Efeitos da transformação de uma variável com distribuição normal em sua inversa sobre os parâmetros de sua distribuição usando técnicas de Monte Carlo

Foram realizados quatro estudos de simulação para verificar a distribuição de inversas de variáveis com distribuição normal, em função de diferentes variâncias, médias, pontos de truncamentos e tamanhos amostrais. As variáveis simuladas foram GMD, com distribuição normal, representando o ganho médio diário e DIAS, obtido a partir da inversa de GMD, representando dias para se obter determinado peso. em todos os estudos, foi utilizado o sistema SAS® (1990) para simulação dos dados e para posterior análise dos resultados. As médias amostrais de DIAS foram dependentes dos desvios-padrão utilizados na simulação. As análises de regressão mostraram redução da média e do desvio-padrão de DIAS em função do aumento na média de GMD. A inclusão de um ponto de truncamento entre 10 e 25% do valor da média de GMD reduziu a média de GMD e aumentou a de DIAS, quando o coeficiente de variação de GMD foi superior a 25%. O efeito do tamanho dos grupos nas médias de GMD e DIAS não foi significativo, mas o desvio-padrão e CV amostrais médios de GMD aumentaram com o tamanho do grupo. em virtude da dependência entre a média e o desvio-padrão e da variação observada nos desvios-padrão de DIAS em função do tamanho do grupo, a utilização de DIAS como critério de seleção pode diminuir a acurácia da variação. Portanto, para a substituição de GMD por DIAS, é necessária a utilização de um método de análise robusto o suficiente para a eliminação da heterogeneidade de variância.

Faixas normais de nutrientes pelos métodos ChM, DRIS e CND e nível crítico pelo método de distribuição normal reduzida para laranjeira-pera

A utilização de métodos de diagnose nutricional para definição de teores ótimos e níveis críticos de nutrientes em tecidos vegetais tem se demonstrado promissora, desde que se conheçam suas limitações. Este trabalho teve como objetivo determinar as faixas normais de nutrientes para a cultura da laranjeira-pera em uma população, utilizando os métodos Chance Matemática (ChM), Sistema Integrado de Diagnose e Recomendação (DRIS) e Diagnose da Composição Nutricional (CND), além do Nível Crítico, pelo método de distribuição normal reduzida. O trabalho foi realizado no município de Bebedouro-SP, na Estação Experimental de Citricultura de Bebedouro. Utilizaram-se como base de dados teores totais de nutrientes de 50 amostras foliares e a produtividade da laranjeira-pera, oriundas de um experimento cujo fator de avaliação foram doses de calcário aplicadas superficialmente. Para o N, maior valor de ChM foi obtido pela classe 2 (23,6 a 24,7 g kg-1), com valores semelhantes aos obtidos pelo DRIS (22,1 a 24,0 g kg-1) e CND (22,1 a 23,9 g kg-1). Os valores inferiores dessas faixas normais concordam com o do nível crítico alcançado (22,7 g kg-1), sendo este muito próximo do proposto pela literatura. Para os nutrientes P, K, Mg, Zn e B, as faixas normais e os níveis críticos não se assemelharam aos descritos na literatura. em relação aos nutrientes Ca, Fe, Mn e Cu, seus valores de faixa normal e nível crítico aproximaram-se dos recomendados, possivelmente devido à maior variação em seus teores. A utilização dos métodos propostos, em uma população, foi mais adequada quando houve maior variação nos teores dos nutrientes, além de possibilitar menor amplitude aos valores de faixas normais, quando comparados aos da faixa de terras suficientes encontrados na literatura.

Power law distribution in education: Effect of economical, teaching, and study conditions in university entrance examination

We studied the statistical distribution of student's performance, which is measured through their marks, in university entrance examination (Vestibular) of UNESP (Universidade Estadual Paulista) with respect to (i) period of study - day versus night period (ii) teaching conditions - private versus public school (iii) economical conditions - high versus low family income. We observed long ubiquitous power law tails in physical and biological sciences in all cases. The mean value increases with better study conditions followed by better teaching and economical conditions. In humanities, the distribution is close to normal distribution with very small tail. This indicates that these power law tails in science subjects axe due to the nature of the subjects themselves. Further and better study, teaching and economical conditions axe more important for physical and biological sciences in comparison to humanities at this level of study. We explain these statistical distributions through Gradually Truncated Power law distributions. We discuss the possible reason for this peculiar behavior.

Univariate and bivariate distribution of growth traits in beef buffaloes from Brazil

The aim of this study was to analyze the weight at birth (BW) and adjusted at 205 (W205), 365 (W365) and 550 (W55O) days in beef buffaloes from Brazil, using two approaches: parametric, by normal distribution, and non-parametric, by kernel function, and thus estimating the genetic, environmental and phenotypic correlation among traits. Information of 5,169 animals at birth (BW), 3,792 at 205 days (W205), 3.883 at 365 days (W365) and 1,524 at 550 days of age (W550) were used. The birth weight distribution presented an evident discrepancy in relation to the normal distribution. However, W205, W365 and W550 presented normal distributions. The birth weight presented weak genetic, environmental, and phenotypic associations with the other weight measurements. on the other hand, the weight traits at 205, 365, 550 days of age showed a high genetic correlation.

Robust Bayesian analysis of heavy-tailed stochastic volatility models using scale mixtures of normal distributions

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Power law distribution in education: University entrance examination

We studied the statistical distribution of candidate's performance which is measured through their marks in university entrance examination (Vestibular) of UNESP (Universidade Estadual Paulista) for years 1998, 1999, and 2000. All students are divided in three groups: Physical, Biological and Humanities. We paid special attention to the examination of Portuguese language which is common for all and examinations for the particular area. We observed long ubiquitous power law tails in Physical and Biological sciences. This indicate the presence of strong positive feedback in sciences. We are able to explain completely these statistical distributions through Gradually Truncated Power law distributions which we developed recently to explain statistical behavior of financial market. The statistical distribution in case of Portuguese language and humanities is close to normal distribution. We discuss the possible reason for this peculiar behavior.

