The equivalence between likelihood ratio test and F-test for testing variance component in a balanced one-way random effects model

In mixed linear models, it is frequently of interest to test hypotheses on the variance components. F-test and likelihood ratio test (LRT) are commonly used for such purposes. Current LRTs available in literature are based on limiting distribution theory. With the development of finite sample distribution theory, it becomes possible to derive the exact test for likelihood ratio statistic. In this paper, we consider the problem of testing null hypotheses on the variance component in a one-way balanced random effects model. We use the exact test for the likelihood ratio statistic and compare the performance of F-test and LRT. Simulations provide strong support of the equivalence between these two tests. Furthermore, we prove the equivalence between these two tests mathematically.     

Some pitfalls of tests of separate families of hypotheses: normality vs. lognormality

It is often desirable to test non-nested hypotheses. Cox (1961, 1962) proposed forming a log-likelihood ratio from their maxima and then comparing this value to its expected value under the null hypothesis. Pitfalls exists when we apply Cox's test to the special case of testing normality versus lognormality. Pesaran (1981) and Kotz (1973) pointed out the slow convergence rate of the Cox's test. In this paper, this fact has been reemphasized; moreover, we propose an alternative likelihood ratio test which remedies problems arising from negative estimates of the asymptotic variance of Cox's test statistic and is uniformly more powerful than most commonly used tests.     

Seme distributions and their implications for an internal pilot study with a univariate linear model

In planning a study, the choice of sample size may depend on a variance value based on speculation or obtained from an earlier study. Scientists may wish to use an internal pilot design to protect themselves against an incorrect choice of variance. Such a design involves collecting a portion of the originally planned sample and using it to produce a new variance estimate. This leads to a new power analysis and increasing or decreasing sample size. For any general linear univariate model, with fixed predictors and Gaussian errors, we prove that the uncorrected fixed sample F-statistic is the likelihood ratio test statistic. However, the statistic does not follow an F distribution. Ignoring the discrepancy may inflate test size. We derive and evaluate properties of the components of the likelihood ratio test statistic in order to characterize and quantify the bias. Most notably, the fixed sample size variance estimate becomes biased downward. The bias may inflate test size for any hypothesis test, even if the parameter being tested was not involved in the sample size re-estimation. Furthermore, using fixed sample size methods may create biased confidence intervals for secondary parameters and the variance estimate.     

Testing Linear Regression Models in Non Regular Case

We propose a new statistic for testing linear hypotheses in the non parametric regression model in the case of a homoscedastic error structure and fixed design. In contrast to most models suggested in the literature, our procedure is applicable in the non parametric model case without regularity condition, and also under either the null or the alternative hypotheses. We show the asymptotic normality of the test statistic under the null hypothesis and the alternative one. A simulation study is conducted to investigate the finite sample properties of the test with application to regime switching.     

A two sample ordered alternative test for means and variances

A two sample test of likelihood ratio type is proposed, assuming normal distribution theory, for testing the hypothesis that two samples come from identical normal populations versus the alternative that the populations are normal but vary in mean value and variance with one population having a smaller mean and smaller variance than the other. The small sample and large sample distribution of the proposed statistic are derived assuming normality. Some computations are presented which show the speed of convergence of small sample critical values to their asymptotic counterparts. Comparisons of local power of the proposed test are made with several potential competing tests. Asymptotics for the test statistic are derived when underlying distributions are not necessarily normal.     

Sample size calculation for the one‐sample log‐rank test

In this paper, an exact variance of the one‐sample log‐rank test statistic is derived under the alternative hypothesis, and a sample size formula is proposed based on the derived exact variance. Simulation results showed that the proposed sample size formula provides adequate power to design a study to compare the survival of a single sample with that of a standard population. Copyright © 2014 John Wiley & Sons, Ltd.     

Structural changes and unit roots in non-stationary time series

The problem of testing hypotheses of a unit root and a structural change in one-dimensional time series is considered. A non-parametric two-step method for solution of the problem is proposed. The method is based upon the modified Kolmogorov-Smirnov statistic. At the first step of this method the hypothesis of stationarity of an obtained sample is tested against a unified alternative of a statistical non-stationarity of a time series (a unit root or a structural change). At the second step of the proposed method, in case of rejecting the stationarity hypothesis at the first step, the hypothesis of an unknown structural change is tested against the alternative of a unit root. We prove that probabilities of errors (false classification of hypotheses) of the proposed method converge to zero as the sample size tends to infinity.     

Likelihood ratio test against simple stochastic ordering among several multinomial populations

Stochastic ordering between probability distributions has been widely studied in the past 50 years. Because it is often easy to make valuable judgments when such orderings exist, it is desirable to recognize their existence and to model distributional structures under them. Likelihood ratio test is the most commonly used method to test hypotheses involving stochastic orderings. Among the various formally defined notions of stochastic ordering, the least stringent is simple stochastic ordering. In this paper, we consider testing the hypothesis that all multinomial populations are identically distributed against the alternative that they are in simple stochastic ordering. We construct likelihood ratio test statistic for this hypothesis test problem, provide limit form of the objective function corresponding to the test statistic and show that the test statistic is asymptotically distributed as a mixture of chi-squared distributions, i.e., a chi-bar-squared distribution.     

