Distributional equivalence and subcompositional coherence in the analysis of contingency tables, ratio-scale measurements and compositional data

We consider two fundamental properties in the analysis of two-way tables of positive data: the principle of distributional equivalence, one of the cornerstones of correspondence analysis of contingency tables, and the principle of subcompositional coherence, which forms the basis of compositional data analysis. For an analysis to be subcompositionally coherent, it suffices to analyse the ratios of the data values. The usual approach to dimension reduction in compositional data analysis is to perform principal component analysis on the logarithms of ratios, but this method does not obey the principle of distributional equivalence. We show that by introducing weights for the rows and columns, the method achieves this desirable property. This weighted log-ratio analysis is theoretically equivalent to spectral mapping , a multivariate method developed almost 30 years ago for displaying ratio-scale data from biological activity spectra. The close relationship between spectral mapping and correspondence analysis is also explained, as well as their connection with association modelling. The weighted log-ratio methodology is applied here to frequency data in linguistics and to chemical compositional data in archaeology.

Measuring subcompositional incoherence

Subcompositional coherence is a fundamental property of Aitchison s approach to compositional data analysis, and is the principal justification for using ratios of components. We maintain, however, that lack of subcompositional coherence, that is incoherence, can be measured in an attempt to evaluate whether any given technique is close enough, for all practical purposes, to being subcompositionally coherent. This opens up the field to alternative methods, which might be better suited to cope with problems such as data zeros and outliers, while being only slightly incoherent. The measure that we propose is based on the distance measure between components. We show that the two-part subcompositions, which appear to be the most sensitive to subcompositional incoherence, can be used to establish a distance matrix which can be directly compared with the pairwise distances in the full composition. The closeness of these two matrices can be quantified using a stress measure that is common in multidimensional scaling, providing a measure of subcompositional incoherence. The approach is illustrated using power-transformed correspondence analysis, which has already been shown to converge to log-ratio analysis as the power transform tends to zero.

Weighted Logratio Biplots, Correspondence Analysis and Spectral Maps

Starting with logratio biplots for compositional data, which are based on the principle of subcompositional coherence, and then adding weights, as in correspondence analysis, we rediscover Lewi's spectral map and many connections to analyses of two-way tables of non-negative data. Thanks to the weighting, the method also achieves the property of distributional equivalence

Ratio maps and correspondence analysis

We compare two methods for visualising contingency tables and developa method called the ratio map which combines the good properties of both.The first is a biplot based on the logratio approach to compositional dataanalysis. This approach is founded on the principle of subcompositionalcoherence, which assures that results are invariant to considering subsetsof the composition. The second approach, correspondence analysis, isbased on the chi-square approach to contingency table analysis. Acornerstone of correspondence analysis is the principle of distributionalequivalence, which assures invariance in the results when rows or columnswith identical conditional proportions are merged. Both methods may bedescribed as singular value decompositions of appropriately transformedmatrices. Correspondence analysis includes a weighting of the rows andcolumns proportional to the margins of the table. If this idea of row andcolumn weights is introduced into the logratio biplot, we obtain a methodwhich obeys both principles of subcompositional coherence and distributionalequivalence.

Weighted Logratio Biplots, Correspondence Analysis and Spectral Maps

Starting with logratio biplots for compositional data, which are based on the principle of subcompositional coherence, and then adding weights, as in correspondence analysis, we rediscover Lewi's spectral map and many connections to analyses of two-way tables of non-negative data. Thanks to the weighting, the method also achieves the property of distributional equivalence

Applications Of Visual Evoked Potentials And Fourier-domain Optical Coherence Tomography In Parkinson's Disease: A Controlled Study.

The goal of this cross-sectional observational study was to quantify the pattern-shift visual evoked potentials (VEP) and the thickness as well as the volume of retinal layers using optical coherence tomography (OCT) across a cohort of Parkinson's disease (PD) patients and age-matched controls. Forty-three PD patients and 38 controls were enrolled. All participants underwent a detailed neurological and ophthalmologic evaluation. Idiopathic PD cases were included. Cases with glaucoma or increased intra-ocular pressure were excluded. Patients were assessed by VEP and high-resolution Fourier-domain OCT, which quantified the inner and outer thicknesses of the retinal layers. VEP latencies and the thicknesses of the retinal layers were the main outcome measures. The mean age, with standard deviation (SD), of the PD patients and controls were 63.1 (7.5) and 62.4 (7.2) years, respectively. The patients were predominantly in the initial Hoehn-Yahr (HY) disease stages (34.8% in stage 1 or 1.5, and 55.8 % in stage 2). The VEP latencies and the thicknesses as well as the volumes of the retinal inner and outer layers of the groups were similar. A negative correlation between the retinal thickness and the age was noted in both groups. The thickness of the retinal nerve fibre layer (RNFL) was 102.7 μm in PD patients vs. 104.2 μm in controls. The thicknesses of retinal layers, VEP, and RNFL of PD patients were similar to those of the controls. Despite the use of a representative cohort of PD patients and high-resolution OCT in this study, further studies are required to establish the validity of using OCT and VEP measurements as the anatomic and functional biomarkers for the evaluation of retinal and visual pathways in PD patients.

