Permutation tests are similar to rank tests, except that we use the observations directly without replacing them by ranks.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Permutation tests are similar to rank tests, except that we use the observations directly without replacing them by ranks."

Transcription

1 Chapter 2 Permutation Tests Permutation tests are similar to rank tests, except that we use the observations directly without replacing them by ranks. 2.1 The two-sample location problem Assumptions: x 1,...,x m are observations on i.i.d. r.vs X 1,...,X m with a c.d.f. F 1, y 1,...,y n are observations on i.i.d. r.vs Y 1,...,Y n with a c.d.f. F 2. Null Hypothesis takes the same form as in the MWW test H 0 : F 1 (x) = F 2 (x) for all x and possible alternative hypotheses are as before: H 1 : F 1 (x) F 2 (x) with inequality for at least one x, H 1 : F 1 (x) F 2 (x) with inequality for at least one x, H 1 : Either F 1 (x) F 2 (x) or F 1 (x) F 2 (x) with inequality for at least one x. For this problem one of the suitable test statistics is Other possibilities are: T = X Ȳ. (2.1) Difference between sample medians, i.e., T = Me 1 Me 2 19

2 20 CHAPTER 2. PERMUTATION TESTS Difference between sample trimmed means, i.e., T = X t Ȳt Any monotonic function of T, say g(t); for example g(t) = T a b, b > 0 Null distribution of T: Under H 0 all of the m + n observations form a single random sample, and every selection of m observations out of m + n is equally likely; there are m+n C m such selections, for every selection of the m observations we get a value of T, say t. Hence, the null distribution of T is given by P(T = t) = #(t,m,n) m+n C m, (2.2) where #(t,m,n) denotes the number of all subsets for which T = t. The p-value is P(T t 0 ) = #(t t 0,m,n) m+n C m, where t 0 is obtained from the sample, #(t t 0,m,n) denotes the number of the subsets for which T t 0. Example: visual acuity McClave and Dietrich(1988) In a comparison of visual acuity (VA) of deaf (D) and hearing (H) children, eye movement rates were taken on eight deaf and ten hearing children. A clinical psychologist believes that deaf children have greater visual acuity than hearing children. The larger a child s eye movement rate, the more visual acuity the child possesses. Test the psychologist s claim using the data given below. VA sample D D D D D D D D H H H H H H H H H H If there is no difference between the two groups with respect to VA then both samples come from a population with a common distribution. Hence the null hypothesis is H 0 : F D (x) = F H (x) for allx against the alternative H 1 : F D (x) F H (x) with inequality for at least one x

3 2.1. THE TWO-SAMPLE LOCATION PROBLEM 21 We will use permutation test function ( 2.1) to verify this hypothesis. GenStat program calculating the value of the statistic T = D H and simulating the null distribution of T will be shown at the lectures.

4 22 CHAPTER 2. PERMUTATION TESTS 2.2 Test for independence of bivariate data Let observations (x 1,y 1 ), (x 2,y 2 ),...,(x n,y n ) be a realization of i.i.d. r.vs (X 1,Y 1 ), (X 2,Y 2 ),...,(X n,y n ) with a c.d.f. F X,Y. As in the rank test of independence, here too, we may state the hypotheses as: H 0 : There is no association between r.vs X and Y H 1 : There is an association between r.vs X and Y. A suitable test statistic for a permutation test is based on the sample correlation coefficient n i=1 ˆρ = X iy i n XȲ n i=1 (X i X) 2 n i=1 (Y i Ȳ )2. For a given sample n xȳ and the denominator are constant values for all permutations of the data. The only part of the coefficient sensitive to changes due to permutations is the sum n i=1 x iy i. Hence, the function V p = n X i Y i (2.3) i=1 may be used as a test statistic for the hypothesis of independence. We get the distribution of V p similarly to that for the rank statistic V. It involves calculating values of V p for all n! ways of pairing y i with x i. 2.3 Matched pairs As in the Wilcoxon signed rank test, let (y 1,...,y n ) and (z 1,...,z n ) be a realization of r.vs (Y 1,...,Y n ) and (Z 1,...,Z n ) such that Y s and Z s do not have to be independent. Then we analyze differences X i = Y i Z i which should be symmetrically distributed about zero if there is no difference between Y and Z. The null and alternative hypotheses have the same form as in the rank test, that is H 0 : X i are symmetrically distributed about zero H 1 : The center of the distribution is not zero

