Preference patterns for skewed gambles in rhesus monkeys

While standard models of risky choice account for the first and second statistical moments of reward outcome distributions (mean and variance, respectively), they often ignore the third moment, skewness. Determining a decision-maker''s attitude about skewness is useful because it can help constrain process models of the mental steps involved in risky choice. We measured three rhesus monkeys’ preferences for gambles whose outcome distributions had almost identical means and variances but differed in skewness. We tested five distributions of skewness: strong negative, weak negative, normal, weak positive and strong positive. Monkeys preferred positively skewed gambles to negatively skewed ones and preferred strongly skewed and normal (i.e. unskewed) gambles to weakly skewed ones. This pattern of preferences cannot be explained solely by monotonic deformations of the utility curve or any other popular single account, but can be accounted for by multiple interacting factors.     

Bayesian QTL mapping using skewed Student-t distributions

In most QTL mapping studies, phenotypes are assumed to follow normal distributions. Deviations from this assumption may lead to detection of false positive QTL. To improve the robustness of Bayesian QTL mapping methods, the normal distribution for residuals is replaced with a skewed Student-t distribution. The latter distribution is able to account for both heavy tails and skewness, and both components are each controlled by a single parameter. The Bayesian QTL mapping method using a skewed Student-t distribution is evaluated with simulated data sets under five different scenarios of residual error distributions and QTL effects.     

Normalization of high dimensional genomics data where the distribution of the altered variables is skewed

Genome-wide analysis of gene expression or protein binding patterns using different array or sequencing based technologies is now routinely performed to compare different populations, such as treatment and reference groups. It is often necessary to normalize the data obtained to remove technical variation introduced in the course of conducting experimental work, but standard normalization techniques are not capable of eliminating technical bias in cases where the distribution of the truly altered variables is skewed, i.e. when a large fraction of the variables are either positively or negatively affected by the treatment. However, several experiments are likely to generate such skewed distributions, including ChIP-chip experiments for the study of chromatin, gene expression experiments for the study of apoptosis, and SNP-studies of copy number variation in normal and tumour tissues. A preliminary study using spike-in array data established that the capacity of an experiment to identify altered variables and generate unbiased estimates of the fold change decreases as the fraction of altered variables and the skewness increases. We propose the following work-flow for analyzing high-dimensional experiments with regions of altered variables: (1) Pre-process raw data using one of the standard normalization techniques. (2) Investigate if the distribution of the altered variables is skewed. (3) If the distribution is not believed to be skewed, no additional normalization is needed. Otherwise, re-normalize the data using a novel HMM-assisted normalization procedure. (4) Perform downstream analysis. Here, ChIP-chip data and simulated data were used to evaluate the performance of the work-flow. It was found that skewed distributions can be detected by using the novel DSE-test (Detection of Skewed Experiments). Furthermore, applying the HMM-assisted normalization to experiments where the distribution of the truly altered variables is skewed results in considerably higher sensitivity and lower bias than can be attained using standard and invariant normalization methods.     

Embracing asymmetry in nature: How to account for skewness in ecological data

Asymmetric regression is an alternative to conventional linear regression that allows us to model the relationship between predictor variables and the response variable while accommodating skewness. Advantages of asymmetric regression include incorporating realistic ecological patterns observed in data, robustness to model misspecification and less sensitivity to outliers. Bayesian asymmetric regression relies on asymmetric distributions such as the asymmetric Laplace (ALD) or asymmetric normal (AND) in place of the normal distribution used in classic linear regression models. Asymmetric regression concepts can be used for process and parameter components of hierarchical Bayesian models and have a wide range of applications in data analyses. In particular, asymmetric regression allows us to fit more realistic statistical models to skewed data and pairs well with Bayesian inference. We first describe asymmetric regression using the ALD and AND. Second, we show how the ALD and AND can be used for Bayesian quantile and expectile regression for continuous response data. Third, we consider an extension to generalize Bayesian asymmetric regression to survey data consisting of counts of objects. Fourth, we describe a regression model using the ALD, and show that it can be applied to add needed flexibility, resulting in better predictive models compared to Poisson or negative binomial regression. We demonstrate concepts by analyzing a data set consisting of counts of Henslow’s sparrows following prescribed fire and provide annotated computer code to facilitate implementation. Our results suggest Bayesian asymmetric regression is an essential component of a scientist’s statistical toolbox.     

