Testing Equality of Order Constrained Inverse Gaussian Means

The inverse Gaussian family of non negative, skewed random variables is analytically simple, and its inference theory is well known to be analogous to the normal theory in numerous ways. Hence, it is widely used for modeling non negative positively skewed data. In this note, we consider the problem of testing homogeneity of order restricted means of several inverse Gaussian populations with a common unknown scale parameter using an approach based on the classical methods, such as Fisher's, for combining independent tests. Unlike the likelihood approach which can only be readily applied to a limited number of restrictions and the settings of equal sample sizes, this approach is applicable to problems involving a broad variety of order restrictions and arbitrary sample size settings, and most importantly, no new null distributions are needed. An empirical power study shows that, in case of the simple order, the test based on Fisher's combination method compares reasonably with the corresponding likelihood ratio procedure.     

Sample Size Determination for Logistic Regression: A Simulation Study

This article considers the different methods for determining sample sizes for Wald, likelihood ratio, and score tests for logistic regression. We review some recent methods, report the results of a simulation study comparing each of the methods for each of the three types of test, and provide Mathematica code for calculating sample size. We consider a variety of covariate distributions, and find that a calculation method based on a first order expansion of the likelihood ratio test statistic performs consistently well in achieving a target level of power for each of the three types of test.     

Tests for symmetry of two-piece normal distribution

In this paper, classical optimum tests for symmetry of two-piece normal distribution is derived. Uniformly most powerful one-sided test for the skewness parameter is obtained when the location and scale parameters are known and is compared with sequential probability ratio test. An ad-hoc test for symmetry and likelihood ratio test for symmetry for large samples, can be found in literature for this distribution. But in this paper, we derive exact likelihood ratio test for symmetry, when location parameter is known. The exact power of the test is evaluated for different sample sizes.     

Some count-based nonparametric tests for circular symmetry of a bivariate distribution

In this paper, we revisit the problem of testing of the hypothesis of circular symmetry of a bivariate distribution. We propose some nonparametric tests based on sector counts. These include tests based on chi-square goodness-of-fit test, the classical likelihood ratio, mean deviation, and the range. The proposed tests are easy to implement and the exact null distributions for small sample sizes of the test statistics are obtained. Two examples with small and large data sets are given to illustrate the application of the tests proposed. For small and moderate sample sizes, the performances of the proposed tests are evaluated using empirical powers (empirical sizes are also reported). Also, we evaluate the performance of these count-based tests with adaptations of several well-known tests such as the Kolmogorov–Smirnov-type tests, tests based on kernel density estimator, and the Wilcoxon-type tests. It is observed that among the count-based tests the likelihood ratio test performs better.     

On bootstrap testing inference in cure rate models

Survival models deal with the time until the occurrence of an event of interest. However, in some situations the event may not occur in part of the studied population. The fraction of the population that will never experience the event of interest is generally called cure rate. Models that consider this fact (cure rate models) have been extensively studied in the literature. Hypothesis testing on the parameters of these models can be performed based on likelihood ratio, gradient, score or Wald statistics. Critical values of these tests are obtained through approximations that are valid in large samples and may result in size distortion in small or moderate sample sizes. In this sense, this paper proposes bootstrap corrections to the four mentioned tests and bootstrap Bartlett correction for the likelihood ratio statistic in the Weibull promotion time model. Besides, we present an algorithm for bootstrap resampling when the data presents cure fraction and right censoring time (random and non-informative). Simulation studies are conducted to compare the finite sample performances of the corrected tests. The numerical evidence favours the corrected tests we propose. We also present an application in an actual data set.     

Statistical analysis of non-inferiority via non-zero risk difference in stratified matched-pair studies

A stratified study is often designed for adjusting several independent trials in modern medical research. We consider the problem of non-inferiority tests and sample size determinations for a nonzero risk difference in stratified matched-pair studies, and develop the likelihood ratio and Wald-type weighted statistics for testing a null hypothesis of non-zero risk difference for each stratum in stratified matched-pair studies on the basis of (1) the sample-based method and (2) the constrained maximum likelihood estimation (CMLE) method. Sample size formulae for the above proposed statistics are derived, and several choices of weights for Wald-type weighted statistics are considered. We evaluate the performance of the proposed tests according to type I error rates and empirical powers via simulation studies. Empirical results show that (1) the likelihood ratio and the Wald-type CMLE test based on harmonic means of the stratum-specific sample size (SSIZE) weight (the Cochran's test) behave satisfactorily in the sense that their significance levels are much closer to the prespecified nominal level; (2) the likelihood ratio test is better than Nam's [2006. Non-inferiority of new procedure to standard procedure in stratified matched-pair design. Biometrical J. 48, 966–977] score test; (3) the sample sizes obtained by using SSIZE weight are smaller than other weighted statistics in general; (4) the Cochran's test statistic is generally much better than other weighted statistics with CMLE method. A real example from a clinical laboratory study is used to illustrate the proposed methodologies.     

