A Five-Decision Testing Procedure to Infer the Value of a Unidimensional Parameter

ABSTRACT

A statistical test can be seen as a procedure to produce a decision based on observed data, where some decisions consist of rejecting a hypothesis (yielding a significant result) and some do not, and where one controls the probability to make a wrong rejection at some prespecified significance level. Whereas traditional hypothesis testing involves only two possible decisions (to reject or not a null hypothesis), Kaiser’s directional two-sided test as well as the more recently introduced testing procedure of Jones and Tukey, each equivalent to running two one-sided tests, involve three possible decisions to infer the value of a unidimensional parameter. The latter procedure assumes that a point null hypothesis is impossible (e.g., that two treatments cannot have exactly the same effect), allowing a gain of statistical power. There are, however, situations where a point hypothesis is indeed plausible, for example, when considering hypotheses derived from Einstein’s theories. In this article, we introduce a five-decision rule testing procedure, equivalent to running a traditional two-sided test in addition to two one-sided tests, which combines the advantages of the testing procedures of Kaiser (no assumption on a point hypothesis being impossible) and Jones and Tukey (higher power), allowing for a nonnegligible (typically 20%) reduction of the sample size needed to reach a given statistical power to get a significant result, compared to the traditional approach.     

Interval-wise testing for functional data

In the framework of null hypothesis significance testing for functional data, we propose a procedure able to select intervals of the domain imputable for the rejection of a null hypothesis. An unadjusted p-value function and an adjusted one are the output of the procedure, namely interval-wise testing. Depending on the sort and level α of type-I error control, significant intervals can be selected by thresholding the two p-value functions at level α. We prove that the unadjusted (adjusted) p-value function point-wise (interval-wise) controls the probability of type-I error and it is point-wise (interval-wise) consistent. To enlighten the gain in terms of interpretation of the phenomenon under study, we applied the interval-wise testing to the analysis of a benchmark functional data set, i.e. Canadian daily temperatures. The new procedure provides insights that current state-of-the-art procedures do not, supporting similar advantages in the analysis of functional data with less prior knowledge.     

A Proposed Hybrid Effect Size Plus p-Value Criterion: Empirical Evidence Supporting its Use

ABSTRACT

When the editors of Basic and Applied Social Psychology effectively banned the use of null hypothesis significance testing (NHST) from articles published in their journal, it set off a fire-storm of discussions both supporting the decision and defending the utility of NHST in scientific research. At the heart of NHST is the p-value which is the probability of obtaining an effect equal to or more extreme than the one observed in the sample data, given the null hypothesis and other model assumptions. Although this is conceptually different from the probability of the null hypothesis being true, given the sample, p-values nonetheless can provide evidential information, toward making an inference about a parameter. Applying a 10,000-case simulation described in this article, the authors found that p-values’ inferential signals to either reject or not reject a null hypothesis about the mean (α?=?0.05) were consistent for almost 70% of the cases with the parameter’s true location for the sampled-from population. Success increases if a hybrid decision criterion, minimum effect size plus p-value (MESP), is used. Here, rejecting the null also requires the difference of the observed statistic from the exact null to be meaningfully large or practically significant, in the researcher’s judgment and experience. The simulation compares performances of several methods: from p-value and/or effect size-based, to confidence-interval based, under various conditions of true location of the mean, test power, and comparative sizes of the meaningful distance and population variability. For any inference procedure that outputs a binary indicator, like flagging whether a p-value is significant, the output of one single experiment is not sufficient evidence for a definitive conclusion. Yet, if a tool like MESP generates a relatively reliable signal and is used knowledgeably as part of a research process, it can provide useful information.     

Multiple testing using the posterior probabilities of directional alternatives,with application to genomic studies

This paper aims to connect Bayesian analysis and frequentist theory in the context of multiple comparisons. The authors show that when testing the equality of two sample means, the posterior probability of the one‐sided alternative hypothesis, defined as a half‐space, shares with the frequentist P‐value the property of uniformity under the null hypothesis. Ultimately, the posterior probability may thus be used in the same spirit as a P‐value in the Benjamini‐Hochberg procedure, or in any of its extensions.     

