Hotelling's T2 Test Research Articles

Heteroscedasticity in a two-factors design model

This paper studies the two-factors design model when the heteroscedasticity of variance is present in errors. As can be observed, testing of hypothesis based on the main effects for this design model can be performed using Hotelling's T2 test. Simultaneous confidence intervals are also proposed. Finally, the proposed methodology is applied to a real-life example.

K0-INAA of Venezuelan ceramics and complete statistical analysis to establish their provenance

A group of 46 archaeological figurines samples (ad 1300 and 1500) from Venezuelan mainland and northern island were analyzed by k0-instrumental neutron activation analysis (k0-INAA) to obtain their elemental content and give a step ahead to establish the provenance of the island figurines. In total 37 elemental concentrations were measured with uncertainties between 3 and 20 %. To make the study of provenance, a complete statistical analysis was achieved; Fisher linear discriminant, principal component analysis, hierarchical clustering and the Hotelling T2 test were used to this end. Furthermore, not only the 46 samples analyzed in this work by k0-INAA were used, but also 40 samples analyzed by PGNAA and reported by Sajo-Bohus et al. (JRNC 265(2):247–256, 2005) were included in the statistical analysis. It was done in order to increase the size of the data set, and then to obtain from the statistical techniques more reliable results. It was found that a very good differentiation exits between the figurines from the island and from the mainland supporting the idea that the raw materials of the figurines come from different places.

Type of Anchor at the Proximal Fusion Level Has a Significant Effect on the Incidence of Proximal Junctional Kyphosis and Outcome in Adults After Long Posterior Spinal Fusion

Study DesignRetrospective review. ObjectivesTo compare the incidence of proximal junctional kyphosis (PJK) and the clinical, radiographic, and functional outcomes in adults undergoing long posterior spinal fusion with transverse process hooks versus pedicle screws at the uppermost instrumented vertebrae. Summary of Background DataProximal junctional kyphosis often occurs after instrumented long spinal fusion. Although there have been numerous studies of PJK development in adolescents with idiopathic scoliosis, few studies have focused on adults. MethodsThis study reviewed data on 47 consecutive adult patients who underwent long spinal fusion (five or more levels) with hooks or screws at the uppermost instrumented vertebrae, from 2004 through 2009, and had 2-year radiographic and clinical follow-up. The hook group (20 patients) and screw group (27 patients) were similar in terms of age, gender, and levels fused. Proximal junctional kyphosis was defined as a sagittal Cobb angle of at least 10° between the lower end plate of the uppermost instrumented vertebrae and the upper end plate of the 2 immediately superior vertebrae, and at least 10° of progression from the previous measurement. The groups' radiographs, complications, and functional outcomes (Scoliosis Research Society–22 Patient Questionnaire and the Oswestry Disability Index) were compared using Hotelling's t2 test (significance, p < .05). ResultsComparing immediate postoperative and final follow-ups, none of the 20 patients in the hook group versus 8 of 27 patients in the screw group (29.6%) developed PJK (p = .01). There were no statistical differences between groups in major or minor complications rates. At final follow-up, patients with hooks had significantly higher functional scores than those with screws (p < .05), and patients with PJK had significantly lower functional scores in all Scoliosis Research Society–22 Patient Questionnaire domains except satisfaction. ConclusionsTransverse process hooks were associated with a lower incidence of PJK and higher functional scores than pedicle screws.

Investigating the single ideal assumption using Ideal Profile Method

Ideal point modeling is a type of multivariate mapping in which consumers are assumed to use internal ideals in their hedonic evaluation of products. In their calculation processes, these techniques typically assume that consumers have a unique ideal for the product set tested. But this assumption is difficult to verify from the liking data alone, and may be violated if different subcategories of products, such as light and dark chocolate, are included in the same experiment. In this paper we propose the use of the Ideal Profile Method (IPM) to test this assumption. In IPM, consumers are asked to rate explicitly their ideals for each product tested. The variability between products of the averaged ideal ratings can then be used to check for the assumption of a single ideal. The procedure we describe involves ANOVA and confidence ellipses associated to the Hotelling T2 test. We then consider three cases to illustrate the use of this methodology in practice.

