Agonistic Display in Betta splendens: Data Analysis I. Betta splendens Research: Parametric or Non-parametric Data?

Size: px
Start display at page:

Download "Agonistic Display in Betta splendens: Data Analysis I. Betta splendens Research: Parametric or Non-parametric Data?"

Transcription

1 Agonistic Display in Betta splendens: Data Analysis By Joanna Weremjiwicz, Simeon Yurek, and Dana Krempels Once you have collected data with your ethogram, you are ready to analyze that data to see whether it indicates rejection or failure to reject your null and alternative experimental hypotheses. This chapter will help you with statistical analysis. I. Betta splendens Research: Parametric or Non-parametric Data? There are essentially three different types of data you might have collected from your Betta splendens subjects. Duration of a particular behavior (seconds; discrete numerical data) o Example: How long were operculum flares in treatment and control groups o Note: Whole seconds are discrete numerical data. Do not use fractions of seconds. Counts/incidences of a particular behavior (integer; discrete numerical data) o Example: How many times did fish in treatment and control groups flare opercula? Time to start/end of a particular behavior (seconds; discrete numerical data) o Example: How long until the first (or n th ) operculum flare in treatment and control groups? o Note: Whole seconds are discrete numerical data. Do not use fractions of seconds. The type of data your team collected will determine the type of statistical analysis to employ. The flow chart shown in Figure 1 will help you decide which test is most appropriate for the data you have collected from your Betta splendens. II. Non-parametric statistics: Mann-Whitney and Kruskal-Wallis Tests If your team collected data that can be counted as integers (e.g., number of incidences of behavior; seconds), then you will use the Mann-Whitney U test if you have only two experimental groups to compare, or a Kruskal-Wallis Test if you have more than two experimental groups to compare. The Mann-Whitney U can be considered a non-parametric analog to the (parametric) t-test, whereas the Kruskal-Wallis Test can be considered a non-parametric analog to (parametric) ANOVA. Neither the Mann-Whitney nor Kruskal-Wallis tests require a normal distribution of data. However, they do require independent observations of data that can be ranked. The tests allow comparison of the rank positions of data points (rather than the values of the data points themselves) to one another, determining the degree of overlap between the groups being compared. A. Mann-Whitney U Test (a.k.a., Wilcoxon rank-sum test) The Mann-Whitney is used to compare data sets from two groups (e.g., treatment and control). It requires that the sample size for each group be less than 30. The test statistic generated by the Mann -Whitney test is known as the U statistic. (The U stands for unbiased, as in unbiased estimate.) (For a sample size of greater than 30, Data Analysis - 1

2 when data approximate a normal distribution, a Z statistic should be calculated, instead of a U statistic.) The experimental hypotheses for the Mann-Whitney test are, generically: HO Two samples have the same rank ordered positions of data. Example: Experimental Group A (e.g., control) has a similar number of fish exhibiting the same number of operculum flares as Experimental Group B (e.g., treatment). HA Two samples come from populations with different rank ordered positions of data. Example: Experimental Group A (e.g., control) has more fish exhibiting a greater number of operculum flares than Experimental Group B (e.g., treatment). The Mann-Whitney test is highly sensitive to the number of interchanges in rank between two experimental groups, but it has less power to detect departures from the null than some other tests. For a full explanation of the logic of the Mann-Whitney U test, go to (Thanks, Vassar Stats!) Figure 1. Flow chart for determining appropriate statistical test. Created by Tom de Jong and Frans Jacobs, Universiteit Leiden, Netherlands. ( Data Analysis - 2

3 Let s say your team has counted the number of operculum flares in two groups of fish being compared, one exposed to a living rival fish, and the other group exposed to a mirror image of itself. An example of raw data appears in Table 1a. The raw data are ranked in Table 1b. Table 1a. Number of operculum flares per minute in male Betta splendens exposed to live rival (Group A) versus mirror image (Group B). Replicate Group A Group B Table 1b. Ranked values for Betta splendens the number of operculum flares per minute. Values should be ranked from lowest to highest number. Each Group A value gets one point for every Group B value that appears below it. Each Group B value gets one point for every Group A value that appears below it. (For example, the first value, 0 for Group B, has six Group A values below it, so it gets 6 points.) In case of a tie: If two values are the same, then the rank of both values is the average of the two ranks. The points assigned each value are equal to the number of values below the tied value plus 0.5 point for the tied value. (See the table for an example.) Rank # of Fish Group Points 1 0 B B B B B A B A A A A A 0 The U statistic for each group is the sum of its points. For our example: U for Group A = = 1.5 U for Group B = = 34.5 If your team chose to record a discrete number of behaviors, then use the Mann- Whitney U to determine if the number of occurrences of the behavior between your two groups shows significant overlap (suggesting the groups are not very different, and you will fail to reject your null hypothesis) or not (suggesting that the groups are different, and you may find a P level consistent with rejecting your null hypothesis). Data Analysis - 3

4 Your final U statistic is the smaller of the two values you will calculate for your two experimental groups. In the above example, the lower value is 1. In general, the lower the value of the U statistic, the less overlap there is between the two groups being compared. In this example, there is only one overlapping value, suggesting that the two groups are very different. Use the table of Critical Values for the Mann-Whitney U (Table 2) to determine the probability value (P) that corresponds to your own team s U statistic. If your U value is smaller than that shown in the table, then there is less than 5% chance that the difference between your two experimental groups is due to chance alone. (If your U value is smaller than the one shown in this table for your two sample sizes, reject your null hypothesis.) If your U value is larger than that shown in the table, fail to reject your null hypothesis. Table 2. Critical values for the Mann-Whitney U statistic. Find the value that corresponds to the sample sizes of your two experimental groups. (From The Open Door Web Site, Data Analysis - 4

