Preliminary Goodness-of-Fit Tests for Normality do not Validate the One-Sample Student t

One of the most basic topics in many introductory statistical methods texts is inference for a population mean, μ. The primary tool for confidence intervals and tests is the Student t sampling distribution. Although the derivation requires independent identically distributed normal random variables with constant variance, σ², most authors reassure the readers about some robustness to the normality and constant variance assumptions. Some point out that if one is concerned about assumptions, one may statistically test these prior to reliance on the Student t. Most software packages provide optional test results for both (a) the Gaussian assumption and (b) homogeneity of variance. Many textbooks advise only informal graphical assessments, such as certain scatterplots for independence, others for constant variance, and normal quantile–quantile plots for the adequacy of the Gaussian model. We concur with this recommendation. As convincing evidence against formal tests of (a), such as the Shapiro–Wilk, we offer a simulation study of the tails of the resulting conditional sampling distributions of the Studentized mean. We analyze the results of systematically screening all samples from normal, uniform, exponential, and Cauchy populations. This pretest does not correct the erroneous significance levels and makes matters worse for the exponential. In practice, we conclude that graphical diagnostics are better than a formal pretest. Furthermore, rank or permutation methods are recommended for exact validity in the symmetric case.     

Comparison of Three Sampling Designs

Three sampling designs are considered for estimating the sum of k population means by the sum of the corresponding sample means. These are (a) the optimal design; (b) equal sample sizes from all populations; and (c) sample sizes that render equal variances to all sample means. Designs (b) and (c) are equally inefficient, and may yield a variance up to k times as large as that of (a). Similar results are true when the cost of sampling is introduced, and they depend on the population sampled.     

The split sample permutation t-tests

Without the exchangeability assumption, permutation tests for comparing two population means do not provide exact control of the probability of making a Type I error. Another drawback of permutation tests is that it cannot be used to test hypothesis about one population. In this paper, we propose a new type of permutation tests for testing the difference between two population means: the split sample permutation t-tests. We show that the split sample permutation t-tests do not require the exchangeability assumption, are asymptotically exact and can be easily extended to testing hypothesis about one population. Extensive simulations were carried out to evaluate the performance of two specific split sample permutation t-tests: the split in the middle permutation t-test and the split in the end permutation t-test. The simulation results show that the split in the middle permutation t-test has comparable performance to the permutation test if the population distributions are symmetric and satisfy the exchangeability assumption. Otherwise, the split in the end permutation t-test has significantly more accurate control of level of significance than the split in the middle permutation t-test and other existing permutation tests.     

More powerful permutation test based on multistage ranked set sampling

Many researches have used ranked set sampling (RSS) method instead of simple random sampling (SRS) to improve power of some nonparametric tests. In this study, the two-sample permutation test within multistage ranked set sampling (MSRSS) is proposed and investigated. The power of this test is compared with the SRS permutation test for some symmetric and asymmetric distributions through Monte Carlo simulations. It has been found that this test is more powerful than the SRS permutation test; its power increased by set size and/or number of cycles and/or number of stages. Symmetric distributions power increased better than asymmetric distributions power.     

A robust I-sample analysis of means type randomization test for variances for unbalanced designs

Two analysis of means type randomization tests for testing the equality of I variances for unbalanced designs 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. A Monte Carlo study is used to verify that the permutation version of the tests exhibit excellent power when compared to other robust tests. The Monte Carlo study also identifies circumstances under which the popular Levene's test fails.     

A permutation alternative and other test procedures for spatially correlated data in one-way anova

Research in many disciplines involves data with spatially correlated observations. Spatial dependence violates the independent errors assumption required for techniques such as the standard one-way analysis of variance for a completely randomized design. The testing methodology within the correlated errors approach has not been investigated within a spatial context. For one-way fixed effects analysis of variance, a permutation test and tests associated with the correlated errors approach are investigated through simulation. No single test was superior with respect to both power and size but the standard Wald F test and a simple adjustment to it performed well overall.     

Some new results on univariate and multivariate permutation tests for ordinal categorical variables under restricted alternatives

In several sciences, especially when dealing with performance evaluation, complex testing problems may arise due in particular to the presence of multidimensional categorical data. In such cases the application of nonparametric methods can represent a reasonable approach. In this paper, we consider the problem of testing whether a “treatment” is stochastically larger than a “control” when univariate and multivariate ordinal categorical data are present. We propose a solution based on the nonparametric combination of dependent permutation tests (Pesarin in Multivariate permutation test with application to biostatistics. Wiley, Chichester, 2001), on variable transformation, and on tests on moments. The solution requires the transformation of categorical response variables into numeric variables and the breaking up of the original problem’s hypotheses into partial sub-hypotheses regarding the moments of the transformed variables. This type of problem is considered to be almost impossible to analyze within likelihood ratio tests, especially in the multivariate case (Wang in J Am Stat Assoc 91:1676–1683, 1996). A comparative simulation study is also presented along with an application example.     

