducted to examine the finite-sample performance of the proposed estimation and variable selection procedures. A real data set is also analysed to demonstrate the usefulness of the proposed method.     

Permutation Tests for Linear Models

Several approximate permutation tests have been proposed for tests of partial regression coefficients in a linear model based on sample partial correlations. This paper begins with an explanation and notation for an exact test. It then compares the distributions of the test statistics under the various permutation methods proposed, and shows that the partial correlations under permutation are asymptotically jointly normal with means 0 and variances 1. The method of Freedman & Lane (1983) is found to have asymptotic correlation 1 with the exact test, and the other methods are found to have smaller correlations with this test. Under local alternatives the critical values of all the approximate permutation tests converge to the same constant, so they all have the same asymptotic power. Simulations demonstrate these theoretical results.     

Multivariate Tests of Mean-Variance Efficiency and Spanning With a Large Number of Assets and Time-Varying Covariances

We develop a finite-sample procedure to test the mean-variance efficiency and spanning hypotheses, without imposing any parametric assumptions on the distribution of model disturbances. In so doing, we provide an exact distribution-free method to test uniform linear restrictions in multivariate linear regression models. The framework allows for unknown forms of nonnormalities as well as time-varying conditional variances and covariances among the model disturbances. We derive exact bounds on the null distribution of joint F statistics to deal with the presence of nuisance parameters, and we show how to implement the resulting generalized nonparametric bounds tests with Monte Carlo resampling techniques. In sharp contrast to the usual tests that are not even computable when the number of test assets is too large, the power of the proposed test procedure potentially increases along both the time and cross-sectional dimensions.     

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.     

Generalized Wald-type tests based on minimum density power divergence estimators

In testing of hypothesis, the robustness of the tests is an important concern. Generally, the maximum likelihood-based tests are most efficient under standard regularity conditions, but they are highly non-robust even under small deviations from the assumed conditions. In this paper, we have proposed generalized Wald-type tests based on minimum density power divergence estimators for parametric hypotheses. This method avoids the use of nonparametric density estimation and the bandwidth selection. The trade-off between efficiency and robustness is controlled by a tuning parameter β. The asymptotic distributions of the test statistics are chi-square with appropriate degrees of freedom. The performance of the proposed tests is explored through simulations and real data analysis.     

Quantile regression for robust estimation and variable selection in partially linear varying-coefficient models

In this paper, we develop a new estimation procedure based on quantile regression for semiparametric partially linear varying-coefficient models. The proposed estimation approach is empirically shown to be much more efficient than the popular least squares estimation method for non-normal error distributions, and almost not lose any efficiency for normal errors. Asymptotic normalities of the proposed estimators for both the parametric and nonparametric parts are established. To achieve sparsity when there exist irrelevant variables in the model, two variable selection procedures based on adaptive penalty are developed to select important parametric covariates as well as significant nonparametric functions. Moreover, both these two variable selection procedures are demonstrated to enjoy the oracle property under some regularity conditions. Some Monte Carlo simulations are conducted to assess the finite sample performance of the proposed estimators, and a real-data example is used to illustrate the application of the proposed methods.     

A resampling procedure for nonparametric combination of several dependent tests

Summary This paper deals with nonparametric methods for combining dependent permutation or randomization tests. Particularly, they are nonparametric with respect to the underlying dependence structure. The methods are based on a without replacement resampling procedure (WRRP) conditional on the observed data, also called conditional simulation, which provide suitable estimates, as good as computing time permits, of the permutational distribution of any statistic. A class C of combining functions is characterized in such a way that all its members, under suitable and reasonable conditions, are found to be consistent and unbiased. Moreover, for some of its members, their almost sure asymptotic equivalence with respect to best tests, in particular cases, is shown. An applicational example to a multivariate permutationalt-paired test is also discussed.     

Robust algorithms for economic designing of a nonparametric control chart for abrupt shift in location

The existing statistical process control procedures typically rely on the fundamental assumption of a parametric distribution of the quality characteristic. However, when there is a lack of knowledge about the underlying distribution (as full knowledge is not available in practice), the performance of these parametric charts is very likely to be heavily degraded. Motivated by this problem, a one-sided nonparametric monitoring procedure using the single sample sign statistic is proposed for detecting a shift in the location parameter of a continuous distribution. An economic model of the control chart is developed to optimize the sample size, sampling interval, and control limits. Three data-dependent estimation approaches for the unknown parameter are evaluated and discussed. Simulation results exhibit that our proposed procedure generally performs well under a great variety of continuous distributions and hence it is recommended as an alternative scheme especially when the knowledge of the underlying distribution is imperfect. Furthermore, beneficial recommendations of estimation approach selection are provided for practical implementation of the control chart.     

Using logistic regression for semiparametric comparison of population means and variances

Abstract

We propose to compare population means and variances under a semiparametric density ratio model. The proposed method is easy to implement by employing logistic regression procedures in many statistical software, and it often works very well when data are not normal. In this paper, we construct semiparametric estimators of the differences of two population means and variances, and derive their asymptotic distributions. We prove that the proposed semiparametric estimators are asymptotically more efficient than the corresponding non parametric ones. In addition, a simulation study and the analysis of two real data sets are presented. Finally, a short discussion is provided.     

