Multivariate Behrens-Fisher Problem Research Articles

Parametric test based on the bootstrapping approach for the MANOVA under a Behrens-Fisher problem

This article presents a comparison of multivariate normal mean vectors under covariance positive definite matrices. We introduce an improved parametric bootstrap (IPB) approach for addressing the multivariate Behrens-Fisher problem, specifically focusing on cases with unequal covariance matrices. Additionally, we evaluate the performance of the IPB test by comparing it with three existing tests: the parametric bootstrap (PB) test, the generalized variable (GV) test, and the Johansen test. Through Monte Carlo simulation, our results demonstrate that both the IPB test and the PB test exhibit superior control over Type I error rates compared to the GV and Johansen tests. Notably, the IPB test outperforms the PB test in terms of controlling Type I error rates. Consequently, our study concludes that the IPB test represents a preferred statistical method for testing the equality of mean vectors in the multivariate Behrens-Fisher problem.

A robust test approach for equality of mean vectors of two independent groups under the multivariate Behrens-Fisher problem

In multivariate statistical inference, the Hotelling T 2 statistic is used to test the equality of mean vectors for two independent groups. This statistic needs the multivariate normality and homogeneous covariance matrices assumptions. However, homogeneous covariance matrices assumption may not be provided in real applications. This case is called the multivariate Behrens-Fisher problem. There are several studies to test the equality of two mean vectors for the independent groups under the multivariate Behrens-fisher problem. But these studies do not interest in outliers at data sets. In this study, we propose solving problems caused by multivariate Behrens-Fisher and outliers in the dataset. We compare our proposed approach with other approaches regarding empirical size and power at simulated data that are both uncontaminated and contaminated. Thus we show that our proposed approach can be used to test the equality of mean vectors for two independent groups under multivariate Behrens-Fisher problem without being affected by outliers in the data. Moreover, we construct an R function in the MVTests package to use our proposed approach for real data applications.

Multivariate Behrens-Fisher problem using means of auxiliary variables

The authors considered the problem of testing equality of two multivariate normal mean vectors when the covariance matrices are unknown and arbitrary. Given auxiliary variables with known means, the authors proposed a pivotal quantity which is similar to the Hotelling T 2 statistic and obtained a satisfying approximation to its distribution. The authors also outlined hypothesis testing and confidence estimation based on the approximate distribution. The merits of the test were studied using Monte Carlo simulation. Monte Carlo studies indicated that the test is very satisfactory even for moderately small samples. At last, the authors illustrated the proposed methods by an example.

Hierarchical Missing Data and Multivariate Behrens–Fisher Problem

This article firstly defines hierarchical data missing pattern, which is a generalization of monotone data missing pattern. Then multivariate Behrens–Fisher problem with hierarchical missing data is considered to illustrate that how ideas in dealing with monotone missing data can be extended to deal with hierarchical missing pattern. A pivotal quantity similar to the Hotelling T 2 is presented, and the moment matching method is used to derive its approximate distribution which is for testing and interval estimation. The precision of the approximation is illustrated through Monte Carlo data simulation. The results indicate that the approximate method is very satisfactory even for moderately small samples.

ON SOME SOLUTIONS OF THE MULTIVARIATE BEHRENS FISHER PROBLEM

Multivariate Behrens-Fisher Problem is a problem that deals with testing the equality of two means from multivariate normal distribution when the covariance matrices are unequal and unknown. However, there is no single procedure served as a better performing solution to this problem, Adebayo (2018). In this study effort is made in selecting five different existing procedures and examined their power and rate to which they control type I error using a different setting and conditions observed from previous studies. To overcome this problem a code was designed via R Statistical Software, to simulate random normal data and independently run 1000 times using MASS package in other to estimate the power and rate at which each procedure control type I error. The simulation result depicts that, in a setting when variance covariance matrices S1 > S2 associated with a sample sizes (n1 > n2) in Table 4.1, 4.2, 4.5, and 4.6, shows that, Adebayos’ procedure performed better but at a sample sizes (n1 = n2 and n1 < n2) Hotelling T2 is recommended in terms of power. For type I error rate where robustness and nominal level matters we found that under some settings none of the procedure maintained nominal level as revealed in Table 4.11 and 4.15. The results presented in Table 4.9 to 4.16 shows that when nominal level matters Krishnamoorthy came first, followed by Adebayos’, Yaos’, Johansons’ then Hotelling T2 were recommended in the sequentially under the settings used in this study.

