Two-sided Alternatives Research Articles

Shifted BH methods for controlling false discovery rate in multiple testing of the means of correlated normals against two-sided alternatives

For simultaneous testing of multivariate normal means with known correlation matrix against two-sided alternatives, this paper introduces new methods with proven finite-sample control of false discovery rate. The methods are obtained by shifting each p-value to the left and considering a Benjamini–Hochberg-type linear step-up procedure based on these shifted p-values. The amount of shift for each p-value is appropriately determined from the correlation matrix to achieve the desired false discovery rate control. Simulation studies and real-data application show favorable performances of the proposed methods when compared with relevant competitors.

A two-sided SPRT control chart for process mean

ABSTRACT A two-sided sequential probability ratio test (SPRT) control chart for monitoring the mean of a normal process is proposed in this paper. Its mechanism is based on the SPRT for the mean of a normal distribution that is extended for a two-sided alternative. The exact statistical performance measures of this chart are obtained using a Markov chain approach. An approach to design the chart such that it has a specified in-control performance and the optimal out-of-control performance for detecting a mean shift of a particular size is provided. The statistical performance of this chart is numerically compared with that of various competing charts in the existing literature. It is found that the proposed chart outperforms all these charts for detecting a mean shift of any size. An example to illustrate the mechanism of the chart is provided.

Fast nonasymptotic testing and support recovery for large sparse Toeplitz covariance matrices

We consider n independent p-dimensional Gaussian vectors with covariance matrix having Toeplitz structure. The aim is two-fold: to test that these vectors have independent components against a stationary distribution with sparse Toeplitz covariance matrix, and also to select the support of non-zero entries under the alternative hypothesis. Our model assumes that the non-zero values occur in the recent past (time-lag less than p/2). We build test procedures that combine a sum and a scan-type procedure, but are computationally fast, and show their non-asymptotic behaviour in both one-sided (only positive correlations) and two-sided alternatives, respectively. We also exhibit a selector of significant lags and bound the Hamming-loss risk of the estimated support. These results can be extended to the case of nearly Toeplitz covariance structure and to sub-Gaussian vectors. Numerical results illustrate the excellent behaviour of both test procedures and support selectors — larger the dimension p, faster are the rates.

Twenty-one-gene recurrence score (RS) in germline (g)CHEK2 mutation-associated versus sporadic breast cancers (BC): A multi-site case-control study.

10531 Background: Genomic assays, such as RS, are used to determine chemotherapy benefit in early-stage, estrogen receptor (ER)- and/or progesterone receptor (PR)-positive, HER2 negative BC patients (pts). Currently, guidelines to use pts’ germline genetic testing results to guide adjuvant therapy are lacking. Several reports have indicated worse outcomes for BC pts with g CHEK2 pathogenic variants (PV). We investigated whether PV in CHEK2 were associated with increased RS. Methods: Patient-level clinical data and RS were derived from electronic medical records of seven medical centers between years 2013-17. Confirmation of RS using the Genomic Health provider portal was performed. 38 pts with germline PV in CHEK2 (15 pts/39.5% with c.1100delC mutation) and RS score (cases) were matched with BC pts whose genetic testing did not identify PV (controls) using a 1:2 matching schema. Pts were matched based on age at diagnosis and lymph node (LN) status. LN negative pts were further matched based on T-stage. A multivariate random intercept linear mixed model of CHEK2 mutation status on RS was performed, adjusting for PR. A secondary ordinal univariate analysis was conducted that categorized RS into low, intermediate and high risk ( < 18, 18-30, and > 30, respectively). P-values were reported based on a null hypothesis of no effect against a two-sided alternative. Results: The median RS for cases was 19.5 (interquartile range [IQR]: 15 to 25) and the median RS for controls was 18 (IQR: 12 to 22). A greater proportion of cases were categorized as high risk (10.5%) compared to controls (5.6%), and a smaller proportion of cases were categorized as low risk (36.8%) compared to controls (49.3%). Cases had higher grade and increased proportion of PR-negative BC as compared with controls (grade 1: 12.1% of cases versus 32.4% of controls; PR-negative: 7.9% of cases versus 5.6% of controls). The variables used to match cases and controls (age, lymph node status, and T-stage) had similar summary statistics. The RS was 1.97-point higher in pts with g CHEK2 PV compared to controls, after adjusting for PR (95% confidence interval [CI]: 1.02-point lower to 4.96-point higher; p = 0.194). The secondary analysis of CHEK2 mutation status on an ordinal RS risk group yielded comparable results; on average, the odds of being high risk compared to the combined intermediate/low risk groups was 1.72 times higher in cases compared to controls (95% CI: 0.77 to 3.80; p = 0.181), but these differences were not significant. Conclusions: Our case-control study did not show a statistically higher RS for BC that develops in pts with g CHEK2 PV. Further studies are warranted to evaluate the association between type of CHEK2 PV (frameshift versus missense) and other modifying genetic variables and RS.

