General Linear Hypothesis Testing Research Articles

The Effects of Post-Surgical Pregnancy on Weight Loss Trajectories after Bariatric Surgery: Are Initial Weight and Age Prognostic Factors?

Introduction: A substantial percentage of patients undergoing bariatric surgery are of childbearing age. Pregnancy outcomes after bariatric surgery are known. However, there are limited data on the impact of pregnancy on weight loss after surgery. Objectives: This study aims to evaluate the effects of pregnancy on post-bariatric surgery weight loss trajectories (WLTs) and to determine the association with age and initial weight. Methods: All who had primary bariatric surgeries (Roux-en-Y gastric bypass or sleeve) between September 2015 and July 2020 were classified into two groups: post-surgery gravid (GG) and post-surgery non-gravid (NG). WLTs were examined using a random intercept mixed-effects model with repeated measures nested within patients. The post-surgery/pre-gravid time phase (PoPG) was modelled using a third-degree polynomial. For GG, two third-degree spline functions modelled the post-surgery while gravid (PoWG) and post-partum (PoPP) time phases. Age and initial weight were used to control for pre-existing differences during PoPG. Weight differences at 6 months PoPP were examined by applying general linear hypothesis testing to the mixed-model results. Results: A total of 508 patients were included, 20 in GG and 488 in NG. The mean age at surgery was 33 years in GG and 37 years in NG. The mean initial BMI was 47 kg/m2 and 43 kg/m2, respectively. During PoPG, adjusted average weight in both groups follows the path across time. For GG, weight decreases and then increases during PoWG. For GG during PoPP, weight immediately decreases after delivery and then increases over time to levels similar to NG. Weight differences at 6 months PoPP for GG and NG were not statistically different. Older age was associated with reduced weight loss during PoPG by Baseline Age, while higher initial weight was associated with increased weight loss during PoPG by Baseline Weight. In both instances, these effects attenuate over time. Conclusions: This model indicates that pregnancy following bariatric surgery affects WLT during PoWG and PoPP, and no difference in weight is expected after 6 months post-gravid. Age and initial weight could be considered prognostic factors during PoPG. Patients wishing to conceive should undergo preconception counselling and be advised to avoid pregnancy during the period of rapid weight loss. They also should be informed that WLT may vary during pregnancy and early post-partum.

A new normal reference test for linear hypothesis testing in high-dimensional heteroscedastic one-way MANOVA

In recent decades, with rapid development of data collecting technologies, high-dimensional data become increasingly prevalent, and much work has been done for hypotheses on mean vectors for high-dimensional data. However, only a few methods have been proposed and studied for the general linear hypothesis testing (GLHT) problem for high-dimensional data which includes many well-studied problems as special cases. A centralized L2-norm based test statistic is proposed and studied for the high-dimensional GLHT problem. It is shown that under some mild conditions, the proposed test statistic and a chi-square-mixture have the same normal or non-normal limiting distributions. It is then justified that the null distribution of the test statistic can be approximated by using that of the chi-square-type mixture. The distribution of the chi-square-type mixture can be well approximated by a three-cumulant matched chi-square approximation with its approximation parameters consistently estimated from the data. Since the chi-square-type mixture is obtained from the test statistic under the null hypothesis and when the data are normally distributed, the resulting test is termed as a normal reference test. The asymptotic power of the proposed test under a local alternative is established. The impact of the data non-normality on the proposed test is also studied. Three simulation studies and a real data example demonstrate that in terms of size control, the proposed test performs well regardless of whether the data are nearly uncorrelated, moderately correlated, or highly correlated and it outperforms two existing competitors substantially.

Linear hypothesis testing in high-dimensional heteroscedastic one-way MANOVA: A normal reference [formula omitted]-norm based test

A general linear hypothesis testing (GLHT) problem in heteroscedastic one-way MANOVA for high-dimensional data is considered and a normal reference L2-norm based test for the problem is proposed. Different from a few existing methodologies on the GLHT problem which impose strong assumptions on the underlying covariance matrices so that the associated tests’ null distributions are asymptotically normal, it is shown that under some regularity conditions, the proposed test statistic under the null hypothesis and a chi-square type mixture have the same normal or non-normal limiting distributions. It is then suggested to approximate the test’s null distribution using the distribution of the chi-square type mixture, which can be further approximated by the Welch–Satterthwaite chi-square-approximation with approximation parameters consistently estimated. Several simulation studies and a real data application are presented to demonstrate the good performance of the proposed test.