A Test on the Multivariate Behrens–Fisher Problem in High–Dimensional Data by Block Covariance Estimation

In this paper, we proposed a new testing statistic for testing the equality of mean vectors from two multivariate normal populations when the covariance matrices are unknown and unequal in high–dimensional data. A new test is proposed based on the idea of keeping more information from the sample covariance matrices as much as possible. A proposed test is invariant under scalar transformations and location shifts. We showed that the asymptotic distribution of proposed statistic is standard normal distribution when number of random variables approach infinity. We also compared the performance of the proposed test with other three existing tests by the simulation study. The simulation results showed that the attained significance level of proposed test close to setting nominal significance level satisfactorily. The attained power of proposed test outperforms as the other comparative tests under form of covariance matrices considered which can be arranged to block diagonal matrix structure. The attained power becomes more powerful when the dimension increases for a given sample size or vice versa, or relationship level between random variables in each sample increases. Finally, the proposed test is also illustrated with an analysis of DNA microarray data.

Some solutions to the multivariate Behrens–Fisher problem for dissimilarity‐based analyses

SummaryThe essence of the generalised multivariate Behrens–Fisher problem (BFP) is how to test the null hypothesis of equality of mean vectors for two or more populations when their dispersion matrices differ. Solutions to the BFP usually assume variables are multivariate normal and do not handle high‐dimensional data. In ecology, species' count data are often high‐dimensional, non‐normal and heterogeneous. Also, interest lies in analysing compositional dissimilarities among whole communities in non‐Euclidean (semi‐metric or non‐metric) multivariate space. Hence, dissimilarity‐based tests by permutation (e.g., PERMANOVA, ANOSIM) are used to detect differences among groups of multivariate samples. Such tests are not robust, however, to heterogeneity of dispersions in the space of the chosen dissimilarity measure, most conspicuously for unbalanced designs. Here, we propose a modification to the PERMANOVA test statistic, coupled with either permutation or bootstrap resampling methods, as a solution to the BFP for dissimilarity‐based tests. Empirical simulations demonstrate that the type I error remains close to nominal significance levels under classical scenarios known to cause problems for the un‐modified test. Furthermore, the permutation approach is found to be more powerful than the (more conservative) bootstrap for detecting changes in community structure for real ecological datasets. The utility of the approach is shown through analysis of 809 species of benthic soft‐sediment invertebrates from 101 sites in five areas spanning 1960 km along the Norwegian continental shelf, based on the Jaccard dissimilarity measure.

Probabilistic clustering via Pareto solutions and significance tests

The present paper proposes a new strategy for probabilistic (often called model-based) clustering. It is well known that local maxima of mixture likelihoods can be used to partition an underlying data set. However, local maxima are rarely unique. Therefore, it remains to select the reasonable solutions, and in particular the desired one. Credible partitions are usually recognized by separation (and cohesion) of their clusters. We use here the p values provided by the classical tests of Wilks, Hotelling, and Behrens–Fisher to single out those solutions that are well separated by location. It has been shown that reasonable solutions to a clustering problem are related to Pareto points in a plot of scale balance vs. model fit of all local maxima. We briefly review this theory and propose as solutions all well-fitting Pareto points in the set of local maxima separated by location in the above sense. We also design a new iterative, parameter-free cutting plane algorithm for the multivariate Behrens–Fisher problem.

Parametric and nonparametric bootstrap methods for general MANOVA

We develop parametric and nonparametric bootstrap methods for multi-factor multivariate data, without assuming normality, and allowing for covariance matrices that are heterogeneous between groups. The newly proposed, general procedure includes several situations as special cases, such as the multivariate Behrens–Fisher problem, the multivariate one-way layout, as well as crossed and hierarchically nested two-way layouts. We derive the asymptotic distribution of the bootstrap tests for general factorial designs and evaluate their performance in an extensive comparative simulation study. For moderate sample sizes, the bootstrap approach provides an improvement to existing methods in particular for situations with nonnormal data and heterogeneous covariance matrices in unbalanced designs. For balanced designs, less computationally intensive alternatives based on approximate sampling distributions of multivariate tests can be recommended.

On the Behrens–Fisher problem from the spatial median point of view

We present a non-parametric affine-invariant test for the multivariate Behrens–Fisher problem. The proposed method based on the spatial medians is asymptotic and does not require normality of the data. To improve its finite sample performance, we apply a correction of the type which was already used in a similar test based on trimmed means, however, our simulations show that in the case of heavy-tailed distributions our method performs better. Also in a simulation comparison with a recently published rank-based test our test yields satisfactory results.