Uniqueness of permuted treatment estimator distributions in the proportional hazards model

We discuss findings regarding the permutation distributions of treatment effect estimators in the proportional hazards model. For fixed sample size n, we will prove that all uncensored and untied event times yield the same permutation distribution of treatment effect estimators in the proportional hazards model. In other words this distribution is irrelevant with respect to the actual event times. We will show several uniqueness properties under different conditions. These properties are useful for small sample permutation tests and also helpful to large sample cases.     

Stratified Double Sampling with Continuous Outcomes: Design and Analysis

We provide a method for estimating the sample mean of a continuous outcome in a stratified population using a double sampling scheme. The stratified sample mean is a weighted average of stratum specific means. It is assumed that the fallible and true outcome data are related by a simple linear regression model in each stratum. The optimal stratified double sampling plan, i.e. , the double sampling plan that minimizes the cost of sampling for fixed variances, or alternatively, minimizes the variance for fixed costs, is found and compared to a standard sampling plan. The design parameters are the total sample size and the number of doubly sampled units in each stratum. We show that the optimal double sampling plan is a function of the between-strata and within-strata cost and variance ratios. The efficiency gains, relative to standard sampling plans, under broad set of conditions, are considerable.     

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.     

A comparison of recent nonparametric methods for testing effects in two-by-two factorial designs

The two-way two-levels crossed factorial design is a commonly used design by practitioners at the exploratory phase of industrial experiments. The F-test in the usual linear model for analysis of variance (ANOVA) is a key instrument to assess the impact of each factor and of their interactions on the response variable. However, if assumptions such as normal distribution and homoscedasticity of errors are violated, the conventional wisdom is to resort to nonparametric tests. Nonparametric methods, rank-based as well as permutation, have been a subject of recent investigations to make them effective in testing the hypotheses of interest and to improve their performance in small sample situations. In this study, we assess the performances of some nonparametric methods and, more importantly, we compare their powers. Specifically, we examine three permutation methods (Constrained Synchronized Permutations, Unconstrained Synchronized Permutations and Wald-Type Permutation Test), a rank-based method (Aligned Rank Transform) and a parametric method (ANOVA-Type Test). In the simulations, we generate datasets with different configurations of distribution of errors, variance, factor's effect and number of replicates. The objective is to elicit practical advice and guides to practitioners regarding the sensitivity of the tests in the various configurations, the conditions under which some tests cannot be used, the tradeoff between power and type I error, and the bias of the power on one main factor analysis due to the presence of effect of the other factor. A dataset from an industrial engineering experiment for thermoformed packaging production is used to illustrate the application of the various methods of analysis, taking into account the power of the test suggested by the objective of the experiment.     

The Use of Permutation Tests for Variance Components in Linear Mixed Models

Standard asymptotic chi-square distribution of the likelihood ratio and score statistics under the null hypothesis does not hold when the parameter value is on the boundary of the parameter space. In mixed models it is of interest to test for a zero random effect variance component. Some available tests for the variance component are reviewed and a new test within the permutation framework is presented. The power and significance level of the different tests are investigated by means of a Monte Carlo simulation study. The proposed test has a significance level closer to the nominal one and it is more powerful.     

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 Approaches for Stochastic Ordering

In many application problems, when dealing with comparisons between two or more groups, the classical parametric inferential statistical methods are used, although in real problems the quite stringent assumptions required by such methods are rarely satisfied. In particular a parametric approach to the test on ordering of C > 2 populations is very difficult. In order to tackle this problem two alternative methods are proposed in the present paper. Both the methods consist in permutation combination based tests: the first is supposed to be more powerful and it is suitable when the main goal of the study is related to the global ordering of the populations; the second is useful when the interest is in the pairwise comparisons between the populations.     

