Tests for Homogeneity of Variances Using Robust Weighted Likelihood Estimates

The classical normal-theory tests for testing the null hypothesis of common variance and the classical estimates of scale have long been known to be quite nonrobust to even mild deviations from normality assumptions for moderate sample sizes. Levene (1960) suggested a one-way ANOVA type statistic as a robust test. Brown and Forsythe (1974) considered a modified version of Levene's test by replacing the sample means with sample medians as estimates of population locations, and their test is computationally the simplest among the three tests recommended by Conover , Johnson , and Johnson (1981) in terms of robustness and power. In this paper a new robust and powerful test for homogeneity of variances is proposed based on a modification of Levene's test using the weighted likelihood estimates (Markatou , Basu , and Lindsay , 1996) of the population means. For two and three populations the proposed test using the Hellinger distance based weighted likelihood estimates is observed to achieve better empirical level and power than Brown-Forsythe's test in symmetric distributions having a thicker tail than the normal, and higher empirical power in skew distributions under the use of F distribution critical values.     

Harmonic Mean Approach to Regression Analysis Under Heteroscedasticity and Nonnormality

In this paper, by combining the harmonic mean approach with the Welch and the James procedure (see WELCH 1951, JAMES, 1951), we develop some robust procedures for testing parallelism in several straight lines under heteroscedasticity and nonnormality. Through Monte Carlo simulations it is shown that these new tests are quite robust with respect to departure from normality. For small sample sizes, however, the TAN-TABATABAI (1984) F_β and F*_β tests appear to be more powerful than the new tests, although when sample sizes are not small, there are hardly any differences between the Tan-Tabatabai F_β and F*_β tests and the new tests.     

TESTING FOR PHYLOGENETIC SIGNAL IN BIOLOGICAL TRAITS: THE UBIQUITY OF CROSS‐PRODUCT STATISTICS

To evaluate rates of evolution, to establish tests of correlation between two traits, or to investigate to what degree the phylogeny of a species assemblage is predictive of a trait value so‐called tests for phylogenetic signal are used. Being based on different approaches, these tests are generally thought to possess quite different statistical performances. In this article, we show that the Blomberg et al. K and K*, the Abouheif index, the Moran's I, and the Mantel correlation are all based on a cross‐product statistic, and are thus all related to each other when they are associated to a permutation test of phylogenetic signal. What changes is only the way phylogenetic and trait similarities (or dissimilarities) among the tips of a phylogeny are computed. The definitions of the phylogenetic and trait‐based (dis)similarities among tips thus determines the performance of the tests. We shortly discuss the biological and statistical consequences (in terms of power and type I error of the tests) of the observed relatedness among the statistics that allow tests for phylogenetic signal. Blomberg et al. K* statistic appears as one on the most efficient approaches to test for phylogenetic signal. When branch lengths are not available or not accurate, Abouheif's C_mean statistic is a powerful alternative to K*.     

A novel cross-shear metric for application in computer simulation of ultra-high molecular weight polyethylene wear

Wear testing of polyethylene in total joint replacements is common and required for any new device. Computational wear modelling has obvious utility in this context as it can be conducted with much greater economy than physical testing. Archard's law has become the accepted standard for wear simulation in total joints but it does not account for cross-shear, which is known to increase wear significantly relative to unidirectional sliding. The purpose of this study was to develop a robust cross-shear model applicable to any interface geometry under any kinematic conditions. The proposed metric, x ^*, is distinguished from existing cross-shear models by the fact that it measures cross-path motion incrementally throughout a motion cycle and quantifies cross-shear based on incremental changes in sliding direction. Validation showed strong support for the predictive capability of x ^* when applied to pin-on-disc test data.     

Tests of Occupancy Rates for Clustered Populations: Case Where the Cluster Sizes are Known

A modified chi-squared statistic Z is proposed for testing hypotheses about category occupancy rates for individuals distributed by clusters, when the cluster sizes are observed. This statistic is the Pearson chi-square statistic based on the individuals' counts divided by 1 + M* where M* is the mean number of other individuals per cluster per individual. The kind of alternative hypothesis for which the Z-based test compares favourably in power with the Pearson chi-square test based on the cluster frequencies is given. However, we prove that this latter test is more powerful than the former one as long as the equidistribution of the random choice vectors is assumed.     