Variance Shifts,Structural Breaks,and Stationarity Tests

This article considers the problem of testing the null hypothesis of stochastic stationarity in time series characterized by variance shifts at some (known or unknown) point in the sample. It is shown that existing stationarity tests can be severely biased in the presence of such shifts, either oversized or undersized, with associated spurious power gains or losses, depending on the values of the breakpoint parameter and on the ratio of the prebreak to postbreak variance. Under the assumption of a serially independent Gaussian error term with known break date and known variance ratio, a locally best invariant (LBI) test of the null hypothesis of stationarity in the presence of variance shifts is then derived. Both the test statistic and its asymptotic null distribution depend on the breakpoint parameter and also, in general, on the variance ratio. Modifications of the LBI test statistic are proposed for which the limiting distribution is independent of such nuisance parameters and belongs to the family of Cramér–von Mises distributions. One such modification is particularly appealing in that it is simultaneously exact invariant to variance shifts and to structural breaks in the slope and/or level of the series. Monte Carlo simulations demonstrate that the power loss from using our modified statistics in place of the LBI statistic is not large, even in the neighborhood of the null hypothesis, and particularly for series with shifts in the slope and/or level. The tests are extended to cover the cases of weakly dependent error processes and unknown breakpoints. The implementation of the tests are illustrated using output, inflation, and exchange rate data series.     

A random-sum Wilcoxon statistic and its application to analysis of ROC and LROC data

The Wilcoxon-Mann-Whitney statistic is commonly used for a distribution-free comparison of two groups. One requirement for its use is that the sample sizes of the two groups are fixed. This is violated in some of the applications such as medical imaging studies and diagnostic marker studies; in the former, the violation occurs since the number of correctly localized abnormal images is random, while in the latter the violation is due to some subjects not having observable measurements. For this reason, we propose here a random-sum Wilcoxon statistic for comparing two groups in the presence of ties, and derive its variance as well as its asymptotic distribution for large sample sizes. The proposed statistic includes the regular Wilcoxon rank-sum statistic. Finally, we apply the proposed statistic for summarizing location response operating characteristic data from a liver computed tomography study, and also for summarizing diagnostic accuracy of biomarker data.     

The generalized Cucconi test statistic for the two-sample problem

When testing hypotheses in two-sample problem, the Lepage test statistic is often used to jointly test the location and scale parameters, and this test statistic has been discussed by many authors over the years. Since two-sample nonparametric testing plays an important role in biometry, the Cucconi test statistic is generalized to the location, scale, and location–scale parameters in two-sample problem. The limiting distribution of the suggested test statistic is derived under the hypotheses. Deriving the exact critical value of the test statistic is difficult when the sample sizes are increased. A gamma approximation is used to evaluate the upper tail probability for the proposed test statistic given finite sample sizes. The asymptotic efficiencies of the proposed test statistic are determined for various distributions. The consistency of the original Cucconi test statistic is shown on the specific cases. Finally, the original Cucconi statistic is discussed in the theory of ties.     

Sample Size Calculation for the Therapeutic Equivalence Problem

A method is proposed for the sample size calculation in the case of therapeutic equivalence of two pharmaceuticals, when the decision is based on post-treatment differences and the post-treatment values are dependent on the pretreatment ones. When the correlation coefficient is large (over 0.7), it is shown that sample size calculation (and the corresponding hypothesis test) based on the sample statistic formed by the mean difference of the post–pre differences of each group has smaller variance and hence leads to smaller sample sizes.     

An analysis of the type I error in the one sample Student t-test

In the one-sample Student t-test, the occurrence of a type-I error is dependent on the estimates of the mean and standard deviation for a fixed sample size, n. The statistic can achieve significance either by the sample mean being too different from the hypothesized mean or by the sample standard deviation being too small. The critical region is partitioned to determine the characteristics of samples in the critical region, assuming the null hypothesis is true. As might be conjectured from the use of the t-statistic, mis-estimation of the mean is shown to be the predominant characteristic of samples in the critical region for sample sizes larger than 20 and significance level greater than 0.01. Underestimation of the variance, unless accompanied by a misestimation of the mean, is a far less frequent characteristic of critical region samples.     

A class of multivariate distribution-free tests of independence based on graphs

A class of distribution-free tests is proposed for the independence of two subsets of response coordinates. The tests are based on the pairwise distances across subjects within each subset of the response. A complete graph is induced by each subset of response coordinates, with the sample points as nodes and the pairwise distances as the edge weights. The proposed test statistic depends only on the rank order of edges in these complete graphs. The response vector may be of any dimensions. In particular, the number of samples may be smaller than the dimensions of the response. The test statistic is shown to have a normal limiting distribution with known expectation and variance under the null hypothesis of independence. The exact distribution free null distribution of the test statistic is given for a sample of size 14, and its Monte-Carlo approximation is considered for larger sample sizes. We demonstrate in simulations that this new class of tests has good power properties for very general alternatives.     