我国农村住户连续性抽样调查方案设计

 在改革开放的新形势下,我国政府统计部门开展了农村住户等一系列农村统计调查,为解决“三农”问题提供了多方面的数据信息。本文通过分析总结现行农村住户抽样调查方案中存在的各种矛盾和问题,利用国际上前沿的连续性抽样调查方法作为理论基础,分别从农村住户抽样框的构建、连续各期调查样本的抽取、二维平衡轮换模式的设计、连续性抽样估计及其方差估计和连续时间序列数据的调整分析等角度提出一系列改革措施,从而设计出更加科学的调查方案,为及时、准确地搜集和提供关于“三农”问题的数据信息服务。关于其它类型的抽样调查方案亦可按照本文研究的思路类似地加以设计和解决。     

Control variates and importance sampling for efficient bootstrap simulations

Importance sampling and control variates have been used as variance reduction techniques for estimating bootstrap tail quantiles and moments, respectively. We adapt each method to apply to both quantiles and moments, and combine the methods to obtain variance reductions by factors from 4 to 30 in simulation examples.We use two innovations in control variates—interpreting control variates as a re-weighting method, and the implementation of control variates using the saddlepoint; the combination requires only the linear saddlepoint but applies to general statistics, and produces estimates with accuracy of order n ^-1/2 B ^-1, where n is the sample size and B is the bootstrap sample size.We discuss two modifications to classical importance sampling—a weighted average estimate and a mixture design distribution. These modifications make importance sampling robust and allow moments to be estimated from the same bootstrap simulation used to estimate quantiles.     

Book Reviews

The Levene test is a widely used test for detecting differences in dispersion. The modified Levene transformation using sample medians is considered in this article. After Levene's transformation the data are not normally distributed, hence, nonparametric tests may be useful. As the Wilcoxon rank sum test applied to the transformed data cannot control the type I error rate for asymmetric distributions, a permutation test based on reallocations of the original observations rather than the absolute deviations was investigated. Levene's transformation is then only an intermediate step to compute the test statistic. Such a Levene test, however, cannot control the type I error rate when the Wilcoxon statistic is used; with the Fisher–Pitman permutation test it can be extremely conservative. The Fisher–Pitman test based on reallocations of the transformed data seems to be the only acceptable nonparametric test. Simulation results indicate that this test is on average more powerful than applying the t test after Levene's transformation, even when the t test is improved by the deletion of structural zeros.     

Permuiation tesis for k-sample binomial data with comparisons of exact and approximate P-levels

Exact ksample permutation tests for binary data for three commonly encountered hypotheses tests are presented,, The tests are derived both under the population and randomization models . The generating function for the number of cases in the null distribution is obtained, The asymptotic distributions of the test statistics are derived . Actual significance levels are computed for the asymptotic test versions , Random sampling of the null distribution is suggested as a superior alternative to the asymptotics and an efficient computer technique for implementing the random sampling is described., finally, some numerical examples are presented and sample size guidelines given for computer implementation of the exact tests.     

Comparison of permutation methods for the partial correlation and partial mantel tests

This study compares empirical type I error and power of different permutation techniques that can be used for partial correlation analysis involving three data vectors and for partial Mantel tests. The partial Mantel test is a form of first-order partial correlation analysis involving three distance matrices which is widely used in such fields as population genetics, ecology, anthropology, psychometry and sociology. The methods compared are the following: (1) permute the objects in one of the vectors (or matrices); (2) permute the residuals of a null model; (3) correlate residualized vector 1 (or matrix A) to residualized vector 2 (or matrix B); permute one of the residualized vectors (or matrices); (4) permute the residuals of a full model. In the partial correlation study, the results were compared to those of the parametric t-test which provides a reference under normality. Simulations were carried out to measure the type I error and power of these permutatio methods, using normal and non-normal data, without and with an outlier. There were 10 000 simulations for each situation (100 000 when n = 5); 999 permutations were produced per test where permutations were used. The recommended testing procedures are the following:(a) In partial correlation analysis, most methods can be used most of the time. The parametric t-test should not be used with highly skewed data. Permutation of the raw data should be avoided only when highly skewed data are combined with outliers in the covariable. Methods implying permutation of residuals, which are known to only have asymptotically exact significance levels, should not be used when highly skewed data are combined with small sample size. (b) In partial Mantel tests, method 2 can always be used, except when highly skewed data are combined with small sample size. (c) With small sample sizes, one should carefully examine the data before partial correlation or partial Mantel analysis. For highly skewed data, permutation of the raw data has correct type I error in the absence of outliers. When highly skewed data are combined with outliers in the covariable vector or matrix, it is still recommended to use the permutation of raw data. (d) Method 3 should never be used.