Robust I-sample analysis of means type randomization tests for variances

Four Analysis of Means (ANOM) type randomization tests for testing the equality of I variances are presented. Randomization techniques for testing statistical hypotheses can be used when parametric tests are inappropriate. Suppose that I independent samples have been collected. Randomization tests are based on shuffles or rearrangements of the (combined) sample. Putting each of the I samples "in a bowl" forms the combined sample. Drawing samples "from the bowl" forms a shuffle. Shuffles can be made with replacement (bootstrap shuffling) or without replacement (permutation shuffling). The tests that are presented offer two advantages. They are robust to non-normality and they allow the user to graphically present the results via a decision chart similar to a Shewhart control chart. The decision chart facilitates easy assessment of both statistical and practical significance. A Monte Carlo study is used to identify robust randomization tests that exhibit excellent power when compared to other robust tests.     

Two permutation tests of equality of variances

The F-ratio test for equality of dispersion in two samples is by no means robust, while non-parametric tests either assume a common median, or are not very powerful. Two new permutation tests are presented, which do not suffer from either of these problems. Algorithms for Monte Carlo calculation of P values and confidence intervals are given, and the performance of the tests are studied and compared using Monte Carlo simulations for a range of distributional types. The methods used to speed up Monte Carlo calculations, e.g. stratification, are of wider applicability.     

Permutation Tests for Comparing Inequality Measures

ABSTRACT

Asymptotic and bootstrap tests for inequality measures are known to perform poorly in finite samples when the underlying distribution is heavy-tailed. We propose Monte Carlo permutation and bootstrap methods for the problem of testing the equality of inequality measures between two samples. Results cover the Generalized Entropy class, which includes Theil’s index, the Atkinson class of indices, and the Gini index. We analyze finite-sample and asymptotic conditions for the validity of the proposed methods, and we introduce a convenient rescaling to improve finite-sample performance. Simulation results show that size correct inference can be obtained with our proposed methods despite heavy tails if the underlying distributions are sufficiently close in the upper tails. Substantial reduction in size distortion is achieved more generally. Studentized rescaled Monte Carlo permutation tests outperform the competing methods we consider in terms of power.     

Dependency and truncated forms of combinations in multivariate combination-based permutation tests and ordered categorical variables

ABSTRACT

Quite an important problem usually occurs in several multi-dimensional hypotheses testing problems when variables are correlated. In this framework the non-parametric combination (NPC) of a finite number of dependent permutation tests is suitable to cover almost all real situations of practical interest since the dependence relations among partial tests are implicitly captured by the combining procedure itself without the need to specify them [Pesarin F, Salmaso L. Permutation tests for complex data: theory, applications and software. Chichester: Wiley; 2010a]. An open problem related to NPC-based tests is the impact of the dependency structure on combined tests, especially in the presence of categorical variables. This paper’s goal is firstly to investigate the impact of the dependency structure on the possible significance of combined tests in cases of ordered categorical responses using Monte Carlo simulations, then to propose some specific procedures aimed at improving the power of multivariate combination-based permutation tests. The results show that an increasing level of correlation/association among responses negatively affects the power of combination-based multivariate permutation tests. The application of special forms of combination functions based on the truncated product method [Zaykin DV, Zhivotovsky LA, Westfall PH, Weir BS. Truncated product method for combining p-values. Genet Epidemiol. 2002;22:170–185; Dudbridge F, Koeleman BPC. Rank truncated product of p-values, with application to genomewide association scans. Genet Epidemiol. 2003;25:360–366] or on Liptak combination allowed us, using Monte Carlo simulations, to demonstrate the possibility of mitigating the negative effect on power of combination-based multivariate permutation tests produced by an increasing level of correlation/association among responses.     