Linear hypothesis testing for weighted functional data with applications

AbstractIn socioeconomic areas, functional observations may be collected with weights, called weighted functional data. In this paper, we deal with a general linear hypothesis testing (GLHT) problem in the framework of functional analysis of variance with weighted functional data. With weights taken into account, we obtain unbiased and consistent estimators of the group mean and covariance functions. For the GLHT problem, we obtain a pointwise F‐test statistic and build two global tests, respectively, via integrating the pointwise F‐test statistic or taking its supremum over an interval of interest. The asymptotic distributions of test statistics under the null and some local alternatives are derived. Methods for approximating their null distributions are discussed. An application of the proposed methods to density function data is also presented. Intensive simulation studies and two real data examples show that the proposed tests outperform the existing competitors substantially in terms of size control and power.

General linear hypothesis testing in functional response model

In this paper, the general linear hypothesis testing and variable selection problems in functional response model are considered. The globalizing pointwise F test and the -test as well as the variable selection methods based on multiple hypothesis testing are adapted to this model. The asymptotic null distributions and consistency under local alternatives of test statistics are established under some standard regularity conditions. Based on these results and nonparametric bootstrap approach, different methods are proposed to approximate the null distribution of test statistics. The performance of the new testing and variable selection procedures is compared with that of the known methods in numerical experiments based on real data set. The results suggest that the new methods often outperform the known ones in terms of size control, power and ability to detect significant variables.

Long‐Term Change of Fish Communities in a Polluted Watershed: Does Cleaner Water “Act” on Fishes?

AbstractPollution is a threat to lotic fishes that enter streams via point source (PS) and nonpoint source (NPS) pathways. The Clean Water Act (CWA) of 1972 and the Surface Mining Control and Reclamation Act (SMCRA) of 1977 reduced PS pollution in the United States, but little information is available concerning the response of fish communities to this legislation. The objectives of this research were to quantify the responses of fishes to the CWA and SMCRA in the Cow Creek watershed of southeastern Kansas, a wadeable stream network that experienced large reductions in PS pollution. Responses were quantified by measuring changes in site occupancy, relative abundance, and species richness across 25 sites sampled in 1963 and again in 2017. Responses were compared among fishes that were intolerant, moderately intolerant, moderately tolerant, and tolerant of pollution using an analysis of deviance coupled with general linear hypothesis testing. We predicted that intolerant fishes would exhibit the most dramatic increases in distribution and abundance and that these changes would be most evident in streams that were directly receiving PS pollution in 1963. However, our predictions were not met, as species tolerant of pollution exhibited greater occupancy, relative abundance, and richness than did intolerant fishes, regardless of time period. Contemporary pH and substrate composition were also quantified and revealed that most streams had recovered from historical acid mine drainage but were dominated by fine substrates. We hypothesized that the limited responses exhibited by intolerant fishes were at least partially attributable to NPS pollution (i.e., sedimentation) resulting from agriculture, which was the dominant land use both historically and currently. We conclude that reductions in both NPS pollution coupled with historical reductions in PS pollution may be necessary to increase the prevalence of intolerant fishes. These management actions may have conservation implications as many intolerant fishes are also imperiled.

Linear Hypothesis Testing With Functional Data

ABSTRACTIn real data analysis, it is often interesting to consider a general linear hypothesis testing (GLHT) problem for functional data, which includes the one-way ANOVA, post hoc, or contrast analysis as special cases. Existing tests for this GLHT problem include an L2-norm-based test and an F-type test but their theoretical properties have not been investigated. In addition, for functional one-way ANOVA, simulation studies in the literature indicate that they are less powerful than the globalizing pointwise F (GPF) test and the Fmax -test. The GPF and Fmax -test enjoy several other good properties. They are scale-invariant in the sense that their test statistics do not change if we multiply each of functional curves with a nonzero function of the observed locations. In this article, the GPF and Fmax -test are adapted to the above GLHT problem. Their theoretical properties, for example, root-n consistency as well as those of the L2-norm-based and F-type tests are established. Intensive simulation studies are carried out to compare the finite-sample behavior of the tests under consideration in scenarios reflecting various practical characteristics of functional data. Simulation results indicate that the GPF test has higher power than other tests when the functional data are less correlated, and the Fmax -test has higher power than other tests when the functional data are moderately or highly correlated. These results are also confirmed by application of the GPF and Fmax tests to the corneal surface data coming from medical industry. This application suggests the new methods may help to make more clear and sure decisions in practice. For a convenient application of the considered testing procedures, their implementation is developed in the R programming language. Supplementary materials for the article are available online.

