Comprehensive Descriptive Analysis and Normality Test 165
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Use of Proc Iml to Calculate L-Moments for the Univariate Distributional Shape Parameters Skewness and Kurtosis
Statistics 573 USE OF PROC IML TO CALCULATE L-MOMENTS FOR THE UNIVARIATE DISTRIBUTIONAL SHAPE PARAMETERS SKEWNESS AND KURTOSIS Michael A. Walega Berlex Laboratories, Wayne, New Jersey Introduction Exploratory data analysis statistics, such as those Gaussian. Bickel (1988) and Van Oer Laan and generated by the sp,ge procedure PROC Verdooren (1987) discuss the concept of robustness UNIVARIATE (1990), are useful tools to characterize and how it pertains to the assumption of normality. the underlying distribution of data prior to more rigorous statistical analyses. Assessment of the As discussed by Glass et al. (1972), incorrect distributional shape of data is usually accomplished conclusions may be reached when the normality by careful examination of the values of the third and assumption is not valid, especially when one-tail tests fourth central moments, skewness and kurtosis. are employed or the sample size or significance level However, when the sample size is small or the are very small. Hopkins and Weeks (1990) also underlying distribution is non-normal, the information discuss the effects of highly non-normal data on obtained from the sample skewness and kurtosis can hypothesis testing of variances. Thus, it is apparent be misleading. that examination of the skewness (departure from symmetry) and kurtosis (deviation from a normal One alternative to the central moment shape statistics curve) is an important component of exploratory data is the use of linear combinations of order statistics (L analyses. moments) to examine the distributional shape characteristics of data. L-moments have several Various methods to estimate skewness and kurtosis theoretical advantages over the central moment have been proposed (MacGillivray and Salanela, shape statistics: Characterization of a wider range of 1988). -
Linear Regression in SPSS
Linear Regression in SPSS Data: mangunkill.sav Goals: • Examine relation between number of handguns registered (nhandgun) and number of man killed (mankill) • Model checking • Predict number of man killed using number of handguns registered I. View the Data with a Scatter Plot To create a scatter plot, click through Graphs\Scatter\Simple\Define. A Scatterplot dialog box will appear. Click Simple option and then click Define button. The Simple Scatterplot dialog box will appear. Click mankill to the Y-axis box and click nhandgun to the x-axis box. Then hit OK. To see fit line, double click on the scatter plot, click on Chart\Options, then check the Total box in the box labeled Fit Line in the upper right corner. 60 60 50 50 40 40 30 30 20 20 10 10 Killed of People Number Number of People Killed 400 500 600 700 800 400 500 600 700 800 Number of Handguns Registered Number of Handguns Registered 1 Click the target button on the left end of the tool bar, the mouse pointer will change shape. Move the target pointer to the data point and click the left mouse button, the case number of the data point will appear on the chart. This will help you to identify the data point in your data sheet. II. Regression Analysis To perform the regression, click on Analyze\Regression\Linear. Place nhandgun in the Dependent box and place mankill in the Independent box. To obtain the 95% confidence interval for the slope, click on the Statistics button at the bottom and then put a check in the box for Confidence Intervals. -
On the Meaning and Use of Kurtosis
Psychological Methods Copyright 1997 by the American Psychological Association, Inc. 1997, Vol. 2, No. 3,292-307 1082-989X/97/$3.00 On the Meaning and Use of Kurtosis Lawrence T. DeCarlo Fordham University For symmetric unimodal distributions, positive kurtosis indicates heavy tails and peakedness relative to the normal distribution, whereas negative kurtosis indicates light tails and flatness. Many textbooks, however, describe or illustrate kurtosis incompletely or incorrectly. In this article, kurtosis is illustrated with well-known distributions, and aspects of its interpretation and misinterpretation are discussed. The role of kurtosis in testing univariate and multivariate normality; as a measure of departures from normality; in issues of robustness, outliers, and bimodality; in generalized tests and estimators, as well as limitations of and alternatives to the kurtosis measure [32, are discussed. It is typically noted in introductory statistics standard deviation. The normal distribution has a kur- courses that distributions can be characterized in tosis of 3, and 132 - 3 is often used so that the refer- terms of central tendency, variability, and shape. With ence normal distribution has a kurtosis of zero (132 - respect to shape, virtually every textbook defines and 3 is sometimes denoted as Y2)- A sample counterpart illustrates skewness. On the other hand, another as- to 132 can be obtained by replacing the population pect of shape, which is kurtosis, is either not discussed moments with the sample moments, which gives or, worse yet, is often described or illustrated incor- rectly. Kurtosis is also frequently not reported in re- ~(X i -- S)4/n search articles, in spite of the fact that virtually every b2 (•(X i - ~')2/n)2' statistical package provides a measure of kurtosis. -
