Skew-normal linear calibration: a Bayesian perspective

In this paper, we present a Bayesian approach for estimation in the skew-normal calibration model, as well as the conditional posterior distributions which are useful for implementing the Gibbs sampler. Data transformation is thus avoided by using the methodology proposed. Model fitting is implemented by proposing the asymmetric deviance information criterion, ADIC, a modification of the ordinary DIC. We also report an application of the model studied by using a real data set, related to the relationship between the resistance and the elasticity of a sample of concrete beams. Copyright (C) 2008 John Wiley & Sons, Ltd.

An extension of the half-normal distribution

In this paper we introduce a new distribution, namely, the slashed half-normal distribution and it can be seen as an extension of the half-normal distribution. It is shown that the resulting distribution has more kurtosis than the ordinary half-normal distribution. Moments and some properties are derived for the new distribution. Moment estimators and maximum likelihood estimators can computed using numerical procedures. Results of two real data application are reported where model fitting is implemented by using maximum likelihood estimation. The applications illustrate the better performance of the new distribution.

Parameter recovery for a skew-normal IRT model under a Bayesian approach: hierarchical framework, prior and kernel sensitivity and sample size

Item response theory (IRT) comprises a set of statistical models which are useful in many fields, especially when there is an interest in studying latent variables (or latent traits). Usually such latent traits are assumed to be random variables and a convenient distribution is assigned to them. A very common choice for such a distribution has been the standard normal. Recently, Azevedo et al. [Bayesian inference for a skew-normal IRT model under the centred parameterization, Comput. Stat. Data Anal. 55 (2011), pp. 353-365] proposed a skew-normal distribution under the centred parameterization (SNCP) as had been studied in [R. B. Arellano-Valle and A. Azzalini, The centred parametrization for the multivariate skew-normal distribution, J. Multivariate Anal. 99(7) (2008), pp. 1362-1382], to model the latent trait distribution. This approach allows one to represent any asymmetric behaviour concerning the latent trait distribution. Also, they developed a Metropolis-Hastings within the Gibbs sampling (MHWGS) algorithm based on the density of the SNCP. They showed that the algorithm recovers all parameters properly. Their results indicated that, in the presence of asymmetry, the proposed model and the estimation algorithm perform better than the usual model and estimation methods. Our main goal in this paper is to propose another type of MHWGS algorithm based on a stochastic representation (hierarchical structure) of the SNCP studied in [N. Henze, A probabilistic representation of the skew-normal distribution, Scand. J. Statist. 13 (1986), pp. 271-275]. Our algorithm has only one Metropolis-Hastings step, in opposition to the algorithm developed by Azevedo et al., which has two such steps. This not only makes the implementation easier but also reduces the number of proposal densities to be used, which can be a problem in the implementation of MHWGS algorithms, as can be seen in [R.J. Patz and B.W. Junker, A straightforward approach to Markov Chain Monte Carlo methods for item response models, J. Educ. Behav. Stat. 24(2) (1999), pp. 146-178; R. J. Patz and B. W. Junker, The applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses, J. Educ. Behav. Stat. 24(4) (1999), pp. 342-366; A. Gelman, G.O. Roberts, and W.R. Gilks, Efficient Metropolis jumping rules, Bayesian Stat. 5 (1996), pp. 599-607]. Moreover, we consider a modified beta prior (which generalizes the one considered in [3]) and a Jeffreys prior for the asymmetry parameter. Furthermore, we study the sensitivity of such priors as well as the use of different kernel densities for this parameter. Finally, we assess the impact of the number of examinees, number of items and the asymmetry level on the parameter recovery. Results of the simulation study indicated that our approach performed equally as well as that in [3], in terms of parameter recovery, mainly using the Jeffreys prior. Also, they indicated that the asymmetry level has the highest impact on parameter recovery, even though it is relatively small. A real data analysis is considered jointly with the development of model fitting assessment tools. The results are compared with the ones obtained by Azevedo et al. The results indicate that using the hierarchical approach allows us to implement MCMC algorithms more easily, it facilitates diagnosis of the convergence and also it can be very useful to fit more complex skew IRT models.