5 B. Kruskal-Wallis Test This test is appropriate if you have more than two experimental groups (For example, Control, Treatment A, and Treatment B). The sample size must be greater than five. The test statistic you will calculate is H (variance), and its distribution follows the same as that of the Chi square (Χ 2 ). The table of Chi Square critical values can be found at the end of this chapter. The generalized hypotheses for a Kruskal-Wallis test are H o : Positions of data from several populations do not significantly differ. (Example: Control, Treatment A, and Treatment B have a similar number of fish exhibiting the same number of operculum flares.) H A : Positions of data from several populations are significantly different. (Example: Control, Treatment A, and Treatment B have different number of fish exhibiting the same number of operculum flares.) As in the Mann-Whitney test, you will not be considering the actual raw values of your data points, but rather their ranks, relative to each other. As before, you wish to determine the degree to which different groups data overlap. The less overlap between groups, the more likely that the difference between them is real, indicating rejection of H O. Consider the imaginary data set shown in Table 3, which shows the duration of operculum flares in a fish subjected to three different experimental conditions. (NOTE: Subjecting a single experimental fish to all three conditions is more powerful than using different individuals, as it reduces variability inherent in using different individuals. However, one must consider the effects of previous treatments on an individual, and thus be sure not only to allow adequate recovery time between the three treatments, but also to randomize the order of the three treatments for experimental replications. The calculations shown below are for independent, not paired/grouped samples.) Table 3. Number of seconds of operculum flaring in a one-minute trial for three experimental groups. The rank of each value (ranking from lowest to highest value) appears in parentheses beside each value. Sums and averages of ranks appear in the bottom two rows. Replicate Treatment A Treatment B Control 1 20 (7) 40 (15) 1 (2) Treatment 2 26 (10) 32 (13) 4 (5.5) 3 30 (11) 47 (17) 3 (4) 4 21 (8) 29 (12) 7 (7) 5 4 (5.5) 41 (16) 2 (2) 6 24 (9) 35 (14) 0 (1) Sum of ranks (T) Mean of ranks (M) A, Treatment B, Control combined 50.5 (T A ) 87 (T B ) 21.5 (T C ) 159 (T all ) 8.4 (M A ) 14.5 (M B ) 3.5 (M C ) 9 (M all ) A measure of the combined degree to which group ranks differ is known as the sum of squared deviates (SS bg, in which bg stands for between groups ). The squared deviate for any particular group (SS grp, in which grp stands for the specific group, A, B, Data Analysis - 5

6 or C) can be calculated as the squared difference between the group s mean rank and the combined mean rank of all groups, multiplied by the sample size of that group: Thus, For Treatment A: SS A = 6(8.4 9) 2 = 2.16 for Treatment B: SS B = 6(14.5 9) 2 = and for Control: SS C = 6(3.5 9) 2 = SS grp = Σ [n grp (M grp M all ) 2 ] The sum of squared deviates, SS bg, is equal to the sum of the squared deviates for all groups. In our example: SS bg = = The logic of the Kruskal-Wallis test is fairly straightforward, and an excellent, easy-tounderstand explanation can be found here: (Thanks again, Vassar Stats!) For the purpose of expediency in this chapter, however, we will cut to the chase, and go straight to the calculation of the Kruskal-Wallis test statistic, H. This statistic represents a ratio with your observed sum of squared deviates as the numerator (in our example, it is 365.2), and the expected sum of squared deviates of a sampling distribution to which your sample belongs (this is represented as N(N+1)/12, in which N is equal to the number of counts. In our example, this would be 18, the sum of the number of replicates in all three experimental groups). H = SS bg [N(N-1)]/12 For our example: H = [18(18-1)]/12 = Conveniently, if each of the experimental groups has yielded at least five (5) observations, the sampling distribution of H is very similar to that of the Chi Square with degrees of freedom = (k 1), in which k is the number of experimental groups (in our example, k = 3). A table of critical values for the Chi square can be found in Table 4. In our example, df = 3-1 = 2. Our value of H (14.32) at 2 degrees of freedom is to the right of the largest value shown (10.597), which is associated with a P value of The probability that this lack of overlap is due to chance is very small (P < 0.005). Hence, we reject the null hypothesis. Data Analysis - 6

7 Armed with these examples, you should now be able to apply these statistical methods to your own data. Table 4. A partial table of the critical values for the Kruskall-Wallis or Chi Square. Data Analysis - 7

Data Analysis: Agonistic Display in Betta splendens I. Betta splendens Research: Parametric or Non-parametric Data?

Data Analysis: Agonistic Display in Betta splendens I. Betta splendens Research: Parametric or Non-parametric Data? Data Analysis: Agonistic Display in Betta splendens By Joanna Weremjiwicz, Simeon Yurek, and Dana Krempels Once you have collected data with your ethogram, you are ready to analyze that data to see whether

More information

CHAPTER 17 CHI-SQUARE AND OTHER NONPARAMETRIC TESTS FROM: PAGANO, R. R. (2007)

CHAPTER 17 CHI-SQUARE AND OTHER NONPARAMETRIC TESTS FROM: PAGANO, R. R. (2007) FROM: PAGANO, R. R. (007) I. INTRODUCTION: DISTINCTION BETWEEN PARAMETRIC AND NON-PARAMETRIC TESTS Statistical inference tests are often classified as to whether they are parametric or nonparametric Parameter

More information

PSY 307 Statistics for the Behavioral Sciences. Chapter 20 Tests for Ranked Data, Choosing Statistical Tests

PSY 307 Statistics for the Behavioral Sciences. Chapter 20 Tests for Ranked Data, Choosing Statistical Tests PSY 307 Statistics for the Behavioral Sciences Chapter 20 Tests for Ranked Data, Choosing Statistical Tests What To Do with Non-normal Distributions Tranformations (pg 382): The shape of the distribution