The Combination Test for Multivariate Normality

A basic graphical approach for checking normality is the Q - Q plot that compares sample quantiles against the population quantiles. In the univariate analysis, the probability plot correlation coefficient test for normality has been studied extensively. We consider testing the multivariate normality by using the correlation coefficient of the Q - Q plot. When multivariate normality holds, the sample squared distance should follow a chi-square distribution for large samples. The plot should resemble a straight line. A correlation coefficient test can be constructed by using the pairs of points in the probability plot. When the correlation coefficient test does not reject the null hypothesis, the sample data may come from a multivariate normal distribution or some other distributions. So, we use the following two steps to test multivariate normality. First, we check the multivariate normality by using the probability plot correction coefficient test. If the test does not reject the null hypothesis, then we test symmetry of the distribution and determine whether multivariate normality holds. This test procedure is called the combination test. The size and power of this test are studied, and it is found that the combination test, in general, is more powerful than other tests for multivariate normality.     

Comparing the overlapping of two independent confidence intervals with a single confidence interval for two normal population parameters

Two overlapping confidence intervals have been used in the past to conduct statistical inferences about two population means and proportions. Several authors have examined the shortcomings of Overlap procedure and have determined that such a method distorts the significance level of testing the null hypothesis of two population means and reduces the statistical power of the test. Nearly all results for small samples in Overlap literature have been obtained either by simulation or by formulas that may need refinement for small sample sizes, but accurate large sample information exists. Nevertheless, there are aspects of Overlap that have not been presented and compared against the standard statistical procedure. This article will present exact formulas for the maximum % overlap of two independent confidence intervals below which the null hypothesis of equality of two normal population means or variances must still be rejected for any sample sizes. Further, the impact of Overlap on the power of testing the null hypothesis of equality of two normal variances will be assessed. Finally, the noncentral t-distribution is used to assess the Overlap impact on type II error probability when testing equality of means for sample sizes larger than 1.     

Inflated statistical significance of student's t test associated with small intersubject correlation

The independence assumption in statistical significance testing becomes increasingly crucial and unforgiving as sample size increases. Seemingly, inconsequential violations of this assumption can substantially increase the probability of a Type I error if sample sizes are large. In the case of Student's t test, it is found that correlations within samples in a range from 0.01 to 0.05 can lead to rejection of a true null hypothesis with high probability, if N is 50, 100 or larger.     

A comparison of alternative unit root tests

In this paper we evaluate the performance of three methods for testing the existence of a unit root in a time series, when the models under consideration in the null hypothesis do not display autocorrelation in the error term. In such cases, simple versions of the Dickey-Fuller test should be used as the most appropriate ones instead of the known augmented Dickey-Fuller or Phillips-Perron tests. Through Monte Carlo simulations we show that, apart from a few cases, testing the existence of a unit root we obtain actual type I error and power very close to their nominal levels. Additionally, when the random walk null hypothesis is true, by gradually increasing the sample size, we observe that p-values for the drift in the unrestricted model fluctuate at low levels with small variance and the Durbin-Watson (DW) statistic is approaching 2 in both the unrestricted and restricted models. If, however, the null hypothesis of a random walk is false, taking a larger sample, the DW statistic in the restricted model starts to deviate from 2 while in the unrestricted model it continues to approach 2. It is also shown that the probability not to reject that the errors are uncorrelated, when they are indeed not correlated, is higher when the DW test is applied at 1% nominal level of significance.     

Small sample performance of large sample tests for tobit versus p -tobit

Large sample tests for the standard To bit model versus the ^p -Tobit model by Deaton and Irish (1984) are studied. The normalized one-tailed score test by Deaton and Irish (1984) is shown to be a version of Neyman's C(α) test that is valid for the non-standard problem of the null hypothesis lying on the boundary of the parameter space. Then, this paper reports the results of Monte Carlo experiments designed to study the small sample performance of large sample tests for the standard Tobit specification versus the ^p -Tobit specification.     

