Normality Test in Clinical Research

( Sang Gyu Kwak ),( Sung-hoon Park ) 대한류마티스학회 2019 대한류마티스학회지 Vol.26 No.1

In data analysis, given that various statistical methods assume that the distribution of the population data is normal distribution, it is essential to check and test whether or not the data satisfy the normality requirement. Although the analytical methods vary depending on whether or not the normality is satisfied, inconsistent results might be obtained depending on the analysis method used. In many clinical research papers, the results are presented and interpreted without checking or testing normality. According to the central limit theorem, the distribution of the sample mean satisfies the normal distribution when the number of samples is above 30. However, in many clinical studies, due to cost and time restrictions during data collection, the number of samples is frequently lower than 30. In this case, a proper statistical analysis method is required to determine whether or not the normality is satisfied by performing a normality test. In this regard, this paper discusses the normality check, several methods of normality test, and several statistical analysis methods with or without normality checks. (J Rheum Dis 2019;26:5-11)

MCEM을 이용한 정규혼합분포 추정

이승찬,이재준,전수영 한국자료분석학회 2017 Journal of the Korean Data Analysis Society Vol.19 No.1

정규혼합분포는 정규분포의 혼합으로 이루어져있으며 위험관리 분야에서 많이 이용되고 있는 극단값분포에 대한 대안으로 활용할 수 있다. 본 연구는 KOSPI 200 수익률의 정규혼합분포를 추정하기 위해 MCEM(Monte Carlo EM) 알고리즘을 적용하고자 한다. MCEM은 각 분포의 성분을 결측치로 보고, 결측치를 메트로폴리스-헤스팅스 알고리즘을 통해 생성하고 EM 알고리즘을 통해 각 분포의 모수를 결정한다. MCEM 알고리즘의 효율성은 여러 성분을 이용한 모의실험과 실증 자료를 통해 알 수 있었다. 본 연구의 실증 자료 표본은 2008년 외환위기 1년 후인 2009년 10월 15일부터 2017년 1월 7일 동안의 일별 수익률 자료이다. MCEM을 통해 분포의 추정을 실시한 결과 대다수의 분포는 2개의 성분을 갖고 있는 정규혼합분포로 나타났다. 실 자료 중 1000개의 표본을 1일씩 연속적으로 이동하여 1017개 표본기간들의 정규혼합분포를 추정하였고, VaR의 추정을 실시한 결과 하나의 정규분포보다 정규혼합분포를 통해 VaR를 더 잘 추정함을 알 수 있었다. 따라서 본 연구 결과는 자료의 특성에 따른 정규혼합분포를 통하여 VaR를 구하는 것이 다중최빈값을 갖는 경우에서 더 활용성이 높으며 충분히 리스크 측정도구로서 사용될 수 있음을 보였다. The normal mixture distribution is a mixture of normal distributions and can be used as an alternative to extreme value distributions which are widely used in risk management. This study proposes a Monte Carlo EM (MCEM) algorithm to estimate the normal mixed distribution of KOSPI 200 returns. MCEM considers the components of each distribution as missing values, generating through a Metropolis-Hastings algorithm, and determines the parameters of each distribution through EM. The effectiveness of MCEM was demonstrated by simulation and empirical data. The empirical data of this study are daily return data from October 15, 2009 to January 7, 2017, one year after the 2008 financial crisis. The result through MCEM indicates that the data follow a normal mixed distribution with two components. We estimated a normal mixed distribution of 1017 sample periods using 1000 samples of real data which are continuously moved by one day. In addition, the result of VaR estimation indicates that a normal mixed distribution is better than one normal distribution. Therefore, obtaining VaR through the normal mixed distribution is more usable in the case of multiple modes and can be used sufficiently as a risk measurement tool.

Estimating Suitable Probability Distribution Function for Multimodal Traffic Distribution Function

유상록,정재용,임정빈 해양환경안전학회 2015 海洋環境安全學會誌 Vol.21 No.3

The purpose of this study is to find suitable probability distribution function of complex distribution data like multimodal. Normal distribution is broadly used to assume probability distribution function. However, complex distribution data like multimodal are very hard to be estimated by using normal distribution function only, and there might be errors when other distribution functions including normal distribution function are used. In this study, we experimented to find fit probability distribution function in multimodal area, by using AIS(Automatic Identification System) observation data gathered in Mokpo port for a year of 2013. By using chi-squared statistic, gaussian mixture model(GMM) is the fittest model rather than other distribution functions, such as extreme value, generalized extreme value, logistic, and normal distribution. GMM was found to the fit model regard to multimodal data of maritime traffic flow distribution. Probability density function for collision probability and traffic flow distribution will be calculated much precisely in the future.

