Degree Of Skewness Research Articles

The Robustness of LISREL Estimates in Structural Equation Models with Categorical Variables

AbstractThis study was an examination of the effect of type of correlation matrix on the robustness of LISREL maximum likelihood and unweighted least squares structural parameter estimates for models with categorical manifest variables. Two types of correlation matrices were analyzed: one containing Pearson product moment correlations and one containing tetrachoric, polyserial, and product moment correlations as appropriate. Using continuous variables generated according to the equations defining the population model, three cases were considered by dichotomizing some of the variables with varying degrees of skewness. When Pearson product moment correlations were used to estimate associations involving dichotomous variables, the structural parameter estimates were biased when skewness was present in the dichotomous variables. Moreover, the degree of bias was consistent for both the maximum likelihood and unweighted least squares estimates. The analysis of mixed matrices produced average estimates that more...

On the skewness of sea-surface elevation

Surface skewness is a statistical measure of the vertical asymmetry of the air-sea interface – exemplified by the sharp crests and rounded troughs of surface gravity waves. Some authors have proposed a constant ratio between surface skewness and the ‘significant slope’ of the waves. Here it is shown theoretically that no such simple relation is to be expected.Wave records are of at least two different types; Eulerian (as made with a fixed probe) or Lagrangian (as with a free-floating buoy). The corresponding statistical properties are examined. At first sight it might appear that the degree of skewness in corresponding records would be different. However it is shown that to lowest order the skewness is invariant; only the apparent mean level is different, at second order.

Measuring Skewness and Kurtosis

The question of how to measure the degree of skewness of a continuous random variable is addressed. In van Zwet (1964) a method for ordering two distributions with regard to skewness is given. Here, using the concept of comparative skewness, we consider properties that a measure of skewness should satisfy. Several extensions of the Bowley measure of skewness taking values on (-1, 1) are discussed. How well these measures reflect one's intuitive idea of skewness is examined. These measures of skewness are extended to measures of kurtosis for symmetric distributions.

Downstream and Lateral Transport of Nymphs of Two Mayfly Species (Ephemeroptera)

By releasing live and dead nymphs of two mayfly species, Baetis tricaudatus Dodds and Ephemerella inermis Eaton, into the water column of a river and catching them downstream in a series of contiguous nets, I estimated settling and lateral dispersal rates. Live animals settled over shorter distances than dead individuals, and became more widely laterally dispersed for any given distance downstream and mean current velocity. Transport of suspended organisms in a river in which mean current velocity increases with increasing distance from shore may produce a skewed cross-river benthic distribution of animals. The degree of skewness will depend on the current velocity gradient and on suspended animals' settling behavior.

Optimal Self-Propulsion of a Deformable Prolate Spheroid

The problem of self-propulsion of an elongated deformable body moving in an infinite medium of inviscid fluid is considered in some detail. A prolate spheroid is chosen as a model shape, and a particular deformation pattern which maximizes the Froude efficiency is sought. The Froude efficiency in this context is defined by the ratio of the kinetic energy of the body to the total kinetic energy of the system comprising the body and the fluid. It is demonstrated that a body can propel itself from rest in a persistent manner even for a periodic surface deformation with zero mean which preserves both the volume and the location of its centroid. Under these constraints the induced forward velocity of the body is of 0(ε2) where ε is the amplitude of the deformation velocity. It is also demonstrated that for a persistent self-propulsion to exist the body should develop a large degree of skewness, resulting from the interaction between the two deformation components—one with fore-and-aft symmetry and one without. It is also essential that the symmetric and asymmetric deformation components should be out of phase.

Diameter Distributions in Even-aged Stands of Shade-tolerant and Midtolerant Tree Species