Examination of Outliers in Bioequivalence Studies

Objective: The existence of outliers in bioequivalence studies is a problem causing important differences in the results of the study. The presence of outliers may lead to the conclusion that formulations are not bioequivalent even if in reality they are or that they are bioequivalent even if in fact they are not. In this study by giving general information about bioequivalence studies and outliers, the methods to examine outliers are analyzed.Methods: Twenty-three participants were included in this study. The likelihood distance test, estimates distance test, Hotelling T2 test, mean-shift test and Liu and Weng's residual test were used for examining outliers.Results: The test and reference drugs were given to healthy participants. The participants were examined by obtaining blood samples at specific time points. The density values of the element in the blood were determined by a reliable method. By the methods such as likelihood distance test, estimates distance test, Hotelling T2 test, mean-shift test and Liu and Weng's residual test, the sixth observation was detected as an outlier.Conclusion: It was concluded that the presence of the outlier reduces the similarities between the test and reference drugs.

Efficacy of specific bioelectrical impedance vector analysis (BIVA) for assessing body composition in the elderly.

This study aimed to ascertain the efficacy of bioelectrical impedance vector analysis (BIVA) in assessing body composition in the elderly by comparing findings with the results of dual-energy X-ray absorptiometry (DXA), and to test an analytical variant of the method (specific BIVA). Cross-sectional study. The sample comprised 207 free-living elderly individuals (75 men and 132 women) aged 65 to 93 years. Anthropometric and bioelectrical measurements were taken according to standard criteria. BIVA was applied using the 'classic' procedure and correcting bioelectrical values for body geometry to obtain an estimate of the whole-body impedivity. DXA was used as a reference body composition assessment method. BIVA (classic and specific values) and DXA findings were compared using Student's t and Hotelling's T2 tests, and Pearson's correlation coefficient. In both sexes, BIVA distinguished between individuals with different amounts of fat and fat-free mass (lean mass including bone mineral content), according to DXA, but not between those with different proportions of fat mass (FM%). Specific bioelectrical values detected changes in body composition. BIVA should be used with caution for evaluating body composition in the elderly. Specific bioelectrical values proved effective, showing promise as a methodological variant of BIVA, suitable for identifying age-related changes in body fatness.

Anterior-Posterior Direction Opponency in the Superficial Mouse Lateral Geniculate Nucleus

We show functional-anatomical organization of motion direction in mouse dorsal lateral geniculate nucleus (dLGN) using two-photon calcium imaging of dense populations in thalamus. Surprisingly, thesuperficial 75μm region contains anterior and posterior direction-selective neurons (DSLGNs) intermingled with nondirection-selective neurons, while upward- and downward-selective neurons are nearlyabsent. Unexpectedly, the remaining neurons encode both anterior and posterior directions, forming horizontal motion-axis selectivity. A model of random wiring consistent with these results makes quantitative predictions about the connectivity of direction-selective retinal ganglion cell (DSRGC) inputs to the superficial dLGN. DSLGNs are more sharply tuned than DSRGCs. These results suggest that dLGN maintains and sharpens retinal direction selectivity and integrates opposing DSRGC subtypes in a functional-anatomical region, perhaps forming a feature representation for horizontal-axis motion, contrary to dLGN being a simple relay. Furthermore, they support recent conjecture that cortical direction and orientation selectivity emerge in part from a previously undescribed motion-selective retinogeniculate pathway.

