On the sample breakdown robustness of some nonparametric tests

In this paper, relying on the sample breakdown points, we investigate the sample breakdown properties of some nonparametric tests. It is shown that the sample breakdown points of the sign test asymptotically dominate those of the Wilcoxon test for one–sided hypotheses, However, the different conclusion is derived in the case of testing some shrinking neighborhood hypotheses. The breakdown behaviors of the Kolmogorov test and X²–test are also explored. These studies unify or refine some existing breakdown analyses of tests.     

A weighted log-rank test and associated effect estimator for cancer trials with delayed treatment effect

The standard log-rank test has been extended by adopting various weight functions. Cancer vaccine or immunotherapy trials have shown a delayed onset of effect for the experimental therapy. This is manifested as a delayed separation of the survival curves. This work proposes new weighted log-rank tests to account for such delay. The weight function is motivated by the time-varying hazard ratio between the experimental and the control therapies. We implement a numerical evaluation of the Schoenfeld approximation (NESA) for the mean of the test statistic. The NESA enables us to assess the power and to calculate the sample size for detecting such delayed treatment effect and also for a more general specification of the non-proportional hazards in a trial. We further show a connection between our proposed test and the weighted Cox regression. Then the average hazard ratio using the same weight is obtained as an estimand of the treatment effect. Extensive simulation studies are conducted to compare the performance of the proposed tests with the standard log-rank test and to assess their robustness to model mis-specifications. Our tests outperform the G^ρ,γ class in general and have performance close to the optimal test. We demonstrate our methods on two cancer immunotherapy trials.     

Modified Stationarity Tests With Data-Dependent Model-Selection Rules

We describe some simple methods for improving the performance of stationarity tests (i.e., tests that have a stationary null and a unit-root alternative). Specifically, we increase the rate of convergence of the test under the unit-root alternative from O _p(T) to O _p (T ²), then suggest an optimal method of selecting the order of the autoregressive component in the fitted autoregressive integrated moving average model on which the test is based. Simulation evidence suggests that these modifications work well. We apply the modified procedure to U.S. monthly macroeconomic data and uncover new evidence of a unit root in unemployment.     

On the Power of Bootstrapped Specification Tests

Abstract

Decisions based on econometric model estimates may not have the expected effect if the model is misspecified. Thus, specification tests should precede any analysis. Bierens' specification test is consistent and has optimality properties against some local alternatives. A shortcoming is that the test statistic is not distribution free, even asymptotically. This makes the test unfeasible. There have been many suggestions to circumvent this problem, including the use of upper bounds for the critical values. However, these suggestions lead to tests that lose power and optimality against local alternatives. In this paper we show that bootstrap methods allow us to recover power and optimality of Bierens' original test. Bootstrap also provides reliable p-values, which have a central role in Fisher's theory of hypothesis testing. The paper also includes a discussion of the properties of the bootstrap Nonlinear Least Squares Estimator under local alternatives.     

A comparative simulation study of the small sample powers of several goodness fit test

Eight goodness of fit tests are compared with respect to their simulated small sample power of detecting an inbreeding alternative to the Hardy-Weinberg null hypothesis. The Pearson's x ² test is found to be most powerful, and the small rample levels of this test are close to the nominal (x ²) significance levels. The use of conditional expectations, rather than expected frequencies based on ML estimates, increases the power and improves thc x ² fit to the true significance level. The small sample powers are also compared to the asymptotic (Pitman) pourer, based on the noncenlral x ² distribution.     

Testing for normality in linear regression models

The importance of the normal distribution for fitting continuous data is well known. However, in many practical situations data distribution departs from normality. For example, the sample skewness and the sample kurtosis are far away from 0 and 3, respectively, which are nice properties of normal distributions. So, it is important to have formal tests of normality against any alternative. D'Agostino et al. [A suggestion for using powerful and informative tests of normality, Am. Statist. 44 (1990), pp. 316–321] review four procedures Z ²(g ₁), Z ²(g ₂), D and K ² for testing departure from normality. The first two of these procedures are tests of normality against departure due to skewness and kurtosis, respectively. The other two tests are omnibus tests. An alternative to the normal distribution is a class of skew-normal distributions (see [A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]). In this paper, we obtain a score test (W) and a likelihood ratio test (LR) of goodness of fit of the normal regression model against the skew-normal family of regression models. It turns out that the score test is based on the sample skewness and is of very simple form. The performance of these six procedures, in terms of size and power, are compared using simulations. The level properties of the three statistics LR, W and Z ²(g ₁) are similar and close to the nominal level for moderate to large sample sizes. Also, their power properties are similar for small departure from normality due to skewness (γ₁≤0.4). Of these, the score test statistic has a very simple form and computationally much simpler than the other two statistics. The LR statistic, in general, has highest power, although it is computationally much complex as it requires estimates of the parameters under the normal model as well as those under the skew-normal model. So, the score test may be used to test for normality against small departure from normality due to skewness. Otherwise, the likelihood ratio statistic LR should be used as it detects general departure from normality (due to both skewness and kurtosis) with, in general, largest power.     