Testing linear hypotheses of mean vectors for high-dimension data with unequal covariance matrices

We propose a new test procedure for testing linear hypothesis on the mean vectors of normal populations with unequal covariance matrices when the dimensionality, p exceeds the sample size N, i.e. p/N→c<∞. Our procedure is based on the Dempster trace criterion and is shown to be consistent in high dimensions.The asymptotic null and non-null distributions of the proposed test statistic are established in the high dimensional setting and improved estimator of the critical point of the test is derived using Cornish–Fisher expansion. As a special case, our testing procedure is applied to multivariate Behrens–Fisher problem. We illustrate the relevance and benefits of the proposed approach via Monte-Carlo simulations which show that our new test is comparable to, and in many cases is more powerful than, the tests for equality of means presented in the recent literature.

Multivariate Behrens–Fisher problem with missing data

Inference about the difference between two normal mean vectors when the covariance matrices are unknown and arbitrary is considered. Assuming that the incomplete data are of monotone pattern, a pivotal quantity, similar to the Hotelling T 2 statistic, is proposed. A satisfactory moment approximation to the distribution of the pivotal quantity is derived. Hypothesis testing and confidence estimation based on the approximate distribution are outlined. The accuracy of the approximation is investigated using Monte Carlo simulation. Monte Carlo studies indicate that the approximate method is very satisfactory even for moderately small samples. The proposed methods are illustrated using an example.

A parametric bootstrap solution to the MANOVA under heteroscedasticity

In this article, we consider the problem of comparing several multivariate normal mean vectors when the covariance matrices are unknown and arbitrary positive definite matrices. We propose a parametric bootstrap (PB) approach and develop an approximation to the distribution of the PB pivotal quantity for comparing two mean vectors. This approximate test is shown to be the same as the invariant test given in [Krishnamoorthy and Yu, Modified Nel and Van der Merwe test for the multivariate Behrens–Fisher problem, Stat. Probab. Lett. 66 (2004), pp. 161–169] for the multivariate Behrens–Fisher problem. Furthermore, we compare the PB test with two existing invariant tests via Monte Carlo simulation. Our simulation studies show that the PB test controls Type I error rates very satisfactorily, whereas other tests are liberal especially when the number of means to be compared is moderate and/or sample sizes are small. The tests are illustrated using an example.

Evaluating the improvement in diagnostic utility from adding new predictors

Multiple diagnostic tests and risk factors are commonly available for many diseases. This information can be either redundant or complimentary. Combining them may improve the diagnostic/predictive accuracy, but also unnecessarily increase complexity, risks, and/or costs. The improved accuracy gained by including additional variables can be evaluated by the increment of the area under (AUC) the receiver-operating characteristic curves with and without the new variable(s). In this study, we derive a new test statistic to accurately and efficiently determine the statistical significance of this incremental AUC under a multivariate normality assumption. Our test links AUC difference to a quadratic form of a standardized mean shift in a unit of the inverse covariance matrix through a properly linear transformation of all diagnostic variables. The distribution of the quadratic estimator is related to the multivariate Behrens-Fisher problem. We provide explicit mathematical solutions of the estimator and its approximate non-central F-distribution, type I error rate, and sample size formula. We use simulation studies to prove that our new test maintains prespecified type I error rates as well as reasonable statistical power under practical sample sizes. We use data from the Study of Osteoporotic Fractures as an application example to illustrate our method.

A Bayesian solution to the multivariate Behrens–Fisher problem

One of the most common problems in applied statistics is to compare two normal population means when the ratio of variances is unknown and not equal to 1. This is the known Behrens–Fisher problem. There are many approaches to the distribution to the t -statistic in the univariate circumstance under the Behrens–Fisher problem. In the multivariate case, most solutions are based on adjusting the degrees of freedom to obtain better approximations to the chi-squared or Hotelling’s T 2 distributions. In both circumstances there are Bayesian solutions proposed by some authors. This work aimed to propose a computational Bayesian solution to the multivariate Behrens–Fisher problem based on the complex analytical solution of Johnson and Weerahandi (1988), to evaluate its performance through Monte Carlo simulation computing the type I error rates and power and to compare it with the modified Nel and Van der Merwe test, that is considered the best frequentist solution. The inferences were made to the population difference δ of the mean vectors. It was used as a conjugate prior distribution to the population mean vector ( μ i ) and covariance matrix ( Σ i ) obtaining a posterior multivariate t distribution to μ i , for i = 1 , 2. In general, the Bayesian test was conservative for samples of different sizes and liberal in some circumstances of equal and small sample sizes and its power was equal to or greater than that of its competitor in large samples and/or in balanced circumstances. The new solution has competitive advantages and in some circumstances surpasses its main competitor, therefore its use in real cases should be recommended.