Effects Of Misalignments In The Retinal Nerve Fiber Layer Thickness Measurements With Spectral Domain Optical Coherence Tomography.

Purpose. To investigate misalignments (MAs) on retinal nerve fiber layer thickness (RNFLT) measurements obtained with Cirrus(©) SD-OCT. Methods. This was a retrospective, observational, cross-sectional study. Twenty-seven healthy and 29 glaucomatous eyes of 56 individuals with one normal exam and another showing MA were included. MAs were defined as an improper alignment of vertical vessels in the en face image. MAs were classified in complete MA (CMA) and partial MA (PMA), according to their site: 1 (superior, outside the measurement ring (MR)), 2 (superior, within MR), 3 (inferior, within MR), and 4 (inferior, outside MR). We compared RNFLT measurements of aligned versus misaligned exams in all 4 sectors, in the superior area (sectors 1 + 2), inferior area (sectors 3 + 4), and within the measurement ring (sectors 2 + 3). Results. RNFLT measurements at 12 clock-hour of eyes with MAs in the superior area (sectors 1 + 2) were significantly lower than those obtained in the same eyes without MAs (P = 0.043). No significant difference was found in other areas (sectors 1 + 2 + 3 + 4, sectors 3 + 4, and sectors 2 + 3). Conclusion. SD-OCT scans with superior MAs may present lower superior RNFLT measurements compared to aligned exams.

Equivalence between Redfield- and master-equation approaches for a time-dependent quantum system and coherence control

We present a derivation of the Redfield formalism for treating the dissipative dynamics of a time-dependent quantum system coupled to a classical environment. We compare such a formalism with the master equation approach where the environments are treated quantum mechanically. Focusing on a time-dependent spin-1/2 system we demonstrate the equivalence between both approaches by showing that they lead to the same Bloch equations and, as a consequence, to the same characteristic times T(1) and T(2) (associated with the longitudinal and transverse relaxations, respectively). These characteristic times are shown to be related to the operator-sum representation and the equivalent phenomenological-operator approach. Finally, we present a protocol to circumvent the decoherence processes due to the loss of energy (and thus, associated with T(1)). To this end, we simply associate the time dependence of the quantum system to an easily achieved modulated frequency. A possible implementation of the protocol is also proposed in the context of nuclear magnetic resonance.

Frequency Domain Connectivity Identification: An Application of Partial Directed Coherence in fMRI

Functional magnetic resonance imaging (fMRI) has become an important tool in Neuroscience due to its noninvasive and high spatial resolution properties compared to other methods like PET or EEG. Characterization of the neural connectivity has been the aim of several cognitive researches, as the interactions among cortical areas lie at the heart of many brain dysfunctions and mental disorders. Several methods like correlation analysis, structural equation modeling, and dynamic causal models have been proposed to quantify connectivity strength. An important concept related to connectivity modeling is Granger causality, which is one of the most popular definitions for the measure of directional dependence between time series. In this article, we propose the application of the partial directed coherence (PDC) for the connectivity analysis of multisubject fMRI data using multivariate bootstrap. PDC is a frequency domain counterpart of Granger causality and has become a very prominent tool in EEG studies. The achieved frequency decomposition of connectivity is useful in separating interactions from neural modules from those originating in scanner noise, breath, and heart beating. Real fMRI dataset of six subjects executing a language processing protocol was used for the analysis of connectivity. Hum Brain Mapp 30:452-461, 2009. (C) 2007 Wiley-Liss, Inc.

Prospective ultramorphological characterization of human hair by optical coherence tomography

Background/purpose The continuous advancement in cosmetic science has led to an increasing demand for the development of non-invasive, reliable scientific techniques directed toward claim substantiation, which is of utmost relevance, to obtain data regarding the efficacy and safety of cosmetic products. Methods In this work, we used the optical coherence tomography (OCT) technique to produce in vitro transversal section-images of human hair. We also compared the OCT signal before and after chemical treatment with an 18% w/w ammonium thioglycolate solution. Results The mean diameter of the medulla was 29 +/- 7 mu m and the hair diameter was 122 +/- 16 mu m in our samples of standard Afro-ethnic hair. A three-dimensional (3D) image was constructed starting from 601 cross-sectional images (slices). Each slice was taken in steps of 6.0 mu m at eight frames per second, and the entire 3D image was constructed in 60 s. Conclusion It was possible to identify, using the A-scan protocol, the principal structures: the cuticle, cortex and medulla. After chemical treatment, it was not possible to identify the main structures of hair fiber due to index matching promoted by deleterious action of the chemical agent.