Proportional subclass numbers in two-factor ANOVA

ABSTRACTThe between-within split of total sum of squares in one-way analysis of variance (ANOVA) is intuitively appealing and computationally simple, whether balanced or not. In the balanced two-factor setting, the same heuristic and computations apply to analyse treatment sum of squares into main effects and interaction effects sums of squares. Accomplishing the same in unbalanced settings is more difficult, requiring development of tests of general linear hypotheses. However, textbooks treat unbalanced settings with proportional subclasss numbers (psn) as essentially equivalent to balanced settings. It is shown here that, while psn permit an ANOVA-like partition of sums of squares, test statistics for main effects of the two factors generally test the wrong hypotheses when the model includes interaction effects.

High-dimensional general linear hypothesis testing under heteroscedasticity

In recent years, high-dimensional data has become increasingly prevalent with rapid development of data collecting technologies. Much work has been done for hypothesis testing on mean vectors, especially for high-dimensional two-sample problems. Rather than considering a specific problem, we are interested in a general linear hypothesis testing problem on mean vectors of several populations, which includes many existing hypotheses about mean vectors as its special cases. We propose a test statistic based on a linear combination of U-statistics which can be quickly calculated without using computationally intensive U-statistics. Asymptotic normality and power of our test are derived under mild conditions without requiring an explicit relation between the data dimension and the sample size. Our test is applicable to non-normal multi-sample high-dimensional data without assuming a common covariance matrix among different samples. It also works well even when different samples follow different distributions, provided that some moment conditions are satisfied. A simple, computationally-efficient, and ratio-consistent estimator of the unknown variance of our test statistic is also provided which is not based on U-statistic calculations. Simulation studies and a real data example are presented to demonstrate the good performance of our test.

Linear hypothesis testing in high-dimensional one-way MANOVA

In recent years, with the rapid development of data collecting technologies, high-dimensional data have become increasingly prevalent. Much work has been done for testing hypotheses on mean vectors, especially for high-dimensional two-sample problems. Rather than considering a specific problem, we are interested in a general linear hypothesis testing (GLHT) problem on mean vectors of several populations, which includes many existing hypotheses about mean vectors as special cases. A few existing methodologies on this important GLHT problem impose strong assumptions on the underlying covariance matrix so that the null distributions of the associated test statistics are asymptotically normal. In this paper, we propose a simple and adaptive test based on the L 2 -norm for the GLHT problem. For normal data, we show that the null distribution of our test statistic is the same as that of a chi-squared type mixture which is generally skewed. Therefore, it may yield misleading results if we blindly approximate the underlying null distribution of our test statistic using a normal distribution. In fact, we show that the null distribution of our test statistic is asymptotically normal only when a necessary and sufficient condition on the underlying covariance matrix is satisfied. This condition, however, is not always satisfied and it is not an easy task to check if it is satisfied in practice. To overcome this difficulty, we propose to approximate the null distribution of our test statistic using the well-known Welch–Satterthwaite chi-squared approximation so that our new test is applicable without any assumption on the underlying covariance matrix. Simple ratio-consistent estimators of the unknown parameters are obtained. The asymptotic and approximate powers of our new test are also investigated. The methodologies are then extended for non-normal data. Four simulation studies and a real data application are presented to demonstrate the good performance of our new test compared with some existing testing procedures available in the literature.

U-Tests of General Linear Hypotheses for High-Dimensional Data Under Nonnormality and Heteroscedasticity

Test statistics are presented for general linear hypotheses, with special focus on the two-sample profile analysis. The statistics are a modification to the classical Hotelling’s T2 statistic, are basically designed for the case when the dimension, p, may exceed the sample sizes, ni, and are valid under the violation of any assumption associated with T2, such as normality, homoscedasticity, or equal sample sizes. Under a few mild assumptions replacing the classical ones, the test statistics are shown to follow a normal limit under both the null and alternative hypothesis. As the test statistics are defined as a linear combination of U-statistics, the limits are correspondingly obtained using the asymptotic theory of degenerate (for null) and nondegenerate (for alternative) U-statistics. Simulation results, under a variety of parameter settings, are used to show the accuracy and robustness of the test statistics. Practical application of the tests is also illustrated using a few real data sets.

Rank-based ridge estimation in multiple linear regression

Multicollinearity and model misspecification are frequently encountered problems in practice that produce undesirable effects on classical ordinary least squares (OLS) regression estimator. The ridge regression estimator is an important tool to reduce the effects of multicollinearity, but it is still sensitive to a model misspecification of error distribution. Although rank-based statistical inference has desirable robustness properties compared to the OLS procedures, it can be unstable in the presence of multicollinearity. This paper introduces a rank regression estimator for regression parameters and develops tests for general linear hypotheses in a multiple linear regression model. The proposed estimator and the tests have desirable robustness features against the multicollinearity and model misspecification of error distribution. Asymptotic behaviours of the proposed estimator and the test statistics are investigated. Real and simulated data sets are used to demonstrate the feasibility and the performance of the estimator and the tests.