Characteristics of diameter distributions in 28 even-aged northern hardwood (Acer-Betula-Tsuga) and upland oak (Quercus spp.) stands were investigated to determine if even-aged stands of shade-tolerant and midtolerant (intermediate) species can be consistently identified without direct age determinations. The diameter distributions of midtolerant species consistently approximated a normal distribution. Those of tolerant species were highly variable, ranging from unimodal to steeply descending in form. Distributions of tolerant species that departed strongly from a unimodal form were those of stands having a large admixture of midtolerant species. Confusion with all-aged stands in such cases can be avoided if classification of age structure is based on the diameter distributions of midtolerant species. Even-aged and all-aged stands can generally be distinguished on the basis of diameter distributions, but even-aged stands cannot be consistently distinguished from multi-aged stands in which young trees are sparse or absent. In such cases, a number of direct age determinations would be necessary. INTRODUCTION Data on age structure of forest stands are often needed in ecological studies, especially for quantifying the role of natural and man-caused disturbances in forested landscapes. The only feasible nondestructive technique for direct interpretation of age structure requires the extraction of cores from trees in sample plots. This procedure is time-consuming and is not feasible when extensive areas must be inventoried. Accuracy in age determination of large trees is limited by the frequent occurrence of heart rot and by the fact that the number of rings above the butt swell does not indicate the total age of the tree. Indirect assessment of age structure has therefore been frequently attempted, of which the most promising method has been the interpretation of diameter frequency distributions. To give a reasonable indication of age structure, diameter distributions must be plotted separately for individual species from small homogeneous stands (Hough, 1932). Even-aged stands of many species typically have unimodal diameter distributions that may show varying degrees of skewness at a young age but gradually approach a more symmetric normal distribution with time (Baker, 1923; Meyer, 1930; Hough, 1932; Schnur, 1934; Nelson, 1964; Mohler et al., 1978). All-aged stands of shade-tolerant species with only light past disturbance, on the other hand, are wellknown to have steeply descending, monotonic diameter distributions that can be approximated by the negative exponential and negative power functions (de Liocourt, 1898; Hough, 1932; Assmann, 1970; Leak, 1973, 1975; Hett and Loucks, 1976; Tubbs, 1977; Lorimer, 1980). A third type of age structure may occur in stands having several age classes that may or may not be of equal prominence. Diameter distributions in such stands may vary from near normal to irregular negative exponential in form depending upon the dispersion of age classes and the degree of shade tolerance of the species (Hough, 1932; Henry and Swan, 1974; Leak, 1975; Oliver and Stephens, 1977; Harcombe and Marks, 1978; Lorimer, 1980, 1983a). Some stands of this latter type are technically uneven-aged or all-aged but nevertheless developed after heavy destruction of the overstory, and the wide range in age merely reflects the range of ages among smaller trees present prior to the disturbance (Henry and Swan, 1974; Lorimer, 1983a). In other cases, the restricted or unbalanced age distribution may be due to other external

Room temperature oxidation of chromium and nickel implanted iron, and bulk alloys

This investigation sets out to directly compare the room temperature oxidation of Cr and Ni implanted Fe with bulk alloys in the range of commercial stainless steels. The samples investigated included pure single crystal (100) Fe and polycrystalline Fe foils implanted with Ni and Cr ions at 40 keV, with doses to match the commercial stainless steel compositions. Analyses of the surfaces were carried out by Auger Electron Spectroscopy (AES) and Ar ion sputter profiling. The samples were oxidized at 7 L and for two weeks at atmospheric pressure and room temperature. Results revealed that both measured and calculated Ni implant distributions in Fe were gaussian, with the measured distribution showing a greater degree of skewness. The measured Cr implant distribution in Fe is significantly different from that for the calculated distribution. The measured Cr distribution peaked sharply at 6 nm (coinciding with the calculated peak in the vacancy distribution produced by implantation), then again with a broader peak at approximately 30 nm. These results were representative of the Ni and Cr distributions in both sets of single crystals and polycrystalline foils. The bulk alloys possessed a greater surface enrichment of Cr than the implanted crystals and foils. The room temperature oxidation of all the samples revealed that the oxygen concentration variation with depth for the bulk alloys was lower compared to the implanted samples. Further, for the implanted samples the oxide thickness was inversely related to the surface Cr concentration.

Growth Patterns in Sand Lance, Ammodytes dubius, from the Grand Bank

Growth patterns in sand lance (Ammodytes dubius) from the Grand Bank were examined using back-calculated lengths from otolith measurements. Lee's phenomenon of apparent change in growth rate with age is clearly demonstrated as is the phenomenon of growth compensation. Lee's phenomenon is explained by discriminatory mortality of the faster-growing sand lance whereas growth compensation is attributed to the differing ages of maturity of the initial faster-growing and slower-growing fish. The significant reduction in the standard deviation and the increasing degree of left skewness in length frequencies of older sand lance is explained by a combination of Lee's phenomenon and growth compensation acting together throughout the growth history of individual sand lance.Key words: sand lance, growth, back-calculation. Grand Bank