5 2.3. MATCHED PAIRS 23 The permutation test statistic is W + p = n Ψ i x i, (2.4) i=1 where the r.v. Ψ i is The null distribution of W + p Ψ i = { 1 if xi > 0 0 otherwise. is given by P(W + p = w) = #(w,n) 2 n, where #(w,n) denotes the number of subsets of the values of the data which give W + p = w. Learning the mechanics example will be given at the lectures

6 24 CHAPTER 2. PERMUTATION TESTS Example: Coca-Cola Advertising. Newbold (1988) The Coca-Cola Company ran a national advertising campaign based on the slogan Twice the cola, twice the fun. To test whether the campaign had improved brand awareness, random samples of 500 people in each of 10 cities were asked to name five soft drinks, both before and after the campaign had run. The accompanying table shows the numbers naming Coca-Cola. Test the hypothesis that the campaign made no difference to the customers awareness of the Coca-Cola brand. City Before After Atlanta Boston Chicago Denver Los Angeles Miami New Orleans New York Philadelphia St. Louis Calculations in GenStat will be shown during the lecture.

7 2.4. RANK TESTS VERSUS PERMUTATION TESTS Rank tests versus permutation tests There are some similarities and some differences between the two kinds of nonparametric tests. Which one to choose depends on the given hypothesistesting problem. Similarities. Both tests are: non-parametric: the underlying distribution does not need to be assumed to belong to any particular family of distributions, based on general principles of statistical hypothesis-testing, exact: for large samples we can get close to the p-value by taking enough simulations, non-informative about the estimates of the parameters. Differences Criterion Rank Tests Permutation Tests Robustness Not sensitive to outliers Quite sensitive to outliers Power Less powerful; loss of More powerful; for large n information by using ranks comparable to parametric tests Computational Null distribution depends Null distribution has to Complexity only on the sample size; be calculated for each critical values can be data set. tabulated. Asymptotic Usually asymptotically In general, not Normality normal (under H 0 ). asymptotically normal. Ties Slight problem. No problem.

. (3.3) n Note that supremum (3.2) must occur at one of the observed values x i or to the left of x i.

. (3.3) n Note that supremum (3.2) must occur at one of the observed values x i or to the left of x i. Chapter 3 Kolmogorov-Smirnov Tests There are many situations where experimenters need to know what is the distribution of the population of their interest. For example, if they want to use a parametric

More information

3.6: General Hypothesis Tests

3.6: General Hypothesis Tests 3.6: General Hypothesis Tests The χ 2 goodness of fit tests which we introduced in the previous section were an example of a hypothesis test. In this section we now consider hypothesis tests more generally.

More information

Permutation Tests for Comparing Two Populations

Permutation Tests for Comparing Two Populations Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. Jae-Wan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of

More information

Nonparametric Statistics

Nonparametric Statistics Nonparametric Statistics References Some good references for the topics in this course are 1. Higgins, James (2004), Introduction to Nonparametric Statistics 2. Hollander and Wolfe, (1999), Nonparametric

More information

Difference tests (2): nonparametric

Difference tests (2): nonparametric NST 1B Experimental Psychology Statistics practical 3 Difference tests (): nonparametric Rudolf Cardinal & Mike Aitken 10 / 11 February 005; Department of Experimental Psychology University of Cambridge

More information

Spearman s correlation

Spearman s correlation Spearman s correlation Introduction Before learning about Spearman s correllation it is important to understand Pearson s correlation which is a statistical measure of the strength of a linear relationship

More information

3. Nonparametric methods

3. Nonparametric methods 3. Nonparametric methods If the probability distributions of the statistical variables are unknown or are not as required (e.g. normality assumption violated), then we may still apply nonparametric tests

More information

Non Parametric Inference

Non Parametric Inference Maura Department of Economics and Finance Università Tor Vergata Outline 1 2 3 Inverse distribution function Theorem: Let U be a uniform random variable on (0, 1). Let X be a continuous random variable

More information

Non-parametric tests I

Non-parametric tests I Non-parametric tests I Objectives Mann-Whitney Wilcoxon Signed Rank Relation of Parametric to Non-parametric tests 1 the problem Our testing procedures thus far have relied on assumptions of independence,