An Adaptive Location‐Scale Test

Increasing locations are often accompanied by an increase in variability. In this case apparent heteroscedasticity can indicate that there are treatment effects and it is appropriate to consider an alternative involving differences in location as well as in scale. As a location‐scale test the sum of a location and a scale test statistic can be used. However, the power can be raised through weighting the sum. In order to select values for this weighting an adaptive design with an interim analysis is proposed: The data of the first stage are used to calculate the weights and with the second stage's data a weighted location‐scale test is carried out. The p‐values of the two stages are combined through Fisher's combination test. With a Lepage‐type location‐scale test it is illustrated that the resultant adaptive test can be more powerful than the ‘optimum’ test with no interim analysis. The principle to calculate weights, which cannot be reasonably chosen a priori, with the data of the first stage may be useful for other tests which utilize weighted statistics, too. Furthermore, the proposed test is illustrated with an example from experimental ecology.     

A remark on a paper by misra

MISRA (1978) sets confidence intervals for a double linear compound of multivariate normal regression coefficients by using ROY'S maximum root test criterion. The exact test statistic to be used is STUDENT'S t. The t statistic gives narrower confidence bounds than those given by ROY's maximum root statistic. A result given by MORRISON (1975, p. 18, equation 10) for profile analysis is also obtained by using the STUDENT'S t test.     

Testing Hypotheses about Regression Parameters,When the Error Term Is Heteroscedastic

The paper considers methods for testing H₀: β₁ = … = β_p = 0, where β₁, … ,β_p are the slope parameters in a linear regression model with an emphasis on p = 2. It is known that even when the usual error term is normal, but heteroscedastic, control over the probability of a type I error can be poor when using the conventional F test in conjunction with the least squares estimator. When the error term is nonnormal, the situation gets worse. Another practical problem is that power can be poor under even slight departures from normality. Liu and Singh (1997) describe a general bootstrap method for making inferences about parameters in a multivariate setting that is based on the general notion of depth. This paper studies the small-sample properties of their method when applied to the problem at hand. It is found that there is a practical advantage to using Tukey's depth versus the Mahalanobis depth when using a particular robust estimator. When using the ordinary least squares estimator, the method improves upon the conventional F test, but practical problems remain when the sample size is less than 60. In simulations, using Tukey's depth with the robust estimator gave the best results, in terms of type I errors, among the five methods studied.     

Properties and Application of the Location-Test V (n,k)

The present paper is concerned with the properties of a test statistic V(n, k) to test location differences in the one-sample case with known hypothetical distribution G(x). The test is similar to the WILCOXON two-sample statistic after replacement of the second sample by quantiles of the hypothetical distribution. A comparison with the exact distribution of V(n, k) shows that an approximation by means of the normal distribution provides good results even for small sample sizes. The V-test is unbiased against one-tailed alternatives and it is consistent with a restriction which is hardly relevant in practical applications. With regard to the application we are interested especially in the power and robustness against extreme observations for small sample size n. It is shown that in a normal distribution with known standard deviation V(n, k) is more powerful than STUDENT's t for small n and more robust in the sense considered here. The test statistic is based on grouping of the observations into classes of equal expected frequency. A generalization to arbitrary classes provides an essential extension of applicability such as to discrete distributions and to situations where only relative frequencies of G(x) in fixed classes are known.     

A Robust Genome‐Wide Scan Statistic of the Wellcome Trust Case–Control Consortium

Summary In genome‐wide association (GWA) studies, test statistics that are efficient and robust across various genetic models are preferable, particularly for studying multiple diseases in the Wellcome Trust Case–Control Consortium ( WTCCC, 2007 , Nature 447 , 661–678). A new test statistic, the minimum of the p‐values of the trend test and Pearson's test, was considered by the WTCCC. It is referred to here as MIN2. Because the minimum of two p‐values is no longer a valid p‐value itself, the WTCCC only used it to rank single nucleotide polymorphisms (SNPs) but did not report the p‐values of the associated SNPs when MIN2 was used for ranking. Given its importance in practice, we derive the asymptotic null distribution of MIN2, study some of its analytical properties related to GWA studies, and compare it with existing methods (the trend test, Pearson's test, MAX3, and the constrained likelihood ratio test [CLRT]) by simulations across a wide range of possible genetic models: the recessive (REC), additive (ADD), multiplicative (MUL), dominant (DOM), and overdominant models. The results show that MAX3 and CLRT have greater efficiency robustness than other tests when the REC, ADD/MUL, and DOM models are possible, whereas Pearson's test and MIN2 have greater efficiency robustness if the possible genetic models also include the overdominant model. We conclude that robust tests (MAX3, MIN2, CLRT, and Pearson's test) are preferable to a single trend test for initial GWA studies. The four robust tests are applied to more than 100 SNPs associated with 11 common diseases identified by the two WTCCC GWA studies.     