A bayesian model for mixed responses and skew laten variable to measure HRQoL in children

The aim of the thesi is to formulate a suitable Item Response Theory (IRT) based model to measure HRQoL (as latent variable) using a mixed responses questionnaire and relaxing the hypothesis of normal distributed latent variable. The new model is a combination of two models already presented in literature, that is, a latent trait model for mixed responses and an IRT model for Skew Normal latent variable. It is developed in a Bayesian framework, a Markov chain Monte Carlo procedure is used to generate samples of the posterior distribution of the parameters of interest. The proposed model is test on a questionnaire composed by 5 discrete items and one continuous to measure HRQoL in children, the EQ-5D-Y questionnaire. A large sample of children collected in the schools was used. In comparison with a model for only discrete responses and a model for mixed responses and normal latent variable, the new model has better performances, in term of deviance information criterion (DIC), chain convergences times and precision of the estimates.

Modeling absolute differences in life expectancy with a censored skew-normal regression approach.

Parameter estimates from commonly used multivariable parametric survival regression models do not directly quantify differences in years of life expectancy. Gaussian linear regression models give results in terms of absolute mean differences, but are not appropriate in modeling life expectancy, because in many situations time to death has a negative skewed distribution. A regression approach using a skew-normal distribution would be an alternative to parametric survival models in the modeling of life expectancy, because parameter estimates can be interpreted in terms of survival time differences while allowing for skewness of the distribution. In this paper we show how to use the skew-normal regression so that censored and left-truncated observations are accounted for. With this we model differences in life expectancy using data from the Swiss National Cohort Study and from official life expectancy estimates and compare the results with those derived from commonly used survival regression models. We conclude that a censored skew-normal survival regression approach for left-truncated observations can be used to model differences in life expectancy across covariates of interest.

Geochemistry, clay mineralogy and grain size distribution in various sediment cores drilled during ODP Leg 119 on the Antarctic continental shelf off Prydz Bay

Drilling was undertaken at five sites (739-743) on ODP Leg 119 on a transect across the continental shelf of Prydz Bay, East Antarctica, to elucidate the long-term glacial history of the area and to examine the importance of the area with respect to the development of the East Antarctic ice sheet as a whole. In addition to providing a record of glaciation spanning 36 m.y. or more, Leg 119 has provided information concerning the development of a continental margin under the prolonged influence of a major ice sheet. This has allowed the development of a sedimentary model that may be applicable not only to other parts of the Antarctic continental margin, but also to northern high-latitude continental shelves. The cored glacial sedimentary record in Prydz Bay consists of three major sequences, dominated by diamictite: 1. An upper flat-lying sequence that ranges in thickness from a few meters in inner and western Prydz Bay to nearly 250 m in the outer or eastern parts of the bay. The uppermost few meters consist of Holocene diatom ooze and diatomaceous mud with a minor ice-rafted component overlying diamicton and diamictite of late Miocene to Quaternary age. The diamictite is mainly massive, but stratified varieties and minor mudstone and diatomite also occur. 2. An upper prograding sequence cored at Sites 739 and 743, unconformly below the flat-lying sequence. This consists of a relatively steep (4° inclination) prograding wedge with a number of discrete sedimentary packages. At Sites 739 and 743 the sequence is dominated by massive and stratified diamictite, some of which shows evidence of slumping and minor debris flowage. 3. A lower, more gently inclined, prograding sequence lies unconformably below the flat-lying sequence at Site 742 and the upper prograding sequence at Site 739. This extends to the base of both sites, to 316 and 487 mbsf, respectively. It is dominated by massive, relatively clast-poor diamictite which is kaolinite-rich, light in color, and contains sporadic carbonate-cemented layers. The lower part of Site 742 includes well-stratified diamictites and very poorly sorted mudstones. The base of this site has indications of large-scale soft-sediment deformation and probably represents proximity to the base of the glacial sequence. Facies analysis of the Prydz Bay glacial sequence indicates a range of depositional environments. Massive diamictite is interpreted largely as waterlain till, deposited close to the grounding line of a floating glacier margin, although basal till and debris flow facies are also present. Weakly stratified diamictite is interpreted as having formed close to or under the floating ice margin and influenced by the input of marine diatomaceous sediment (proximal glaciomarine setting). Well-stratified diamictite has a stronger marine input, being more diatom-rich, and probably represents a proximal-distal glaciomarine sediment with the glaciogenic component being supplied by icebergs. Other facies include a variety of mudstones and diatom-rich sediments of marine origin, in which an ice-rafted component is still significant. None of the recovered sediments are devoid of a glacial influence. The overall depositional setting of the prograding sequence is one in which the grounded ice margin is situated close to the shelf edge. Progradation was achieved primarily by deposition of waterlain till. The flat-lying sequence illustrates a complex sequence of advances and retreats across the outer part of the shelf, with intermittent phases of ice loading and erosion. The glacial chronology is based largely on diatom stratigraphy, which has limited resolution. It appears that ice reached the paleoshelf break by earliest Oligocene, suggesting full-scale development of the East Antarctic ice sheet by that time. The ice sheet probably dominated the continental margin for much of Oligocene to middle Miocene time. Retreat, but not total withdrawal of the ice sheet, took place in late Miocene to mid-Pliocene time. The late Pliocene to Pleistocene was characterized by further advances across, and progradation of, the continental shelf. Holocene time has been characterized by reduced glacial conditions and a limited influence of glacial processes on sedimentation.