More information

NONPARAMETRIC STATISTICS 1. depend on assumptions about the underlying distribution of the data (or on the Central Limit Theorem)

NONPARAMETRIC STATISTICS 1. depend on assumptions about the underlying distribution of the data (or on the Central Limit Theorem) NONPARAMETRIC STATISTICS 1 PREVIOUSLY parametric statistics in estimation and hypothesis testing... construction of confidence intervals computing of p-values classical significance testing depend on assumptions

More information

Non-Parametric Two-Sample Analysis: The Mann-Whitney U Test

Non-Parametric Two-Sample Analysis: The Mann-Whitney U Test Non-Parametric Two-Sample Analysis: The Mann-Whitney U Test When samples do not meet the assumption of normality parametric tests should not be used. To overcome this problem, non-parametric tests can

More information

On Small Sample Properties of Permutation Tests: A Significance Test for Regression Models

On Small Sample Properties of Permutation Tests: A Significance Test for Regression Models On Small Sample Properties of Permutation Tests: A Significance Test for Regression Models Hisashi Tanizaki Graduate School of Economics Kobe University (tanizaki@kobe-u.ac.p) ABSTRACT In this paper we

More information

Chapter 16 Appendix. Nonparametric Tests with Excel, JMP, Minitab, SPSS, CrunchIt!, R, and TI-83-/84 Calculators

Chapter 16 Appendix. Nonparametric Tests with Excel, JMP, Minitab, SPSS, CrunchIt!, R, and TI-83-/84 Calculators The Wilcoxon Rank Sum Test Chapter 16 Appendix Nonparametric Tests with Excel, JMP, Minitab, SPSS, CrunchIt!, R, and TI-83-/84 Calculators These nonparametric tests make no assumption about Normality.

More information

The Modified Kolmogorov-Smirnov One-Sample Test Statistic

The Modified Kolmogorov-Smirnov One-Sample Test Statistic Thailand Statistician July 00; 8() : 43-55 http://statassoc.or.th Contributed paper The Modified Kolmogorov-Smirnov One-Sample Test Statistic Jantra Sukkasem epartment of Mathematics, Faculty of Science,

More information

QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS

QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS This booklet contains lecture notes for the nonparametric work in the QM course. This booklet may be online at http://users.ox.ac.uk/~grafen/qmnotes/index.html.

More information

CHAPTER 14 NONPARAMETRIC TESTS

CHAPTER 14 NONPARAMETRIC TESTS CHAPTER 14 NONPARAMETRIC TESTS Everything that we have done up until now in statistics has relied heavily on one major fact: that our data is normally distributed. We have been able to make inferences

More information

Statistiek (WISB361)

Statistiek (WISB361) Statistiek (WISB361) Final exam June 29, 2015 Schrijf uw naam op elk in te leveren vel. Schrijf ook uw studentnummer op blad 1. The maximum number of points is 100. Points distribution: 23 20 20 20 17

More information

The Wilcoxon Rank-Sum Test

The Wilcoxon Rank-Sum Test 1 The Wilcoxon Rank-Sum Test The Wilcoxon rank-sum test is a nonparametric alternative to the twosample t-test which is based solely on the order in which the observations from the two samples fall. We

More information

MEASURES OF LOCATION AND SPREAD

MEASURES OF LOCATION AND SPREAD Paper TU04 An Overview of Non-parametric Tests in SAS : When, Why, and How Paul A. Pappas and Venita DePuy Durham, North Carolina, USA ABSTRACT Most commonly used statistical procedures are based on the

More information

Nonparametric tests these test hypotheses that are not statements about population parameters (e.g.,

Nonparametric tests these test hypotheses that are not statements about population parameters (e.g., CHAPTER 13 Nonparametric and Distribution-Free Statistics Nonparametric tests these test hypotheses that are not statements about population parameters (e.g., 2 tests for goodness of fit and independence).