손해보험사의 재보험료 산출을 위한 손해분포의 적합

박상균 ( Sangkyun Park ),송성주 ( Seongjoo Song ) 한국보험학회 2019 保險學會誌 Vol.119 No.-

본 논문에서는 손해보험사의 다양한 보험종목 손해액의 개별적인 특성을 고려하면서 동시에 그들 사이에 존재하는 의존성을 반영하여 손해액의 합에 대한 결합 재보험료를 산출하고자 하였다. 의존성 반영을 위해 코퓰라 함수를 이용하였고, 각 보험종목 손해액에 대한 주변분포로는 normal inverse Gaussian 분포와 일반화 파레토분포, 와이블 분포를 사용하였다. 모의실험과 삼성화재, 현대해상의 월별 손해액 자료를 통해 단일 보험종목 재보험료 산출결과를 살펴보고, 삼성화재와 현대해상 자료에 대한 다변량 분포를 적합하여 재보험료를 산출하였다. 보험종목 간의 의존성을 고려한 경우 재보험료가 크게 추정되었으며, 두꺼운 꼬리를 잘 설명할 수 있는 일반화 파레토분포와 normal inverse Gaussian 분포를 사용하였을 때 정규분포나 와이불 분포를 사용했을 때에 비해 높은 재보험료가 얻어졌다. Normal inverse Gaussian 분포는 일변량 손해액 자료에 높은 적합도를 보이면서 일반화 파레토분포와 크게 다르지 않는 재보험료를 산출하고 있으며, 일반화 파레토분포와 달리 임계치를 따로 설정할 필요가 없고 수치적분도 상대적으로 빠르게 수행되어 보다 편리하게 사용할 수 있었다. In this paper, we calculate the reinsurance premiums on the sum of several insurance lines for non-life insurance companies considering the dependency among loss amounts of lines as well as their individual characteristics. We try several copula functions to reflect the dependency among loss amounts, and apply normal inverse Gaussian distribution, generalized Pareto distribution, and Weibull distribution for the marginal distribution of the monthly loss amount of each insurance field. Through simulation studies and real data analysis with loss amounts from Samsung Fire&Marine Insurance and Hyundai Insurace, we calculate the reinsurance premiums for individual insurance field. Also, we calculate enterprise-wide reinsurance premiums by fitting multivariate distributions to monthly losses from Samsung Fire&Marine Insurance and Hyundai Insurace. When we take into account the dependence structure among different insurance fields, reinsurance premiums were calculated greater compared to those based on the independence assumption. Moreover, the generalized Pareto distribution and the normal inverse Gaussian distribution provided larger premiums than the normal or Weibull distribution when they were used as marginal loss distribution . The normal inverse Gaussian distribution yields a good fit to the marginal loss data and produces premiums that do not differ significantly from the generalized Pareto distribution. Unlike the generalized Pareto distribution, there is no need to set a threshold value, which makes it more convenient to use.

Estimating Suitable Probability Distribution Function for Multimodal Traffic Distribution Function

Sang-Lok Yoo,Jae-Yong Jeong,Jeong-Bin Yim 해양환경안전학회 2015 海洋環境安全學會誌 Vol.21 No.3

The purpose of this study is to find suitable probability distribution function of complex distribution data like multimodal. Normal distribution is broadly used to assume probability distribution function. However, complex distribution data like multimodal are very hard to be estimated by using normal distribution function only, and there might be errors when other distribution functions including normal distribution function are used. In this study, we experimented to find fit probability distribution function in multimodal area, by using AIS(Automatic Identification System) observation data gathered in Mokpo port for a year of 2013. By using chi-squared statistic, gaussian mixture model(GMM) is the fittest model rather than other distribution functions, such as extreme value, generalized extreme value, logistic, and normal distribution. GMM was found to the fit model regard to multimodal data of maritime traffic flow distribution. Probability density function for collision probability and traffic flow distribution will be calculated much precisely in the future.

Some counterexamples of a skew-normal distribution

Zhao, Jun,Lee, Sang Kyu,Kim, Hyoung-Moon The Korean Statistical Society 2019 Communications for statistical applications and me Vol.26 No.6

Counterexamples of a skew-normal distribution are developed to improve our understanding of this distribution. Two examples on bivariate non-skew-normal distribution owning marginal skew-normal distributions are first provided. Sum of dependent skew-normal and normal variables does not follow a skew-normal distribution. Continuous bivariate density with discontinuous marginal density also exists in skew-normal distribution. An example presents that the range of possible correlations for bivariate skew-normal distribution is constrained in a relatively small set. For unified skew-normal variables, an example about converging in law are discussed. Convergence in distribution is involved in two separate examples for skew-normal variables. The point estimation problem, which is not a counterexample, is provided because of its importance in understanding the skew-normal distribution. These materials are useful for undergraduate and/or graduate teaching courses.