More information

Parametric versus Nonparametric Statistics-when to use them and which is more powerful? Dr Mahmoud Alhussami

Parametric versus Nonparametric Statistics-when to use them and which is more powerful? Dr Mahmoud Alhussami Parametric versus Nonparametric Statistics-when to use them and which is more powerful? Dr Mahmoud Alhussami Parametric Assumptions The observations must be independent. Dependent variable should be continuous

More information

4/6/16. Non-parametric Test. Overview. Stephen Opiyo. Distinguish Parametric and Nonparametric Test Procedures

4/6/16. Non-parametric Test. Overview. Stephen Opiyo. Distinguish Parametric and Nonparametric Test Procedures Non-parametric Test Stephen Opiyo Overview Distinguish Parametric and Nonparametric Test Procedures Explain commonly used Nonparametric Test Procedures Perform Hypothesis Tests Using Nonparametric Procedures

More information

Chapter 15: Nonparametric Statistics Section 15.1: An Overview of Nonparametric Statistics

Chapter 15: Nonparametric Statistics Section 15.1: An Overview of Nonparametric Statistics Section 15.1: An Overview of Nonparametric Statistics Understand Difference between Parametric and Nonparametric Statistical Procedures Parametric statistical procedures inferential procedures that rely

More information

Lecture Slides. Elementary Statistics. by Mario F. Triola. and the Triola Statistics Series

Lecture Slides. Elementary Statistics. by Mario F. Triola. and the Triola Statistics Series Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 13 Nonparametric Statistics 13-1 Overview 13-2 Sign Test 13-3 Wilcoxon Signed-Ranks

More information

Lecture Slides. Section 13-1 Overview. Elementary Statistics Tenth Edition. Chapter 13 Nonparametric Statistics. by Mario F.

Lecture Slides. Section 13-1 Overview. Elementary Statistics Tenth Edition. Chapter 13 Nonparametric Statistics. by Mario F. Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 13 Nonparametric Statistics 13-1 Overview 13-2 Sign Test 13-3 Wilcoxon Signed-Ranks

More information

CHI SQUARE ANALYSIS 8/18/2011 HYPOTHESIS TESTS SO FAR PARAMETRIC VS. NON-PARAMETRIC

CHI SQUARE ANALYSIS 8/18/2011 HYPOTHESIS TESTS SO FAR PARAMETRIC VS. NON-PARAMETRIC CHI SQUARE ANALYSIS I N T R O D U C T I O N T O N O N - P A R A M E T R I C A N A L Y S E S HYPOTHESIS TESTS SO FAR We ve discussed One-sample t-test Dependent Sample t-tests Independent Samples t-tests

More information

Statistical Analysis for QBIC Genetics Adapted by Ellen G. Dow 2017

Statistical Analysis for QBIC Genetics Adapted by Ellen G. Dow 2017 Statistical Analysis for QBIC Genetics Adapted by Ellen G. Dow 2017 I. χ 2 or chi-square test Objectives: Compare how close an experimentally derived value agrees with an expected value. One method to

More information

Nonparametric Statistics. Leah Wright, Tyler Ross, Taylor Brown

Nonparametric Statistics. Leah Wright, Tyler Ross, Taylor Brown Nonparametric Statistics Leah Wright, Tyler Ross, Taylor Brown Before we get to nonparametric statistics, what are parametric statistics? These statistics estimate and test population means, while holding

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

Analysis of variance (ANOVA) Comparing the means of more than two groups

Analysis of variance (ANOVA) Comparing the means of more than two groups Analysis of variance (ANOVA) Comparing the means of more than two groups Example: Cost of mating in male fruit flies Drosophila Treatments: place males with and without unmated (virgin) females Five treatments

More information

ST4241 Design and Analysis of Clinical Trials Lecture 7: N. Lecture 7: Non-parametric tests for PDG data

ST4241 Design and Analysis of Clinical Trials Lecture 7: N. Lecture 7: Non-parametric tests for PDG data ST4241 Design and Analysis of Clinical Trials Lecture 7: Non-parametric tests for PDG data Department of Statistics & Applied Probability 8:00-10:00 am, Friday, September 2, 2016 Outline Non-parametric

More information

Biostatistics 270 Kruskal-Wallis Test 1. Kruskal-Wallis Test

Biostatistics 270 Kruskal-Wallis Test 1. Kruskal-Wallis Test Biostatistics 270 Kruskal-Wallis Test 1 ORIGIN 1 Kruskal-Wallis Test The Kruskal-Wallis is a non-parametric analog to the One-Way ANOVA F-Test of means. It is useful when the k samples appear not to come

More information

Mitosis Data Analysis: Testing Statistical Hypotheses By Dana Krempels, Ph.D. and Steven Green, Ph.D.

Mitosis Data Analysis: Testing Statistical Hypotheses By Dana Krempels, Ph.D. and Steven Green, Ph.D. Mitosis Data Analysis: Testing Statistical Hypotheses By Dana Krempels, Ph.D. and Steven Green, Ph.D. The number of cells in various stages of mitosis in your treatment and control onions are your raw

More information

Non-parametric (Distribution-free) approaches p188 CN

Non-parametric (Distribution-free) approaches p188 CN Week 1: Introduction to some nonparametric and computer intensive (re-sampling) approaches: the sign test, Wilcoxon tests and multi-sample extensions, Spearman s rank correlation; the Bootstrap. (ch14

More information

STAT 135 Lab 9 Multiple Testing, One-Way ANOVA and Kruskal-Wallis

STAT 135 Lab 9 Multiple Testing, One-Way ANOVA and Kruskal-Wallis STAT 135 Lab 9 Multiple Testing, One-Way ANOVA and Kruskal-Wallis Rebecca Barter April 6, 2015 Multiple Testing Multiple Testing Recall that when we were doing two sample t-tests, we were testing the equality

More information

ANOVA - analysis of variance - used to compare the means of several populations.

