Effect of Positively Skewed Distribution on the Two sample t-test: Based on Chi-square Distribution

Heo, Sunyeong The Basic Science Institute Chosun University 2021 조선자연과학논문집 Vol.14 No.3

This research examines the effect of positively skewed population distribution on the two sample t-test through simulation. For simulation work, two independent samples were selected from the same chi-square distributions with 3, 5, 10, 15, 20, 30 degrees of freedom and sample sizes 3, 5, 10, 15, 20, 30, respectively. Chi-square distribution is largely skewed to the right at small degrees of freedom and getting symmetric as the degrees of freedom increase. Simulation results show that the sampled populations are distributed positively skewed like chi-square distribution with small degrees of freedom, the F-test for the equality of variances shows poor performances even at the relatively large degrees of freedom and sample sizes like 30 for both, and so it is recommended to avoid using F-test. When two population variances are equal, the skewness of population distribution does not affect on the t-test in terms of the confidence level. However even though for the highly positively skewed distribution and small sample sizes like three or five the t-test achieved the nominal confidence level, the error limits are very large at small sample size. Therefore, if the sampled population is expected to be highly skewed to the right, it will be recommended to use relatively large sample size, at least 20.

A Study of Mann-Whitney Test and Median Test in Several Distributions

박찬근,신성민,허태영 한국자료분석학회 2008 Journal of the Korean Data Analysis Society Vol.10 No.2

The Mann-Whitney test and the Median test are two tests that can be used to test for a difference in location parameters. This paper compared powers of the two tests under a variety of population distributions through a simulation study. Both tests require that two underlying populations have the same variance, but this assumption was relaxed in some of the comparisons. In every cases, equal sizes of 10 and 20 were used. We compared the those two nonparametric tests when underlying distributions were mixed normal, lognormal, beta, gamma and chi-square distributions. Each test was performed 5,000 times and the SAS/MACRO used.

A Note on the Nonparametric Tests for the Grouped Data

홍승만,박효일 한국자료분석학회 2014 Journal of the Korean Data Analysis Society Vol.16 No.6

In this research, we consider to compare efficiency among tests used for the grouped data through a simulation study. We consider two kinds of the chi-square tests used for the categorical data and the nonparametric linear rank test used for the grouped data. For this, first of all, we formulize the grouped data by introducing several notation and review the test statistics with their corresponding asymptotic distributions. Then we obtain the empirical powers to compare efficiency under the location translation model by varying the value of location translation parameter with five different distributions and three sets of sample sizes through a simulation study. Finally, we discuss some interesting features for the nonparametric test procedures for the grouped data and comments briefly on the grouped data as concluding remarks.

시추공 영상자료와 카이제곱 검정을 이용한 절리 방향성의 수직적 변화양상에 관한 정량적 평가

김기석(Kiseog Kim),박영도(Youngdo Park),박연준(Yeonjun Park) 한국암반공학회 2008 터널과지하공간 Vol.18 No.3

이 연구에서는 시추공 분석 작업을 통해 획득된 암반절리 방향성이 심도에 따라 변화하는 양상의 확인을 위해 피어슨 카이제곱 통계검정이 실시되었다.대상 암반은 모암이 화강암질 편마암인 두 지역으로서,이와 같은 엽리가 발달하지 않은 괴상의 암상 선정은, 엽리가 존재하는 암석의 경우 절리 방향성이 엽리에 의해 영향을 받고 엽리의 방향은 습곡 등의 지질작용에 의해 심도에 따라 다를 수가 있기 때문이다. 암반절리들의 방향파악을 위해 시추공 영상이 이용되었다.획득된 방향자료를 천부구간과 심부구간의 자료로 분류한 후21영역으로 구성된 분할망에 각각 투영 후,분류표를 작성하여 통계검정을 실시하였으며,분석결과 두 지역중 한 지역의 자료는 비동질로 나타났다. 이러한 결과는 터널과 같은 지하구조물의 설계를 위한 암반공학적으로 중요한 절리면의 방향성에 대한 조사시 원위치 조사가 바람직함을 시사한다. We have carried out Pearson-type chi-square tests on the orientation data of joints from different depths in order to estimate the homogeneity of joint orientations obtained from a borehole. The orientation data of joints were collected from two non-foliated massive rocks of granitic gneisses in South Korea since orientations of joints in folded metamorphic rocks, for example, are controlled by foliation and also changes as the orientations of foliation change by folding. Borehole images were used for the analysis of the orientations of individual joints. The orientation data were subdivided into the upper level data and lower level data. The data from these two levels are plotted on the patch net consisting of 21 orientation patches. Then, the two patterns on the patch net were analyzed using a contingency table. From the chi-square test on the data collected from two sites, we found that some data sets show statistically meaningful differences in orientations of joints. Since joints are one of the important parameters in determining the physical properties of rock masses, in situ investigation of joints are desirable in the geotechnical investigation and also in design of subsurface structures (e.g. tunnels and underground storages).