Estimating Suitable Probability Distribution Function for Multimodal Traffic Distribution Function

Yoo, Sang-Lok,Jeong, Jae-Yong,Yim, Jeong-Bin The Korean Society of Marine Environment and safet 2015 海洋環境安全學會誌 Vol.21 No.3

The purpose of this study is to find suitable probability distribution function of complex distribution data like multimodal. Normal distribution is broadly used to assume probability distribution function. However, complex distribution data like multimodal are very hard to be estimated by using normal distribution function only, and there might be errors when other distribution functions including normal distribution function are used. In this study, we experimented to find fit probability distribution function in multimodal area, by using AIS(Automatic Identification System) observation data gathered in Mokpo port for a year of 2013. By using chi-squared statistic, gaussian mixture model(GMM) is the fittest model rather than other distribution functions, such as extreme value, generalized extreme value, logistic, and normal distribution. GMM was found to the fit model regard to multimodal data of maritime traffic flow distribution. Probability density function for collision probability and traffic flow distribution will be calculated much precisely in the future.

수생태 독성자료의 정규성 분포 특성 확인을 통해 통계분석 시 분포 특성 적용에 대한 타당성 확인 연구

옥승엽,문효방,나진성 한국환경보건학회 2019 한국환경보건학회지 Vol.45 No.2

Objectives: According to the central limit theorem, the samples in population might be considered to follow normal distribution if a large number of samples are available. Once we assume that toxicity dataset follow normal distribution, we can treat and process data statistically to calculate genus or species mean value with standard deviation. However, little is known and only limited studies are conducted to investigate whether toxicity dataset follows normal distribution or not. Therefore, the purpose of study is to evaluate the generally accepted normality hypothesis of aquatic toxicity datasetMethods: We selected the 8 chemicals, which consist of 4 organic and 4 inorganic chemical compounds considering data availability for the development of species sensitivity distribution. Toxicity data were collected at the US EPA ECOTOX Knowledgebase by simple search with target chemicals. Toxicity data were re-arranged to a proper format based on the endpoint and test duration, where we conducted normality test according to the Shapiro-Wilk test. Also we investigated the degree of normality by simple log transformation of toxicity dataResults: Despite of the central limit theorem, only one large dataset (n>25) follow normal distribution out of 25 large dataset. By log transforming, more 7 large dataset show normality. As a result of normality test on small dataset (n<25), log transformation of toxicity value generally increases normality. Both organic and inorganic chemicals show normality growth for 26 species and 30 species, respectively. Those 56 species shows normality growth by log transformation in the taxonomic groups such as amphibian (1), crustacean (21), fish (22), insect (5), rotifer (2), and worm (5). In contrast, mollusca shows normality decrease at 1 species out of 23 that originally show normality. Conclusions: The normality of large toxicity dataset was not always satisfactory to the central limit theorem. Normality of those data could be improved through log transformation. Therefore, care should be taken when using toxicity data to induce, for example, mean value for risk assessment.