ANOVA - analysis of variance - used to compare the means of several populations. 12.1 One-Way Analysis of Variance ANOVA - analysis of variance - used to compare the means of several populations. Assumptions for One-Way ANOVA: 1. Independent samples are taken using a randomized design.

More information

Glossary. The ISI glossary of statistical terms provides definitions in a number of different languages:

Glossary. The ISI glossary of statistical terms provides definitions in a number of different languages: Glossary The ISI glossary of statistical terms provides definitions in a number of different languages: http://isi.cbs.nl/glossary/index.htm Adjusted r 2 Adjusted R squared measures the proportion of the

More information

Data analysis and Geostatistics - lecture VII

Data analysis and Geostatistics - lecture VII Data analysis and Geostatistics - lecture VII t-tests, ANOVA and goodness-of-fit Statistical testing - significance of r Testing the significance of the correlation coefficient: t = r n - 2 1 - r 2 with

More information

MATH Notebook 3 Spring 2018

MATH Notebook 3 Spring 2018 MATH448001 Notebook 3 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2010 2018 by Jenny A. Baglivo. All Rights Reserved. 3 MATH448001 Notebook 3 3 3.1 One Way Layout........................................

More information

Lecture 14: ANOVA and the F-test

Lecture 14: ANOVA and the F-test Lecture 14: ANOVA and the F-test S. Massa, Department of Statistics, University of Oxford 3 February 2016 Example Consider a study of 983 individuals and examine the relationship between duration of breastfeeding

More information

Degrees of freedom df=1. Limitations OR in SPSS LIM: Knowing σ and µ is unlikely in large

Degrees of freedom df=1. Limitations OR in SPSS LIM: Knowing σ and µ is unlikely in large Z Test Comparing a group mean to a hypothesis T test (about 1 mean) T test (about 2 means) Comparing mean to sample mean. Similar means = will have same response to treatment Two unknown means are different

More information

Nonparametric statistic methods. Waraphon Phimpraphai DVM, PhD Department of Veterinary Public Health

Nonparametric statistic methods. Waraphon Phimpraphai DVM, PhD Department of Veterinary Public Health Nonparametric statistic methods Waraphon Phimpraphai DVM, PhD Department of Veterinary Public Health Measurement What are the 4 levels of measurement discussed? 1. Nominal or Classificatory Scale Gender,

More information

Non-parametric tests, part A:

Non-parametric tests, part A: Two types of statistical test: Non-parametric tests, part A: Parametric tests: Based on assumption that the data have certain characteristics or "parameters": Results are only valid if (a) the data are

More information

Unit 14: Nonparametric Statistical Methods

Unit 14: Nonparametric Statistical Methods Unit 14: Nonparametric Statistical Methods Statistics 571: Statistical Methods Ramón V. León 8/8/2003 Unit 14 - Stat 571 - Ramón V. León 1 Introductory Remarks Most methods studied so far have been based

More information

Data are sometimes not compatible with the assumptions of parametric statistical tests (i.e. t-test, regression, ANOVA)

Data are sometimes not compatible with the assumptions of parametric statistical tests (i.e. t-test, regression, ANOVA) BSTT523 Pagano & Gauvreau Chapter 13 1 Nonparametric Statistics Data are sometimes not compatible with the assumptions of parametric statistical tests (i.e. t-test, regression, ANOVA) In particular, data

More information

Glossary for the Triola Statistics Series

Glossary for the Triola Statistics Series Glossary for the Triola Statistics Series Absolute deviation The measure of variation equal to the sum of the deviations of each value from the mean, divided by the number of values Acceptance sampling

More information

Non-parametric Hypothesis Testing

Non-parametric Hypothesis Testing Non-parametric Hypothesis Testing Procedures Hypothesis Testing General Procedure for Hypothesis Tests 1. Identify the parameter of interest.. Formulate the null hypothesis, H 0. 3. Specify an appropriate

More information

SEVERAL μs AND MEDIANS: MORE ISSUES. Business Statistics

SEVERAL μs AND MEDIANS: MORE ISSUES. Business Statistics SEVERAL μs AND MEDIANS: MORE ISSUES Business Statistics CONTENTS Post-hoc analysis ANOVA for 2 groups The equal variances assumption The Kruskal-Wallis test Old exam question Further study POST-HOC ANALYSIS

More information

Statistics Handbook. All statistical tables were computed by the author.

Statistics Handbook. All statistical tables were computed by the author. Statistics Handbook Contents Page Wilcoxon rank-sum test (Mann-Whitney equivalent) Wilcoxon matched-pairs test 3 Normal Distribution 4 Z-test Related samples t-test 5 Unrelated samples t-test 6 Variance

More information

Basic Business Statistics, 10/e

Basic Business Statistics, 10/e Chapter 1 1-1 Basic Business Statistics 11 th Edition Chapter 1 Chi-Square Tests and Nonparametric Tests Basic Business Statistics, 11e 009 Prentice-Hall, Inc. Chap 1-1 Learning Objectives In this chapter,

More information

Nonparametric Statistics

Nonparametric Statistics Nonparametric Statistics Nonparametric or Distribution-free statistics: used when data are ordinal (i.e., rankings) used when ratio/interval data are not normally distributed (data are converted to ranks)

More information

HYPOTHESIS TESTING II TESTS ON MEANS. Sorana D. Bolboacă

HYPOTHESIS TESTING II TESTS ON MEANS. Sorana D. Bolboacă HYPOTHESIS TESTING II TESTS ON MEANS Sorana D. Bolboacă OBJECTIVES Significance value vs p value Parametric vs non parametric tests Tests on means: 1 Dec 14 2 SIGNIFICANCE LEVEL VS. p VALUE Materials and

More information

What is a Hypothesis?