Quantifying pulmonary perfusion from noncontrast computed tomography.

PurposeComputed tomography (CT)‐derived ventilation methods compute respiratory induced volume changes as a surrogate for pulmonary ventilation. Currently, there are no known methods to derive perfusion information from noncontrast CT. We introduce a novel CT‐Perfusion (CT‐P) method for computing the magnitude mass changes apparent on dynamic noncontrast CT as a surrogate for pulmonary perfusion.Methods CT‐Perfusion is based on a mass conservation model which describes the unknown mass change as a linear combination of spatially corresponding inhale and exhale HU estimated voxel densities. CT‐P requires a deformable image registration (DIR) between the inhale/exhale lung CT pair, a preprocessing lung volume segmentation, and an estimate for the Jacobian of the DIR transformation. Given this information, the CT‐P image, which provides the magnitude mass change for each voxel within the lung volume, is formulated as the solution to a constrained linear least squares problem defined by a series of subregional mean magnitude mass change measurements. Similar to previous robust CT‐ventilation methods, the amount of uncertainty in a subregional sample mean measurement is related to measurement resolution and can be characterized with respect to a tolerance parameter τ. Spatial Spearman correlation between single photon emission CT perfusion (SPECT‐P) and the proposed CT‐P method was assessed in two patient cohorts via a parameter sweep of τ. The first cohort was comprised of 15 patients diagnosed with pulmonary embolism (PE) who had SPECT‐P and 4DCT imaging acquired within 24 h of PE diagnosis. The second cohort was comprised of 15 nonsmall cell lung cancer patients who had SPECT‐P and 4DCT images acquired prior to radiotherapy. For each test case, CT‐P images were computed for 30 different uncertainty parameter values, uniformly sampled from the range [0.01, 0.125], and the Spearman correlation between the SPECT‐P and the resulting CT‐P images were computed.ResultsThe median correlations between CT‐P and SPECT‐P taken over all 30 test cases ranged between 0.49 and 0.57 across the parameter sweep. For the optimal tolerance τ = 0.0385, the CT‐P and SPECT‐P correlations across all 30 test cases ranged between 0.02 and 0.82. A one‐sample sign test was applied separately to the PE and lung cancer cohorts. A low Spearmen correlation of 15% was set as the null median value and two‐sided alternative was tested. The PE patients showed a median correlation of 0.57 (IQR = 0.305). One‐sample sign test was statistically significant with 96.5 % confidence interval: 0.20–0.63, P < 0.00001. Lung cancer patients had a median correlation of 0.57(IQR = 0.230). Again, a one‐sample sign test for median was statistically significant with 96.5 percent confidence interval: 0.45–0.71, P < 0.00001.Conclusion CT‐Perfusion is the first mechanistic model designed to quantify magnitude blood mass changes on noncontrast dynamic CT as a surrogate for pulmonary perfusion. While the reported correlations with SPECT‐P are promising, further investigation is required to determine the optimal CT acquisition protocol and numerical method implementation for CT‐P imaging.

A random-perturbation-based rank estimator of the number of factors

Summary We introduce a random-perturbation-based rank estimator of the number of factors of a large-dimensional approximate factor model. An expansion of the rank estimator demonstrates that the random perturbation reduces the biases due to the persistence of the factor series and the dependence between the factor and error series. A central limit theorem for the rank estimator with convergence rate higher than root $n$ gives a new hypothesis-testing procedure for both one-sided and two-sided alternatives. Simulation studies verify the performance of the test.

Comparing two nonparametric regression curves in the presence of long memory in covariates and errors

This paper discusses the problem of testing the equality of two nonparametric regression functions against two-sided alternatives in the presence of long memory in the common covariate and errors. The proposed test is based on a marked empirical process of the differences between the response variables. We discuss asymptotic null distribution of this process and consistency of the test for a class of general alternatives. We also conduct a Monte Carlo simulation study to evaluate the finite sample level and power behavior of the test at some alternatives.