Hypothesis testing in smoothing spline models

Nonparametric regression models are often used to check or suggest a parametric model. Several methods have been proposed to test the hypothesis of a parametric regression function against an alternative smoothing spline model. Some tests such as the locally most powerful (LMP) test by Cox et al. (Cox, D., Koh, E., Wahba, G. and Yandell, B. (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Stat., 16, 113–119.), the generalized maximum likelihood (GML) ratio test and the generalized cross validation (GCV) test by Wahba (Wahba, G. (1990). Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.) were developed from the corresponding Bayesian models. Their frequentist properties have not been studied. We conduct simulations to evaluate and compare finite sample performances. Simulation results show that the performances of these tests depend on the shape of the true function. The LMP and GML tests are more powerful for low frequency functions while the GCV test is more powerful for high frequency functions. For all test statistics, distributions under the null hypothesis are complicated. Computationally intensive Monte Carlo methods can be used to calculate null distributions. We also propose approximations to these null distributions and evaluate their performances by simulations.     

A Correction Note on Improvement in Variance Estimation Using Auxiliary Information

ABSTRACT

In this paper, the testing problem for homogeneity in the mixture exponential family is considered. The model is irregular in the sense that each interest parameter forms a part of the null hypothesis (sub-null hypothesis) and the null hypothesis is the union of the sub-null hypotheses. The generalized likelihood ratio test does not distinguish between the sub-null hypotheses. The Supplementary Score Test is proposed by combining two orthogonalized score tests obtained corresponding to the two sub-null hypotheses after proper reparameterization. The test is easy to design and performs better than the generalized likelihood ratio test and other alternative tests by numerical comparisons.     

On pooling data summaries in the absence of interactions “response-by-study”

The standard hypothesis testing procedure in meta-analysis (or multi-center clinical trials) in the absence of treatment-by-center interaction relies on approximating the null distribution of the standard test statistic by a standard normal distribution. For relatively small sample sizes, the standard procedure has been shown by various authors to have poor control of the type I error probability, leading to too many liberal decisions. In this article, two test procedures are proposed, which rely on thet—distribution as the reference distribution. A simulation study indicates that the proposed procedures attain significance levels closer to the nominal level compared with the standard procedure.     

Ordered Alternatives and the 1 1/2-Tail Test

Under proper conditions, two independent tests of the null hypothesis of homogeneity of means are provided by a set of sample averages. One test, with tail probability P ₁, relates to the variation between the sample averages, while the other, with tail probability P ₂, relates to the concordance of the rankings of the sample averages with the anticipated rankings under an alternative hypothesis. The quantity G = P ₁ P ₂ is considered as the combined test statistic and, except for the discreteness in the null distribution of P ₂, would correspond to the Fisher statistic for combining probabilities. Illustration is made, for the case of four means, on how to get critical values of G or critical values of P ₁ for each possible value of P ₂, taking discreteness into account. Alternative measures of concordance considered are Spearman's ρ and Kendall's τ. The concept results, in the case of two averages, in assigning two-thirds of the test size to the concordant tail, one-third to the discordant tail.     

A generalized p-value approach to inference on the performance measures of an M/Ek/1 queueing system

Abstract

The hypothesis tests of performance measures for an M/E_k/1 queueing system are considered. With pivotal models deduced from sufficient statistics for the unknown parameters, a generalized p-value approach to derive tests about parametric functions are proposed. The focus is on derivation of the p-values of hypothesis testing for five popular performance measures of the system in the steady state. Given a sample T, let p(T) be the p values we developed. We derive a closed form expression to show that, for small samples, the probability P(p(T) ? γ) is approximately equal to γ, for 0 ? γ ? 1.     