What is a Hypothesis? What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population mean Example: The mean monthly cell phone bill in this city is μ = $42 population proportion Example:

More information

THE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE

THE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE THE ROYAL STATISTICAL SOCIETY 004 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER II STATISTICAL METHODS The Society provides these solutions to assist candidates preparing for the examinations in future

More information

DETAILED CONTENTS PART I INTRODUCTION AND DESCRIPTIVE STATISTICS. 1. Introduction to Statistics

DETAILED CONTENTS PART I INTRODUCTION AND DESCRIPTIVE STATISTICS. 1. Introduction to Statistics DETAILED CONTENTS About the Author Preface to the Instructor To the Student How to Use SPSS With This Book PART I INTRODUCTION AND DESCRIPTIVE STATISTICS 1. Introduction to Statistics 1.1 Descriptive and

More information

Module 9: Nonparametric Statistics Statistics (OA3102)

Module 9: Nonparametric Statistics Statistics (OA3102) Module 9: Nonparametric Statistics Statistics (OA3102) Professor Ron Fricker Naval Postgraduate School Monterey, California Reading assignment: WM&S chapter 15.1-15.6 Revision: 3-12 1 Goals for this Lecture

More information

HYPOTHESIS TESTING: THE CHI-SQUARE STATISTIC

HYPOTHESIS TESTING: THE CHI-SQUARE STATISTIC 1 HYPOTHESIS TESTING: THE CHI-SQUARE STATISTIC 7 steps of Hypothesis Testing 1. State the hypotheses 2. Identify level of significant 3. Identify the critical values 4. Calculate test statistics 5. Compare

More information

Analysis of variance (ANOVA) ANOVA. Null hypothesis for simple ANOVA. H 0 : Variance among groups = 0

Analysis of variance (ANOVA) ANOVA. Null hypothesis for simple ANOVA. H 0 : Variance among groups = 0 Analysis of variance (ANOVA) ANOVA Comparing the means of more than two groups Like a t-test, but can compare more than two groups Asks whether any of two or more means is different from any other. In

More information

Non-parametric Tests

Non-parametric Tests Statistics Column Shengping Yang PhD,Gilbert Berdine MD I was working on a small study recently to compare drug metabolite concentrations in the blood between two administration regimes. However, the metabolite

More information

Chapte The McGraw-Hill Companies, Inc. All rights reserved.

Chapte The McGraw-Hill Companies, Inc. All rights reserved. er15 Chapte Chi-Square Tests d Chi-Square Tests for -Fit Uniform Goodness- Poisson Goodness- Goodness- ECDF Tests (Optional) Contingency Tables A contingency table is a cross-tabulation of n paired observations

More information

S D / n t n 1 The paediatrician observes 3 =

S D / n t n 1 The paediatrician observes 3 = Non-parametric tests Paired t-test A paediatrician measured the blood cholesterol of her patients and was worried to note that some had levels over 00mg/100ml To investigate whether dietary regulation

More information

Lecture 7: Hypothesis Testing and ANOVA

Lecture 7: Hypothesis Testing and ANOVA Lecture 7: Hypothesis Testing and ANOVA Goals Overview of key elements of hypothesis testing Review of common one and two sample tests Introduction to ANOVA Hypothesis Testing The intent of hypothesis

More information

Kruskal-Wallis and Friedman type tests for. nested effects in hierarchical designs 1

Kruskal-Wallis and Friedman type tests for. nested effects in hierarchical designs 1 Kruskal-Wallis and Friedman type tests for nested effects in hierarchical designs 1 Assaf P. Oron and Peter D. Hoff Department of Statistics, University of Washington, Seattle assaf@u.washington.edu, hoff@stat.washington.edu

More information

Introduction to Nonparametric Statistics

Introduction to Nonparametric Statistics Introduction to Nonparametric Statistics by James Bernhard Spring 2012 Parameters Parametric method Nonparametric method µ[x 2 X 1 ] paired t-test Wilcoxon signed rank test µ[x 1 ], µ[x 2 ] 2-sample t-test

More information

An Analysis of College Algebra Exam Scores December 14, James D Jones Math Section 01

An Analysis of College Algebra Exam Scores December 14, James D Jones Math Section 01 An Analysis of College Algebra Exam s December, 000 James D Jones Math - Section 0 An Analysis of College Algebra Exam s Introduction Students often complain about a test being too difficult. Are there

More information

Introduction and Descriptive Statistics p. 1 Introduction to Statistics p. 3 Statistics, Science, and Observations p. 5 Populations and Samples p.

Introduction and Descriptive Statistics p. 1 Introduction to Statistics p. 3 Statistics, Science, and Observations p. 5 Populations and Samples p. Preface p. xi Introduction and Descriptive Statistics p. 1 Introduction to Statistics p. 3 Statistics, Science, and Observations p. 5 Populations and Samples p. 6 The Scientific Method and the Design of

More information

Department of Economics. Business Statistics. Chapter 12 Chi-square test of independence & Analysis of Variance ECON 509. Dr.

Department of Economics. Business Statistics. Chapter 12 Chi-square test of independence & Analysis of Variance ECON 509. Dr. Department of Economics Business Statistics Chapter 1 Chi-square test of independence & Analysis of Variance ECON 509 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should be able

More information

ST4241 Design and Analysis of Clinical Trials Lecture 9: N. Lecture 9: Non-parametric procedures for CRBD

ST4241 Design and Analysis of Clinical Trials Lecture 9: N. Lecture 9: Non-parametric procedures for CRBD ST21 Design and Analysis of Clinical Trials Lecture 9: Non-parametric procedures for CRBD Department of Statistics & Applied Probability 8:00-10:00 am, Friday, September 9, 2016 Outline Nonparametric tests

More information

The goodness-of-fit test Having discussed how to make comparisons between two proportions, we now consider comparisons of multiple proportions.