A random decision for testing of the homogeneity of normal means against the tree order alternative

Consider the problem of testing the homogeneity of k + 1 normal means under the tree order restriction. In this paper, we propose new multiple comparisons procedures for testing of the tree order constraint. New test procedures with the corresponding simultaneous confidence intervals are motivated by some new estimation methods which are constructed based on a random decision and the Bayesian approach. Also, these procedures are developed for two-sided tree order alternatives. We compare the performance of the proposed methods with some existing test procedures, such as likelihood ratio test and some multiple comparisons tests for the tree order constraint. In some cases, the gains in power due to the proposed procedures can be substantial. The results for two sided alternative are similar to the one-sided hypotheses and two new procedures perform well for almost every configuration. We illustrate the efficiency of the proposed methods by analyzing of the two bioassay numerical examples.

Resampling-Based Inference Methods for Comparing Two Coefficients Alpha.

The two-sample problem for Cronbach's coefficient [Formula: see text], as an estimate of test or composite score reliability, has attracted little attention compared to the extensive treatment of the one-sample case. It is necessary to compare the reliability of a test for different subgroups, for different tests or the short and long forms of a test. In this paper, we study statistical procedures of comparing two coefficients [Formula: see text] and [Formula: see text]. The null hypothesis of interest is [Formula: see text], which we test against one-or two-sided alternatives. For this purpose, resampling-based permutation and bootstrap tests are proposed for two-group multivariate non-normal models under the general asymptotically distribution-free (ADF) setting. These statistical tests ensure a better control of the type-I error, in finite or very small sample sizes, when the state-of-affairs ADF large-sample test may fail to properly attain the nominal significance level. By proper choice of a studentized test statistic, the resampling tests are modified in order to be valid asymptotically even in non-exchangeable data frameworks. Moreover, extensions of this approach to other designs and reliability measures are discussed as well. Finally, the usefulness of the proposed resampling-based testing strategies is demonstrated in an extensive simulation study and illustrated by real data applications.

Extend TOPSIS-Based Two-Sided Matching Decision in Incomplete Indifferent Order Relations Setting Considering Matching Aspirations

This paper develops a method for two-sided matching decision in the environment of incomplete indifferent order relations. The two-sided matching decision problem with incomplete indifferent order relations and matching aspirations is firstly described. In order to solve this problem, the incomplete indifferent order relations are converted into the generalized Borda number matrices. The matching aspiration matrix can be determined based on the model calculation on the reciprocal differences of generalized Borda numbers. On this basis, the weighted satisfaction degree matrices are set up. The extended relative closeness matrices are determined by using an extended TOPSIS technique. Moreover, a two-sided matching model is developed. The two-sided matching alternative can be obtained by solving the model. For the purpose of illustration, an example including sensitivity analysis is presented.

Extended likelihood approach to multiple testing with directional error control under a hidden Markov random field model

Current multiple testing procedures are often based on assumptions of independence of observations. However, the observations in genomics and neuroimaging are correlated and ignoring such a correlation can severely distort the conclusions of a test. Moreover, most tests investigate two-sided alternatives only as a two-action problem and do not worry about directional errors. Misspecifications in signs of effects should not be regarded as power. In this study, we derive an optimal multiple testing procedure to incorporate dependence among tests, controlling directional false discovery rates. Real data examples for gene expression and neuroimaging using hidden Markov random field models show that an appropriate model is crucial for the efficiency of tests. Proper modeling of the correlation structure and model selection tools in the likelihood approach enhance the performance of a test. Reporting the estimates of various error rates is useful for the test’s validity.

Multivariate tests based on interpoint distances with application to magnetic resonance imaging.

The multivariate location problem is addressed. The most familiar method to address the problem is the Hotelling test. When the hypothesis of normal distributions holds, the Hotelling test is optimal. Unfortunately, in practice the distributions underlying the samples are generally unknown and without assuming normality the finite sample unbiasedness of the Hotelling test is not guaranteed. Moreover, high-dimensional data are increasingly encountered when analyzing medical and biological problems, and in these situations the Hotelling test performs poorly or cannot be computed. A test that is unbiased for non-normal data, for small sample sizes as well as for two-sided alternatives and that can be computed for high-dimensional data has been recently proposed and is based on the ranks of the interpoint Euclidean distances between observations. Five modifications of this test are proposed and compared to the original test and the Hotelling test. Unbiasedness and consistency of the tests are proven and the problem of power computation is addressed. It is shown that two of the modified interpoint distance-based tests are always more powerful than the original test. Particularly, the modified test based on the Tippett criterium is suggested when the assumption of normality is not tenable and/or in case of high-dimensional data with complex dependence structure which are typical in molecular biology and medical imaging. A practical application to a case-control study where functional magnetic resonance imaging is used is discussed.