QuickMMCTest: quick multiple Monte Carlo testing

Multiple hypothesis testing is widely used to evaluate scientific studies involving statistical tests. However, for many of these tests, p values are not available and are thus often approximated using Monte Carlo tests such as permutation tests or bootstrap tests. This article presents a simple algorithm based on Thompson Sampling to test multiple hypotheses. It works with arbitrary multiple testing procedures, in particular with step-up and step-down procedures. Its main feature is to sequentially allocate Monte Carlo effort, generating more Monte Carlo samples for tests whose decisions are so far less certain. A simulation study demonstrates that for a low computational effort, the new approach yields a higher power and a higher degree of reproducibility of its results than previously suggested methods.     

Parametric bootstrap tests for unbalanced nested designs under heteroscedasticity

In this article, we consider the two-factor unbalanced nested design model without the assumption of equal error variance. For the problem of testing ‘main effects’ of both factors, we propose a parametric bootstrap (PB) approach and compare it with the existing generalized F (GF) test. The Type I error rates of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test performs better than the GF test. The PB test performs very satisfactorily even for small samples while the GF test exhibit poor Type I error properties when the number of factorial combinations or treatments goes up. It is also noted that the same tests can be used to test the significance of the random effect variance component in a two-factor mixed effects nested model under unequal error variances.     

Small sample size comparisons of tests for homogeneity of variances by Monte-Carlo

A number of tests are available for testing the equality of several population variances. Some are claimed to be robust. We compared six of those claimed robust procedures by Monte Carlo simulated experiments, particularly for cases of small and unequal sample sizes. Our results show that the jack-knife test compares favorably with the other tests.     

Permutation tests for homogeneity based on some characterizations

In this work, non parametric tests are proposed for testing the homogeneity of two or more populations. The tests are based on recently obtained characterizations. The test procedure is based on the permutation bootstrap technique. For the two-sample case the new tests are compared with permutation tests based on the empirical characteristic function and some other tests. The comparison is fulfilled via a Monte Carlo simulation.     

A test for the presence of conditional heteroskedasticity within arch-m framework

This paper is concerned with testing the presence of ARCH within the ARCH-M model as the alternative hypothesis. Standard testing procedures are inapplicable since a nuisance parameter is unidentified under the null hypothesis. Nonetheless, the diagnostic tests for the presence of the conditional variance is very important since any misspecification in the conditional variance equation leads to inconsistent estimates of the conditional mean parameters. BTo resolve the problem of unidentified nuisance parameter, ‘Ne apply Davies’ approach, and investigate its finite sample performance through a Monte Carlo study.     

Evaluating the performance of simple estimators for probit models with two dummy endogenous regressors

This study considers the small sample performance of approximate but simple two-stage estimators for probit models with two endogenous binary covariates. Monte Carlo simulations show that all the considered estimators, including the simulated maximum-likelihood (SML) estimation, of the trivariate probit model are biased in very small samples (N=100). With moderately small samples (N=500), some of the approximations perform as well as the SML estimator when the degree of endogeneity is not very large. Some of the approximations seem robust with higher correlations and are also promising for testing the exogeneity of binary covariates. The methods are used to estimate the impact of employment-based health insurance and health care (HC) on HC use, where the approximations seem to work at least as well as the SML and in some cases better.     

Semiparametric Estimation for Two-Sample Location-Scale Models under Type I Censorship

To compare two samples under Type I censorship, this article proposes a method of semiparametric inference for the two-sample location-scale problem when the model for two samples is characterized by an unknown distribution and two unknown parameters. Simultaneous estimators for both the location shift and scale change parameters are given. It is shown that the two estimators are strongly consistent and asymptotically normal. The approach in this article can also be used for scale-shape models. Monte Carlo studies indicate that the proposed estimation procedure performs well in finite and heavily censored samples, maintains high relative efficiencies for a wide range of censoring proportions and is robust to the model misspecification, and also outperforms other competitive estimators.     

Wild-bootstrapped variance-ratio test for autocorrelation in the presence of heteroskedasticity

The Breusch–Godfrey LM test is one of the most popular tests for autocorrelation. However, it has been shown that the LM test may be erroneous when there exist heteroskedastic errors in a regression model. Recently, remedies have been proposed by Godfrey and Tremayne [9] and Shim et al. [21]. This paper suggests three wild-bootstrapped variance-ratio (WB-VR) tests for autocorrelation in the presence of heteroskedasticity. We show through a Monte Carlo simulation that our WB-VR tests have better small sample properties and are robust to the structure of heteroskedasticity.     