The goodness-of-fit test Having discussed how to make comparisons between two proportions, we now consider comparisons of multiple proportions. The goodness-of-fit test Having discussed how to make comparisons between two proportions, we now consider comparisons of multiple proportions. A common problem of this type is concerned with determining

More information

Statistics for Managers Using Microsoft Excel Chapter 10 ANOVA and Other C-Sample Tests With Numerical Data

Statistics for Managers Using Microsoft Excel Chapter 10 ANOVA and Other C-Sample Tests With Numerical Data Statistics for Managers Using Microsoft Excel Chapter 10 ANOVA and Other C-Sample Tests With Numerical Data 1999 Prentice-Hall, Inc. Chap. 10-1 Chapter Topics The Completely Randomized Model: One-Factor

More information

Statistical Significance of Ranking Paradoxes

Statistical Significance of Ranking Paradoxes Statistical Significance of Ranking Paradoxes Anna E. Bargagliotti and Raymond N. Greenwell 1 February 28, 2009 1 Anna E. Bargagliotti is an Assistant Professor in the Department of Mathematical Sciences

More information

Comparing the means of more than two groups

Comparing the means of more than two groups Comparing the means of more than two groups Chapter 15 Analysis of variance (ANOVA) Like a t-test, but can compare more than two groups Asks whether any of two or more means is different from any other.

More information

Statistical Inference Theory Lesson 46 Non-parametric Statistics

Statistical Inference Theory Lesson 46 Non-parametric Statistics 46.1-The Sign Test Statistical Inference Theory Lesson 46 Non-parametric Statistics 46.1 - Problem 1: (a). Let p equal the proportion of supermarkets that charge less than $2.15 a pound. H o : p 0.50 H

More information

T.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS

T.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS In our work on hypothesis testing, we used the value of a sample statistic to challenge an accepted value of a population parameter. We focused only

More information

Summary of Chapter 7 (Sections ) and Chapter 8 (Section 8.1)

Summary of Chapter 7 (Sections ) and Chapter 8 (Section 8.1) Summary of Chapter 7 (Sections 7.2-7.5) and Chapter 8 (Section 8.1) Chapter 7. Tests of Statistical Hypotheses 7.2. Tests about One Mean (1) Test about One Mean Case 1: σ is known. Assume that X N(µ, σ

More information

Assignment #7. Chapter 12: 18, 24 Chapter 13: 28. Due next Friday Nov. 20 th by 2pm in your TA s homework box

Assignment #7. Chapter 12: 18, 24 Chapter 13: 28. Due next Friday Nov. 20 th by 2pm in your TA s homework box Assignment #7 Chapter 12: 18, 24 Chapter 13: 28 Due next Friday Nov. 20 th by 2pm in your TA s homework box Lab Report Posted on web-site Dates Rough draft due to TAs homework box on Monday Nov. 16 th

More information

Intro to Parametric & Nonparametric Statistics

Intro to Parametric & Nonparametric Statistics Kinds of variable The classics & some others Intro to Parametric & Nonparametric Statistics Kinds of variables & why we care Kinds & definitions of nonparametric statistics Where parametric stats come

More information

STAT Section 5.8: Block Designs

STAT Section 5.8: Block Designs STAT 518 --- Section 5.8: Block Designs Recall that in paired-data studies, we match up pairs of subjects so that the two subjects in a pair are alike in some sense. Then we randomly assign, say, treatment

More information

One-way ANOVA. Experimental Design. One-way ANOVA

One-way ANOVA. Experimental Design. One-way ANOVA Method to compare more than two samples simultaneously without inflating Type I Error rate (α) Simplicity Few assumptions Adequate for highly complex hypothesis testing 09/30/12 1 Outline of this class

More information

Frequency table: Var2 (Spreadsheet1) Count Cumulative Percent Cumulative From To. Percent <x<=

Frequency table: Var2 (Spreadsheet1) Count Cumulative Percent Cumulative From To. Percent <x<= A frequency distribution is a kind of probability distribution. It gives the frequency or relative frequency at which given values have been observed among the data collected. For example, for age, Frequency

More information

Background to Statistics

Background to Statistics FACT SHEET Background to Statistics Introduction Statistics include a broad range of methods for manipulating, presenting and interpreting data. Professional scientists of all kinds need to be proficient

More information

Chapter 8 Student Lecture Notes 8-1. Department of Economics. Business Statistics. Chapter 12 Chi-square test of independence & Analysis of Variance

Chapter 8 Student Lecture Notes 8-1. Department of Economics. Business Statistics. Chapter 12 Chi-square test of independence & Analysis of Variance Chapter 8 Student Lecture Notes 8-1 Department of Economics Business Statistics Chapter 1 Chi-square test of independence & Analysis of Variance ECON 509 Dr. Mohammad Zainal Chapter Goals After completing

More information

STATISTIKA INDUSTRI 2 TIN 4004

STATISTIKA INDUSTRI 2 TIN 4004 STATISTIKA INDUSTRI 2 TIN 4004 Pertemuan 11 & 12 Outline: Nonparametric Statistics Referensi: Walpole, R.E., Myers, R.H., Myers, S.L., Ye, K., Probability & Statistics for Engineers & Scientists, 9 th

More information

Contents Kruskal-Wallis Test Friedman s Two-way Analysis of Variance by Ranks... 47

Contents Kruskal-Wallis Test Friedman s Two-way Analysis of Variance by Ranks... 47 Contents 1 Non-parametric Tests 3 1.1 Introduction....................................... 3 1.2 Advantages of Non-parametric Tests......................... 4 1.3 Disadvantages of Non-parametric Tests........................