More information

Homework Zero. Max H. Farrell Chicago Booth BUS41100 Applied Regression Analysis. Complete before the first class, do not turn in

Homework Zero. Max H. Farrell Chicago Booth BUS41100 Applied Regression Analysis. Complete before the first class, do not turn in Homework Zero Max H. Farrell Chicago Booth BUS41100 Applied Regression Analysis Complete before the first class, do not turn in This homework is intended as a self test of your knowledge of the basic statistical

More information

UNIVERSITY OF NAIROBI

UNIVERSITY OF NAIROBI UNIVERSITY OF NAIROBI MASTERS IN PROJECT PLANNING AND MANAGEMENT NAME: SARU CAROLYNN ELIZABETH REGISTRATION NO: L50/61646/2013 COURSE CODE: LDP 603 COURSE TITLE: RESEARCH METHODS LECTURER: GAKUU CHRISTOPHER

More information

Non-Parametric Tests (I)

Non-Parametric Tests (I) Lecture 5: Non-Parametric Tests (I) KimHuat LIM lim@stats.ox.ac.uk http://www.stats.ox.ac.uk/~lim/teaching.html Slide 1 5.1 Outline (i) Overview of Distribution-Free Tests (ii) Median Test for Two Independent

More information

Statistical Significance and Bivariate Tests

Statistical Significance and Bivariate Tests Statistical Significance and Bivariate Tests BUS 735: Business Decision Making and Research 1 1.1 Goals Goals Specific goals: Re-familiarize ourselves with basic statistics ideas: sampling distributions,

More information

Nonparametric Test Procedures

Nonparametric Test Procedures Nonparametric Test Procedures 1 Introduction to Nonparametrics Nonparametric tests do not require that samples come from populations with normal distributions or any other specific distribution. Hence

More information

Stat 5102 Notes: Nonparametric Tests and. confidence interval

Stat 5102 Notes: Nonparametric Tests and. confidence interval Stat 510 Notes: Nonparametric Tests and Confidence Intervals Charles J. Geyer April 13, 003 This handout gives a brief introduction to nonparametrics, which is what you do when you don t believe the assumptions

More information

Data Analysis. Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) SS Analysis of Experiments - Introduction

Data Analysis. Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) SS Analysis of Experiments - Introduction Data Analysis Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) Prof. Dr. Dr. h.c. Dieter Rombach Dr. Andreas Jedlitschka SS 2014 Analysis of Experiments - Introduction

More information

Statistical tests for SPSS

Statistical tests for SPSS Statistical tests for SPSS Paolo Coletti A.Y. 2010/11 Free University of Bolzano Bozen Premise This book is a very quick, rough and fast description of statistical tests and their usage. It is explicitly

More information

Exact Nonparametric Tests for Comparing Means - A Personal Summary

Exact Nonparametric Tests for Comparing Means - A Personal Summary Exact Nonparametric Tests for Comparing Means - A Personal Summary Karl H. Schlag European University Institute 1 December 14, 2006 1 Economics Department, European University Institute. Via della Piazzuola

More information

Unit 24 Hypothesis Tests about Means

Unit 24 Hypothesis Tests about Means Unit 24 Hypothesis Tests about Means Objectives: To recognize the difference between a paired t test and a two-sample t test To perform a paired t test To perform a two-sample t test A measure of the amount

More information

Hypothesis testing S2

Hypothesis testing S2 Basic medical statistics for clinical and experimental research Hypothesis testing S2 Katarzyna Jóźwiak k.jozwiak@nki.nl 2nd November 2015 1/43 Introduction Point estimation: use a sample statistic to

More information

Chapter G08 Nonparametric Statistics

Chapter G08 Nonparametric Statistics G08 Nonparametric Statistics Chapter G08 Nonparametric Statistics Contents 1 Scope of the Chapter 2 2 Background to the Problems 2 2.1 Parametric and Nonparametric Hypothesis Testing......................

More information

Hypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test...

Hypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test... Hypothesis Testing COMP 45 STATISTICS Dr N A Heard Contents 1 Hypothesis Testing 1.1 Introduction........................................ 1. Error Rates and Power of a Test.............................