Comparison of unweighted and weighted rank based tests for an ordered alternative in randomized complete block designs

In randomized complete block designs, a monotonic relationship among treatment groups may already be established from prior information, e.g., a study with different dose levels of a drug. The test statistic developed by Page and another from Jonckheere and Terpstra are two unweighted rank based tests used to detect ordered alternatives when the assumptions in the traditional two-way analysis of variance are not satisfied. We consider a new weighted rank based test by utilizing a weight for each subject based on the sample variance in computing the new test statistic. The new weighted rank based test is compared with the two commonly used unweighted tests with regard to power under various conditions. The weighted test is generally more powerful than the two unweighted tests when the number of treatment groups is small to moderate.     

Statistical inference of risk difference in K correlated 2×2 tables with structural zero

K correlated 2×2 tables with structural zero are commonly encountered in infectious disease studies. A hypothesis test for risk difference is considered in K independent 2×2 tables with structural zero in this paper. Score statistic, likelihood ratio statistic and Wald‐type statistic are proposed to test the hypothesis on the basis of stratified data and pooled data. Sample size formulae are derived for controlling a pre‐specified power or a pre‐determined confidence interval width. Our empirical results show that score statistic and likelihood ratio statistic behave better than Wald‐type statistic in terms of type I error rate and coverage probability, sample sizes based on stratified test are smaller than those based on the pooled test in the same design. A real example is used to illustrate the proposed methodologies. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust weighted one-way ANOVA: Improved approximation and efficiency

A robust test for the one-way ANOVA model under heteroscedasticity is developed in this paper. The data are assumed to be symmetrically distributed, apart from some outliers, although the assumption of normality may be violated. The test statistic to be used is a weighted sum of squares similar to the Welch [1951. On the comparison of several mean values: an alternative approach. Biometrika 38, 330-336.] test statistic, but any of a variety of robust measures of location and scale for the populations of interest may be used instead of the usual mean and standard deviation. Under the commonly occurring condition that the robust measures of location and scale are asymptotically normal, we derive approximations to the distribution of the test statistic under the null hypothesis and to its distribution under alternative hypotheses. An expression for relative efficiency is derived, thus allowing comparison of the efficiency of the test as a function of the choice of the location and scale estimators used in the test statistic. As an illustration of the theory presented here, we apply it to three commonly used robust location–scale estimator pairs: the trimmed mean with the Winsorized standard deviation; the Huber Proposal 2 estimator pair; and the Hampel robust location estimator with the median absolute deviation.     

Testtesting for the equality of scale parameters of k(> 2) exponential populations based on complete and type ii censored samples

This paper is concerned with testing the equality of scale parameters of K(> 2) two-parameter exponential distributions in presence of unspecified location parameters based on complete and type II censored samples. We develop a marginal likelihood ratio statistic, a quadratic statistic (Qu) (Nelson, 1982) based on maximum marginal likelihood estimates of the scale parameters under the null and the alternative hypotheses, a C(a) statistic (CPL) (Neyman, 1959) based on the profile likelihood estimate of the scale parameter under the null hypothesis and an extremal scale parameter ratio statistic (ESP) (McCool, 1979). We show that the marginal likelihood ratio statistic is equivalent to the modified Bartlett test statistic. We use Bartlett's small sample correction to the marginal likelihood ratio statistic and call it the modified marginal likelihood ratio statistic (MLB). We then compare the four statistics, MLBi Qut CPL and ESP in terms of size and power by using Monte Carlo simulation experiments. For the variety of sample sizes and censoring combinations and nominal levels considered the statistic MLB holds nominal level most accurately and based on empirically calculated critical values, this statistic performs best or as good as others in most situations. Two examples are given.     

Testing transition probability matrix of a multi-state model with censored data

In this paper, we develop procedures to test hypotheses concerning transition probability matrices arising from certain nonhomogeneous Markov processes. It is assumed that the data consist of sample paths, some of which are observed until a certain terminal state, and the other paths are censored. Problems of this type arise in the context of multi-state models relevant to Health Related Quality of Life (HRQoL) and Competing Risks. The test statistic is based on the estimator for the associated intensity matrix. We show that the asymptotic null distribution of the proposed statistic is Gaussian, and demonstrate how the procedure can be adopted for HRQoL studies and competing risks model using real data sets. Finally, we establish that the test statistic for the HRQoL has greatest local asymptotic power against a sequence of proportional hazards alternatives converging to the null hypothesis.     

Confidence interval estimation and hypothesis testing for a common correlation coefficient

Several procedures have been proposed for testing the hypothesis that all off-diagonal elements of the correlation matrix of a multivariate normal distribution are equal. If the hypothesis of equal correlation can be accepted, it is then of interest to estimate and perhaps test hypotheses for the common correlation. In this paper, two versions of five different test statistics are compared via simulation in terms of adequacy of the normal approximation, coverage probabilities of confidence intervals, control of Type I error, and power. The results indicate that two test statistics based on the average of the Fisher z-transforms of the sample correlations should be used in most cases. A statistic based on the sample eigenvalues also gives reasonable results for confidence intervals and lower-tailed tests.