Polychlorinated dibenzo-p-dioxins, polychlorinated dibenzofurans and non-ortho-PCBs in eggs of white-tailed sea eagles collected along the Swedish coast in the Baltic Sea

Concentrations of polychlorinated dibenzo-p-dioxins (PCDD), polychlorinated dibenzofurans (PCDF), and dioxin-like non-ortho-PCBs were measured in white-tailed sea eagle (WTSE) eagle eggs collected along the Swedish coast of the Baltic Sea during the period 1992–2004. The eggs represent two different subpopulations with significantly different nestling brood sizes; the Baltic Proper (BP) with an approximately normal nestling brood size (1.62), and the south Bothnian Sea (sBS) with reduced nestling brood size (1.22) combined with a significantly higher rate of dead eggs. The aim was to investigate if this difference in reproductive outcome was linked to differences in exposure to dioxin-like compounds. Three eggs collected in Greenland in 2000 were included in the study to provide a reference sea eagle population outside of the Baltic Sea region. The concentrations of ∑PCDD, ∑PCDF and ∑non-ortho-PCB in the two subpopulations from the Baltic Sea (BS) region ranged from 0.41–4.1, 1.2–5.3 and 180–970ng/g lipids, respectively, while in the Greenland population the ranges were 0.11–0.16, 0.22–0.33 and 57–83ng/g lipid, respectively. 2,3,4,7,8-PCDF was the predominant congener in all areas and accounted for on average 31–49% of the total ∑PCDD/F concentrations. The total toxic equivalents (TEQ) in sBS WTSEs were higher (approximately 39ng TEQ/g lipid) than reported in eggs for many other birds, and the major contributors to the TEQ in the Baltic Sea were the non-ortho-PCBs. A principal component analysis (PCA) showed a difference in congener pattern between the two Baltic regions that was statistically significant (Hotelling's T2 test). We found no significant differences in the total TEQ between the two populations (sBS–BP) and thus no evidence was found linking the reproductive impairment in WTSE in sBS to the concentrations of PCDD/Fs or non-ortho-PCBs in the eggs.

Differences between uni- and multidimensional scales for assessing pain in term newborn infants at the bedside

OBJECTIVES:This study sought to determine the level of agreement between behavioral and multidimensional pain assessment scales in term newborn infants submitted to an acute nociceptive stimulus.METHODS:This cross-sectional study was performed on 400 healthy term newborns who received an intramuscular injection of vitamin K during the first 6 hours of life. Two behavioral pain scales (the Neonatal Facial Coding System and the Behavioral Indicators of Infant Pain) and one multidimensional tool (the Premature Infant Pain Profile) were applied by a single observer before the procedure, during cleansing, during injection and two minutes after injection. The Cochran Q, McNemar and kappa tests were used to compare the presence and degree of agreement between the three scales. The Hotelling T2 test was used to compare the groups of newborns for which the scales showed agreement or disagreement. A generalized linear regression was used to compare the results of the Neonatal Facial Coding System and the Behavioral Indicators of Infant Pain across the four study time points.RESULTS:The neonates studied had a gestational age of 39±1 weeks, a birth weight of 3169±316 g and and postnatal age of 67±45 minutes. During the stimulus procedure, 80% of the newborns exhibited pain behaviors according to the Neonatal Facial Coding System and the Behavioral Indicators of Infant Pain, and 70% experienced pain according to the Premature Infant Pain Profile (p<0.001). The frequencies of the detection of pain using the Behavioral Indicators of Infant Pain and the Neonatal Facial Coding System were similar. The characteristics of the neonates were not associated with the level of agreement between the scales.CONCLUSION:The Neonatal Facial Coding System and the Behavioral Indicators of Infant Pain behavioral scales are more sensitive for the identification of pain in healthy term newborn infants than the multidimensional Premature Infant Pain Profile scale.

An Approximate Hotelling T&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;-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.