A new test for the mean vector in large dimension and small samples

In this article, we consider the problem of testing the mean vector in the multivariate normal distribution, where the dimension p is greater than the sample size N. We propose a new test T_Block and obtain its asymptotic distribution. We also compare the proposed test with other two tests. The simulation results suggest that the performance of the new test is comparable to the existing two tests, and under some circumstances it may have higher power. Therefore, the new statistic can be employed in practice as an alternative choice.     

C334. On the neutron and proton masses: a numerological case study

The intercomponent rank test suggested by Thompson (1991a) for the bivariate two sample problem is compared with the intracomponent rank test discussed by Puri and Sen (1971) and Hettmansperger (1984) and with the Hotelling T ² test. Asymptotic relative efficiencies are discussed and the results of a simulation study are presented. Power studies show that for small sample sizes and small Type 1 error rates, say n = 5 and α = .01, the intercomponent rank test of Thompson (1991a) is somewhat liberal and the intracomponent test is quite conservative. For larger sample sizes and larger Type 1 error rates, both rank tests have improved properties under the null hypothesis. In almost all simulated cases, the intercomponent test is more powerful. In light of these studies it is suggested that the intercomponent rank test of Thompson (1991a), which has the added advantage of being easily computed with standard statistical software, is a strong competitor to the intracomponent rank test.     

On the Exact Size of Tests of Treatment Effects in Multi‐Arm Clinical Trials

When testing treatment effects in multi‐arm clinical trials, the Bonferroni method or the method of Simes 1986) is used to adjust for the multiple comparisons. When control of the family‐wise error rate is required, these methods are combined with the close testing principle of Marcus et al. (1976). Under weak assumptions, the resulting p‐values all give rise to valid tests provided that the basic test used for each treatment is valid. However, standard tests can be far from valid, especially when the endpoint is binary and when sample sizes are unbalanced, as is common in multi‐arm clinical trials. This paper looks at the relationship between size deviations of the component test and size deviations of the multiple comparison test. The conclusion is that multiple comparison tests are as imperfect as the basic tests at nominal size α/m where m is the number of treatments. This, admittedly not unexpected, conclusion implies that these methods should only be used when the component test is very accurate at small nominal sizes. For binary end‐points, this suggests use of the parametric bootstrap test. All these conclusions are supported by a detailed numerical study.     

An observation on regression-based specification tests

The use of regression-based specification tests, such as the nR² form of the Lagrange Multiplier test, has become quite widespread over the last 20 years. The popularization of the nR² form of the Lagrange Multiplier (LM) test, perhaps the most widely used class of regression-based tests, has come about in large part from the ease of its application to many tests of nonlinear restrictions and its asymptotic equivalence to Likelihood Ratio and Wald tests. Properly performed, these regression-based tests invariably include regressors which are orthogonal by construction to the dependent variable of the regression. The purpose of this paper is to motivate the inclusion of such variables by investigating implications for the test size and power if these regressors are erroneously omitted. It is straightforward to show that both the size and power of the test are adversely affected by omitting these regressors.     

A new adaptive test for paired data for small to moderate sample sizes

When carrying out data analysis, a practitioner has to decide on a suitable test for hypothesis testing, and as such, would look for a test that has a high relative power. Tests for paired data tests are usually conducted using t-test, Wilcoxon signed-rank test or the sign test. Some adaptive tests have also been suggested in the literature by O'Gorman, who found that no single member of that family performed well for all sample sizes and different tail weights, and hence, he recommended that choice of a member of that family be made depending on both the sample size and the tail weight. In this paper, we propose a new adaptive test. Simulation studies for n=25 and n=50 show that it works well for nearly all tail weights ranging from the light-tailed beta and uniform distributions to t(4) distributions. More precisely, our test has both robustness of level (in keeping the empirical levels close to the nominal level) and efficiency of power. The results of our study contribute to the area of statistical inference.     