More information

How to choose a statistical test. Francisco J. Candido dos Reis DGO-FMRP University of São Paulo

How to choose a statistical test. Francisco J. Candido dos Reis DGO-FMRP University of São Paulo How to choose a statistical test Francisco J. Candido dos Reis DGO-FMRP University of São Paulo Choosing the right test One of the most common queries in stats support is Which analysis should I use There

More information

Rank-Based Non-Parametric Tests

Rank-Based Non-Parametric Tests Rank-Based Non-Parametric Tests Reminder: Student Instructional Rating Surveys You have until May 8 th to fill out the student instructional rating surveys at https://sakai.rutgers.edu/portal/site/sirs

More information

EPS 625 INTERMEDIATE STATISTICS FRIEDMAN TEST

EPS 625 INTERMEDIATE STATISTICS FRIEDMAN TEST EPS 625 INTERMEDIATE STATISTICS The Friedman test is an extension of the Wilcoxon test. The Wilcoxon test can be applied to repeated-measures data if participants are assessed on two occasions or conditions

More information

Statistics: revision

Statistics: revision NST 1B Experimental Psychology Statistics practical 5 Statistics: revision Rudolf Cardinal & Mike Aitken 3 / 4 May 2005 Department of Experimental Psychology University of Cambridge Slides at pobox.com/~rudolf/psychology

More information

Tutorial 5: Hypothesis Testing

Tutorial 5: Hypothesis Testing Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrc-lmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................

More information

Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

More information

Inference for two Population Means

Inference for two Population Means Inference for two Population Means Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison October 27 November 1, 2011 Two Population Means 1 / 65 Case Study Case Study Example

More information

T adult = 96 T child = 114.

T adult = 96 T child = 114. Homework Solutions Do all tests at the 5% level and quote p-values when possible. When answering each question uses sentences and include the relevant JMP output and plots (do not include the data in your

More information

Tests of relationships between variables Chi-square Test Binomial Test Run Test for Randomness One-Sample Kolmogorov-Smirnov Test.

Tests of relationships between variables Chi-square Test Binomial Test Run Test for Randomness One-Sample Kolmogorov-Smirnov Test. N. Uttam Singh, Aniruddha Roy & A. K. Tripathi ICAR Research Complex for NEH Region, Umiam, Meghalaya uttamba@gmail.com, aniruddhaubkv@gmail.com, aktripathi2020@yahoo.co.in Non Parametric Tests: Hands

More information

Invariance and optimality Linear rank statistics Permutation tests in R. Rank Tests. Patrick Breheny. October 7. STA 621: Nonparametric Statistics

Invariance and optimality Linear rank statistics Permutation tests in R. Rank Tests. Patrick Breheny. October 7. STA 621: Nonparametric Statistics Rank Tests October 7 Power Invariance and optimality Permutation testing allows great freedom to use a wide variety of test statistics, all of which lead to exact level-α tests regardless of the distribution

More information

Outline of Topics. Statistical Methods I. Types of Data. Descriptive Statistics

Outline of Topics. Statistical Methods I. Types of Data. Descriptive Statistics Statistical Methods I Tamekia L. Jones, Ph.D. (tjones@cog.ufl.edu) Research Assistant Professor Children s Oncology Group Statistics & Data Center Department of Biostatistics Colleges of Medicine and Public

More information

Nonparametric tests, Bootstrapping

Nonparametric tests, Bootstrapping Nonparametric tests, Bootstrapping http://www.isrec.isb-sib.ch/~darlene/embnet/ Hypothesis testing review 2 competing theories regarding a population parameter: NULL hypothesis H ( straw man ) ALTERNATIVEhypothesis

More information

1.5 Oneway Analysis of Variance

1.5 Oneway Analysis of Variance Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

More information

Nonparametric Statistics

Nonparametric Statistics 1 14.1 Using the Binomial Table Nonparametric Statistics In this chapter, we will survey several methods of inference from Nonparametric Statistics. These methods will introduce us to several new tables

More information

NCSS Statistical Software

NCSS Statistical Software Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

More information

StatCrunch and Nonparametric Statistics

StatCrunch and Nonparametric Statistics StatCrunch and Nonparametric Statistics You can use StatCrunch to calculate the values of nonparametric statistics. It may not be obvious how to enter the data in StatCrunch for various data sets that

More information

SAS/STAT. 9.2 User s Guide. Introduction to. Nonparametric Analysis. (Book Excerpt) SAS Documentation

SAS/STAT. 9.2 User s Guide. Introduction to. Nonparametric Analysis. (Book Excerpt) SAS Documentation SAS/STAT Introduction to 9.2 User s Guide Nonparametric Analysis (Book Excerpt) SAS Documentation This document is an individual chapter from SAS/STAT 9.2 User s Guide. The correct bibliographic citation