Grain-size distribution and Folk's statistics at DSDP Leg 64 Holes

Grain size of 139 unconsolidated sediment samples from seven DSDP sites in the Guaymas Basin and the southeastern tip of the Baja California Peninsula was determined by sieve and pipette techniques. Shepard (1954) classification and Inman (1952) parameters correlation were used for all samples. Sediment texture ranged from sand to silty clay. On the basis of grain-size parameter, the sediments can be divided into three broad groups: (1) very fine sands and coarse silts; (2) medium- to very fine silts; and (3) clays and coarse silts.

Grain-size distribution of sediments from DSDP Leg 65 Holes

The grain-size distribution of 223 unconsolidated sediment samples from four DSDP sites at the mouth of the Gulf of California was determined using sieve and pipette techniques. Shepard's (1954) and Inman's (1952) classification schemes were used for all samples. Most of the sediments are hemipelagic with minor turbidites of terrigenous origin. Sediment texture ranges from silty sand to silty clay. On the basis of grain-size parameters, the sediments can be divided into the following groups: (1) poorly to very poorly sorted coarse and medium sand; and (2) poorly to very poorly sorted fine to very fine sand and clay.

Grains size and components distribution in turbidite layesr of ODP Leg 101

Textural and compositional differences were found between gravity-flow sheets in an open-ocean environment on the northern slope of Little Bahama Bank (Site 628, Pliocene turbidite sequence) and in a closed-basin depositional setting (Site 632, Quaternary turbidite sequence). Mud-supported debris-flow sheets were cored at Site 628. Average mean grain size of the turbidite samples was lower, mud content was higher, and sorting was poorer than in comparable samples from Site 632. This reflects the deposition of proximal, low-energy turbidity currents and debris flows on a base-ofslope carbonate apron. No mud-supported debris-flow sheets were deposited in the investigated sediment sequence of Hole 632A. Many larger turbidity currents from around the margins of Exuma Sound may have reached this central basin setting, depositing sediments that had been transported over longer distances. Planktonic components dominate in the grain-sized fraction (500-1000 µm) of turbidite samples from Hole 628A, while platform detritus is rare. We interpreted this as resulting from the erosion and reworking of a large area of open-ocean slope sediments by gravity flows. In contrast, large amounts of benthic and platform components were found in the turbidite samples of Hole 632A. This may be explained by the fact that the slopes of the enclosed Exuma Sound are steep, and turbidity currents bypassed much of these slopes through pronounced channels, delivering more shallow-water detritus to the deep basin. Erosion of slope sediments, a possible source area of planktonic detritus, is assumed to be low. The small slope area in relation to the larger surrounding platform areas and lower production of planktonic components in the enclosed waters of Exuma Sound may also explain the observed low number of planktonic components at Hole 632A. Turbidite material from both open-ocean and enclosed-basin environments was deposited at Site 635.