Tests of Linear Hypotheses in the ANOVA under Heteroscedasticity

It is often of interest to undertake a general linear hypothesis testing (GLHT) problem in the one-way ANOVA without assuming the equality of the group variances. When the equality of the group variances is valid, it is well known that the GLHT problem can be solved by the classical F-test. The classical Ftest, however, may lead to misleading conclusions when the variance homogeneity assumption is seriously violated since it does not take the group variance heteroscedasticity into account. To our knowledge, little work has been done for this heteroscedastic GLHT problem except for some special cases. In this paper, we propose a simple approximate Hotelling T 2 (AHT) test. We show that the AHT test is invariant under a‐ne-transformations, difierent choices of the coe‐cient matrix used to deflne the same hypothesis, and difierent labeling schemes of the group means. Simulations and real data applications indicate that the AHT test is comparable with or outperforms some well-known approximate solutions proposed for the k-sample Behrens-Fisher problem which is a special case of the heteroscedastic GLHT problem.

R Estimates and Associated Inferences for Mixed Models With Covariates in a Multicenter Clinical Trial

Robust rank-based methods are proposed for the analysis of data from multicenter clinical trials using a mixed model (including covariates) in which the treatment effects are assumed to be fixed and the center effects are assumed to be random. These rank-based methods are developed under the usual mixed-model structure but without the normality assumption of the random components in the model. For this mixed model, our proposed estimation includes R estimation of the fixed effects, robust estimation of the variance componets, and studentized residuals. Our accompanying inference includes estimates of the standard errors of the fixed-effects estimators and tests of general linear hypotheses concerning fixed effects. While the development is for general scores function, the Wilcoxon linear scores are emphasized. A discussion of the relative efficiency results shows that the R estimates are highly efficient compared to the traditional maximum likelihood (ML) estimates. A small Monte Carlo study confirms the validity of the analysis and its gain in power over the ML analysis for heavy-tailed distributions. We further develop a rank-based test for center by treatment interactions. We discuss the results of our analysis for an example of a multicenter clinical trial which shows the robustness of our procedure.

Rfit: Rank-based Estimation for Linear Models

In the nineteen seventies, Jureckova and Jaeckel proposed rank estimation for linear models. Since that time, several authors have developed inference and diagnostic methods for these estimators. These rank-based estimators and their associated inference are highly efficient and are robust to outliers in response space. The methods include estimation of standard errors, tests of general linear hypotheses, confidence intervals, diagnostic procedures including studentized residuals, and measures of influential cases. We have developed an R package, Rfit, for computing of these robust procedures. In this paper we highlight the main features of the package. The package uses standard linear model syntax and includes many of the main inference and diagnostic functions.

An Approximate Hotelling T<sup>2</sup>-Test for Heteroscedastic One-Way MANOVA

In this paper, we consider the general linear hypothesis testing (GLHT) problem in heteroscedastic one-way MANOVA. The well-known Wald-type test statistic is used. Its null distribution is approximated by a Hotelling T2 distribution with one parameter estimated from the data, resulting in the so-called approximate Hotelling T2 (AHT) test. The AHT test is shown to be invariant under affine transformation, different choices of the contrast matrix specifying the same hypothesis, and different labeling schemes of the mean vectors. The AHT test can be simply conducted using the usual F-distribution. Simulation studies and real data applications show that the AHT test substantially outperforms the test of [1] and is comparable to the parametric bootstrap (PB) test of [2] for the multivariate k-sample Behrens-Fisher problem which is a special case of the GLHT problem in heteroscedastic one-way MANOVA.

History of the Peoples of the Great Hungarian Plain in the First Millennium: A Craniometric Point of View