More information

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

More information

Nonparametric Two-Sample Tests. Nonparametric Tests. Sign Test

Nonparametric Two-Sample Tests. Nonparametric Tests. Sign Test Nonparametric Two-Sample Tests Sign test Mann-Whitney U-test (a.k.a. Wilcoxon two-sample test) Kolmogorov-Smirnov Test Wilcoxon Signed-Rank Test Tukey-Duckworth Test 1 Nonparametric Tests Recall, nonparametric

More information

Pearson s correlation

Pearson s correlation Pearson s correlation Introduction Often several quantitative variables are measured on each member of a sample. If we consider a pair of such variables, it is frequently of interest to establish if there

More information

Wilcoxon Rank Sum or Mann-Whitney Test Chapter 7.11

Wilcoxon Rank Sum or Mann-Whitney Test Chapter 7.11 STAT Non-Parametric tests /0/0 Here s a summary of the tests we will look at: Setting Normal test NonParametric Test One sample One-sample t-test Sign Test Wilcoxon signed-rank test Matched pairs Apply

More information

On Importance of Normality Assumption in Using a T-Test: One Sample and Two Sample Cases

On Importance of Normality Assumption in Using a T-Test: One Sample and Two Sample Cases On Importance of Normality Assumption in Using a T-Test: One Sample and Two Sample Cases Srilakshminarayana Gali, SDM Institute for Management Development, Mysore, India. E-mail: lakshminarayana@sdmimd.ac.in

More information

Robust nonparametric tests for the two-sample location problem SFB 823. Discussion Paper. Roland Fried, Herold Dehling

Robust nonparametric tests for the two-sample location problem SFB 823. Discussion Paper. Roland Fried, Herold Dehling SFB 823 Robust nonparametric tests for the two-sample location problem Discussion Paper Roland Fried, Herold Dehling Nr. 19/2011 Robust nonparametric tests for the two-sample location problem Roland Fried

More information

Lesson19: Comparing Predictive Accuracy of two Forecasts: Th. Diebold-Mariano Test

Lesson19: Comparing Predictive Accuracy of two Forecasts: Th. Diebold-Mariano Test Lesson19: Comparing Predictive Accuracy of two Forecasts: The Diebold-Mariano Test Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@univaq.it

More information

5. Linear Regression

5. Linear Regression 5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4

More information

NCSS Statistical Software

NCSS Statistical Software Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

More information

Inferential Statistics

Inferential Statistics Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

More information

Parametric and Nonparametric: Demystifying the Terms

Parametric and Nonparametric: Demystifying the Terms Parametric and Nonparametric: Demystifying the Terms By Tanya Hoskin, a statistician in the Mayo Clinic Department of Health Sciences Research who provides consultations through the Mayo Clinic CTSA BERD

More information

Median of the p-value Under the Alternative Hypothesis

Median of the p-value Under the Alternative Hypothesis Median of the p-value Under the Alternative Hypothesis Bhaskar Bhattacharya Department of Mathematics, Southern Illinois University, Carbondale, IL, USA Desale Habtzghi Department of Statistics, University

More information

NCSS Statistical Software. One-Sample T-Test

NCSS Statistical Software. One-Sample T-Test Chapter 205 Introduction This procedure provides several reports for making inference about a population mean based on a single sample. These reports include confidence intervals of the mean or median,

More information

7. Tests of association and Linear Regression

7. Tests of association and Linear Regression 7. Tests of association and Linear Regression In this chapter we consider 1. Tests of Association for 2 qualitative variables. 2. Measures of the strength of linear association between 2 quantitative variables.