Screening analysis of seston from a domestic wastewater treatment plant using digital images

This paper concerns the use of digital images of seston to monitor different ponds of a domestic wastewater treatment plant. For this purpose, a commercial camera was employed to photograph the seston samples and an automatic procedure based on the Hough Circle Transform was used to segment the resulting picture. Multivariate techniques were then employed to analyze the seston images in terms of their RGB colour components. The results indicate that such colour components undergo a significant change after each stage of the wastewater treatment process, according to a Hotelling T2 test (p-value smaller than 10−4). This finding was corroborated by the use of quadratic discriminant analysis (QDA), which resulted in the correct classification of all samples according to the treatment stage. In addition, it was found that differences in the illumination of the samples can be standardized by converting the RGB data to the YCbCr format and correcting the luminance (Y) component. This correction may unveil changes in the seston features that were previously masked by the illumination variability.

Rational management of food intolerance in an elite soccer club

Background Gastric intestinal, skeletal muscle and neurological symptoms are just some of the possible effects of alimentary intolerances that may represent a risk for one’s health and may frustrate the athlete’s practice benefits and thwart the performance. The immunological tolerance recovery and the re-establishment of a normal diet are generally reached by means of strict dietetic schemes (a turnover or elimination diet) requiring a strong effort into changing one’s diet habit. In the elite soccer athlete, an intense competitive schedule including long transfers represents another risk to these dietetic therapies fulfilment that may even worsen the symptomatology once the allergens responsible for the intolerances are again within the diet.The purpose of this study was to objectively estimate an food intolerance in elite soccer players who have foodintolerance symptoms, by arranging a nutritional plan for the immunological tolerance recovery, the symptomatology improvement and to test the effectiveness in terms of nutritional situation by means of the BIVA, assuming a close relation among food intolerances, inflammatory reactions and nutritional status. Methods Eight males 32.5 ± 1.9 years old soccer players and BMI 24.9 ± 1.1 (Average ± DS) with symptoms of possible food intolerance (gastralgia, headache, intestinal meteorism, diarrhoea, constipation, nausea) of an Italian Serie A soccer team were subjected to the ALCAT test (IMGeP, Milan, Italy) before and after eight months of a personalized nutritional treatment. The athletes body composition was basally valued and at the end of the BIVA analysis (50 kHz, BIA 101 RJL, Akern Bioresearch, Florence, Italy). Results The athletes tested, with food intolerance symptoms, were ALCAT test variously positive. The personalized nutritional treatment based on moderation rather than on drastic elimination of reactive foods and complying with the specific nutritional needs of the elite soccer player led to a nearly complete resolution of the first symptoms as the clinical evaluation and the post-treatment ALCAT test results demonstrate. Parallel to these results a significant shift of the mean impedance vector was observed (Hotelling T2 test, p < 0.0001), so indicating a more favourable condition of the soft tissues (hydration and/or mass) with no BMI variation (p<0.05). Conclusions The ALCAT test seems to be able to detect the food intolerance reactions when it is applied to patients with initial specific symptoms. A personalized and flexible nutritional therapy based on moderation and rational elimination of reactive foods seems to be working and be suitable for the elite athlete whose specific logistic necessities ( for example long travels) discourage the classic dietary regime. An efficient handling of the food intolerances seems to lead to a nutritional condition improvement, maybe reducing the concerned inflammatory situation as observed in body composition changing, which may influence the sports performance.

Two-Way MANOVA With Unequal Cell Sizes and Unequal Cell Covariance Matrices

In this article, we propose and study an approximate Hotelling T2 (AHT) test for heteroscedastic two-way MANOVA. The AHT test is shown to be invariant under affine-transformations, different choices of the contrast matrix used to define the same hypothesis, and different labeling schemes of the cell mean vectors. We demonstrate via intensive simulations that the AHT test generally performs well and outperforms two existing approaches in terms of size and power. An extension of the AHT test for heteroscedastic multi-way MANOVA is briefly described. A dataset from a smoking cessation trial is analyzed to illustrate the methodologies. This article has supplementary material online in a single archive.