麻城黄山松天然林直径分布

为研究黄山松天然次生林直径分布特征,以麻城市黄山松天然次生林为研究对象,采用标准样地调查,计算林分直径的偏度、峰度,林分直径分布的Shannon-Weiner和Simpson指数,运用负指数分布、normal分布、lognormal分布、logistic分布和Weibull分布等5种概率密度函数对黄山松天然次生林林分直...     

Selection on skewed characters and the paradox of stasis

Observed phenotypic responses to selection in the wild often differ from predictions based on measurements of selection and genetic variance. An overlooked hypothesis to explain this paradox of stasis is that a skewed phenotypic distribution affects natural selection and evolution. We show through mathematical modeling that, when a trait selected for an optimum phenotype has a skewed distribution, directional selection is detected even at evolutionary equilibrium, where it causes no change in the mean phenotype. When environmental effects are skewed, Lande and Arnold's (1983) directional gradient is in the direction opposite to the skew. In contrast, skewed breeding values can displace the mean phenotype from the optimum, causing directional selection in the direction of the skew. These effects can be partitioned out using alternative selection estimates based on average derivatives of individual relative fitness, or additive genetic covariances between relative fitness and trait (Robertson–Price identity). We assess the validity of these predictions using simulations of selection estimation under moderate sample sizes. Ecologically relevant traits may commonly have skewed distributions, as we here exemplify with avian laying date — repeatedly described as more evolutionarily stable than expected — so this skewness should be accounted for when investigating evolutionary dynamics in the wild.     

Economic values under inappropriate normal distribution assumptions

The objectives of this study were to quantify the errors in economic values (EVs) for traits affected by cost or price thresholds when skewed or kurtotic distributions of varying degree are assumed to be normal and when data with a normal distribution is subject to censoring. EVs were estimated for a continuous trait with dichotomous economic implications because of a price premium or penalty arising from a threshold ranging between −4 and 4 standard deviations from the mean. In order to evaluate the impacts of skewness, positive and negative excess kurtosis, standard skew normal, Pearson and the raised cosine distributions were used, respectively. For the various evaluable levels of skewness and kurtosis, the results showed that EVs can be underestimated or overestimated by more than 100% when price determining thresholds fall within a range from the mean that might be expected in practice. Estimates of EVs were very sensitive to censoring or missing data. In contrast to practical genetic evaluation, economic evaluation is very sensitive to lack of normality and missing data. Although in some special situations, the presence of multiple thresholds may attenuate the combined effect of errors at each threshold point, in practical situations there is a tendency for a few key thresholds to dominate the EV, and there are many situations where errors could be compounded across multiple thresholds. In the development of breeding objectives for non-normal continuous traits influenced by value thresholds, it is necessary to select a transformation that will resolve problems of non-normality or consider alternative methods that are less sensitive to non-normality.     

A commingling analysis of quantitative neuromuscular performance in a Kansas Mennonite community

The method of commingling analysis is applied to distributions of six quantitative neuromuscular traits. Results show that only two of these traits may be described by the single normal distribution commonly associated with quantitative variables. The remaining four traits show significant skewness which is best accounted for by a mixture of three component distributions. The pattern of commingling found suggests the presence of a major (megaphenic) effect operating in these traits. Further, evidence is found which links the major effect in three of the four commingled traits to a single process which may be related to neurological control. The etiology of the major effect (i.e., whether genetic or environmental) cannot be determined from commingling analysis, but some suggestions are offered based upon the nature of the traits themselves and the major effects.     

Evidence of Commingling in Human Eating Behavior

This investigation tested whether distributions of certain aspects of eating behavior were consistent with the notion of a “mixture model;” that is, two or more distinct Commingled component distributions, consistent with the possibility of major gene action. Undergraduates (n=901) completed self-report trait measures of hunger, disinhibition, and dietary restraint. Variables were residualized for gender and age and transformed to remove skewness. Residualized transformed distributions were tested for departure from unimodality with Hartigan's (14) dip statistic. The distributions of all three aspects of eating behavior were significantly non-unimodal. Next, component multivariate normal distributions were estimated via maximum likelihood. Likelihood ratio tests were employed to compare nested models. A mixture of four distributions with unequal variance-covariance matrices tit significantly better than any more parsimonious model. In sum, these data strongly suggest that the distributions of several measures of eating behavior are composed of four component distributions. This finding is consistent with the possibility of major gene effects for eating behavior.     

