Univariate Analysis and Normality Test Using SAS, STATA, and SPSS
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Section 13 Kolmogorov-Smirnov Test
Section 13 Kolmogorov-Smirnov test. Suppose that we have an i.i.d. sample X1; : : : ; Xn with some unknown distribution P and we would like to test the hypothesis that P is equal to a particular distribution P0; i.e. decide between the following hypotheses: H : P = P ; H : P = P : 0 0 1 6 0 We already know how to test this hypothesis using chi-squared goodness-of-fit test. If dis tribution P0 is continuous we had to group the data and consider a weaker discretized null hypothesis. We will now consider a different test for H0 based on a very different idea that avoids this discretization. 1 1 men data 0.9 0.9 'men' fit all data women data 0.8 normal fit 0.8 'women' fit 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 Cumulative probability 0.3 Cumulative probability 0.3 0.2 0.2 0.1 0.1 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Data Data Figure 13.1: (a) Normal fit to the entire sample. (b) Normal fit to men and women separately. Example.(KS test) Let us again look at the normal body temperature dataset. Let ’all’ be a vector of all 130 observations and ’men’ and ’women’ be vectors of length 65 each corresponding to men and women. First, we fit normal distribution to the entire set ’all’. -
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). -
Efficient Estimation of Parameters of the Negative Binomial Distribution
E±cient Estimation of Parameters of the Negative Binomial Distribution V. SAVANI AND A. A. ZHIGLJAVSKY Department of Mathematics, Cardi® University, Cardi®, CF24 4AG, U.K. e-mail: SavaniV@cardi®.ac.uk, ZhigljavskyAA@cardi®.ac.uk (Corresponding author) Abstract In this paper we investigate a class of moment based estimators, called power method estimators, which can be almost as e±cient as maximum likelihood estima- tors and achieve a lower asymptotic variance than the standard zero term method and method of moments estimators. We investigate di®erent methods of implementing the power method in practice and examine the robustness and e±ciency of the power method estimators. Key Words: Negative binomial distribution; estimating parameters; maximum likelihood method; e±ciency of estimators; method of moments. 1 1. The Negative Binomial Distribution 1.1. Introduction The negative binomial distribution (NBD) has appeal in the modelling of many practical applications. A large amount of literature exists, for example, on using the NBD to model: animal populations (see e.g. Anscombe (1949), Kendall (1948a)); accident proneness (see e.g. Greenwood and Yule (1920), Arbous and Kerrich (1951)) and consumer buying behaviour (see e.g. Ehrenberg (1988)). The appeal of the NBD lies in the fact that it is a simple two parameter distribution that arises in various di®erent ways (see e.g. Anscombe (1950), Johnson, Kotz, and Kemp (1992), Chapter 5) often allowing the parameters to have a natural interpretation (see Section 1.2). Furthermore, the NBD can be implemented as a distribution within stationary processes (see e.g. Anscombe (1950), Kendall (1948b)) thereby increasing the modelling potential of the distribution. -
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. -
Univariate and Multivariate Skewness and Kurtosis 1
Running head: UNIVARIATE AND MULTIVARIATE SKEWNESS AND KURTOSIS 1 Univariate and Multivariate Skewness and Kurtosis for Measuring Nonnormality: Prevalence, Influence and Estimation Meghan K. Cain, Zhiyong Zhang, and Ke-Hai Yuan University of Notre Dame Author Note This research is supported by a grant from the U.S. Department of Education (R305D140037). However, the contents of the paper do not necessarily represent the policy of the Department of Education, and you should not assume endorsement by the Federal Government. Correspondence concerning this article can be addressed to Meghan Cain ([email protected]), Ke-Hai Yuan ([email protected]), or Zhiyong Zhang ([email protected]), Department of Psychology, University of Notre Dame, 118 Haggar Hall, Notre Dame, IN 46556. UNIVARIATE AND MULTIVARIATE SKEWNESS AND KURTOSIS 2 Abstract Nonnormality of univariate data has been extensively examined previously (Blanca et al., 2013; Micceri, 1989). However, less is known of the potential nonnormality of multivariate data although multivariate analysis is commonly used in psychological and educational research. Using univariate and multivariate skewness and kurtosis as measures of nonnormality, this study examined 1,567 univariate distriubtions and 254 multivariate distributions collected from authors of articles published in Psychological Science and the American Education Research Journal. We found that 74% of univariate distributions and 68% multivariate distributions deviated from normal distributions. In a simulation study using typical values of skewness and kurtosis that we collected, we found that the resulting type I error rates were 17% in a t-test and 30% in a factor analysis under some conditions. Hence, we argue that it is time to routinely report skewness and kurtosis along with other summary statistics such as means and variances. -
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. -
Characterization of the Bivariate Negative Binomial Distribution James E