Detecting global and local hippocampal shape changes in Alzheimer's disease using statistical shape models

The hippocampus is affected at an early stage in the development of Alzheimer's disease (AD). With the use of structural magnetic resonance (MR) imaging, we can investigate the effect of AD on the morphology of the hippocampus. The hippocampal shape variations among a population can be usually described using statistical shape models (SSMs). Conventional SSMs model the modes of variations among the population via principal component analysis (PCA). Although these modes are representative of variations within the training data, they are not necessarily discriminative on labeled data or relevant to the differences between the subpopulations. We use the shape descriptors from SSM as features to classify AD from normal control (NC) cases. In this study, a Hotelling's T2 test is performed to select a subset of landmarks which are used in PCA. The resulting variation modes are used as predictors of AD from NC. The discrimination ability of these predictors is evaluated in terms of their classification performances with bagged support vector machines (SVMs). Restricting the model to landmarks with better separation between AD and NC increases the discrimination power of SSM. The predictors extracted on the subregions also showed stronger correlation with the memory-related measurements such as Logical Memory, Auditory Verbal Learning Test (AVLT) and the memory subscores of Alzheimer Disease Assessment Scale (ADAS).

Assessment of quality of life in patients with chronic urticaria and reliability of the Dermatology Life Quality Index(DLQI)

Objective To evaluate the influence of chronic urticaria on the quality of life in patients by the DLQI, and to assess the factor structure and reliability of the DLQI. Methods The DLQI (Chinese version) was used to evaluate the quality of life of outpatients with CU. The reliability and factor structure of the DLQI were evaluated by reliability analysis and factor analysis. Results Eighty-five patients were asked to complete the DLQI and 13 patients were excluded for other complicated disorders or unanswered items in the DLQI. The mean age was 33.43 ± 0.90 years in the 72 patients with a median clinical course of 6 months. The DLQI score, which varied from 1 to 23 with an average of 9.46 ± 0.61, was uncorrelated to the patients' age,sex, or clinical course of CU (all P ＞ 0.05). The impact of CU was observed mainly on the synptoms and feeling (50%), followed by work and study (40.67%). Factor analysis indicated that there were two dimensions with eigenvalues greater than 1.0, accounting for 53.7% of the variances. Reliability analysis showed that the Cronbach's α value was 0.836, and reached 0.849 if the item 1 in the DLQI was deleted. An analysis of variance showed that the F value was 31.88 (P ＜ 0.01 ), and Hotelling T2 test yielded an F value of 29.87 (P ＜0.01 ). Conclusions CU has a moderate impact on the quality of life in patients. DLQI can be applied to the evaluation of quality of life in patients with CU with a high reliability, but more attention should be paid to the interference from item 1 to the reliability of the assessment. Key words: Urticaria; Quality of life; Questionnaires

Testing Equality of Two Mean Vectors with Two-Step Monotone Missing Data

SYNOPTIC ABSTRACTWe consider the problem of testing for two normal mean vectors when the data have two-step monotone pattern missing observations. Under the assumption that the population covariance matrices are equal, we obtain two test statistics for this problem: a generalized Hotelling's T2 test statistic and the likelihood ratio test statistic. We propose the approximate upper percentiles of these statistics. The accuracy of the approximation is investigated by Monte Carlo simulation. The proposed method is illustrated using an example.

Inverse Correlation Between Plasma Beta-Carotene and Interleukin-6 in Patients with Advanced Coronary Artery Disease

The interrelationships between plasma beta-carotene, alpha-tocopherol, and the level of systemic inflammation and oxidative stress were investigated in patients with advanced coronary artery disease (CAD). Plasma beta-carotene, alpha-tocopherol, malondialdehyde, free radicals, interleukin-6, high sensitive C-reactive protein levels, and other risk factors of CAD were determined in a group of patients with advanced CAD [significant stenosis according to coronarographic examination (n=91) and a control group of examined patients with coronary arteries with no stenosis (n=49)]. Between-group differences in continuous variables were analyzed with the Hotelling T2-test (software NCSS2000), analyses of correlation matrix with the software STATISTICA. Advanced CAD coincided with significantly lower plasma concentrations of high-density lipoprotein (HDL)-cholesterol and beta-carotene as well as with elevated levels of all inflammatory markers, but only with mild increase of oxidative stress. Beta-carotene significantly inversely correlated with interleukin-6. This inverse correlation could suggest potential protective effect of beta-carotene on atherosclerosis due to the inhibition of inflammatory processes.

