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"Student" and Small-Sample Theory E
Statistical Science 1999, Vol. 14, No.4, 418- 426 "Student" and Small-Sample Theory E. L. Lehmann Abstract. The paper discusses the contributions Student (W. S. Gosset) made to the three stages in which small-sample methodology was established in the period 1908-1933: (i) the distributions of the test statistics under the assumption of normality, (ii) the robustness of these distributions against nonnormality, (iii) the optimal choice of test statis tics. The conclusions are based on a careful reading of the correspon dence of Gosset with Fisher and E. S. Pearson. Key words and phrases: History of statistics, "exact" distribution theory, assumption of normality, robustness, hypothesis testing, Neyman-Pear son theory. 1. INTRODUCTION determined the distributions of the statistics used to test means, variances, correlation and regression In an interview published in Statistical Science coefficients under the assumption of normality. At [Laird (1989)], F. N. David talks about statistics in the second stage (Pearson), the robustness of these the 1920s and 30s as developed by Gosset (Student), distributions under nonnormality was investigated. Fisher, Egon Pearson and Neyman. [In the remain Finally, at the last stage Neyman and Pearson laid der of this paper we shall usually refer to E(gon) S. the foundation for a rational choice of test statistics Pearson simply as Pearson and to his father as not only in the normal case but quite generally. In Karl Pearson or occasionally asK. P.] She describes the following sections we shall consider the contri herself as a contemporary observer who saw "all butions Gosset made to each of these stages. -
STATISTICS in the WORLD DATABASE of HAPPINESS
World Database of Happiness, Correlational Findings, Introductory text Chapter 4 STATISTICS USED IN THE EXCERPTS ________________________________________________________________________ 4/1 Some basics of statistical analysis 4/2 Statistics of distribution 4/3 Statistics of association 4/3.1 bi-variate scheme 4/3.2 multi-variate scheme 4/4 Statistics for testing a hypothesis 4/5 Survey of statistics and methods _________________________________________________________________________ Text by Wim Kalmijn Findings on happiness can be quantified in various ways. There are many statistical techniques, which apply in different situations and which have their strong and weak points in that various situations. The finding-excerpts specify the statistical techniques that have been used. Current summary statistics are coded and appear in the field ‘statistics’. Further measures and methods are mentioned in the field ‘remarks’. This chapter starts with a short ‘refresher’ on some basic notions of descriptive statistics and statistical analysis (section 4/1). Next the three kind of statistical measures used in the excerpts are discussed. The statistics that are used for describing how happy people are in a particular population are dealt with in section 4/2. The statistics used for characterising/quantifying the relationship with other variables (‘correlates’) are enumerated in section 4/3. The statistics for testing hypotheses about associations (mostly the null hypothesis of no association at all in the population from which the sample has been drawn) are depicted in section 4/4. Finally a survey of the various statistics and statistical techniques that have been used in empirical studies on happiness is provided in section 4/5. This overview is alphabetically ordered and consists of shorthand descriptions of the statistics. -
Measures of Effect Size
Behavior Research Methods, Instruments, & Computers 1996,28 (1),12-22 Measures ofeffect size JOHNT. E. RICHARDSON Brunei University, Uxbridge, England Twodifferent approaches have been used to derive measures of effect size. One approach is based on the comparison of treatment means, The standardized mean difference is an appropriate measure of effect size when one is merely comparing two treatments, but there is no satisfactory analogue for comparing more than two treatments. The second approach is based on the proportion of variance in the dependent variable that is explained by the independent variable. Estimates have been pro posed for both fixed-factor and random-factor designs, but their sampling properties are not well un derstood. Nevertheless, measures of effect size can allow quantitative comparisons to be made across different studies, and they can be a useful adjunct to more traditional outcome measures such as test statistics and significance levels. Most psychological researchers appreciate in abstract signs, where the treatments employed exhaust the popu terms at least that statements describing the outcomes lation ofinterest. The second approach is typically used of tests of statistical inference need to be distinguished in random-effects designs, in which the treatments are from statements describing the importance of the rele regarded as a sample from some indefinite population of vant findings in theoretical or practical terms. The latter treatments, and in which it makes little sense to compute may have more to do with the magnitude ofthe effects in an effect size index by comparing the particular treat question than their level of statistical significance. ments that happened to be sampled. -