정규분포 신용리스크 모형의 추정 성과 시뮬레이션

최필선 ( Pil Sun Choi ),민인식 ( In Sik Min ) 한국금융학회 2009 금융연구 Vol.23 No.1

바젤II와 KMV(R) 및 CreditMetrics(TM) 등 전통적 신용리스크 모형은 다변량 정규분포를 이용하여 차주의 자산 변동을 모형화하고 거기에서 손실분포를 도출한다. 그러나 자산수익률을 정규분포로 설정하는 것에 대해서는 여러 선행연구들이 그 한계를 지적해 왔다. 자산수익률이 정규분포가 아니라 예를 들어 t 분포일 경우 부도확률 및 상관계수가 동일하더라도 도출된 손실분포가 크게 달라지며, 특히 정규분포 모형이 신용리스크를 상당히 과소평가하는 경향이 있다는 점이 자주 지적되었다. 이에 대해 Hamerle and Rosch(2005)는 한 가지 흥미로운 작업을 통해 전통적인 정규분포 모형의 유용성을 옹호했는데, 즉 자산수익률의 실제 분포가 다변량 t 분포일 때 이를 정규분포 모형을 사용하여 추정하는 경우 비록 상관계수에 대한 추정치는 실제치와 크게 다르지만, 그 추정치들을 이용하여 도출한 손실분포 및 VaR은 실제값과 매우 유사하다는 것이다. 본 논문은 그들의 연구를 한 단계 확장시킨 것으로서 다변량 t 분포뿐만 아니라 그 이외의 상황에서도 정규분포 모형이 여전히 유용한지를 검토한다. 즉 단일요인모형의 구성요소인 체계적 요인과 개별 교란항의 분포를 다양하게 변화시켜 가면서 그것을 정규분포 모형으로 추정하여 실제치와 비교한다. 시뮬레이션 결과 Hamerle and Rosch(2005)가 보인 것과는 달리 일반적으로 정규분포 모형이 실제 신용리스크를 왜곡되게 평가할 가능성이 큰 것으로 나타났다. The traditional credit risk models such as Basel II, KMV(R) and CreditMetrics(TM) derive the credit loss distribution by assuming that obligors` asset returns follow a multivariate Gaussian distribution. Previous researches have shown, however, that there are crucial differences between loss distributions generated from a multivariate Gaussian distribution and a multivariate Student`s t distribution, when the linear correlation coefficient and probability of default in both models are, respectively, the same. In particular, the Gaussian model tends to underestimate the credit risk. Against the critiques, Hamerle and Rosch (2005) show some interesting estimation results that support the validity of the Gaussian model in estimating credit risks. According to their work, when the true distribution of portfolio return is a multivariate t distribution, the estimation of loss distribution and VaR (Value at Risk) based on Gaussian model may still be adequate, even though the estimate of correlation coefficient may be biased. Our study extends Hamerle and Rosch`s simulation scheme, and investigates if the Gaussian model is still adequate even under the assumption that true distribution of the portfolio returns is non-normal other than the t distribution. Consider a portfolio consisting of homogeneous N obligors. R(i), the obligor i`s asset return, is assumed to be described as the following one-factor model. where idiosyncratic shocks εi are assumed independent from a systematic factor Y and independent from different obligors. Y and εi are normally distributed with mean zero and variance one. ρ is the correlation coefficient parameter between Ri and Rj. Under the assumption of the one-factor model and infinitely fine-grained portfolios, the cumulative distribution function F(x) of the loss fraction x, and α×100% quantile xα are given by where G and H are cumulative distribution functions of Y and εi , respectively, and K is a pre-specified default barrier. As Hamerle and Rosch (2005) present, we confirm again the fact that when the true distribution of portfolio returns is a multivariate t distribution, the loss distribution VaRs estimated by a Gaussian model are very close to the true values. However, when we apply other non-normal distributions to the portfolio returns, the performance of the Gaussian model deteriorates dramatically. The major findings of the estimations are as follows. First, when we generate the true values by applying Student`s t variable(s) to Y or/and εi, on the contrary to Hamerle and Rosch (2005), the Gaussian model underestimates the loss distribution VaR for all levels. This pattern becomes more obvious when the degrees of freedom parameter of the t distribution are lower and/or the confidence level for VaR is higher. Second, when we generate the true values by applying S(U)-normal variable(s) to Y or/and εi, the overall performance of the Gaussian model becomes even more serious: the Gaussian model usually underestimates, but sometimes overestimates the true VaR values.

정규분포에 대한 교수학적 변환 방식과 학생들의 이해 분석

신보미 대한수학교육학회 2012 수학교육학연구 Vol.22 No.2

The goal of this study is to investigate a didactic transposition method of text books and high school students' understanding for the Normal Distribution. To accomplish this goal, framework descriptors were developed to analyse the didactic transposition method and interpret the students' understanding based on the Historico-Genetic process of the Normal Distribution, the meaning of the Normal Distribution as a scholarly knowledge and the results of previous studies on students' understanding for the Normal Distribution. This study presented several recommendations for instruction of the Normal Distribution according to the results of analysing the didactic transposition method and interpreting the students' understanding in terms of the developed framework. 이 연구는 정규분포에 대한 교수학적 변환 방식과 학생들의 이해의 특징을 분석함으로써 고등학교 통계 단원에서 정규분포를 지도하는 것과 관련된 시사점을 얻는데 목적을 두었다. 이를 위해 정규분포의 역사 발생 과정과 학문적 지식으로서 정규분포의 의미를 확인하여 정규분포의 교수학적 변환 방식을 살펴보기 위한 4개의 분석 관점을 추출하고 이를 토대로 미국, 영국, 우리나라 교과서의 정규분포에 대한 교수학적 변환 방식을 분석하였다. 또한 정규분포에 대한 학생들의 이해 특징과 관련된 선행 연구 결과를 살펴봄으로써 고등학생들의 정규분포에 대한 이해의 특징을 기술하는데 필요한 분석 관점을 4가지로 구체화하였으며, 이를 토대로 분석한 학생들의 이해 특징을 4가지로 요약하였다.