Economic and economic statistical design of T 2 control chart with two adaptive sample sizes

The Hotelling's T ² control chart, a direct analogue of the univariate Shewhart X¯ chart, is perhaps the most commonly used tool in industry for simultaneous monitoring of several quality characteristics. Recent studies have shown that using variable sampling size (VSS) schemes results in charts with more statistical power when detecting small to moderate shifts in the process mean vector. In this paper, we build a cost model of a VSS T ² control chart for the economic and economic statistical design using the general model of Lorenzen and Vance [The economic design of control charts: A unified approach, Technometrics 28 (1986), pp. 3–11]. We optimize this model using a genetic algorithm approach. We also study the effects of the costs and operating parameters on the VSS T ² parameters, and show, through an example, the advantage of economic design over statistical design for VSS T ² charts, and measure the economic advantage of VSS sampling versus fixed sample size sampling.     

On the distributions of multivariate sample skewness

In this paper, we consider the multivariate normality test based on measure of multivariate sample skewness defined by Srivastava (1984). Srivastava derived asymptotic expectation up to the order N⁻¹ for the multivariate sample skewness and approximate χ²${\chi }^2$ test statistic, where N   is sample size. Under normality, we derive another expectation and variance for Srivastava's multivariate sample skewness in order to obtain a better test statistic. From this result, improved approximate χ²${\chi }^2$ test statistic using the multivariate sample skewness is also given for assessing multivariate normality. Finally, the numerical result by Monte Carlo simulation is shown in order to evaluate accuracy of the obtained expectation, variance and improved approximate χ²${\chi }^2$ test statistic. Furthermore, upper and lower percentiles of χ²${\chi }^2$ test statistic derived in this paper are compared with those of χ²${\chi }^2$ test statistic derived by Mardia (1974) which is used multivariate sample skewness defined by Mardia (1970).     

Shapiro–Francia test compared to other normality test using expected p-value

The Shapiro–Francia (SF) normality test is an important test in statistical modelling. However, little has been done by researchers to compare the performance of this test to other normality tests. This paper therefore measures the performance of the SF and other normality tests by studying the distribution of their p-values. For the purpose of this study, we selected eight well-known normality tests to compare with the SF test: (i) Kolmogorov–Smirnov (KS), (ii) Anderson–Darling (AD), (iii) Cramer von Mises (CM), (iv) Lilliefors (LF), (v) Shapiro–Wilk (SW), (vi) Pearson chi-square (PC), (vii) Jarque– Bera (JB) and (viii) D'Agostino (DA). The distribution of p-values of these normality tests were obtained by generating data from normal distribution and well-known symmetric non-normal distribution at various sample sizes (small, medium and large). Our simulation results showed that the SF normality test was the best test statistic in detecting deviation from normality among the nine tests considered at all sample sizes.     

Admissible tests for the mean with additional information

Let X be a po-normal random vector with unknown µ and unknown covariance matrix ∑ and let X be partitioned as X = (X^′ ₍₁₎, …, X^′ _(r))′ where X_(j)is a subvector of X with dimension p_jsuch that ∑^r _j=1Pj = P0. Some admissible tests are derived for testing H₀: μ = 0 versus H₁: μ ¦0 based on a sample drawn from the whole vector X of dimension p and r additional samples drawn from X₍₁₎, X₍₂₎, …, X(r) respectively, All (r+1) samples are assumed to be independent. The distribution of some of the tests' statistics involved are also derived.     

A Bayesian algorithm for sample size determination for equivalence and non-inferiority test

Bayesian sample size estimation for equivalence and non-inferiority tests for diagnostic methods is considered. The goal of the study is to test whether a new screening test of interest is equivalent to, or not inferior to the reference test, which may or may not be a gold standard. Sample sizes are chosen by the model performance criteria of average posterior variance, length and coverage probability. In the absence of a gold standard, sample sizes are evaluated by the ratio of marginal probabilities of the two screening tests; whereas in the presence of gold standard, sample sizes are evaluated by the measures of sensitivity and specificity.     