Distributions of Amenorrhoea and Anovulation

Most statistics of postpartum amenorrhea are based on retrospective rather than prospective reporting and except when short (averaging less than 6 months) exhibit bimodality negative skewness and gross heaping on multiples of 6 months. The 2 best prospective series 1 representing moderately long and the other very long postpartum amenorrhea exhibit modest degrees of bimodality and negative skewness. Bimodality is plausibly explained in terms of sample heterogeneity that is a combination of lengths of amenorrhea of successful lactators which might be expected to be unimodal with the brief amenorrhea of nonlactators and of mother experiencing early infant deaths and sometimes of respondents experiencing pregnancy losses or confusing postpartum bleeding for genuine onset of menstruation. To cope with the problems of bimodality and heaping on multiples of 6 months Lesthaeghe and Page have used an ingenious logit regression amenorrhea approach. Logit-fitted curves exhibit positive skewness when the retrospective amenorrhea series is short or very long and negative skewness when the series is of intermediate length. For retrospective series averaging between 6 and 9 months the negative skewness is especially marked with a 2nd mode several months later than the mean. A key issue is whether this feature reflects a real phenomenon related to extreme sample heterogeneity or is an artifact of partially surpressed heaping. Hitherto in the construction of family building models postpartum anovulation has been conventionally represented by Barretts modified Pascal distribution. It is shown than when amenorrhea series are long averaging over a year this distribution generates excessive variances. A generalization of Barretts curve the negative binomial yields control over the variance as well as the mean but also unimodal positively skewed curves of amenorrhea that grossly underestimate the left tail of the distribution. A mixed geometric-negative binomial distribution is proposed which ensures bimodality avoids the deficit of short length and produces a much more moderate degree of positive skewness for series of intermediate length. The present analysis makes clear that considerably more prospective data are needed to prove decisively that the degree of bimodality depends primarily on sample heterogeneity; or to decide whether the strong negative skewness found in some logit-fitted curves is artifactual or real. Correspondingly more prospective data including information about the timing of infant deaths are a prerequisite to exploring how well mixed distributions can simulate bimodal distributions of amenorrhea. (Authors)

POPULATION PROFILE DEVELOPMENT AND MORPHOTYPIC DIFFERENTIATION IN THE GIANT FRESHWATER PRAWN Macrobrachium rosenbergii (de Man)

AbstractPost‐larval populations of Macrobrachium rosenbergii are characterized by a homogenous size distribution curve which changes, with time, to a positively skewed curve, with some individuals demonstrating a higher growth rate than the rest of the population. No morphotypic variation is observed at the post‐larval and juvenile stages, and the ratio of males to females is 1:1, equally distributed along the whole size distribution curve. Both the mean weight and the development of the skewed distribution curve are inversely affected by increasing density.In a mature population, three male morphotypes can be clearly differentiated by claw color, relative size within the population and the ratio of claw length to body length. The Small Male (SM) morphotype is characterized by short clear claws, while the two larger morphotypes are characterized by Orange Claws (OC) and Blue Claws (BC). Their relative occurrence in the population is SM 20–25%, OC 15–25%, and BC 5–10%. No apparent morphotypic variation exists in females and their size distribution is generally homogenous. Increased population density does not change the relative position, nor the relative proportions of the three male morphotypes and the females, however, the degree of skewness of the curve is inversely related to the density.

The distribution of intraocular pressures in the general population

A variety of studies are reviewed in which intraocular pressure (IOP) was determined in defined, general populations. The studies exhibit a remarkably consistent pattern for the frequency distribution of IOP. The distribution conforms well to a normal curve for pressures up to 21 mm Hg, after which a distinct skewness to the right begins to appear. When the data are grouped by age and sex, the degree of right skewness intensifies with age. Mean IOP increases with age. Some studies also show higher mean levels for women than men, but not all do. The arbitrary nature of specifying upper limits of normality is stressed. In addition, the literature is reviewed concerning the arguments for a mixture of two overlapping normal (or lognormal) curves vs a continuum for the distribution of IOP.

Linear transformation and the interpretation of MMPI T scores

Raw scores for the clinical scales of the MMPI can be changed to T scores by linear transformation. However, if the underlying distributions of raw scores are not distributed normally, one cannot assume that the traditional point of significance, T score 70, falls at the 97.7 percentile. Our analysis of the improved Minnesota normal sample described by Hathaway and Briggs in 1957 revealed statistically significant degrees of skewness on most of the clinical scales. We suggest that clinicians consider more conservative interpretation for scales 1 (Hs), 2 (D), 7 (PT), 8 (SC), and 9 (MA) for females and for scales 1 (Hs), 2 (D), 4 (PD), 7 (PT), 8 (SC), and 9 (MA) for males.

Fractiles for the nonsymmetric stable distributions

The paper presents a table of fractiles for the nonsymmetric stable distributions with index a =1.0(0.25)2.0 and varying degrees of skewness, and is an extension of the tabulation of Fama and Roll (1968). The application to Monte Carlo studies is briefly discussed.