A Note on the Simple Chi-Squared Test of Multivariate Normality

Park, Cheol-Yong 한국데이터정보과학회 2004 한국데이터정보과학회지 Vol.15 No.2

We provide the exact form of a Rao-Robson version of the chi-squared test of multivariate normality suggested by Park(2001). This test is easy to apply in practice since it is easily computed and has a limiting chi-squared distribution under multivariate normality. A self-contained formal argument is provided that it has the limiting chi-squared distribution. A simulation study is provided to study the accuracy, in finite samples, of the limiting distribution. Finally, a simulation study in a nonnormal distribution is conducted in order to compare the power of our test with those of other popular normality tests.

A Note on the Simple Chi-Squared Test of Multivariate Normality

Cheol Yong Park 한국데이터정보과학회 2004 한국데이터정보과학회지 Vol.15 No.2

We provide the exact form of a Rao-Robson version of the chi-squared test of multivariate normality suggested by Park(2001). This test is easy to apply in practice since it is easily computed and has a limiting chi-squared distribution under multivariate normality. A self-contained formal argument is provided that it has the limiting chi-squared distribution. A simulation study is provided to study the accuracy, in finite samples, of the limiting distribution. Finally, a simulation study in a nonnormal distribution is conducted in order to compare the power of our test with those of other popular normality tests.

Likelihood ratio in estimating Chi-square parameter

Mezbahur Rahman 한국데이터정보과학회 2009 한국데이터정보과학회지 Vol.20 No.3

The most frequent use of the chi-square distribution is in the area of goodness-of-fit of a distribution. The likelihood ratio test is a commonly used test statistic as the maximum likelihood estimate in statistical inferences. The recently revised versions of the likelihood ratio test statistics are used in estimating the parameter in the chi-square distribution. The estimates are compared with the commonly used method of moments and the maximum likelihood estimate.

Likelihood ratio in estimating Chi-square parameter

Rahman, Mezbahur The Korean Data and Information Science Society 2009 한국데이터정보과학회지 Vol.20 No.3

The most frequent use of the chi-square distribution is in the area of goodness-of-t of a distribution. The likelihood ratio test is a commonly used test statistic as the maximum likelihood estimate in statistical inferences. The recently revised versions of the likelihood ratio test statistics are used in estimating the parameter in the chi-square distribution. The estimates are compared with the commonly used method of moments and the maximum likelihood estimate.

A note on the simple chi-squared test of multivariate normality

박철용 한국데이터정보과학회 2004 한국데이터정보과학회지 Vol.15 No.2

We provide the exact form of a Rao-Robson version of the chi-squared test of multivariate normality suggested by Park(2001). This test is easy to apply in practice since it is easily computed and has a limiting chi-squared distribution under multivariate normality. A self-contained formal argument is provided that it has the limiting chi-squared distribution. A simulation study is provided to study the accuracy, in finite samples, of the limiting distribution. Finally, a simulation study in a nonnormal distribution is conducted in order to compare the power of our test with those of other popular normality tests.