More information

Lecture 13 More on hypothesis testing

Lecture 13 More on hypothesis testing Lecture 13 More on hypothesis testing Thais Paiva STA 111 - Summer 2013 Term II July 22, 2013 1 / 27 Thais Paiva STA 111 - Summer 2013 Term II Lecture 13, 07/22/2013 Lecture Plan 1 Type I and type II error

More information

Chapter 12 Nonparametric Tests. Chapter Table of Contents

Chapter 12 Nonparametric Tests. Chapter Table of Contents Chapter 12 Nonparametric Tests Chapter Table of Contents OVERVIEW...171 Testing for Normality...... 171 Comparing Distributions....171 ONE-SAMPLE TESTS...172 TWO-SAMPLE TESTS...172 ComparingTwoIndependentSamples...172

More information

AP STATISTICS 2012 SCORING GUIDELINES

AP STATISTICS 2012 SCORING GUIDELINES 2012 SCORING GUIDELINES Question 4 Intent of Question The primary goal of this question was to assess students ability to identify, set up, perform, and interpret the results of an appropriate hypothesis

More information

Session 1.6 Measures of Central Tendency

Session 1.6 Measures of Central Tendency Session 1.6 Measures of Central Tendency Measures of location (Indices of central tendency) These indices locate the center of the frequency distribution curve. The mode, median, and mean are three indices

More information

Hypothesis testing. c 2014, Jeffrey S. Simonoff 1

Hypothesis testing. c 2014, Jeffrey S. Simonoff 1 Hypothesis testing So far, we ve talked about inference from the point of estimation. We ve tried to answer questions like What is a good estimate for a typical value? or How much variability is there

More information

Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 2000: Page 1:

Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 2000: Page 1: Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 000: Page 1: NON-PARAMETRIC TESTS: What are non-parametric tests? Statistical tests fall into two kinds: parametric tests assume that

More information

Test of Hypotheses. Since the Neyman-Pearson approach involves two statistical hypotheses, one has to decide which one

Test of Hypotheses. Since the Neyman-Pearson approach involves two statistical hypotheses, one has to decide which one Test of Hypotheses Hypothesis, Test Statistic, and Rejection Region Imagine that you play a repeated Bernoulli game: you win $1 if head and lose $1 if tail. After 10 plays, you lost $2 in net (4 heads

More information

Part I. Simple Linear Regression

Part I. Simple Linear Regression Part I Simple Linear Regression 8 Chapter 1 Introduction This course is designed to provide a broad overview of many common procedures encountered when performing a regression analysis. Roughly, the first

More information

CONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE

CONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE 1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,

More information

Sample Size and Power in Clinical Trials

Sample Size and Power in Clinical Trials Sample Size and Power in Clinical Trials Version 1.0 May 011 1. Power of a Test. Factors affecting Power 3. Required Sample Size RELATED ISSUES 1. Effect Size. Test Statistics 3. Variation 4. Significance

More information

Statistics for Sports Medicine

Statistics for Sports Medicine Statistics for Sports Medicine Suzanne Hecht, MD University of Minnesota (suzanne.hecht@gmail.com) Fellow s Research Conference July 2012: Philadelphia GOALS Try not to bore you to death!! Try to teach

More information

BIOSTATISTICS QUIZ ANSWERS

BIOSTATISTICS QUIZ ANSWERS BIOSTATISTICS QUIZ ANSWERS 1. When you read scientific literature, do you know whether the statistical tests that were used were appropriate and why they were used? a. Always b. Mostly c. Rarely d. Never

More information

1 Nonparametric Statistics

1 Nonparametric Statistics 1 Nonparametric Statistics When finding confidence intervals or conducting tests so far, we always described the population with a model, which includes a set of parameters. Then we could make decisions

More information

Recall this chart that showed how most of our course would be organized:

Recall this chart that showed how most of our course would be organized: Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

More information

Combining Paired and Two-Sample Data Using a Permutation Test

Combining Paired and Two-Sample Data Using a Permutation Test Journal of Data Science 11(2013), 767-779 Combining Paired and Two-Sample Data Using a Permutation Test Richard L. Einsporn and Desale Habtzghi University of Akron Abstract: This paper presents a permutation

More information

Module 5: Multiple Regression Analysis

Module 5: Multiple Regression Analysis Using Statistical Data Using to Make Statistical Decisions: Data Multiple to Make Regression Decisions Analysis Page 1 Module 5: Multiple Regression Analysis Tom Ilvento, University of Delaware, College

More information

The Variability of P-Values. Summary

The Variability of P-Values. Summary The Variability of P-Values Dennis D. Boos Department of Statistics North Carolina State University Raleigh, NC 27695-8203 boos@stat.ncsu.edu August 15, 2009 NC State Statistics Departement Tech Report