Overnight Dexamethasone Suppression Test: A Reliable Screen for Cushing's Syndrome in the Obese

Objective: Reevaluation of the validity of the 1-mg overnight dexamethasone suppression test (ODST) as a screening test for Cushing's syndrome in obese patients. Research Methods and Procedures: Eighty-six obese patients (body mass index, 30 to 53 kg/m²) that were referred to a general endocrine outpatient clinic for evaluation of simple obesity, diabetes mellitus, hypertension, polycystic ovary disease, or pituitary tumor. One milligram dexamethasone was administered orally at 11:00 pm , and serum cortisol levels were measured the following morning between 8:00 am and 9:00 am . Suppression of serum cortisol to <80 nM (3 μg/dL) was chosen as the cut-off point for normal suppression. Patients with serum cortisol levels ≥80 nM were evaluated for Cushing's syndrome. Results: Suppression of morning cortisol levels to <80 nM occurred in 79 of the 86 obese patients. Seven patients had serum cortisol levels higher than 80 nM; five were eventually diagnosed with Cushing's syndrome and two were considered false positive results in view of normal 24-hour free urinary cortisol and normal suppression on a low dose dexamethasone suppression test (0.5 mg of dexamethasone every 6 hours for 2 days). We found a false positive rate of 2.3% for the ODST using a cut-off serum cortisol of 80 nM. Discussion: The ODST is a valid screening test for Cushing's syndrome in the obese population. The false positive rate was 2.3%, even when using a strict cut-off serum cortisol of 80 nM. Abnormal cortisol suppression in obese patients should be investigated and not be considered false positive results.     

Multiple Comparisons of Polynomial Distributions

Asymptotically correct 90 and 95 percentage points are given for multiple comparisons with control and for all pair comparisons of several independent samples of equal size from polynomial distributions. Test statistics are the maxima of the X²-statistics for single comparisons. For only two categories the asymptotic distributions of these test statistics result from DUNNETT'S many-one tests and TUKEY'S range test (cf. MILLER, 1981). The percentage points for comparisons with control are computed from the limit distribution of the test statistic under the overall hypothesis H₀. To some extent the applicability of these bounds is investigated by simulation. The bounds can also be used to improve Holm's sequentially rejective Bonferroni test procedure (cf. HOLM, 1979). The percentage points for all pair comparisons are obtained by large simulations. Especially for 3×3-tables the limit distribution of the test statistic under H₀ is derived also for samples of unequal size. Also these bounds can improve the corresponding Bonferroni-Holm procedure. Finally from SKIDÁK's probability inequality for normal random vectors (cf. SKIDÁK, 1967) a similar inequality is derived for dependent X²-variables applicable to simultaneous X²-tests.     

On Testing Equality of Means of Correlated Variates with Incomplete Data

A statistic is proposed for testing the hypothesis of equality of the means of a bivariate normal distribution with unknown common variance and correlation coefficient when observations are missing on one of the variates. Expressions for the second and fourth central moments of the statistic are obtained. These moments are used to approximate the distribution of the statistic by a Student's t distribution under the null hypothesis. The powers of the test are computed and compared with those of the conventional paired t and the other known statistics.     

Sample Size Determination for Designing a Strata-Matched Case-Control Study to Detect Multiple Risk Factors

Investigations of sample size for planning case-control studies have usually been limited to detecting a single factor. In this paper, we investigate sample size for multiple risk factors in strata-matched case-control studies. We construct an omnibus statistic for testing M different risk factors based on the jointly sufficient statistics of parameters associated with the risk factors. The statistic is non-iterative, and it reduces to the Cochran statistic when M = 1. The asymptotic power function of the test is a non-central chi-square with M degrees of freedom and the sample size required for a specific power can be obtained by the inverse relationship. We find that the equal sample allocation is optimum. A Monte Carlo experiment demonstrates that an approximate formula for calculating sample size is satisfactory in typical epidemiologic studies. An approximate sample size obtained using Bonferroni's method for multiple comparisons is much larger than that obtained using the omnibus test. Approximate sample size formulas investigated in this paper using the omnibus test, as well as the individual tests, can be useful in designing case-control studies for detecting multiple risk factors.     