More information

Chapter 12. Analysis of variance

Chapter 12. Analysis of variance Serik Sagitov, Chalmers and GU, January 9, 016 Chapter 1. Analysis of variance Chapter 11: I = samples independent samples paired samples Chapter 1: I 3 samples of equal size J one-way layout two-way layout

More information

We know from STAT.1030 that the relevant test statistic for equality of proportions is:

We know from STAT.1030 that the relevant test statistic for equality of proportions is: 2. Chi 2 -tests for equality of proportions Introduction: Two Samples Consider comparing the sample proportions p 1 and p 2 in independent random samples of size n 1 and n 2 out of two populations which

More information

SBAOD Statistical Methods & their Applications - II. Unit : I - V

SBAOD Statistical Methods & their Applications - II. Unit : I - V SBAOD Statistical Methods & their Applications - II Unit : I - V SBAOD Statistical Methods & their applications -II 2 Unit I - Syllabus Random Variable Mathematical Expectation Moments Moment generating

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

Chapter 7 Comparison of two independent samples

Chapter 7 Comparison of two independent samples Chapter 7 Comparison of two independent samples 7.1 Introduction Population 1 µ σ 1 1 N 1 Sample 1 y s 1 1 n 1 Population µ σ N Sample y s n 1, : population means 1, : population standard deviations N

More information

Inferential Statistics

Inferential Statistics Inferential Statistics Eva Riccomagno, Maria Piera Rogantin DIMA Università di Genova riccomagno@dima.unige.it rogantin@dima.unige.it Part G Distribution free hypothesis tests 1. Classical and distribution-free

More information

Statistics: revision

Statistics: revision NST 1B Experimental Psychology Statistics practical 5 Statistics: revision Rudolf Cardinal & Mike Aitken 29 / 30 April 2004 Department of Experimental Psychology University of Cambridge Handouts: Answers

More information

= 1 i. normal approximation to χ 2 df > df

= 1 i. normal approximation to χ 2 df > df χ tests 1) 1 categorical variable χ test for goodness-of-fit ) categorical variables χ test for independence (association, contingency) 3) categorical variables McNemar's test for change χ df k (O i 1

More information

Dr. Maddah ENMG 617 EM Statistics 10/12/12. Nonparametric Statistics (Chapter 16, Hines)

Dr. Maddah ENMG 617 EM Statistics 10/12/12. Nonparametric Statistics (Chapter 16, Hines) Dr. Maddah ENMG 617 EM Statistics 10/12/12 Nonparametric Statistics (Chapter 16, Hines) Introduction Most of the hypothesis testing presented so far assumes normally distributed data. These approaches

More information

My data doesn t look like that..

My data doesn t look like that.. Testing assumptions My data doesn t look like that.. We have made a big deal about testing model assumptions each week. Bill Pine Testing assumptions Testing assumptions We have made a big deal about testing

More information

GROUPED DATA E.G. FOR SAMPLE OF RAW DATA (E.G. 4, 12, 7, 5, MEAN G x / n STANDARD DEVIATION MEDIAN AND QUARTILES STANDARD DEVIATION

GROUPED DATA E.G. FOR SAMPLE OF RAW DATA (E.G. 4, 12, 7, 5, MEAN G x / n STANDARD DEVIATION MEDIAN AND QUARTILES STANDARD DEVIATION FOR SAMPLE OF RAW DATA (E.G. 4, 1, 7, 5, 11, 6, 9, 7, 11, 5, 4, 7) BE ABLE TO COMPUTE MEAN G / STANDARD DEVIATION MEDIAN AND QUARTILES Σ ( Σ) / 1 GROUPED DATA E.G. AGE FREQ. 0-9 53 10-19 4...... 80-89

More information

8.1-4 Test of Hypotheses Based on a Single Sample

8.1-4 Test of Hypotheses Based on a Single Sample 8.1-4 Test of Hypotheses Based on a Single Sample Example 1 (Example 8.6, p. 312) A manufacturer of sprinkler systems used for fire protection in office buildings claims that the true average system-activation

More information

4.1. Introduction: Comparing Means

4.1. Introduction: Comparing Means 4. Analysis of Variance (ANOVA) 4.1. Introduction: Comparing Means Consider the problem of testing H 0 : µ 1 = µ 2 against H 1 : µ 1 µ 2 in two independent samples of two different populations of possibly

More information

Lecture 41 Sections Mon, Apr 7, 2008

Lecture 41 Sections Mon, Apr 7, 2008 Lecture 41 Sections 14.1-14.3 Hampden-Sydney College Mon, Apr 7, 2008 Outline 1 2 3 4 5 one-proportion test that we just studied allows us to test a hypothesis concerning one proportion, or two categories,

More information

Analysis of Variance

Analysis of Variance Analysis of Variance Blood coagulation time T avg A 62 60 63 59 61 B 63 67 71 64 65 66 66 C 68 66 71 67 68 68 68 D 56 62 60 61 63 64 63 59 61 64 Blood coagulation time A B C D Combined 56 57 58 59 60 61

More information

Statistics for Managers Using Microsoft Excel Chapter 9 Two Sample Tests With Numerical Data

Statistics for Managers Using Microsoft Excel Chapter 9 Two Sample Tests With Numerical Data Statistics for Managers Using Microsoft Excel Chapter 9 Two Sample Tests With Numerical Data 999 Prentice-Hall, Inc. Chap. 9 - Chapter Topics Comparing Two Independent Samples: Z Test for the Difference