A likelihood ratio test of a homoscedastic normal mixture against a heteroscedastic normal mixture

It is generally assumed that the likelihood ratio statistic for testing the null hypothesis that data arise from a homoscedastic normal mixture distribution versus the alternative hypothesis that data arise from a heteroscedastic normal mixture distribution has an asymptotic χ ² reference distribution with degrees of freedom equal to the difference in the number of parameters being estimated under the alternative and null models under some regularity conditions. Simulations show that the χ ² reference distribution will give a reasonable approximation for the likelihood ratio test only when the sample size is 2000 or more and the mixture components are well separated when the restrictions suggested by Hathaway (Ann. Stat. 13:795–800, 1985) are imposed on the component variances to ensure that the likelihood is bounded under the alternative distribution. For small and medium sample sizes, parametric bootstrap tests appear to work well for determining whether data arise from a normal mixture with equal variances or a normal mixture with unequal variances.     

The Slippage Problem for the Circular Normal Distribution

The slippage problem occurs when an unspecified observation in a given random sample is from a distribution other than that for all the remaining observations. This paper studies the problem in terms of the 'slip' in the mean direction of a circular normal distribution. The slippage problem is first treated as a multiple decision problem with a prior which is invariant under the permutations of the hypotheses. The probabilities of accepting the various hypotheses for the Bayes rule with respect to this prior are explicitly obtained. The likelihood ratio tests for this slippage problem, for the cases when the mean directions are both known and unknown, are shown to be easily computable. The tests are illustrated through two well-known datasets. The performances of a range of tests are compared using extensive simulation.     

Rényi statistics for testing hypotheses in mixed linear regression models

Rényi divergences are used to propose some statistics for testing general hypotheses in mixed linear regression models. The asymptotic distribution of these tests statistics, of the Kullback–Leibler and of the likelihood ratio statistics are provided, assuming that the sample size and the number of levels of the random factors tend to infinity. A simulation study is carried out to analyze and compare the behavior of the proposed tests when the sample size and number of levels are small.     

Testing for multivariate heteroscedasticity

In this article, we propose a testing technique for multivariate heteroscedasticity, which is expressed as a test of linear restrictions in a multivariate regression model. Four test statistics with known asymptotical null distributions are suggested, namely the Wald, Lagrange multiplier (LM), likelihood ratio (LR) and the multivariate Rao F-test. The critical values for the statistics are determined by their asymptotic null distributions, but bootstrapped critical values are also used. The size, power and robustness of the tests are examined in a Monte Carlo experiment. Our main finding is that all the tests limit their nominal sizes asymptotically, but some of them have superior small sample properties. These are the F, LM and bootstrapped versions of Wald and LR tests.     

Optimal testing in a three way anova model

We consider likelihood ratio, score and Wald tests for a three-way random effects ANOVA model. Competitor tests are compared using criteria such as small sample power, asymptotic relative efficiency, and convenient null distribution. The final choice is between a new test and two tests long used in practice.     

How does the DerSimonian and Laird procedure for random effects meta-analysis compare with its more efficient but harder to compute counterparts?

The procedure suggested by DerSimonian and Laird is the simplest and most commonly used method for fitting the random effects model for meta-analysis. Here it is shown that, unless all studies are of similar size, this is inefficient when estimating the between-study variance, but is remarkably efficient when estimating the treatment effect. If formal inference is restricted to statements about the treatment effect, and the sample size is large, there is little point in implementing more sophisticated methodology. However, it is further demonstrated, for a simple special case, that use of the profile likelihood results in actual coverage probabilities for 95% confidence intervals that are closer to nominal levels for smaller sample sizes. Alternative methods for making inferences for the treatment effect may therefore be preferable if the sample size is small, but the DerSimonian and Laird procedure retains its usefulness for larger samples.     

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.     