Brain Perfusion SPECT(Diamox) 검사의 수행결과에 영향을 주는 요인

이진형,김상언,박현수,박영재,이인원,Lee, Jin-Hyeong,Kim, Sang-Eon,Park, Hyeon-Soo,Park, Yeoung-Jae,Lee, In-Won 대한핵의학기술학회 2011 핵의학 기술 Vol.15 No.2

Brain Perfusion SPECT (Diamox)는 아세타졸아미드를 사용하여 혈관예비능을 평가하는데 아주 유용한 검사이다. 1day-protocol method는 영상의 질이 뛰어나고 한 번에 두 가지 영상을 얻을 수 있는 이점이 있어, 검사실과 내원 환자에 모두 유리하다. 하지만 긴 검사시간으로 인해 다른 검사보다 재검사율이 높은 문제점을 가지고 있다. 이에 시행한 환자의 성별, 나이, 상병, 머리고정기구 사용 여부에 따른 재검사율을 파악하여 검사의 최적화를 도모하고자 하였다. 2010년 3월부터 2011년 2월까지 시행한 환자 676명(남:359명, 여:317명)을 대상으로 하였으며, 머리의 움직임을 유발하는 요인을 성별, 연령, 상병의 종류, 머리고정기구 적용여부로 나누어 검사결과에 기여하는 여부와 정도를, 카이제곱 검정기법과 이분형 로지스틱 회귀분석기법을 사용하여 검증하였다. 남성이 3.4%, 여성이 1.5%로 남성이 여성보다 실패율이 높았으며, 70대가 1.0%로 가장 높은 실패율을 보였다. 카이제곱 검정결과 p-value는 0.001미만으로 통계적으로 유의한 차이가 있는 것으로 나타났다. 이분형 로지스틱 회귀분석결과에서는 성별, 상병(CI), 머리고정기구 사용이 통계적으로 유의한 수준에서 영향을 끼치는 요인으로 평가 되었다. 각각의 Wald value는 3.3, 3.7, 9.3으로 머리고정기구의 사용이 가장 큰 영향을 주는 것으로 나타났다. 연구결과를 통해 연령을 제외한 성별과 상병 그리고 머리고정기구 사용이 검사수행결과에 통계적으로 유의한 수준의 영향력을 가지는 요인들임을 알게 되었으며. 그 중 머리고정기구 사용은 검사수행결과에 끼치는 영향이 가장 큰 요인임을 알 수 있었다. 검사의 시행에 있어서 머리고정기구를 필수적으로 사용한다면 Brain Perfusion SPECT(Diamox)검사의 재검사율을 크게 낮출 수 있을 것으로 기대된다. Purpose: Head movement during brain perfusion SPECT (Diamox) scan is a one of important issues which decreases image quality. It also causes repeated scans. This study was designed to evaluate variable factors causing scan failures. Materials and Methods: 676 patients (359 men, 317 women, age average $54.5{\pm}18.4$) for brain perfusion SPECT (Diamox) scan from March, 2010 to Feb. 2011 were used as a subject. Age data and the kind of disease(Moyamoya disease (MMD), None moyamoya disease (NMMD), Cerebral infarction (CI)), test performance outcome (success,failure) were collected. The head movement factors(gender, disease, age, head fixation device) were evaluated by chi-square test and logistic regression analysis Results: The result showed that men had higher scan failure rate than women. Seniors in seventies(men 3.4%, women 1.5%) showed the most highest failure rate. Using head fixation device increased scan success rate up to 94.4~97.7%. The scan success rate is dependent upon gender, head fixation device by chi-square test(${\chi}^2$=3.8 (df=1, p<0.05), ${\chi}^2$=10.4 (df=1, p<0.001)) Gender, disease(CI), head fixation device showed very effective result in logistic regression analysis.(Wald=3.3 (p<0.07), Wald=3.7 (p<0.05), Wald=9.3 (p<0.05) Conclusion: It is demonstrated that gender, disease, using head fixation device is statistically very useful factors. Especially, head fixation device is a main key minimizing repeated scan.