More information

Summary of Chapters 7-9

Summary of Chapters 7-9 Summary of Chapters 7-9 Chapter 7. Interval Estimation 7.2. Confidence Intervals for Difference of Two Means Let X 1,, X n and Y 1, Y 2,, Y m be two independent random samples of sizes n and m from two

More information

B.N.Bandodkar College of Science, Thane. Random-Number Generation. Mrs M.J.Gholba

B.N.Bandodkar College of Science, Thane. Random-Number Generation. Mrs M.J.Gholba B.N.Bandodkar College of Science, Thane Random-Number Generation Mrs M.J.Gholba Properties of Random Numbers A sequence of random numbers, R, R,., must have two important statistical properties, uniformity

More information

Exam details. Final Review Session. Things to Review

Exam details. Final Review Session. Things to Review Exam details Final Review Session Short answer, similar to book problems Formulae and tables will be given You CAN use a calculator Date and Time: Dec. 7, 006, 1-1:30 pm Location: Osborne Centre, Unit

More information

Lecture 28 Chi-Square Analysis

Lecture 28 Chi-Square Analysis Lecture 28 STAT 225 Introduction to Probability Models April 23, 2014 Whitney Huang Purdue University 28.1 χ 2 test for For a given contingency table, we want to test if two have a relationship or not

More information

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution

More information

QUANTITATIVE TECHNIQUES

QUANTITATIVE TECHNIQUES UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION (For B Com. IV Semester & BBA III Semester) COMPLEMENTARY COURSE QUANTITATIVE TECHNIQUES QUESTION BANK 1. The techniques which provide the decision maker

More information

Selection should be based on the desired biological interpretation!

Selection should be based on the desired biological interpretation! Statistical tools to compare levels of parasitism Jen_ Reiczigel,, Lajos Rózsa Hungary What to compare? The prevalence? The mean intensity? The median intensity? Or something else? And which statistical

More information

One-Way ANOVA. Some examples of when ANOVA would be appropriate include:

One-Way ANOVA. Some examples of when ANOVA would be appropriate include: One-Way ANOVA 1. Purpose Analysis of variance (ANOVA) is used when one wishes to determine whether two or more groups (e.g., classes A, B, and C) differ on some outcome of interest (e.g., an achievement

More information

This is particularly true if you see long tails in your data. What are you testing? That the two distributions are the same!

This is particularly true if you see long tails in your data. What are you testing? That the two distributions are the same! Two sample tests (part II): What to do if your data are not distributed normally: Option 1: if your sample size is large enough, don't worry - go ahead and use a t-test (the CLT will take care of non-normal

More information

Solutions to Final STAT 421, Fall 2008

Solutions to Final STAT 421, Fall 2008 Solutions to Final STAT 421, Fall 2008 Fritz Scholz 1. (8) Two treatments A and B were randomly assigned to 8 subjects (4 subjects to each treatment) with the following responses: 0, 1, 3, 6 and 5, 7,

More information

Introduction to hypothesis testing

Introduction to hypothesis testing Introduction to hypothesis testing Review: Logic of Hypothesis Tests Usually, we test (attempt to falsify) a null hypothesis (H 0 ): includes all possibilities except prediction in hypothesis (H A ) If

More information

Rank-Based Methods. Lukas Meier

Rank-Based Methods. Lukas Meier Rank-Based Methods Lukas Meier 20.01.2014 Introduction Up to now we basically always used a parametric family, like the normal distribution N (µ, σ 2 ) for modeling random data. Based on observed data

More information

2 and F Distributions. Barrow, Statistics for Economics, Accounting and Business Studies, 4 th edition Pearson Education Limited 2006

2 and F Distributions. Barrow, Statistics for Economics, Accounting and Business Studies, 4 th edition Pearson Education Limited 2006 and F Distributions Lecture 9 Distribution The distribution is used to: construct confidence intervals for a variance compare a set of actual frequencies with expected frequencies test for association

More information

Note: k = the # of conditions n = # of data points in a condition N = total # of data points

Note: k = the # of conditions n = # of data points in a condition N = total # of data points The ANOVA for2 Dependent Groups -- Analysis of 2-Within (or Matched)-Group Data with a Quantitative Response Variable Application: This statistic has two applications that can appear very different, but

More information

THE PAIR CHART I. Dana Quade. University of North Carolina. Institute of Statistics Mimeo Series No ~.:. July 1967

THE PAIR CHART I. Dana Quade. University of North Carolina. Institute of Statistics Mimeo Series No ~.:. July 1967 . _ e THE PAR CHART by Dana Quade University of North Carolina nstitute of Statistics Mimeo Series No. 537., ~.:. July 1967 Supported by U. S. Public Health Service Grant No. 3-Tl-ES-6l-0l. DEPARTMENT

More information

22s:152 Applied Linear Regression. Chapter 8: 1-Way Analysis of Variance (ANOVA) 2-Way Analysis of Variance (ANOVA)

22s:152 Applied Linear Regression. Chapter 8: 1-Way Analysis of Variance (ANOVA) 2-Way Analysis of Variance (ANOVA) 22s:152 Applied Linear Regression Chapter 8: 1-Way Analysis of Variance (ANOVA) 2-Way Analysis of Variance (ANOVA) We now consider an analysis with only categorical predictors (i.e. all predictors are

More information

STAT 135 Lab 8 Hypothesis Testing Review, Mann-Whitney Test by Normal Approximation, and Wilcoxon Signed Rank Test.

STAT 135 Lab 8 Hypothesis Testing Review, Mann-Whitney Test by Normal Approximation, and Wilcoxon Signed Rank Test. STAT 135 Lab 8 Hypothesis Testing Review, Mann-Whitney Test by Normal Approximation, and Wilcoxon Signed Rank Test. Rebecca Barter March 30, 2015 Mann-Whitney Test Mann-Whitney Test Recall that the Mann-Whitney

More information