The distribution of the intervals between neural impulses in the maintained discharges of retinal ganglion cells

Simulated neural impulse trains were generated by a digital realization of the integrate-and-fire model. The variability in these impulse trains had as its origin a random noise of specified distribution. Three different distributions were used: the normal (Gaussian) distribution (no skew, normokurtic), a first-order gamma distribution (positive skew, leptokurtic), and a uniform distribution (no skew, platykurtic). Despite these differences in the distribution of the variability, the distributions of the intervals between impulses were nearly indistinguishable. These inter-impulse distributions were better fit with a hyperbolic gamma distribution than a hyperbolic normal distribution, although one might expect a better approximation for normally distributed inverse intervals. Consideration of why the inter-impulse distribution is independent of the distribution of the causative noise suggests two putative interval distributions that do not depend on the assumed noise distribution: the log normal distribution, which is predicated on the assumption that long intervals occur with the joint probability of small input values, and the random walk equation, which is the diffusion equation applied to a random walk model of the impulse generating process. Either of these equations provides a more satisfactory fit to the simulated impulse trains than the hyperbolic normal or hyperbolic gamma distributions. These equations also provide better fits to impulse trains derived from the maintained discharges of ganglion cells in the retinae of cats or goldfish. It is noted that both equations are free from the constraint that the coefficient of variation (CV) have a maximum of unity. The concluding discussion argues against the random walk equation because it embodies a constraint that is not valid, and because it implies specific parameters that may be spurious.     

Circular distributions based on nonnegative trigonometric sums

A new family of distributions for circular random variables is proposed. It is based on nonnegative trigonometric sums and can be used to model data sets which present skewness and/or multimodality. In this family of distributions, the trigonometric moments are easily expressed in terms of the parameters of the distribution. The proposed family is applied to two data sets, one related with the directions taken by ants and the other with the directions taken by turtles, to compare their goodness of fit versus common distributions used in the literature.     

Differentiation and frequency distributions of body weights in plants and animals

The basic and simplest system that one can consider in ecology is a group of individuals of equal age and representing one species, that is, a cohort. This paper is an attempt to show that analysis of such a system may be of great importance to understanding basic ecological problems, such as, intraspecific competition and the dynamics of a single population. It is easy to observe that in even-aged populations individuals differ in weights. A close look can show that weight distributions in even-aged populations may have different skewness. Most common are distributions with coefficients of skewness greater than zero. Sometimes weight distributions are symmetrical or with skewness coefficients less than zero. In a cohort of growing individuals the coefficient of skewness changes with time: most often starting from zero (symmetrical distribution), it increases in time; sometimes after an initial increase it can decrease in the final stage of growth, which is related to an increased mortality of individuals. The rate of change in skewness, and the skewness itself depend on the density of individuals in a cohort and on food conditions. They are greater at higher densities and increase with deteriorating food conditions. Weight distributions are symmetrical at low densities and optimal food conditions. The differences in individual weights measured by variance of weight distributions or coefficient of variation follow the same pattern, but observed changes with time, density and food conditions are not so clear. These conclusions rest upon the review of numerous papers concerning both plants and animals, which is presented in this paper. In the past, the properties of weight distributions in even-aged populations were explained not by interactions between individuals, but rather as a natural outcome of the growth process of non-interacting individuals. The exponential equation of growth, with relative growth rate having a normal distribution in populations, was used to support this hypothesis. Obtained weight distributions were of positive skewness; however, this model, which in fact is able to describe the growth process only in its initial stage, cannot explain the changes of skewness of weight distributions with density and food conditions. A model has been developed which includes competitive interactions among members of even-aged populations to explain observed properties of weight distributions in them. The basic assumption is that intraspecific competition leads to uneven partitioning of resources, which are the object of competition. Functions describing resource partitioning among individuals are included into the model.(ABSTRACT TRUNCATED AT 400 WORDS)     

Analytical Quantile Solution for the S‐distribution,Random Number Generation and Statistical Data Modeling