Some Simple Statistics to Test Against Ordered Alternatives in Complete Block Designs

Consider testing the hypothesis of no treatment effects against a postulated ranking of the m treatments, given data from n Complete Blocks. A suitable test statistic is the weighted average rank correlation w = σb_QiC_i where C_i is the correlation between the postulated ranking and the ranking observed within the ith block, Q_i is the rank of the ith block with respect to credibility, and the b_i's are weights such that 0 ≦ b¹ ≦ … ≦ bn. In this paper we introduce some simple statistics: the first extends the signed-rank statistic to m ≦ 3, the second uses a simple measure of correlation based on the antirank, and the third a statistic based on Spearman's footrule. Tables for critical values are provided and the normal approximation is investigated.     

Nonparametric Tests for Trend: Jonckheere's Test,a Modification and a Maximum Test

Jonckheere's test is a frequently used nonparametric trend test for the evaluation of preclinical studies and clinical dose-finding trials. In this paper, a modification of Jonckheere's test is proposed. If the exact permutation distribution is used for inference, the modified test can fill out the level of the type I error in a much more complete way and is substantially more powerful than the common Jonckheere test. If the asymptotic normality is used for inference, the modified test is slightly more powerful. In addition, a maximum test is investigated which is more robust concerning an a priori unknown dose-response shape. The robustness is advantageous, especially in a closed testing procedure. The different tests are applied to two example data sets.     

A powerful and robust test statistic for randomization inference in group-randomized trials with matched pairs of groups

For group-randomized trials, randomization inference based on rank statistics provides robust, exact inference against nonnormal distributions. However, in a matched-pair design, the currently available rank-based statistics lose significant power compared to normal linear mixed model (LMM) test statistics when the LMM is true. In this article, we investigate and develop an optimal test statistic over all statistics in the form of the weighted sum of signed Mann-Whitney-Wilcoxon statistics under certain assumptions. This test is almost as powerful as the LMM even when the LMM is true, but it is much more powerful for heavy tailed distributions. A simulation study is conducted to examine the power.     

Chop-Lump Tests for Vaccine Trials

Summary This article proposes new tests to compare the vaccine and placebo groups in randomized vaccine trials when a small fraction of volunteers become infected. A simple approach that is consistent with the intent‐to‐treat principle is to assign a score, say W, equal to 0 for the uninfecteds and some postinfection outcome X > 0 for the infecteds. One can then test the equality of this skewed distribution of W between the two groups. This burden of illness (BOI) test was introduced by Chang, Guess, and Heyse (1994, Statistics in Medicine 13 , 1807–1814). If infections are rare, the massive number of 0s in each group tends to dilute the vaccine effect and this test can have poor power, particularly if the X's are not close to zero. Comparing X in just the infecteds is no longer a comparison of randomized groups and can produce misleading conclusions. Gilbert, Bosch, and Hudgens (2003, Biometrics 59 , 531–541) and Hudgens, Hoering, and Self (2003, Statistics in Medicine 22 , 2281–2298) introduced tests of the equality of X in a subgroup—the principal stratum of those “doomed” to be infected under either randomization assignment. This can be more powerful than the BOI approach, but requires unexaminable assumptions. We suggest new “chop‐lump” Wilcoxon and t‐tests (CLW and CLT) that can be more powerful than the BOI tests in certain situations. When the number of volunteers in each group are equal, the chop‐lump tests remove an equal number of zeros from both groups and then perform a test on the remaining W's, which are mostly >0. A permutation approach provides a null distribution. We show that under local alternatives, the CLW test is always more powerful than the usual Wilcoxon test provided the true vaccine and placebo infection rates are the same. We also identify the crucial role of the “gap” between 0 and the X's on power for the t‐tests. The chop‐lump tests are compared to established tests via simulation for planned HIV and malaria vaccine trials. A reanalysis of the first phase III HIV vaccine trial is used to illustrate the method.     

Testing the Equality of Dependent Variances

The likelihood ratio test for testing equality of vgE;2 correlated variables is developed. In general, evaluation of the test statistic involves the iterative optimization of a likelihood function with 1 + v(v – 1)/2 parameters. The explicit form of the test statistic is derived in the bivariate case, and an iterative algorithm for determining the maximum likelihood estimates is suggested. A limited Monte Carlo study determines the behavior of the proposed procedure under the null hypothesis and variety of parameter values.     

Nonparametric,GINI-like Measures of Location and Dispersion

GINI's nonparametric measure of dispersion, defined as the mean of (n/2) pairs of a sample of n measurements, is modified for triplets. That way a GINI-like measure of location G* is defined as the mean of the (n/3) triplet medians, and a GINI-like measure of dispersion is defined as the mean of the (n/3) triplet ranges. G* and g* are less sensitive to outlier measurements than are the corresponding parametric measures, arithmetic mean and standard deviation, in small samples. Applications and application-related problems are discussed.