Correcting Four Similar Correlational Measures for Attenuation Due to Errors of Measurement in the Dependent Variable; Eta, Epsilon, Cmega, and Intraclass R
DOCUMENT RESUME ED 050 151 TM 000 540 AUTHOR Stanley, Julian C.; Livingston, Samuel A. TITLE Correcting Four Similar Correlational Measures for Attenuation Due to Errors of Measurement in the Dependent Variable; Eta, Epsilon, Cmega, and Intraclass r. INSTITUTION Johns Hopkins Univ., Baltimore, Md. Center for the Study ot Social Organization of Schools. PUB LATE Feb 71 NOTE 10p.; Part of a symposium entitled "Some Attenuating Effects of Errors cf Measurement," presented at the Annual Meeting of the American Educational Research Association, Neu York, New York, February 1971 ERRS PRICE EDRS Price MF-$0.65 HC-$3.29 DESCRIPTORS *Analysis of Variance, *Cluster Grouping, *Correlation, Data Analysis, *Reliability, Sampling, *Standard Error of Measurement, Statistical Analysis, Surveys, True Scores IIENTIFTERS *Attenuation (Correction) ABSTRACT Besides the ubiquitous Pearson product - moment r, there are a nutrter c± other measures cf relationship that are attenuated by errors of measurement and for which the relationship between true treasures can be estimated. Among these are the correlation ratio (eta squared), Kelley's unbiased correlation ratio (epsilon squared), Hays' omega squared, and the intraciass coefficient of correlation, expressed as a ratio of variance components by Ronald Fisher. This paper shows how tc correct each of these for attenuation. Such corrections permit estimates of relationships between true scores to be made when estimates of relevant reliability coefficients are available. (Author) Correcting Four Similar Correlational Measures for Attenuation Due to Errors of Measurement in the Dependent Variable: Eta, Epsilon, Omega, and Intraclassr U.S. DEPARTMENT OF HEALTH, EDUCATION & WELFARE OFFICE OF EDUCATION THIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVED FROM THE PERSON OR Julian C. -
Statistical Methods for Research Workers BIOLOGICAL MONOGRAPHS and MANUALS
BIOLOGICAL MONOGRAPHS AND MANUALS General Editors: F. A. E. CREW, Edinburgh D. WARD CUTLER, Rothamsted No. V Statistical Methods for Research Workers BIOLOGICAL MONOGRAPHS AND MANUALS THE PIGMENTARY EFFECTOR SYSTEM By L. T. Hogben, London School of Economics. THE ERYTHROCYTE AND THE ACTION OF SIMPLE HlEl\IOLYSINS By E. Ponder, New York University. ANIMAL GENETICS: AN INTRODUCTION TO THE SCIENCE OF ANIMAL BREEDING By F. A. E. Crew, Edinburgh University. REPRODUCTION IN THE RABBIT By J. Hammon~, School of Agriculture, Cambridge. STATISTICAL METHODS FOR RESI<:ARCH WORKERS-· By R. A. Fisher, University of London. .., THE COMPARATIVE ANATOMY, HISTOLOGY, AND DE~ELOPMENT OF THIe PITUITARY BODY By G. R. de Beer, Oxford University. THE COMPOSITION AND DISTRIBUTION OF THE PROTOZOAN FAUNA OF THE SOIL By H. Sandon, Egyptian University, Cairo. THE SPECIES PROBLEM By G. C. Robson, British Museum (Natural History). • THE MIGRATION OF BUTTERFLIES By C. B. Williams, Rothamsted Experimental Station. GROWTH AND THE DEVELOP:\IENT OF MUTTON QUALITIES IN THE SHEEP By John Hammond, Cambridge. MATERNAL BEHAVIOUR IN THE RAT By Bertold P. Wiesner and Norah M. Sheard. And other volumes are in course 0/ puJilication. Statistical Methods for Research Workers BY R. A. FISHER, Sc.D., F.R.S. Formerly Fellow oj Conville and Caius Colle/;e, Cambridge Honorary Member, American Statistical Assoc~f!tl0!1-.._~ .. Calton Professor, ziver~¥YA'l0 LOI!,ffn I 1,.'· I, j FIFTH EDITION-REVISED AND ENLARGED - ANGRAU Central Library Hydl'rabad G3681 /11.1111.1111111111111111111111 1111 OL~C~r;-.~OYD EDINBURGH:" TWEEDDALE COURT LONDON: 33 PATERNOSTER ROW, E.C. I934 }IADE; IN OLIVER AND :R~AT BRITAIN BY OYD LTD ., EDINBURGH EDITORS' PREFACE THE increasing specialisation in biological inquiry has made it impossible for anyone author to deal adequately with current advances in knowledge. -
Reporting Intraclass Correlation Coe