Estimation and inference for varying coefficient partially nonlinear errors-in-variables models

In this article, we study the varying coefficient partially nonlinear model with measurement errors in the nonparametric part. A local corrected profile nonlinear least-square estimation procedure is proposed and the asymptotic properties of the resulting estimators are established. Further, a generalized likelihood ratio (GLR) statistic is proposed to test whether the varying coefficients are constant. The asymptotic null distribution of the statistic is obtained and a residual-based bootstrap procedure is employed to compute the p-value of the statistic. Some simulations are conducted to evaluate the performance of the proposed methods. The results show that the estimating and testing procedures work well in finite samples.     

Contributions to two-sample statistics

When testing the equality of the means from two independent normally distributed populations given that the variances of the two populations are unknown but assumed equal, the classical Student's two-sample t-test is recommended. If the underlying population distributions are normal with unequal and unknown variances, either Welch's t-statistic or Satterthwaite's approximate F test is suggested. However, Welch's procedure is non-robust under most non-normal distributions. There is a variable tolerance level around the strict assumptions of data independence, homogeneity of variances, and identical and normal distributions. Few textbooks offer alternatives when one or more of the underlying assumptions are not defensible. While there are more than a few non-parametric (rank) procedures that provide alternatives to Student's t-test, we restrict this review to the promising alternatives to Student's two-sample t-test in non-normal models.     

Empirical smoothing lack-of-fit tests for variance function

This paper discusses a nonparametric empirical smoothing lack-of-fit test for the functional form of the variance in regression models. The proposed test can be treated as a nontrivial modification of Zheng's nonparametric smoothing test, Koul and Ni's minimum distance test for the mean function in the classic regression models. The paper establishes the asymptotic normality of the proposed test under the null hypothesis. Consistency at some fixed alternatives and asymptotic power under some local alternatives are also discussed. A simulation study is conducted to assess the finite sample performance of the proposed test. Simulation study also shows that the proposed test is more powerful and computationally more efficient than some existing tests.     

A CONSISTENT MODEL SPECIFICATION TEST BASED ON THE KERNEL SUM OF SQUARES OF RESIDUALS

This paper constructs a consistent model specification test based on the difference between the nonparametric kernel sum of squares of residuals and the sum of squares of residuals from a parametric null model. We establish the asymptotic normality of the proposed test statistic under the null hypothesis of correct parametric specification and show that the wild bootstrap method can be used to approximate the null distribution of the test statistic. Results from a small simulation study are reported to examine the finite sample performance of the proposed tests.     

Complete ranking procedures with appropriate loss functions

This paper treats the problem of comparing different evaluations of procedures which rank the variances of k normal populations. Procedures are evaluated on the basis of appropriate loss functions for a particular goal. The goal considered involves ranking the variances of k independent normal populations when the corresponding population means are unknown. The variances are ranked by selecting samples of size n from each population and using the sample variances to obtain the ranking. Our results extend those of various authors who looked at the narrower problem of evaluating the standard proceduv 3 associated with selecting the smallest of the population variances (see e.g.,P. Somerville (1975)).

Different loss functions (both parametric and non-parametric) appropriate to the particular goal under consideration are proposed. Procedures are evaluated by the performance of their risk over a particular preference zone. The sample size n, the least favorable parametric configuration, and the maximum value of the risk are three quantities studied for each procedure. When k is small these quantities, calculated by numerical simulation, show which loss functions respond better and which respond worse to increases in sample size. Loss functions are compared with one another according to the extent of this response. Theoretical results are given for the case of asymptotically large k. It is shown that for certain cases the error incurred by using these asymptotic results is small when k is only moderately large.

This work is an outgrowth of and extends that of J. Reeves and M.J. Sobel (1987) by comparing procedures on the basis of the sample size (perpopulation) required to obtain various bounds on the associated risk functions. New methodologies are developed to evaluate complete ranking procedures in different settings.     

A Permutation Approach to Split-Plot Experiments

Split-plot design may be refer to a common experimental setting where a particular type of restricted randomization has occurred during a planned experiment. The aim of this article is to suggest a new method to perform inference on split-plot experiments by combination-based permutation tests. This novel nonparametric approach has been studied and validated using a Monte Carlo simulation study where we compared it with the parametric and nonparametric procedures proposed in the literature. Results suggest that in each experimental situation where normality is hard to justify and especially when errors have heavy-tailed distribution, the proposed nonparametric procedure can be considered as a valid solution.     

The Multivariate Randomized Complete Block Design: A Novel Permutation Solution in Case of Ordered Categorical Variables

In this article, multivariate extensions of the combination-based test statistics for the comparison of several treatments in the multivariate Randomized Complete Block designs are introduced in case of categorical response variables. Several tests for the multivariate Randomized Complete Block designs, including MANOVA procedure, are compared with the method proposed via a Monte Carlo simulation study. The method has also been applied to a real case study in the field of sensorial testing studies. Results suggest that in each experimental situation where normality of the supposed underlying continuous model is hard to justify and especially when errors have heavy-tailed distributions, the proposed nonparametric procedure can be considered as a valid solution.