An Empirical Study of Error Characteristics in Audit Populations

Most modeling and analysis of audit populations has been based on surprisingly little formal study of actual empirical evidence. The single major empirical investigation to date has been the pioneering work of Neter and Loebbecke [1975; 1977]. This study examined the behavior of many statistical procedures commonly used in auditing, based on simulated audit populations constructed by extrapolating characteristics of errors found in audits of four actual populations. Neter and Loebbecke made a thorough examination of the four audit populations and reported on both the shapes of the book value distributions and error characteristics including error rate, magnitude, and changes in both rate and magnitude as a function of book value. In spite of their limited number, these four populations represented considerable variety in terms of type of account, degree of skewness, error rate, and balance of errors between underand overstatements. The effective size of the empirical data base was greatly expanded by the authors' careful control of these error characteristics in creating simulated populations with varying error rates. A major conclusion of the study was that a number of widely used statistical procedures, particularly those adopted from the sample survey literature, can be seriously misleading in the auditing context. This conclusion concurs with that of Kaplan [1973], who based his analysis on

On the Assumptions Concerning the Assumptions of atTest

Summary A computer simulation which generated various sample distributions was performed. The equal interval assumption and degree of skewness were systematically manipulated to examine the sensitivity of the t and Mann-Whitney tests to varying degrees of violations of these t test assumptions. Results show that of the 348 sets (2436 t tests), 94 (27%) contained shifts from significant to nonsignificant t test values under mild violations of the equal-interval and normality assumption and would have led to decision errors. Of these, 55% would have been Type I and 45% would have been Type II. The implications of these results are discussed relative to Types I and II errors, to the use of parametric and nonparametric statistics, and to the likelihood of encountering such problems in sampled distributions.

Estimating Means Via Multiple Matrix Sampling: a Note On the Effects of Selected Data Base Characteristics

In this paper, Shoemaker's (1972, 1973b) conclusions related to the influence of various data base characteristics (reliability, variability of item difficulty indices, and degree of skewness in the normative distribution) on the standard error of a mean estimated via Multiple Matrix Sampling (MMS) procedures are examined. It is noted that these conclusions are true under fairly restrictive conditions. Also, it is shown that even under these conditions it is more reasonable to attribute changes in the standard error to changes in examinee variability rather than to changes in these other data base characteristics.

Effects of Coarse Grouping and Skewed Marginal Distributions on the Pearson Product Moment Correlation Coefficient

Twelve non-normal bivariate distributions were constructed to investigate (a) the effects which coarsely grouped, skewed marginal distributions have on the correlation coefficient and (b) the accuracy of current methods of correction for grouping. The investigation revealed that the correlation coefficient is severely limited in the values it can assume when the underlying marginals are highly, oppositely skewed and one or both of the marginals has only two class intervals. The study further revealed that the correction formula (which is based on the assumption of underlying, normal marginals) is extremely accurate, regardless of the degree of marginal skewness, except where one or both of the marginals has only two class intervals. The major conclusions of the study were that the use of asymmetrical, dichotomous variables should be avoided in correlation analysis, and that the correction for coarse grouping should be applied, on an experimental basis, to correlations computed for joint distributions of coarsely categorized variables.

The median voter rule and the theory of political choice

This paper examines the relationship between voter preferences and political decisions. The behavior of political candidates is investigated under conditions where either the expected number of votes or the probability of election is maximized. The primary conclusion is that distributions of preferences which are skewed to the right are likely to be associated with political decisions which understate the point of median preferences. However, under reasonable assumptions regarding the degree of skewness, the extent of the difference may be relatively small so that the median voter rule remains a useful means of approximating the results obtained from the process of political choice.

The electroencephalogram in the prediction of human reaction time during growth and development

Characteristics of the distributions of electroencephalographic (EEG) half waves recorded in children during performance of a simple auditory reaction time (RT) task were investigated. The purpose was to determine the extent to which difference in these distributions could account for children's slow RT. The durations of a sample of 760 EEG half waves in each of a group of 41 healthy children aged 5–17 yr were measured and distributed into an interval histogram, and the first four central moments of the 41 distributions were computed. All four of the moments — which measure the central tendency, dispersion, skewness and kurtosis of the distributions — proved to be significantly correlated ( p < 0.01) with RT. The multiple correlation ( R) between RT and the four moments was equal to 0.68 and was statistically significant. R was unchanged when only the measures of dispersion, skewness and kurtosis were used as predictor variables, suggesting that the predictive value of the central tendency is low. The predictive capability of a multiple regression equation based on these three predictor variables was tested in an independent group of 42 children also aged 5–17 yr. EEGs and RTs were recorded and this group was treated in exactly the same way as the other group. The correlation between these children's average RTs and their RTs as estimated from the other group's regression equation was 0.53. These findings suggested that knowing the degree of dispersion, skewness and kurtosis of the children's EEG distributions reduces from chance by about 28 % the error of predicting their RTs.

Non-linear fitting method for recorded chromatographic peaks

A method for fitting mathematical functions to experimental chromatographic peaks is described, with special reference to those gas-solid chromatographic peaks that have particular fitting difficulties owing to their high degree of skewness. The method was used with different functions and with peaks of diverse characteristies. The computer program used for the calculation and the different results obtained are also discussed.