Implementation of a New Solution to the Multivariate Behrens-Fisher Problem

Krishnamoorthy and Yu (2004, Statistics and Probability Letters 66: 161–169) published a new approximate solution to the multivariate Behrens–Fisher problem. It is a modification of Nel and Van der Merwe's (1986, Communications in Statistics, Theory and Methods 15: 3719–3735) test. The test is invariant and identical to Welch's test for one-dimensional data. In this article, I describe an implementation of the test in Stata. The hotelmnm command allows you to perform the test easily and returns computed values for possible further computations.

Inferences on a linear combination of K multivariate normal mean vectors

In this paper, the hypothesis testing and confidence region construction for a linear combination of mean vectors for K independent multivariate normal populations are considered. A new generalized pivotal quantity and a new generalized test variable are derived based on the concepts of generalized p-values and generalized confidence regions. When only two populations are considered, our results are equivalent to those proposed by Gamage et al. [Generalized p-values and confidence regions for the multivariate Behrens–Fisher problem and MANOVA, J. Multivariate Aanal. 88 (2004), pp. 117–189] in the bivariate case, which is also known as the bivariate Behrens–Fisher problem. However, in some higher dimension cases, these two results are quite different. The generalized confidence region is illustrated with two numerical examples and the merits of the proposed method are numerically compared with those of the existing methods with respect to their expected areas, coverage probabilities under different scenarios.

Some Aspects of Multivariate Behrens-Fisher Problem

In this paper we discuss the well known multivariate Behrens-Fisher problem which deals with testing the equality of two normal mean vectors under heteroscedasticity of dispersion matrices. Some existing tests are reviewed and a new test based on Roy's union-intersection principle coupled with the generalized P-value is proposed. The tests are compared with respect to size and power based on simulation, and applied to a few useful data sets. AMS (2000) Subject Classification : 62H10, 62H15.

Hotelling's one-sample and two-sample [formula omitted] tests and the multivariate Behrens–Fisher problem under nonnormality

This paper provides a powerful method for obtaining an asymptotic expansion for the expectation of a function of the normalized sample mean vector N 1 / 2 U ¯ and the sample covariance matrix S U under general distributions, without using the Edgeworth expansion of N - 1 / 2 ∑ i = 1 N U i and N - 1 / 2 ∑ i = 1 N vech ( U i U i ′ - Σ ) . It is shown that asymptotic expansions of the nonnull distributions of some multivariate test statistics on mean vectors under general distributions are derived in a unified way, by finding the differential operator associated with the expectation according to situations under consideration. Unlike Kano [1995. An asymptotic expansion of the distribution of Hotelling's T 2 -statistic under general distributions. Amer. J. Math. Manage. Sci. 15, 317–341] and Fujikoshi [1997. An asymptotic expansion for the distribution of Hotelling's T 2 -statistic under nonnormality. J. Multivariate Anal. 61, 187–193], our routine for getting asymptotic expansions is to collect some patterned derivatives, without constructing the Edgeworth expansion of some basic statistics. In this sense, our approach seems to be more convenient, at least, for Hotelling's T 2 -type statistics and other related statistics on mean vectors.

On Selecting Tests for Equality of Two Normal Mean Vectors

The conventional approach for testing the equality of two normal mean vectors is to test first the equality of covariance matrices, and if the equality assumption is tenable, then use the two-sample Hotelling T 2 test. Otherwise one can use one of the approximate tests for the multivariate Behrens–Fisher problem. In this article, we study the properties of the Hotelling T 2 test, the conventional approach, and one of the best approximate invariant tests (Krishnamoorthy & Yu, 2004) for the Behrens–Fisher problem. Our simulation studies indicated that the conventional approach often leads to inflated Type I error rates. The approximate test not only controls Type I error rates very satisfactorily when covariance matrices were arbitrary but was also comparable with the T 2 test when covariance matrices were equal.