Testing linear hypotheses in high-dimensional regressions

For a multivariate linear model, Wilk's likelihood ratio test (LRT) constitutes one of the cornerstone tools. However, the computation of its quantiles under the null or the alternative hypothesis requires complex analytic approximations, and more importantly, these distributional approximations are feasible only for moderate dimension of the dependent variable, say p≤20. On the other hand, assuming that the data dimension p as well as the number q of regression variables are fixed while the sample size n grows, several asymptotic approximations are proposed in the literature for Wilk's Λ including the widely used chi-square approximation. In this paper, we consider necessary modifications to Wilk's test in a high-dimensional context, specifically assuming a high data dimension p and a large sample size n. Based on recent random matrix theory, the correction we propose to Wilk's test is asymptotically Gaussian under the null hypothesis and simulations demonstrate that the corrected LRT has very satisfactory size and power, surely in the large p and large n context, but also for moderately large data dimensions such as p=30 or p=50. As a byproduct, we give a reason explaining why the standard chi-square approximation fails for high-dimensional data. We also introduce a new procedure for the classical multiple sample significance test in multivariate analysis of variance which is valid for high-dimensional data.     

Approximate testing in two-stage nonlinear mixed models

We investigate here small sample properties of approximate F-tests about fixed effects parameters in nonlinear mixed models. For estimation of population fixed effects parameters as well as variance components, we apply the two-stage approach. This method is useful and popular when the number of observations per sampling unit is large enough. The approximate F-test is constructed based on large-sample approximation to the distribution of nonlinear least-squares estimates of subject-specific parameters. We recommend a modified test statistic that takes into consideration approximation to the large-sample Fisher information matrix (See [Volaufova J, Burton JH. Note on hypothesis testing in mixed models. Oral presentation at: LINSTAT 2012/21st IWMS; 2012; Bedlewo, Poland]). Our main focus is on comparing finite sample properties of broadly used approximate tests (Wald test and likelihood ratio test) and the modified F-test under the null hypothesis, especially accuracy of p-values (See [Volaufova J, LaMotte L. Comparison of approximate tests of fixed effects in linear repeated measures design models with covariates. Tatra Mountains. 2008;39:17–25]). For that purpose two extensive simulation studies are conducted based on pharmacokinetic models (See [Hartford A, Davidian M. Consequences of misspecifying assumptions in nonlinear mixed effects models. Comput Stat and Data Anal. 2000;34:139–164; Pinheiro J, Bates D. Approximations to the log-likelihood function in the non-linear mixed-effects model. J Comput Graph Stat. 1995;4(1):12–35]).     

Negative exponential disparity based family of goodness-of-fit tests for multinomial models

This paper investigates a new family of goodness-of-fit tests based on the negative exponential disparities. This family includes the popular Pearson's chi-square as a member and is a subclass of the general class of disparity tests (Basu and Sarkar, 1994) which also contains the family of power divergence statistics. Pitman efficiency and finite sample power comparisons between different members of this new family are made. Three asymptotic approximations of the exact null distributions of the negative exponential disparity famiiy of tests are discussed. Some numerical results on the small sample perfomance of this family of tests are presented for the symmetric null hypothesis. It is shown that the negative exponential disparity famiiy, Like the power divergence family, produces a new goodness-of-fit test statistic that can be a very attractive alternative to the Pearson's chi-square. Some numerical results suggest that, application of this test statistic, as an alternative to Pearson's chi-square, could be preferable to the I ^2/3 statistic of Cressie and Read (1984) under the use of chi-square critical values.     

The exact unconditional z-pooled test for equality of two binomial probabilities: optimal choice of the Berger and Boos confidence coefficient

Exact unconditional tests for comparing two binomial probabilities are generally more powerful than conditional tests like Fisher's exact test. Their power can be further increased by the Berger and Boos confidence interval method, where a p-value is found by restricting the common binomial probability under H ₀ to a 1?γ confidence interval. We studied the average test power for the exact unconditional z-pooled test for a wide range of cases with balanced and unbalanced sample sizes, and significance levels 0.05 and 0.01. The detailed results are available online on the web. Among the values 10^?3, 10^?4, …, 10^?10, the value γ=10^?4 gave the highest power, or close to the highest power, in all the cases we looked at, and can be given as a general recommendation as an optimal γ.