A história da origem da curva normal

Pós-graduação em Educação Matemática - IGCE

Ecological distribution of the shrimp Pleoticus muelleri (Bate, 1888) and Artemesia longinaris Bate, 1888 (Decapoda, Penaeoidea) in the southeastern Brazilian littoral

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

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.

Distribution pattern of 68Ga-DOTATATE in disease-free patients

68Ga-DOTA-DPhe1,Tyr3-octreotate (68Ga-DOTATATE) is a somatostatin analogue that shows high affinity for somatostatin receptor subtype 2 (sst2) and has been used for imaging neuroendocrine tumours. However, normal uptake patterns and potential pitfalls have not been described with this high-sensitivity radiotracer. The aim of this study was therefore to outline the normal distribution pattern of 68Ga-DOTATATE in disease-free patients, to provide standardized uptake values (SUVs) of various organs and to compare our results with the current knowledge on sst2 receptor expression in vitro.

Model Evaluation Based on the Distribution of Estimated Absolute Prediction Error

The construction of a reliable, practically useful prediction rule for future response is heavily dependent on the "adequacy" of the fitted regression model. In this article, we consider the absolute prediction error, the expected value of the absolute difference between the future and predicted responses, as the model evaluation criterion. This prediction error is easier to interpret than the average squared error and is equivalent to the mis-classification error for the binary outcome. We show that the distributions of the apparent error and its cross-validation counterparts are approximately normal even under a misspecified fitted model. When the prediction rule is "unsmooth", the variance of the above normal distribution can be estimated well via a perturbation-resampling method. We also show how to approximate the distribution of the difference of the estimated prediction errors from two competing models. With two real examples, we demonstrate that the resulting interval estimates for prediction errors provide much more information about model adequacy than the point estimates alone.

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.

Size frequency distribution of the β-amyloid (aβ) deposits in dementia with Lewy bodies with associated Alzheimer’s disease pathology

The objective is to study beta-amyloid (Abeta) deposition in dementia with Lewy bodies (DLB) with Alzheimer's disease (AD) pathology (DLB/AD). The size frequency distributions of the Abeta deposits were studied and fitted by log-normal and power-law models. Patients were ten clinically and pathologically diagnosed DLB/AD cases. Size distributions had a single peak and were positively skewed and similar to those described in AD and Down's syndrome. Size distributions had smaller means in DLB/AD than in AD. Log-normal and power-law models were fitted to the size distributions of the classic and diffuse deposits, respectively. Size distributions of Abeta deposits were similar in DLB/AD and AD. Size distributions of the diffuse deposits were fitted by a power-law model suggesting that aggregation/disaggregation of Abeta was the predominant factor, whereas the classic deposits were fitted by a log-normal distribution suggesting that surface diffusion was important in the pathogenesis of the classic deposits.

Factors determining the size frequency distribution of β-amyloid (Aβ) deposits in Alzheimer's disease

The size frequency distributions of discrete β-amyloid (Aβ) deposits were studied in single sections of the temporal lobe from patients with Alzheimer's disease. The size distributions were unimodal and positively skewed. In 18/25 (72%) tissues examined, a log normal distribution was a good fit to the data. This suggests that the abundances of deposit sizes are distributed randomly on a log scale about a mean value. Three hypotheses were proposed to account for the data: (1) sectioning in a single plane, (2) growth and disappearance of Aβ deposits, and (3) the origin of Aβ deposits from clusters of neuronal cell bodies. Size distributions obtained by serial reconstruction through the tissue were similar to those observed in single sections, which would not support the first hypothesis. The log normal distribution of Aβ deposit size suggests a model in which the rate of growth of a deposit is proportional to its volume. However, mean deposit size and the ratio of large to small deposits were not positively correlated with patient age or disease duration. The frequency distribution of Aβ deposits which were closely associated with 0, 1, 2, 3, or more neuronal cell bodies deviated significantly from a log normal distribution, which would not support the neuronal origin hypothesis. On the basis of the present data, growth and resolution of Aβ deposits would appear to be the most likely explanation for the log normal size distributions.

A quantitative study of abnormally enlarged neurons in cognitively normal brain and neurodegenerative disease

Abnormally enlarged neurons (AEN) occur in many neurodegenerative diseases. To define AEN more objectively, the frequency distribution of the ratio of greatest cell diameter(CD) to greatest nuclear diameter (ND) was studied in populations of cortical neurons in tissue sections of seven cognitively normal brains. The frequency distribution of CD/ND deviated from a normal distribution in 15 out of 18 populations of neurons studied and hence, the 95th percentile (95P) was used to define a limit of the CD/ND ratio excluding the5% most extreme observations. The 95P of the CD/ ND ratio varied from 2.0 to 3.0 in different cases and regions and a value of 95P = 3.0 was chosen to define the limit for normalneurons under non-pathological conditions. Based on the 95P = 3.0 criterion, the proportion of AEN with a CD/ND ≥ 3 varied from 2.6% in Alzheimer's disease (AD) to 20.3% in Pick's disease (PiD). The data suggest: (1) that a CL/ND ≥ 3.0 may be a useful morphological criterion for defining AEN, and (2) AEN were most numerous in PiD and corticobasal degeneration (CBD) and least abundant in AD and in dementia with Lewy bodies (DLB). © 2013 Dustri-Verlag Dr. K. Feistle.