Monte carlo results for conditional survival tests under randomly censored data

Within a Monte Carlo study finite sample results are obtained for different generalized rank tests based on randomly censored life time data. It is pointed out that conditional tests should be applied in practice whenever drastic differences between the censoring distributions for the underlying groups do not appear. The tests are slight modifications of known permutation tests for censored data.     

A comparative study of tests for paired lifetime data

In this paper, we investigate different procedures for testing the equality of two mean survival times in paired lifetime studies. We consider Owen’s M-test and Q-test, a likelihood ratio test, the paired t-test, the Wilcoxon signed rank test and a permutation test based on log-transformed survival times in the comparative study. We also consider the paired t-test, the Wilcoxon signed rank test and a permutation test based on original survival times for the sake of comparison. The size and power characteristics of these tests are studied by means of Monte Carlo simulations under a frailty Weibull model. For less skewed marginal distributions, the Wilcoxon signed rank test based on original survival times is found to be desirable. Otherwise, the M-test and the likelihood ratio test are the best choices in terms of power. In general, one can choose a test procedure based on information about the correlation between the two survival times and the skewness of the marginal survival distributions.     

Non-nested hypothesis testing inference for GAMLSS models

Two or more regression models are said to be non-nested if neither can be obtained from the remaining models when parametric restrictions are imposed. Tests for choosing between linear non-nested regression models are found in literature, such as J and MJ tests. In this paper we propose variants of these two tests for the GAMLSS (Generalized Additive Models for Location, Scale and Shape) class of models. We report Monte Carlo evidence on finite sample behaviour of the proposed tests. Bootstrap-based testing inference is also considered. Overall, bootstrap MJ test had the best performance. An empirical application is presented and discussed.     

On the power of the t-test and some rank tests when outliers may be present

The power of some rank tests, used for testing the hypothesis of shift, is found when the underlying distributions contain outliers. The outliers are assumed to occur as the result of mixing two normal distributions with common variance. A small sample case shows how the scores for the rank tests are found and the exact power is computed for each of these rank tests. A Monte Carlo study provides an estimate of the power of the usual two sample t-test.     

On the theory of modified randomization tests for nonparametric hypotheses

A general randomization test for nonparametric hypotheses which is a modification of permutation tests in proposed. The exact level of the test is derived and under mild gegularity conditions, a general result on the consistency of the power function is obtained. Applications to several testing problems are considered. Asymptotic expansions of the power of this test are derived with respect to contiguous alternatives thus test are derived with respect to contiguous alternatives thus enabling us to make deficiency comparisons with permutation tests. The paper concludes with some Monte Carlo simulations verifying the theoretical results derived.     

Testing Inference in Inflated Beta Regressions under Model Misspecification

We consider testing inference in inflated beta regressions subject to model misspecification. In particular, quasi-z tests based on sandwich covariance matrix estimators are described and their finite sample behavior is investigated via Monte Carlo simulations. The numerical evidence shows that quasi-z testing inference can be considerably more accurate than inference made through the usual z tests, especially when there is model misspecification. Interval estimation is also considered. We also present an empirical application that uses real (not simulated) data.     

A re-appraisal of tests for normality

Tests for normality can be divided into two groups - those based upon a function of the empirical distribution function and those based upon a function of the original observations. The latter group of statistics test spherical symmetry and not necessarily normality. If the distribution is completely specified then the first group can be used to test for ‘spherical’ normality. However, if the distribution is incompletely specified and F‘‘x_i - x’/s’ is used these test statistics also test sphericity rather than normality. A Monte Carlo study was conducted for the completely specified case, to investigate the sensitivity of the distance tests to departures from normality when the alternative distributions are non-normal spherically symmetric laws. A “new” test statistic is proposed for testing a completely specified normal distribution