Reporting intraclass correlation coe Continue Descriptive statistics are a point graph showing a set of data with a high intra-class correlation. Values from the same group tend to be similar. A dot graph showing a data set with low intra-class correlation. There is no tendency for the values of the same group to be the same. In statistics, intra-class correlation, or intra-class correlation ratio (ICC), representing 1, is a narrative statistic that can be used to quantify units organized into groups. It describes how much units in the same group are similar to each other. Although it is seen as a type of correlation, unlike most other correlation indicators it works on data structured as a group rather than data structured as a paired observation. Intraclass correlation is usually used to quantify the extent to which people with a fixed degree of kinship (e.g. full siblings) are similar in terms of quantitative traits (see hired). Another notable application is the assessment of the consistency or reproducibility of quantitative measurements taken by different observers, measuring the same number. The ICC's early definition: the unbiased but complex formula For the earliest work on intra-class correlations, focused on the case of pair measurements, and the first intra-class correlation (ICC) statistics to be proposed were changes in interclass correlation (pearson correlation). Consider a data set consisting of N pair data values (xn.1, xn.2), for n No. 1, ..., N. The intra- class correlation r, originally proposed by Ronald Fischer, is r 1 N s 2 ∑ n n 1 N (x n, 1 x ̄) (x n, 2 x ̄) , displaystyle rfrac {1}'N'{2} Sum yn-1 (x_ yn 1-bar x) (x_ n ,2-bar (x) where x ̄ No 1 2 N ∑ n No 1 N (x n) , 1 x n, 2) , display style barxfrac {1} 2N amount (n'1) (x_ n,1'x_.2), from 2 x 1 2 N - ∑ n y 1 H (h, 1 - x ̄) 2 x ∑ n No 1 H (h, 2 x ̄) 2 . -
ST-101 Statistics
DIRECTORATE OF DISTANCE EDUCATION UNIVERSITY OF JAMMU JAMMU SELF LEARNING MATERIAL B.A. SEMESTER – I SUBJECT : STATISTICS UNIT : I-V COURSE NO. : ST - 101 LESSON NO.: 1-23 COURSE CO-ORDINATOR Proof Reading by : Mr. Stanzin Shakya Dr. Vikas Sharma http:/www.distanceeducationju.in Printed and Published on behalf of the Directorate of Distance Education, University of Jammu, Jammu by the Director, DDE, University of Jammu, Jammu 1 DESCRIPTIVE STATISTICS AND PROBABALITY THEORY © Directorate of Distance Education, University of Jammu, Jammu, 2020 • All rights reserved. No part of this work may be reproduced in any form, by mimeograph or any other means, without permission in writing from the DDE, University of Jammu. • The script writer shall be responsible for the lesson/script submitted to the DDE and any plagiarism shall be his/her entire responsibility. Printed by S K Printers/ 2020 / 100 2 DETAILED SYLLABUS OF B.A. SEMESTER - I STATISTICS PAPER TITLE : DESCRIPTIVE STATISTICS AND PROBABILITY TFEORY Objectives : The objectives of this course is to impart students the basic knowledge of measures of central tendencies and measures of dispersion along with the introduction to concept of probability and its basic theory. UNIT-I Definition, Scope & importance of statistics, General nature of statistical data, qualitative & quantitative data, discrete & continuous data, Primary & Secondary data, classification & Tabulation, frequency distribution & their graphical& diagrammatic representations Histogram, frequency Curves, Bar diagram, Ogive & measure of central tendency (A.M.,G.M., H.M.) Median & mode, their merits& demerits. UNIT- II Measures of Dispersion : Range, Inter-Quartile range, Mean Deviation,Standard Deviation, Variance & Coefficient of Variation, Partition values, Moments (raw & central moments up to order four. -
Effect Size and Statistical Power
Effect Size and Statistical Power Joseph Stevens, Ph.D., University of Oregon (541) 346-2445, [email protected] © Stevens, 2007 1 An Introductory Problem or Two: Which Study is Stronger? Answer: Study B Study A: t (398) = 2.30, p = .022 ω2 for Study A = .01 Study B: t (88) = 2.30, p = .024 ω2 for Study B = .05 Examples inspired by Rosenthal & Gaito (1963) 2 Study C Shows a Highly Significant Result Study C: F = 63.62, p < .0000001 η2 for Study C = .01, N = 6,300 Study D: F = 5.40, p = .049 η2 for Study D = .40, N = 10 Correct interpretation of statistical results requires consider- ation of statistical significance, effect size, and statistical power 3 Three Fundamental Questions Asked in Science Is there a relationship? Answered by Null Hypothesis Significance Tests (NHST; e.g., t tests, F tests, χ2, p-values, etc.) What kind of relationship? Answered by testing if relationship is linear, curvilinear, etc. How strong is the relationship? Answered by effect size measures, not NHST’s (e.g., R2, r2, η2, ω2, Cohen’s d) 4 The Logic of Inferential Statistics Three Distributions Used in Inferential Statistics: Population: the entire universe of individuals we are interested in studying (µ, σ, ∞) Sample: the selected subgroup that is actually observed and measured ( X , sˆ , N) Sampling Distribution of the Statistic: A theoretical distribution that describes how a statistic behaves µ across a large number of samples ( X , sˆ X , ∞) 5 The Three Distributions Used in Inferential Statistics I. Population Inference Selection III. Sampling Distribution of the Statistic II. -