We carried out an examination relying on six dimensions of 1,573 crania coming from the Great Hungarian Plain. The crania represent seven archeological periods: Sarmatian age (1-4th century), the period of transition (about 400-420), Hun and Gepidic epochs (about 420-455 and 455-567, respectively), early Avar age (about 568-670), late Avar period (about 670-895), the epoch of the Hungarian conquest and settlement (about 895-1000), and the Arpadian age (about 1000-1301). We were curious about the anatomical background behind cultural changes of the various populations that inhabited this area. After having noticed some discontinuities between the populations, as revealed by univariate analysis of single dimensions, we performed a principal-components analysis to see whether or not the diverse components showed eventual breaks in the sequence of the populations. Knowing that all the dominant populations had Asian roots, except for the Gepids of Germanic origin, we expected a considerable difference between the Gepidic population and all the other inhabitants. We also assumed that a conquest itself with a large-scale assimilation was unlikely to leave breaklike traits in anatomical patterns, except for aggressive conquests. We found that the second principal component (which correlated with cranial breadth and partly with height) showed a remarkable hiatus in both sexes between Gepids and early Avars. Having done a statistical proof (simultaneous tests for general linear hypotheses) of the observed phenomenon, we found that the gap referring to subsequent populations was significant only in males. A possible reason for this result is that the Avar conquest was much more radical than has been thought. In addition, considering that men were more likely to die in wars, women survived and were assimilated into the conquerors' populations with higher probability, so it is not surprising that the results of multicomparison tests are significant only in men.

History of the Peoples of the Great Hungarian Plain in the First Millennium: A Craniometric Point of View

We carried out an examination relying on six dimensions of 1,573 crania coming from the Great Hungarian Plain. The crania represent seven archeological periods: Sarmatian age (1–4th century), the period of transition (about 400–420), Hun and Gepidic epochs (about 420–455 and 455–567, respectively), early Avar age (about 568–670), late Avar period (about 670–895), the epoch of the Hungarian conquest and settlement (about 895–1000), and the Arpadian age (about 1000–1301). We were curious about the anatomical background behind cultural changes of the various populations that inhabited this area. After having noticed some discontinuities between the populations, as revealed by univariate analysis of single dimensions, we performed a principal-components analysis to see whether or not the diverse components showed eventual breaks in the sequence of the populations. Knowing that all the dominant populations had Asian roots, except for the Gepids of Germanic origin, we expected a considerable difference between the Gepidic population and all the other inhabitants. We also assumed that a conquest itself with a large-scale assimilation was unlikely to leave breaklike traits in anatomical patterns, except for aggressive conquests. We found that the second principal component (which correlated with cranial breadth and partly with height) showed a remarkable hiatus in both sexes between Gepids and early Avars. Having done a statistical proof (simultaneous tests for general linear hypotheses) of the observed phenomenon, we found that the gap referring to subsequent populations was significant only in males. A possible reason for this result is that the Avar conquest was much more radical than has been thought. In addition, considering that men were more likely to die in wars, women survived and were assimilated into the conquerors’ populations with higher probability, so it is not surprising that the results of multicomparison tests are significant only in men.

Power of double-sampling tests for General Linear Hypotheses

In this paper, testing procedures based on double-sampling are proposed that yield gains in terms of power for the tests of General Linear Hypotheses. The distribution of a test statistic, involving both the measurements of the outcome on the smaller sample and of the covariates on the wider sample, is first derived. Then, approximations are provided in order to allow for a formal comparison between the powers of double-sampling and single-sampling strategies. Furthermore, it is shown how to allocate the measurements of the outcome and the covariates in order to maximize the power of the tests for a given experimental cost.

다층 신경 망을 이용한 비중심F분포 확률계산

ANOVA 검정에서 검정통계량은 단일 또는 이중 비중심F분포를 따르며 비중심F분포는 일반적인 선형 가설 검정에서 검정함수 계산에 적용되고 있다. 기존 비중심F분포의 함수 계산에 대한 연구로 여러 접근 방법이 제시되었지만, 하나의 정확한 함수값을 구하는데도 많은 시간이 소요되는 문제점이 발생되었다. 본 논문에서는 기존 함수 계산의 문제점을 해결하기 위하여 다층 퍼셉트론 네트워크로부터 역전파 학습 알고리즘을 적용하여 비중심F분포의 함수값을 구하는 방법을 제안하였다. 제안된 신경망에 의한 함수값과 기존 Patnaik이 제시한 분포식에 의한 함수값의 차이를 표와 그림을 통하여 비교하였으며, 정확성과 계산속도를 고려할 때 Patnaik의 함수식에 의한 방법보다 신경망을 이용한 방법이 효율적임을 알 수가 있다. The test statistic in ANOVA tests has a single or doubly noncentral F distribution and the noncentral F distribution is applied to the calculation of the power functions of tests of general linear hypotheses. Although various approximations of noncentral F distribution are suggested, they are troublesome to compute. In this paper, the calculation of noncentral F distribution is applied to the neural network theory, to solve the computation problem. The neural network consists of the multi-layer perceptron structure and learning process has the algorithm of the backpropagation. Using fables and figs, comparisons are made between the results obtained by neural network theory and the Patnaik's values. Regarding of accuracy and calculation, the results by neural network are efficient than the Patnaik's values.