More information

Ranked Set Sampling: an Approach to More Efficient Data Collection

Ranked Set Sampling: an Approach to More Efficient Data Collection Raned Set Sampling: an Approach to More Efficient Data Collection by Douglas A. Wolfe Department of Statistics Ohio State University 1 Abstract. This paper is intended to provide the reader with an introduction

More information

Bivariate Statistics Session 2: Measuring Associations Chi-Square Test

Bivariate Statistics Session 2: Measuring Associations Chi-Square Test Bivariate Statistics Session 2: Measuring Associations Chi-Square Test Features Of The Chi-Square Statistic The chi-square test is non-parametric. That is, it makes no assumptions about the distribution

More information

Non-Inferiority Tests for One Mean

Non-Inferiority Tests for One Mean Chapter 45 Non-Inferiority ests for One Mean Introduction his module computes power and sample size for non-inferiority tests in one-sample designs in which the outcome is distributed as a normal random

More information

II. DISTRIBUTIONS distribution normal distribution. standard scores

II. DISTRIBUTIONS distribution normal distribution. standard scores Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

More information

Hypothesis testing - Steps

Hypothesis testing - Steps Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =

More information

7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

More information

Contents 1. Contents

Contents 1. Contents Contents 1 Contents 3 K-sample Methods 2 3.1 Setup............................ 2 3.2 Classic Method Based on Normality Assumption..... 3 3.3 Permutation F -test.................... 5 3.4 Kruskal-Wallis

More information

, where f(x,y) is the joint pdf of X and Y. Therefore

, where f(x,y) is the joint pdf of X and Y. Therefore INDEPENDENCE, COVARIANCE AND CORRELATION M384G/374G Independence: For random variables and, the intuitive idea behind " is independent of " is that the distribution of shouldn't depend on what is. This

More information

Null Hypothesis H 0. The null hypothesis (denoted by H 0

Null Hypothesis H 0. The null hypothesis (denoted by H 0 Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property

More information

Analysis of Questionnaires and Qualitative Data Non-parametric Tests

Analysis of Questionnaires and Qualitative Data Non-parametric Tests Analysis of Questionnaires and Qualitative Data Non-parametric Tests JERZY STEFANOWSKI Instytut Informatyki Politechnika Poznańska Lecture SE 2013, Poznań Recalling Basics Measurment Scales Four scales

More information

Math 62 Statistics Sample Exam Questions

Math 62 Statistics Sample Exam Questions Math 62 Statistics Sample Exam Questions 1. (10) Explain the difference between the distribution of a population and the sampling distribution of a statistic, such as the mean, of a sample randomly selected

More information

Multivariate Analysis of Ecological Data

Multivariate Analysis of Ecological Data Multivariate Analysis of Ecological Data MICHAEL GREENACRE Professor of Statistics at the Pompeu Fabra University in Barcelona, Spain RAUL PRIMICERIO Associate Professor of Ecology, Evolutionary Biology

More information

t Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon

t Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon t-tests in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com www.excelmasterseries.com

More information

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

More information

Joint Exam 1/P Sample Exam 1

Joint Exam 1/P Sample Exam 1 Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question

More information

arxiv: v1 [stat.ot] 28 Jun 2011

arxiv: v1 [stat.ot] 28 Jun 2011 arxiv:116.5598v1 [stat.ot] 28 Jun 211 Some notes on biasedness and unbiasedness of two-sample Kolmogorov-Smirnov test P. Bubeliny e-mail: bubeliny@karlin.mff.cuni.cz Charles University, Faculty of Mathematics

More information

Sample Exam #1 Elementary Statistics

Sample Exam #1 Elementary Statistics Sample Exam #1 Elementary Statistics Instructions. No books, notes, or calculators are allowed. 1. Some variables that were recorded while studying diets of sharks are given below. Which of the variables

More information

Multivariate normal distribution and testing for means (see MKB Ch 3)

Multivariate normal distribution and testing for means (see MKB Ch 3) Multivariate normal distribution and testing for means (see MKB Ch 3) Where are we going? 2 One-sample t-test (univariate).................................................. 3 Two-sample t-test (univariate).................................................

More information

Permutation & Non-Parametric Tests

Permutation & Non-Parametric Tests Permutation & Non-Parametric Tests Statistical tests Gather data to assess some hypothesis (e.g., does this treatment have an effect on this outcome?) Form a test statistic for which large values indicate

More information