Monte carlo sampling approach to testing nonnested hypothesis: monte carlo results

Alternative ways of using Monte Carlo methods to implement a Cox-type test for separate families of hypotheses are considered. Monte Carlo experiments are designed to compare the finite sample performances of Pesaran and Pesaran's test, a RESET test, and two Monte Carlo hypothesis test procedures. One of the Monte Carlo tests is based on the distribution of the log-likelihood ratio and the other is based on an asymptotically pivotal statistic. The Monte Carlo results provide strong evidence that the size of the Pesaran and Pesaran test is generally incorrect, except for very large sample sizes. The RESET test has lower power than the other tests. The two Monte Carlo tests perform equally well for all sample sizes and are both clearly preferred to the Pesaran and Pesaran test, even in large samples. Since the Monte Carlo test based on the log-likelihood ratio is the simplest to calculate, we recommend using it.     

Simple and Exact Empirical Likelihood Ratio Tests for Normality Based on Moment Relations

The empirical likelihood (EL) technique is a powerful nonparametric method with wide theoretical and practical applications. In this article, we use the EL methodology in order to develop simple and efficient goodness-of-fit tests for normality based on the dependence between moments that characterizes normal distributions. The new empirical likelihood ratio (ELR) tests are exact and are shown to be very powerful decision rules based on small to moderate sample sizes. Asymptotic results related to the Type I error rates of the proposed tests are presented. We present a broad Monte Carlo comparison between different tests for normality, confirming the preference of the proposed method from a power perspective. A real data example is provided.     

A likelihood ratio test of quasi-independence for sparse two-way contingency tables

We consider a likelihood ratio test of independence for large two-way contingency tables having both structural (non-random) and sampling (random) zeros in many cells. The solution of this problem is not available using standard likelihood ratio tests. One way to bypass this problem is to remove the structural zeroes from the table and implement a test on the remaining cells which incorporate the randomness in the sampling zeros; the resulting test is a test of quasi-independence of the two categorical variables. This test is based only on the positive counts in the contingency table and is valid when there is at least one sampling (random) zero. The proposed (likelihood ratio) test is an alternative to the commonly used ad hoc procedures of converting the zero cells to positive ones by adding a small constant. One practical advantage of our procedure is that there is no need to know if a zero cell is structural zero or a sampling zero. We model the positive counts using a truncated multinomial distribution. In fact, we have two truncated multinomial distributions; one for the null hypothesis of independence and the other for the unrestricted parameter space. We use Monte Carlo methods to obtain the maximum likelihood estimators of the parameters and also the p-value of our proposed test. To obtain the sampling distribution of the likelihood ratio test statistic, we use bootstrap methods. We discuss many examples, and also empirically compare the power function of the likelihood ratio test relative to those of some well-known test statistics.     

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.     

The likelihood ratio test for high-dimensional linear regression model

The paper considers a significance test of regression variables in the high-dimensional linear regression model when the dimension of the regression variables p, together with the sample size n, tends to infinity. Under two sightly different cases, we proved that the likelihood ratio test statistic will converge in distribution to a Gaussian random variable, and the explicit expressions of the asymptotical mean and covariance are also obtained. The simulations demonstrate that our high-dimensional likelihood ratio test method outperforms those using the traditional methods in analyzing high-dimensional data.     

SIZE CHARACTERISTICS OF TESTS FOR SAMPLE SELECTION BIAS: A MONTE CARLO COMPARISON AND EMPIRICAL EXAMPLE

The t-test of an individual coefficient is used widely in models of qualitative choice. However, it is well known that the t-test can yield misleading results when the sample size is small. This paper provides some experimental evidence on the finite sample properties of the t-test in models with sample selection biases, through a comparison of the t-test with the likelihood ratio and Lagrange multiplier tests, which are asymptotically equivalent to the squared t-test. The finite sample problems with the t-test are shown to be alarming, and much more serious than in models such as binary choice models. An empirical example is also presented to highlight the differences in the calculated test statistics.     

BARTLETT CORRECTED LIKELIHOOD RATIO TESTS IN LOCATION-SCALE NONLINEAR MODELS

The paper derives Bartlett corrections for improving the chisquare approximation to the likelihood ratio statistics in a class of location-scale family of distributions, which encompasses the elliptical family of distributions and also asymmetric distributions such as the extreme value distributions. We present, in matrix notation, a Bartlett corrected likelihood ratio statistic for testing that a subset of the nonlinear regression coefficients in this class of models equals a given vector of constants. The formulae derived are simple enough to be used analytically to obtain several Bartlett corrections in a variety of important models. We show that these formulae generalize a number of previously published results. We also present simulation results comparing the sizes and powers of the usual likelihood ratio tests and their Bartlett corrected versions when the scale parameter is considered known and when this parameter is uncorrectly specified.