Journal of the Arkansas Academy of Science Volume 21 Article 17 1967 Characterization of the Bivariate Negative Binomial Distribution James E. Dunn University of Arkansas, Fayetteville Follow this and additional works at: http://scholarworks.uark.edu/jaas Part of the Other Applied Mathematics Commons Recommended Citation Dunn, James E. (1967) "Characterization of the Bivariate Negative Binomial Distribution," Journal of the Arkansas Academy of Science: Vol. 21 , Article 17. Available at: http://scholarworks.uark.edu/jaas/vol21/iss1/17 This article is available for use under the Creative Commons license: Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0). Users are able to read, download, copy, print, distribute, search, link to the full texts of these articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the author. This Article is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Journal of the Arkansas Academy of Science by an authorized editor of ScholarWorks@UARK. For more information, please contact [email protected], [email protected]. Journal of the Arkansas Academy of Science, Vol. 21 [1967], Art. 17 77 Arkansas Academy of Science Proceedings, Vol.21, 1967 CHARACTERIZATION OF THE BIVARIATE NEGATIVE BINOMIAL DISTRIBUTION James E. Dunn INTRODUCTION The univariate negative binomial distribution (also known as Pascal's distribution and the Polya-Eggenberger distribution under vari- ous reparameterizations) has recently been characterized by Bartko (1962). Its broad acceptance and applicability in such diverse areas as medicine, ecology, and engineering is evident from the references listed there. -
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MARCH 2007 NOTES AND CORRESPONDENCE 1151 NOTES AND CORRESPONDENCE A Cautionary Note on the Use of the Kolmogorov–Smirnov Test for Normality DAG J. STEINSKOG Nansen Environmental and Remote Sensing Center, and Bjerknes Centre for Climate Research, and Geophysical Institute, University of Bergen, Bergen, Norway DAG B. TJØSTHEIM Department of Mathematics, University of Bergen, Bergen, Norway NILS G. KVAMSTØ Geophysical Institute, University of Bergen, and Bjerknes Centre for Climate Research, Bergen, Norway (Manuscript received and in final form 28 April 2006) ABSTRACT The Kolmogorov–Smirnov goodness-of-fit test is used in many applications for testing normality in climate research. This note shows that the test usually leads to systematic and drastic errors. When the mean and the standard deviation are estimated, it is much too conservative in the sense that its p values are strongly biased upward. One may think that this is a small sample problem, but it is not. There is a correction of the Kolmogorov–Smirnov test by Lilliefors, which is in fact sometimes confused with the original Kolmogorov–Smirnov test. Both the Jarque–Bera and the Shapiro–Wilk tests for normality are good alternatives to the Kolmogorov–Smirnov test. A power comparison of eight different tests has been undertaken, favoring the Jarque–Bera and the Shapiro–Wilk tests. The Jarque–Bera and the Kolmogorov– Smirnov tests are also applied to a monthly mean dataset of geopotential height at 500 hPa. The two tests give very different results and illustrate the danger of using the Kolmogorov–Smirnov test. 1. Introduction estimates of skewness and kurtosis. Such moments can be used to diagnose nonlinear processes and to provide The Kolmogorov–Smirnov test (hereafter the KS powerful tools for validating models (see also Ger- test) is a much used goodness-of-fit test. -
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. -
UNIT 1 INTRODUCTION to STATISTICS Introduction to Statistics
UNIT 1 INTRODUCTION TO STATISTICS Introduction to Statistics Structure 1.0 Introduction 1.1 Objectives 1.2 Meaning of Statistics 1.2.1 Statistics in Singular Sense 1.2.2 Statistics in Plural Sense 1.2.3 Definition of Statistics 1.3 Types of Statistics 1.3.1 On the Basis of Function 1.3.2 On the Basis of Distribution of Data 1.4 Scope and Use of Statistics 1.5 Limitations of Statistics 1.6 Distrust and Misuse of Statistics 1.7 Let Us Sum Up 1.8 Unit End Questions 1.9 Glossary 1.10 Suggested Readings 1.0 INTRODUCTION The word statistics has different meaning to different persons. Knowledge of statistics is applicable in day to day life in different ways. In daily life it means general calculation of items, in railway statistics means the number of trains operating, number of passenger’s freight etc. and so on. Thus statistics is used by people to take decision about the problems on the basis of different type of quantitative and qualitative information available to them. However, in behavioural sciences, the word ‘statistics’ means something different from the common concern of it. Prime function of statistic is to draw statistical inference about population on the basis of available quantitative information. Overall, statistical methods deal with reduction of data to convenient descriptive terms and drawing some inferences from them. This unit focuses on the above aspects of statistics. 1.1 OBJECTIVES After going through this unit, you will be able to: Define the term statistics; Explain the status of statistics; Describe the nature of statistics; State basic concepts used in statistics; and Analyse the uses and misuses of statistics.