'Student': a Statistical Biography of William Sealy Gosset
(OVStudent 9 A Statistical Biography of William Sealy Gosset Based on writings by E.S. Pearson Edited and augmented by R.L. Plackett With the assistance of G.A. Barnard CLARENDONPRESS : OXFORD 1990 Oxford University Press, Walton Street, Oxford Ox2 6DP Oxford New York Toronto Delhi Bombay Calcutta Madras Karachi Petaling Jaya Singapore Hong Kong Tokyo Nairobi Dar es Salaam Cape Town Melbourne Auckland and associated companies in Berlin Ibadan Oxford is a trade mark of Oxford University Press Published in the United states by Oxford University Press, New York © G.A. Barnard, R.L. Plackett, and S. Pearson, 1990 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in anyform or by any means,electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press British Library Cataloguing in Publication Data Pearson, E. S. (Egon Sharpe) “Student”: a statistical biography of William Sealy Gosset: based on writings by E. S. Pearson. 1. Statistical mathematics. Gosset, William Sealy, d1937 I. Title I. Plackett, R. L. (Robert Lewis) Ill. Barnard, G. A. (George Alfred), 1915-— 519.5092 ISBN O0-19-852227-4 Library of Congress Cataloging in Publication Data Pearson, E. S. (Egon Sharpe), 1895- Student: a statistical biography of William Sealy Gosset/based on the writings by E. S. Pearson; edited and augmented by R.L. Plackett with the assistance of G. A. Barnard. p. cm. Includes bibliographical references. 1. Gosset, William Sealy, d. 1937. 2. Statisticians—England— Biography. 3. Statistics—Great Britain—History. I. Gosset, William Sealy, d. -
Correlation and Dependence 1 Correlation and Dependence
Correlation and dependence 1 Correlation and dependence In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence. Familiar examples of dependent phenomena include the correlation between the physical statures of parents and their offspring, and the correlation between the demand for a product and its price. Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example there is a causal relationship, because extreme weather causes people to use more electricity for heating or cooling; however, statistical dependence is not sufficient to demonstrate the presence of such a causal relationship. Formally, dependence refers to any situation in which random variables do not satisfy a mathematical condition of probabilistic independence. In loose usage, correlation can refer to any departure of two or more random variables from independence, but technically it refers to any of several more specialized types of relationship between mean values. There are several correlation coefficients, often denoted ρ or r, measuring the degree of correlation. The most common of these is the Pearson correlation coefficient, which is sensitive only to a linear relationship between two variables (which may exist even -
Methodological Aspects in Using Pearson Coefficient in Analyzing Social and Economical Phenomena
European Research Studies, Volume XI, Special Issue (3-4) 2007 Methodological Aspects in Using Pearson Coefficient in Analyzing Social and Economical Phenomena By Daniela-Emanuela D ;n;cic ;1 Ana-Gabriela Babucea 2 Abstract: The authors illustrate in this paper a series of methodological aspects generated by the use of Pearson correlation coefficient in analyzing social and economical phenomena. Pearson correlation coefficient is largely used in economics and social sciences; however, the diversified nature and subtle nuances of this concept raises significant methodological issues. This article deals with aspects concerning the factors that impact on the size and interpretation of Pearson correlation coefficient, as well as special cases of this coefficient. Keywords: Peasron correlation, tetrachoric correlation coefficient. 1 A ssistant Professor, Faculty of Economics, Constantin Brâncu 2i University of Târgu-Jiu 2Professor, Faculty of Economics, Constantin Brâncu 2i University of Târgu-Jiu 90 European Research Studies, Volume XI, Special Issue (3-4) 2007 1. Introduction In the socio-economic field phenomenon variability is not exclusively determined by the action of a singular factor, but in most cases it is the result of the action of a pluralism of factors. This gives us the possibility, that by the means of the known variation of a factor, to determine the level of another variable it is in certain dependence with. The causality ratios between social and economical phenomena can be quantified, analyzed and interpreted by the means of correlation and regression analysis. Within it there is studied the dependence between a resultative variable (characteristic), usually marked with Y and one or more independent variables (characteristics), marked with X X , X .......