Variances of the Foster–Stuart round-trip test statistics

The Foster–Stuart test is often used in the analysis of time series to determine trends. The test is based on calculating the lower and upper time series records x1, …, xn. Unlike other tests of randomness, tests based on records are not invariant under a reversal of the direction of the time variable. To construct invariant round-trip tests, it is necessary to count the records in both forward and backward time variables. Thus far, the construction of this test has been impossible because the variances of test statistics were not known when the null hypothesis was true. Variances for Foster–Stuart round-trip test statistics D and S have been found in the present paper. The test was designed to identify positive and negative trends in means and variances of x1, …, xn time series. Dispersions of S and D test statistics have been found under the H0 randomness null hypothesis. Asymptotic approximations were obtained for dispersions. Thus, it has turned out to be possible to construct a full-fledged invariant test for two-sided alternatives. The Foster–Stuart round-trip test application example has been reviewed.

Some Properties of a Recently Introduced Approach to Ordinal Regression

The statistical properties of a novel approach to ordinal regression which was only recently introduced in the literature are discussed. It is shown that for ordinal explanatory variables the approach is equivalent to isotonic regression, with some advantages when dealing with two-sided alternatives. For ordinal response variables the procedure behaves very differently and is asymptotically equivalent to a two-sample t-test between the extremecategories. A penalized version is introduced to improve power, and the procedureis evaluated using Monte Carlo simulations. Finally the method is applied to microarray gene expression data on prostate cancer.

Convexity issues in multivariate multiple testing of treatments vs. control

The problem of multiple testing of each of several treatment mean vectors versus a control mean vector is considered. Both one-sided and two-sided alternatives are treated. It is shown that typical choices for marginal test procedures will lead to step-down procedures that do not have convex acceptance regions. This lack of convexity has both intuitive and theoretical disadvantages. The only exception being linear tests in the one-sided problem. Although such a procedure is atypical, it not only has convex acceptance regions but is such that critical values are obtainable so that the overall procedure can control FDR or FWER. For both one-sided and two-sided alternatives, two other stepwise multiple testing methods are presented that do have convex acceptance regions.

On the GLR and UMP tests in the family with support dependent on the parameter

Some general results about the GLR tests, for testing simple hypothesis versus two-sided hypothesis, in the family with support dependent on the parameter, are obtained. In addition, we show that such GLR tests are equivalent to the UMP tests in the same problems. Moreover, we derive the general form of the UMP tests for testing an interval hypothesis versus two-sided alternative.

On the GLR and UMP tests in the family with support dependent on the parameter

Some general results about the GLR tests, for testing simple hypothesis versus two-sided hypothesis, in the family with support dependent on the parameter, are obtained. In addition, we show that such GLR tests are equivalent to the UMP tests in the same problems. Moreover, we derive the general form of the UMP tests for testing an interval hypothesis versus two-sided alternative.

Point optimal testing: A survey of the post 1987 literature

In the absence of uniformly most powerful (UMP) tests or uniformly most powerful invariant (UMPI) tests, King [80] suggested the use of Point Optimal (PO) tests, which are most powerful at a chosen point under the alternative hypothesis. This paper surveys the literature and major developments on point optimal testing since 1987 and suggests some areas for future research. Topics include tests for which all nuisance parameters have been eliminated and dealing with nuisance parameters via (i) a weighted average of p values, (ii) approximate point optimal tests, (iii) plugging in estimated parameter values, (iv) using asymptotics and (v) integration. Progress on using point-optimal testing principles for two-sided testing and multi-dimensional alternatives is also reviewed. The paper concludes with thoughts on how best to deal with nuisance parameters under both the null and alternative hypotheses, as well as the development of a new class of point optimal tests for multi-dimensional testing.

A comparison of aggregate p-value methods and multivariate statistics for self-contained tests of metabolic pathway analysis.

For pathway analysis of genomic data, the most common methods involve combining p-values from individual statistical tests. However, there are several multivariate statistical methods that can be used to test whether a pathway has changed. Because of the large number of variables and pathway sizes in genomics data, some of these statistics cannot be computed. However, in metabolomics data, the number of variables and pathway sizes are typically much smaller, making such computations feasible. Of particular interest is being able to detect changes in pathways that may not be detected for the individual variables. We compare the performance of both the p-value methods and multivariate statistics for self-contained tests with an extensive simulation study and a human metabolomics study. Permutation tests, rather than asymptotic results are used to assess the statistical significance of the pathways. Furthermore, both one and two-sided alternatives hypotheses are examined. From the human metabolomic study, many pathways were statistically significant, although the majority of the individual variables in the pathway were not. Overall, the p-value methods perform at least as well as the multivariate statistics for these scenarios.

Union–intersection permutation solution for two-sample equivalence testing

One of the well-known problems with testing for sharp null hypotheses against two-sided alternatives is that, when sample sizes diverge, every consistent test rejects the null with a probability co...