Directional multivariate tests rejecting null and negative effects in all variables

This paper suggests two directional multivariate tests that aim at establishing superiority of a treatment over a control in at least one of several endpoints that are assumed to have a multivariate normal distribution. One of these tests is a one-sided, scale-invariant version of the classical Hotelling T²-test. The other is based on a summary score with weights derived from the data. Both tests overcome an important shortcoming of previous "one-sided" multivariate suggestions, namely that the null hypothesis was restricted to a single point in the multidimensional parameter space. The derivation of the tests is supplemented by simulations investigating their performance and by the application in an osteoporosis trial.

A Practical Introduction to Landmark-Based Geometric Morphometrics

Landmark-based geometric morphometrics is a powerful approach to quantifying biological shape, shape variation, and covariation of shape with other biotic or abiotic variables or factors. The resulting graphical representations of shape differences are visually appealing and intuitive. This paper serves as an introduction to common exploratory and confirmatory techniques in landmark-based geometric morphometrics. The issues most frequently faced by (paleo)biologists conducting studies of comparative morphology are covered. Acquisition of landmark and semilandmark data is discussed. There are several methods for superimposing landmark configurations, differing in how and in the degree to which among-configuration differences in location, scale, and size are removed. Partial Procrustes superimposition is the most widely used superimposition method and forms the basis for many subsequent operations in geometric morphometrics. Shape variation among superimposed configurations can be visualized as a scatter plot of landmark coordinates, as vectors of landmark displacement, as a thin-plate spline deformation grid, or through a principal components analysis of landmark coordinates or warp scores. The amount of difference in shape between two configurations can be quantified as the partial Procrustes distance; and shape variation within a sample can be quantified as the average partial Procrustes distance from the sample mean. Statistical testing of difference in mean shape between samples using warp scores as variables can be achieved through a standard Hotelling's T2test, MANOVA, or canonical variates analysis (CVA). A nonparametric equivalent to MANOVA or Goodall's F-test can be used in analysis of Procrustes coordinates or Procrustes distance, respectively. CVA can also be used to determine the confidence with which a priori specimen classification is supported by shape data, and to assign unclassified specimens to pre-defined groups (assuming that the specimen actually belongs in one of the pre-defined groups).Examples involving Cambrian olenelloid trilobites are used to illustrate how the various techniques work and their practical application to data. Mathematical details of the techniques are provided as supplemental online material. A guide to conducting the analyses in the free Integrated Morphometrics Package software is provided in the appendix.

A simple powerful bivariate test for two sample location problems in experimental and observational studies

BackgroundIn many areas of medical research, a bivariate analysis is desirable because it simultaneously tests two response variables that are of equal interest and importance in two populations. Several parametric and nonparametric bivariate procedures are available for the location problem but each of them requires a series of stringent assumptions such as specific distribution, affine-invariance or elliptical symmetry.The aim of this study is to propose a powerful test statistic that requires none of the aforementioned assumptions. We have reduced the bivariate problem to the univariate problem of sum or subtraction of measurements. A simple bivariate test for the difference in location between two populations is proposed.MethodIn this study the proposed test is compared with Hotelling's T2 test, two sample Rank test, Cramer test for multivariate two sample problem and Mathur's test using Monte Carlo simulation techniques. The power study shows that the proposed test performs better than any of its competitors for most of the populations considered and is equivalent to the Rank test in specific distributions.ConclusionsUsing simulation studies, we show that the proposed test will perform much better under different conditions of underlying population distribution such as normality or non-normality, skewed or symmetric, medium tailed or heavy tailed. The test is therefore recommended for practical applications because it is more powerful than any of the alternatives compared in this paper for almost all the shifts in location and in any direction.