The selection of a specific statistical distribution as a model for describing the population behavior of a given variable is seldom a simple problem. One strategy consists in testing different distributions (normal, lognormal, Weibull, etc.), and selecting the one providing the best fit to the observed data and being the most parsimonious. Alternatively, one can make a choice based on theoretical arguments and simply fit the corresponding parameters to the observed data. In either case, different distributions can give similar results and provide almost equivalent models for a given data set. Model selection can be more complicated when the goal is to describe a trend in the distribution of a given variable. In those cases, changes in shape and skewness are difficult to represent by a single distributional form. As an alternative to the use of complicated families of distributions as models for data, the S‐distribution [Voit, E. O. (1992) Biom. J. 7 , 855–878] provides a highly flexible mathematical form in which the density is defined as a function of the cumulative. S‐distributions can accurately approximate many known continuous and unimodal distributions, preserving the well known limit relationships between them. Besides representing well‐known distributions, S‐distributions provide an infinity of new possibilities that do not correspond with known classical distributions. Although the utility and performance of this general form has been clearly proved in different applications, its definition as a differential equation is a potential drawback for some problems. In this paper we obtain an analytical solution for the quantile equation that highly simplifies the use of S‐distributions. We show the utility of this solution in different applications. After classifying the different qualitative behaviors of the S‐distribution in parameter space, we show how to obtain different S‐distributions that accomplish specific constraints. One of the most interesting cases is the possibility of obtaining distributions that acomplish P(X ≤ X_c) = 0. Then, we demonstrate that the quantile solution facilitates the use of S‐distributions in Monte‐Carlo experiments through the generation of random samples. Finally, we show how to fit an S‐distribution to actual data, so that the resulting distribution can be used as a statistical model for them.     

The size distribution of conspecific populations: the peoples of New Guinea

The size distribution of the language populations in New Guinea, which represent over 15% of the world's languages, is analysed using models analogous to the resource division models of species abundance distribution in ecological communities. A model distribution of resource segments reflecting population size is created by repeated selection of an existing resource segment and its division into two. We found that any dependency of the selection probability on the size of the segment generated negatively skewed abundance distributions after log transformation. Asymmetric segment division further exacerbated the negative skewness. Size-independent selection produced lognormal abundance distributions, irrespective of the segment division method. Size-dependent selection and asymmetric division were deemed reasonable assumptions since large language populations are more likely to generate isolates, which develop into new populations, than small ones, and these isolates are likely to be small relative to the progenitor population. A negatively skewed distribution of the log-transformed population sizes was therefore expected. However, the observed distributions were lognormal, scale invariant for areas containing between 100 and over 1000 language populations. The dynamics of language differentiation, as reflected by the models, may therefore be unimportant relative to the effect of variable growth rates among populations. All lognormal distributions from resource division models had a higher variance than the observed one, where half of the 1053 populations had between 350 and 3000 individuals. The possible mechanisms maintaining such a low variance around a modal population size of 1000 are discussed.     

Commingling analysis of obesity in twins

Evidence is presented for multiple components in the distribution of human fatness across several large twin samples, after removing age effects and allowing for residual skewness in component distributions. The upper component distributions corresponded to overweight or obesity in samples of middle-aged or older individuals. A bivariate analysis demonstrated that, while monozygotic co-twins appeared to be drawn from the same component distributions (normal or overweight), the twin correlations varied across components, with the lowest correlation in the overweight group. While these analyses cannot provide a definitive test of competing genetic and environmental hypotheses, this approach is useful for generating hypotheses about the causes of obesity. When combined with other published literature, our results suggest that the genetic background largely determines the propensity to become obese. Whether a predisposed person becomes obese and the extent of obesity depend on environmental exposures that are largely independent of early family experience. Both genes and environment appear to be important in obesity, but it appears that some genotypes may be much more sensitive to the environment than are others.     

Biotic and abiotic effects on cohort size distributions in fish

Intraspecific variation in body size is common in animals and plants. Body size affects trophic interactions like foraging ability and vulnerability to predation, which in turn affect individual fitness as well as population stability and extinction risk. Experimental and theoretical work has shown that the size distribution of individuals within cohorts is strongly influenced by intraspecific competition for resources, often leading to skewed frequency distributions. However, little is known about the effects of environmental factors such as climate and eutrophication on the cohort size‐structure of natural populations. We use a long‐term time series of scientific monitoring of a freshwater fish (European perch Perca fluviatilis) to investigate the effects of density dependence, predation, nutrient availability, climate and the timing of spawning on the cohort size distributions. We find that the mean length of the fish is best predicted by the extrinsic factors phosphorus concentration and summer temperature, and the densities of the different age‐classes, whereas the skewness of the length distribution is best predicted by phosphorus concentration, summer temperature, abundance of small fish, and the timing of spawning. Higher nutrient levels, temperatures and densities of small fish increase food availability and thus reduce competition, which is reflected in increased mean length and decreased skewness. The timing of spawning affects skewness presumably through changes in the initial size variation of the cohort and the length of the first growth season. Our results indicate that higher temperatures increase the mean length and decrease skewness due to the concurrent eutrophication of the lake. The study thereby highlights the potential impact of human‐induced environmental change on the size structure of fish populations. More studies are needed to understand better the complex mechanisms through which these factors alter the intensity of intraspecific competition in fish communities.     