Kinetic Model for Micro Algae Cell Concentration and Size Distribution: Application to Nannochloropsis gaditana

The cell concentration and size distribution of the microalgae Nannochloropsis gaditana were studied over the whole growth process. Various samples were taken during the light and dark periods the algae were exposed to. The distributions obtained exhibited positive skew, and no change in the type of distribution was observed during the growth process. The size distribution shifted to lower diameters in dark periods while in light periods the opposite occurred. The overall trend during the growth process was one where the size distribution shifted to larger cell diameters, with differences between initial and final distributions of individual cycles becoming smaller. A model based on the Logistic model for cell concentration as a function of time in the dark period that also takes into account cell respiration and growth processes during dark and light periods, respectively, was proposed and successfully applied. This model provides a picture that is closer to the real growth and evolution of cultures, and reveals a clear effect of light and dark periods on the different ways in which cell concentration and diameter evolve with time.

Alzheimer's disease: size class frequency distribution of senile plaques: do they indicate when a brain tissue was affected?

The size class frequency distribution of a sample of senile plaques (SP) was determined in a total of 20 brain regions from 5 elderly cases of Alzheimer's disease (AD). The purpose of the study was to determine whether a comparison of the frequency distributions could be used to determine the chronology of SP development in the AD brain. SP from 10 microns to a maximum diameter of 160 microns were present in the tissue and the size class frequency distributions were positively skewed. The frequency distributions varied between brain regions in: (1) the size class containing the mode, (2) the degree of positive skew, and (3) the ratio of large to small SP. In most patients the ratio of large to small SP was higher in the hippocampus or adjacent gyrus compared with temporal, parietal and frontal neocortex. If the diameter of a SP reflects its age in the tissue than the data suggest that SP formed earlier either in the hippocampus or adjacent gyrus compared with the other neocortical tissues. However, this conclusion rests on a number of assumptions including: (1) that SP diameter is directly related to age, (2) that SP development occurs at similar rates in different brain regions and (3) that, once formed, SP are not removed from the tissue by astrocytes.

Size frequency distribution of beta-amyloid (Abeta) deposits in dementia with Lewy bodies

In Alzheimer's disease (AD) and Down's syndrome (DS), the size frequency distribution of the beta-amyloid (Abeta) deposits can be described by a log-normal model and may indictae the growth of the deposits. This study determined the size frequency distribution of the Abeta deposits in the temporal lobe in 8 casaes of dementia with Lewy bodies (DLB) with associated AD pathology (DLB/AD. The size distributions of Abeta deposits were unimodal and positively skewed; the mean size of deposi and the degree of skew varying with deposit type and brain region. Size distributions of the primitive deposits had lower means and were less skewed compared with the diffuse and classic deposits. In addition, size distributions in the hippocampus and parahippocampal gyrus (PHG) had larger means and a greater degree of skew compared with other cortical gyri. All size distributions deviated significantly from a log-normal model. There were more Abeta deposits than expected in the smaller size classes and fewer than expected near the mean and in the larger size classes. The data suggest thatthe pattern of growth of the Abeta deposits in DLB/AD depends both on deposit morphology and brain area. In addition, Abeta deposits in DLB appear to grow to within a more restricted size range than predicted and hence, to have less potential for growth compared with cases of 'pure' AD and DS.