The Normal Probability Distribution
BIOSTATISTICS BIOL 4343 Assessing normality Not all continuous random variables are normally distributed. It is important to evaluate how well the data set seems to be adequately approximated by a normal distribution. In this section some statistical tools will be presented to check whether a given set of data is normally distributed. 1. Previous knowledge of the nature of the distribution Problem: A researcher working with sea stars needs to know if sea star size (length of radii) is normally distributed. What do we know about the size distributions of sea star populations? 1. Has previous work with this species of sea star shown them to be normally distributed? 2. Has previous work with a closely related species of seas star shown them to be normally distributed? 3. Has previous work with seas stars in general shown them to be normally distributed? If you can answer yes to any of the above questions and you do not have a reason to think your population should be different, you could reasonably assume that your population is also normally distributed and stop here. However, if any previous work has shown non-normal distribution of sea stars you had probably better use other techniques. 2. Construct charts For small- or moderate-sized data sets, the stem-and-leaf display and box-and- whisker plot will look symmetric. For large data sets, construct a histogram or polygon and see if the distribution bell-shaped or deviates grossly from a bell-shaped normal distribution. Look for skewness and asymmetry. Look for gaps in the distribution – intervals with no observations. -
A Robust Rescaled Moment Test for Normality in Regression
Journal of Mathematics and Statistics 5 (1): 54-62, 2009 ISSN 1549-3644 © 2009 Science Publications A Robust Rescaled Moment Test for Normality in Regression 1Md.Sohel Rana, 1Habshah Midi and 2A.H.M. Rahmatullah Imon 1Laboratory of Applied and Computational Statistics, Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia 2Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, USA Abstract: Problem statement: Most of the statistical procedures heavily depend on normality assumption of observations. In regression, we assumed that the random disturbances were normally distributed. Since the disturbances were unobserved, normality tests were done on regression residuals. But it is now evident that normality tests on residuals suffer from superimposed normality and often possess very poor power. Approach: This study showed that normality tests suffer huge set back in the presence of outliers. We proposed a new robust omnibus test based on rescaled moments and coefficients of skewness and kurtosis of residuals that we call robust rescaled moment test. Results: Numerical examples and Monte Carlo simulations showed that this proposed test performs better than the existing tests for normality in the presence of outliers. Conclusion/Recommendation: We recommend using our proposed omnibus test instead of the existing tests for checking the normality of the regression residuals. Key words: Regression residuals, outlier, rescaled moments, skewness, kurtosis, jarque-bera test, robust rescaled moment test INTRODUCTION out that the powers of t and F tests are extremely sensitive to the hypothesized error distribution and may In regression analysis, it is a common practice over deteriorate very rapidly as the error distribution the years to use the Ordinary Least Squares (OLS) becomes long-tailed. -
Testing Normality: a GMM Approach Christian Bontempsa and Nour
Testing Normality: A GMM Approach Christian Bontempsa and Nour Meddahib∗ a LEEA-CENA, 7 avenue Edouard Belin, 31055 Toulouse Cedex, France. b D´epartement de sciences ´economiques, CIRANO, CIREQ, Universit´ede Montr´eal, C.P. 6128, succursale Centre-ville, Montr´eal (Qu´ebec), H3C 3J7, Canada. First version: March 2001 This version: June 2003 Abstract In this paper, we consider testing marginal normal distributional assumptions. More precisely, we propose tests based on moment conditions implied by normality. These moment conditions are known as the Stein (1972) equations. They coincide with the first class of moment conditions derived by Hansen and Scheinkman (1995) when the random variable of interest is a scalar diffusion. Among other examples, Stein equation implies that the mean of Hermite polynomials is zero. The GMM approach we adopt is well suited for two reasons. It allows us to study in detail the parameter uncertainty problem, i.e., when the tests depend on unknown parameters that have to be estimated. In particular, we characterize the moment conditions that are robust against parameter uncertainty and show that Hermite polynomials are special examples. This is the main contribution of the paper. The second reason for using GMM is that our tests are also valid for time series. In this case, we adopt a Heteroskedastic-Autocorrelation-Consistent approach to estimate the weighting matrix when the dependence of the data is unspecified. We also make a theoretical comparison of our tests with Jarque and Bera (1980) and OPG regression tests of Davidson and MacKinnon (1993). Finite sample properties of our tests are derived through a comprehensive Monte Carlo study. -