Random Phenotypic Variation of Yeast (Saccharomyces cerevisiae) Single-Gene Knockouts Fits a Double Pareto-Lognormal Distribution

Background

Distributed robustness is thought to influence the buffering of random phenotypic variation through the scale-free topology of gene regulatory, metabolic, and protein-protein interaction networks. If this hypothesis is true, then the phenotypic response to the perturbation of particular nodes in such a network should be proportional to the number of links those nodes make with neighboring nodes. This suggests a probability distribution approximating an inverse power-law of random phenotypic variation. Zero phenotypic variation, however, is impossible, because random molecular and cellular processes are essential to normal development. Consequently, a more realistic distribution should have a y-intercept close to zero in the lower tail, a mode greater than zero, and a long (fat) upper tail. The double Pareto-lognormal (DPLN) distribution is an ideal candidate distribution. It consists of a mixture of a lognormal body and upper and lower power-law tails.

Objective and Methods

If our assumptions are true, the DPLN distribution should provide a better fit to random phenotypic variation in a large series of single-gene knockout lines than other skewed or symmetrical distributions. We fit a large published data set of single-gene knockout lines in Saccharomyces cerevisiae to seven different probability distributions: DPLN, right Pareto-lognormal (RPLN), left Pareto-lognormal (LPLN), normal, lognormal, exponential, and Pareto. The best model was judged by the Akaike Information Criterion (AIC).

Results

Phenotypic variation among gene knockouts in S. cerevisiae fits a double Pareto-lognormal (DPLN) distribution better than any of the alternative distributions, including the right Pareto-lognormal and lognormal distributions.

Conclusions and Significance

A DPLN distribution is consistent with the hypothesis that developmental stability is mediated, in part, by distributed robustness, the resilience of gene regulatory, metabolic, and protein-protein interaction networks. Alternatively, multiplicative cell growth, and the mixing of lognormal distributions having different variances, may generate a DPLN distribution.     

Mass extinctions do not explain skew in interspecific body size distributions

In several higher animal taxa, such as mammals and birds, the distribution of species body sizes is heavily skewed towards small size. Previous studies have suggested that small‐bodied organisms are less prone to extinction than large‐bodied species. If small body size is favourable during mass extinction events, a post mass extinction excess of small‐bodied species may proliferate and maintain skewed body size distributions sometime after. Here, we modelled mass extinctions and found that even unrealistically strong body mass selection has little effect on the skew of interspecific body size distributions. Moreover, selection against large body size may, counter intuitively, skew size distributions towards large body size. In any case, subsequent evolutionary diversification rapidly erases these rather small effects mass extinctions may have on size distributions. Next, we used body masses of extant species and phylogenetic methods to investigate possible changes in body size distributions across the Cretaceous–Paleogene (K‐Pg) mass extinction. Body size distributions of extant clades that originated during the Cretaceous are on average more skewed than their subclades that originated during the Paleogene, but the difference is only minor in mammals, and in birds, it can be explained by a positive relationship between species richness and skewness that is also present in clades that originated after the transition. Hence, we cannot infer from extant species whether the K‐Pg mass extinctions were size‐selective, but they are not the reason why most extant bird and mammal species are small‐bodied.     

The effect of habitat stability on benthic invertebrate communities: the utility of species abundance distributions

Spatial and temporal patterns in the species abundance distribution of benthic invertebrate communities of 11 freshwater habitats (10 streams and a wind-swept lake shore) were examined with respect to habitat stability. Abundance patterns varied markedly between seasons at most sites. However, mean abundance distributions at 4 of the 5 unstable sites and the 2 most stable sites were dominated by one or two taxa with a large number of rare species, whereas sites of intermediate stability had more equitable distributions. Both the log series and log normal distributions were statistically indistinguishable, at the 5% level, from all the observed mean abundance patterns. In contrast, graphical comparisons of the observed and fitted distributions suggested the log series may be the better fit at most of the unstable sites and the two most stable sites, whereas the more equitable distribution at sites of intermediate stability suggested the log normal distribution was the better fit. If conditions at a site favoured one or two species, either through severe physical conditions, or through competitive superiority in the absence of disturbance then the log series distribution may result. However, if no species in the community was strongly advantaged over others, a log normal distribution should result. Given the discriminating power of the appropriate statistical test it may not, however, be possible to pick up these differences without graphical comparisons as well.