Refractive index tomography of turbid media by bifocal optical coherence refractometry

We demonstrate tomographic imaging of the refractive index of turbid media using bifocal optical coherence refractometry (BOCR). The technique, which is a variant of optical coherence tomography, is based on the measurement of the optical pathlength difference between two foci simultaneously present in a medium of interest. We describe a new method to axially shift the bifocal optical pathlength that avoids the need to physically relocate the objective lens or the sample during an axial scan, and present an experimental realization based on an adaptive liquid-crystal lens. We present experimental results, including video clips, which demonstrate refractive index tomography of a range of turbid liquid phantoms, as well as of human skin in vivo.

Gamma-band oscillations in fronto-central areas during performance of a sensorimotor integration task: A qEEG coherence study

This study aimed to elucidate electrophysiological and cortical mechanisms involved in anticipatory actions when 23 healthy right-handed subjects had to catch a free falling object by qEEG gamma-band (30-100 Hz). It is involved in cognitive processes, memory, spatial/temporal and proprioceptive factors. Our hypothesis is that an increase in gamma coherence in frontal areas will be observed during moment preceding ball drop, due to their involvement in attention, planning, selection of movements, preparation and voluntary control of action and in central areas during moment after ball drop, due to their involvement in motor preparation, perception and execution of movement. However, through a paired t-test, we found an increase in gamma coherence for F3-F4 electrode pair during moment preceding ball drop and confirmed our hypothesis for C3-C4 electrode pair. We conclude that gamma plays an important role in reflecting binding of several brain areas in a complex motor task as observed in our results. Moreover, for selection of movements, preparation and voluntary control of action, motor preparation, perception and execution of movement, the integration of somatosensory and visual information is mandatory. (C) 2010 Elsevier Ireland Ltd. All rights reserved.

Evaluation of retinal nerve fiber layer thickness measurements using optical coherence tomography in patients with tobacco-alcohol-induced toxic optic neuropathy

Three patients with progressive visual loss, chronic alcoholism and tabagism were submitted to a complete neuro-ophthalmic examination and to retinal nerve fiber layer (RNFL) measurements using optical coherence tomography (OCT) scanning. Two patients showed marked RNFL loss in the temporal sector of the optic disc. However, a third patient presented RNFL measurements within or above normal limits, based on the Stratus-OCT normative database. Such findings may be due to possible RNFL edema similar to the one that may occur in the acute phase of toxic optic neuropathies. Stratus-OCT was able to detect RNFL loss in the papillomacular bundle of patients with tobacco-alcohol-induced toxic optic neuropathy. However, interpretation must be careful when OCT does not show abnormality in order to prevent diagnostic confusion, since overestimation of RNFL thickness measurements is possible in such cases.

Comparison of Fourier-Domain and Time-Domain Optical Coherence Tomography in the Detection of Band Atrophy of the Optic Nerve

PURPOSE: To compare the ability of Fourier-domain (FD) optical coherence tomography (3D OCT-1000; Top, con, Tokyo, Japan) and time domain (TD) OCT (Stratus; Carl Zeiss Meditec Inc, Dublin, California, USA) to detect axonal loss in eyes with band atrophy (BA) of the optic nerve. DESIGN: Cross-sectional study. METHODS: Thirty-six eyes from 36 patients with BA and temporal visual field (VF) defect from chiasmal compression and 36 normal eyes were studied. Subjects were submitted to standard automated perimetry and macular and retinal nerve fiber layer (RNFL) measurements were taken using 3D OCT-1000 and Stratus OCT. Receiver operating characteristic (ROC) curves were calculated for each parameter. Spearman correlation coefficients were obtained to evaluate the relationship between RNFL and macular thickness parameters and severity of VF loss. Measurements from the two devices were compared. RESULTS: Regardless of OCT device, all RNFL and macular thickness parameters were significantly lower in eyes with BA compared with normal eyes, but no statistically significant difference was found with regard to the area under the ROC curve. Structure-function relationships were also similar for the two devices. In both groups, RNFL and macular thickness measurements were generally and in some cases significantly smaller with 3D OCT-1000 than with Stratus OCT. CONCLUSIONS: The introduction of FD technology did not lead to better discrimination ability for detecting BA of the optic nerve compared with TD technology when using the software currently provided by the manufacturer. 3D OCT-1000 FD OCT RNFL and macular measurements were generally smaller than TD Stratus OCT measurements. Investigators should be aware of this fact when comparing measurements obtained with these two devices. (Am J Oplathalmol 2009;147: 56-63. (c) 2009 by Elsevier Inc. All rights reserved.)