Power Comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling Tests
Journal ofStatistical Modeling and Analytics Vol.2 No.I, 21-33, 2011 Power comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling tests Nornadiah Mohd Razali1 Yap Bee Wah 1 1Faculty ofComputer and Mathematica/ Sciences, Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia E-mail: nornadiah@tmsk. uitm. edu.my, yapbeewah@salam. uitm. edu.my ABSTRACT The importance of normal distribution is undeniable since it is an underlying assumption of many statistical procedures such as I-tests, linear regression analysis, discriminant analysis and Analysis of Variance (ANOVA). When the normality assumption is violated, interpretation and inferences may not be reliable or valid. The three common procedures in assessing whether a random sample of independent observations of size n come from a population with a normal distribution are: graphical methods (histograms, boxplots, Q-Q-plots), numerical methods (skewness and kurtosis indices) and formal normality tests. This paper* compares the power offour formal tests of normality: Shapiro-Wilk (SW) test, Kolmogorov-Smirnov (KS) test, Lillie/ors (LF) test and Anderson-Darling (AD) test. Power comparisons of these four tests were obtained via Monte Carlo simulation of sample data generated from alternative distributions that follow symmetric and asymmetric distributions. Ten thousand samples ofvarious sample size were generated from each of the given alternative symmetric and asymmetric distributions. The power of each test was then obtained by comparing the test of normality statistics with the respective critical values. Results show that Shapiro-Wilk test is the most powerful normality test, followed by Anderson-Darling test, Lillie/ors test and Kolmogorov-Smirnov test. However, the power ofall four tests is still low for small sample size. -
2.8 Probability-Weighted Moments 33 2.8.1 Exact Variances and Covariances of Sample Probability Weighted Moments 35 2.9 Conclusions 36
Durham E-Theses Probability distribution theory, generalisations and applications of l-moments Elamir, Elsayed Ali Habib How to cite: Elamir, Elsayed Ali Habib (2001) Probability distribution theory, generalisations and applications of l-moments, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/3987/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk 2 Probability distribution theory, generalisations and applications of L-moments A thesis presented for the degree of Doctor of Philosophy at the University of Durham Elsayed Ali Habib Elamir Department of Mathematical Sciences, University of Durham, Durham, DHl 3LE. The copyright of this thesis rests with the author. No quotation from it should be published in any form, including Electronic and the Internet, without the author's prior written consent. All information derived from this thesis must be acknowledged appropriately. -
A Quantitative Validation Method of Kriging Metamodel for Injection Mechanism Based on Bayesian Statistical Inference
metals Article A Quantitative Validation Method of Kriging Metamodel for Injection Mechanism Based on Bayesian Statistical Inference Dongdong You 1,2,3,* , Xiaocheng Shen 1,3, Yanghui Zhu 1,3, Jianxin Deng 2,* and Fenglei Li 1,3 1 National Engineering Research Center of Near-Net-Shape Forming for Metallic Materials, South China University of Technology, Guangzhou 510640, China; [email protected] (X.S.); [email protected] (Y.Z.); fl[email protected] (F.L.) 2 Guangxi Key Lab of Manufacturing System and Advanced Manufacturing Technology, Guangxi University, Nanning 530003, China 3 Guangdong Key Laboratory for Advanced Metallic Materials processing, South China University of Technology, Guangzhou 510640, China * Correspondence: [email protected] (D.Y.); [email protected] (J.D.); Tel.: +86-20-8711-2933 (D.Y.); +86-137-0788-9023 (J.D.) Received: 2 April 2019; Accepted: 24 April 2019; Published: 27 April 2019 Abstract: A Bayesian framework-based approach is proposed for the quantitative validation and calibration of the kriging metamodel established by simulation and experimental training samples of the injection mechanism in squeeze casting. The temperature data uncertainty and non-normal distribution are considered in the approach. The normality of the sample data is tested by the Anderson–Darling method. The test results show that the original difference data require transformation for Bayesian testing due to the non-normal distribution. The Box–Cox method is employed for the non-normal transformation. The hypothesis test results of the calibrated kriging model are more reliable after data transformation. The reliability of the kriging metamodel is quantitatively assessed by the calculated Bayes factor and confidence. -
Swilk — Shapiro–Wilk and Shapiro–Francia Tests for Normality