Goodness‐of‐fit Test for Directional Data

In this paper, we study the problem of testing the hypothesis on whether the density f of a random variable on a sphere belongs to a given parametric class of densities. We propose two test statistics based on the L² and L¹ distances between a non‐parametric density estimator adapted to circular data and a smoothed version of the specified density. The asymptotic distribution of the L² test statistic is provided under the null hypothesis and contiguous alternatives. We also consider a bootstrap method to approximate the distribution of both test statistics. Through a simulation study, we explore the moderate sample performance of the proposed tests under the null hypothesis and under different alternatives. Finally, the procedure is illustrated by analysing a real data set based on wind direction measurements.     

Statistical evidence and surprise unified under possibility theory

Sander Greenland argues that reported results of hypothesis tests should include the surprisal, the base-2 logarithm of the reciprocal of a p-value. The surprisal measures how many bits of evidence in the data warrant rejecting the null hypothesis. A generalization of surprisal also can measure how much the evidence justifies rejecting a composite hypothesis such as the complement of a confidence interval. That extended surprisal, called surprise, quantifies how many bits of astonishment an agent believing a hypothesis would experience upon observing the data. While surprisal is a function of a point in hypothesis space, surprise is a function of a subset of hypothesis space. Satisfying the conditions of conditional min-plus probability, surprise inherits a wealth of tools from possibility theory. The equivalent compatibility function has been recently applied to the replication crisis, to adjusting p-values for prior information, and to comparing scientific theories.     

TESTS FOR DETECTING MORE NBU-NESS AT A SPECIFIC AGE

This paper proposes a class of non‐parametric test procedures for testing the null hypothesis that two distributions, F and G, are equal versus the alternative hypothesis that F is ‘more NBU (new better than used) at specified age t₀’ than G. Using Hoeffding's two‐sample U‐statistic theorem, it establishes the asymptotic normality of the test statistics and produces a class of asymptotically distribution‐free tests. Pitman asymptotic efficacies of the proposed tests are calculated with respect to the location and shape parameters. A numerical example is provided for illustrative purposes.     

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.     

A Projection-Based Nonparametric Test of Conditional Quantile Independence

Abstract

This paper proposes a nonparametric procedure for testing conditional quantile independence using projections. Relative to existing smoothed nonparametric tests, the resulting test statistic: (i) detects the high frequency local alternatives that converge to the null hypothesis in probability at faster rate and, (ii) yields improvements in the finite sample power when a large number of variables are included under the alternative. In addition, it allows the researcher to include qualitative information and, if desired, direct the test against specific subsets of alternatives without imposing any functional form on them. We use the weighted Nadaraya-Watson (WNW) estimator of the conditional quantile function avoiding the boundary problems in estimation and testing and prove weak uniform consistency (with rate) of the WNW estimator for absolutely regular processes. The procedure is applied to a study of risk spillovers among the banks. We show that the methodology generalizes some of the recently proposed measures of systemic risk and we use the quantile framework to assess the intensity of risk spillovers among individual financial institutions.     

Evaluations of FWER-controlling methods in multiple hypothesis testing

Simultaneously testing a family of n null hypotheses can arise in many applications. A common problem in multiple hypothesis testing is to control Type-I error. The probability of at least one false rejection referred to as the familywise error rate (FWER) is one of the earliest error rate measures. Many FWER-controlling procedures have been proposed. The ability to control the FWER and achieve higher power is often used to evaluate the performance of a controlling procedure. However, when testing multiple hypotheses, FWER and power are not sufficient for evaluating controlling procedure’s performance. Furthermore, the performance of a controlling procedure is also governed by experimental parameters such as the number of hypotheses, sample size, the number of true null hypotheses and data structure. This paper evaluates, under various experimental settings, the performance of some FWER-controlling procedures in terms of five indices, the FWER, the false discovery rate, the false non-discovery rate, the sensitivity and the specificity. The results can provide guidance on how to select an appropriate FWER-controlling procedure to meet a study’s objective.