Title stata.com swilk — Shapiro–Wilk and Shapiro–Francia tests for normality Description Quick start Menu Syntax Options for swilk Options for sfrancia Remarks and examples Stored results Methods and formulas Acknowledgment References Also see Description swilk performs the Shapiro–Wilk W test for normality for each variable in the specified varlist. Likewise, sfrancia performs the Shapiro–Francia W 0 test for normality. See[ MV] mvtest normality for multivariate tests of normality. Quick start Shapiro–Wilk test of normality Shapiro–Wilk test for v1 swilk v1 Separate tests of normality for v1 and v2 swilk v1 v2 Generate new variable w containing W test coefficients swilk v1, generate(w) Specify that average ranks should not be used for tied values swilk v1 v2, noties Test that v3 is distributed lognormally generate lnv3 = ln(v3) swilk lnv3 Shapiro–Francia test of normality Shapiro–Francia test for v1 sfrancia v1 Separate tests of normality for v1 and v2 sfrancia v1 v2 As above, but use the Box–Cox transformation sfrancia v1 v2, boxcox Specify that average ranks should not be used for tied values sfrancia v1 v2, noties 1 2 swilk — Shapiro–Wilk and Shapiro–Francia tests for normality Menu swilk Statistics > Summaries, tables, and tests > Distributional plots and tests > Shapiro-Wilk normality test sfrancia Statistics > Summaries, tables, and tests > Distributional plots and tests > Shapiro-Francia normality test Syntax Shapiro–Wilk normality test swilk varlist if in , swilk options Shapiro–Francia normality test sfrancia varlist if in , sfrancia options swilk options Description Main generate(newvar) create newvar containing W test coefficients lnnormal test for three-parameter lognormality noties do not use average ranks for tied values sfrancia options Description Main boxcox use the Box–Cox transformation for W 0; the default is to use the log transformation noties do not use average ranks for tied values by and collect are allowed with swilk and sfrancia; see [U] 11.1.10 Prefix commands. -
Tests of Normality and Other Goodness-Of-Fit Tests
Statistics 2 7 3 TESTS OF NORMALITY AND OTHER GOODNESS-OF-FIT TESTS Ralph B. D'Agostino, Albert J. Belanger, and Ralph B. D' Agostino Jr. Boston University Mathematics Department, Statistics and Consulting Unit Probability plots and goodness-of-fit tests are standardized deviates, :z;. on the horizontal axis. In table useful tools in detennining the underlying disnibution of I, we list the formulas for the disnibutions we plot in our a population (D'Agostino and Stephens, 1986. chapter 2). macro. If the underlying distribution is F(x), the resulting Probability plotting is an informal procedure for describing plot will be approximately a straight line. data and for identifying deviations from the hypothesized disnibution. Goodness-of-fit tests are formal procedures TABLE 1 which can be used to test for specific hypothesized disnibutions. We will present macros using SAS for Plotting Fonnulas for the six distributions plotted in our macro. creating probability plots for six disnibutions: the uniform. (PF(i-0.5)/n) normal, lognonnai, logistic, Weibull. and exponential. In Distribution cdf F(x) Vertical Horizontal addition these macros will compute the skewness <.pi;> . Axis Axis 7, kurtosis (b2), and D'Agostino-Pearson Kz statistics for testing if the underlying disttibution is normal (or Unifonn x-11 for 1'<%<1' +o ~ P... ~ ·lognormal) and the Anderson-Darling (EDF) A2 statistic G II for testing for the normal, log-normal, and exponential disnibutions. The latter can be modified for general ··I( ~;j i-3/8) distributions. 11+1(4 PROBABILITY PLOTS In(~~ ·-1( 1-3/8) 11+1{4 Say we desire to investigate if the underlying cumulative disnibution of a population is F(x) where this Weibull 1-exp( -(.!.t) ln(lw) In( -In( 1-p,)) disnibution depends upon a location parameter )l and scale 6 plll31lleter a. -
Univariate Analysis and Normality Test Using SAS, STATA, and SPSS
© 2002-2006 The Trustees of Indiana University Univariate Analysis and Normality Test: 1 Univariate Analysis and Normality Test Using SAS, STATA, and SPSS Hun Myoung Park This document summarizes graphical and numerical methods for univariate analysis and normality test, and illustrates how to test normality using SAS 9.1, STATA 9.2 SE, and SPSS 14.0. 1. Introduction 2. Graphical Methods 3. Numerical Methods 4. Testing Normality Using SAS 5. Testing Normality Using STATA 6. Testing Normality Using SPSS 7. Conclusion 1. Introduction Descriptive statistics provide important information about variables. Mean, median, and mode measure the central tendency of a variable. Measures of dispersion include variance, standard deviation, range, and interquantile range (IQR). Researchers may draw a histogram, a stem-and-leaf plot, or a box plot to see how a variable is distributed. Statistical methods are based on various underlying assumptions. One common assumption is that a random variable is normally distributed. In many statistical analyses, normality is often conveniently assumed without any empirical evidence or test. But normality is critical in many statistical methods. When this assumption is violated, interpretation and inference may not be reliable or valid. Figure 1. Comparing the Standard Normal and a Bimodal Probability Distributions Standard Normal Distribution Bimodal Distribution .4 .4 .3 .3 .2 .2 .1 .1 0 0 -5 -3 -1 1 3 5 -5 -3 -1 1 3 5 T-test and ANOVA (Analysis of Variance) compare group means, assuming variables follow normal probability distributions. Otherwise, these methods do not make much http://www.indiana.edu/~statmath © 2002-2006 The Trustees of Indiana University Univariate Analysis and Normality Test: 2 sense.