1 NONPARAMETRIC STATISTICS (Adapted from J. Hurley Notes
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12.6 Sign Test (Web)
12.4 Sign test (Web) Click on 'Bookmarks' in the Left-Hand menu and then click on the required section. 12.6 Sign test (Web) 12.6.1 Introduction The Sign Test is a one sample test that compares the median of a data set with a specific target value. The Sign Test performs the same function as the One Sample Wilcoxon Test, but it does not rank data values. Without the information provided by ranking, the Sign Test is less powerful (9.4.5) than the Wilcoxon Test. However, the Sign Test is useful for problems involving 'binomial' data, where observed data values are either above or below a target value. 12.6.2 Sign Test The Sign Test aims to test whether the observed data sample has been drawn from a population that has a median value, m, that is significantly different from (or greater/less than) a specific value, mO. The hypotheses are the same as in 12.1.2. We will illustrate the use of the Sign Test in Example 12.14 by using the same data as in Example 12.2. Example 12.14 The generation times, t, (5.2.4) of ten cultures of the same micro-organisms were recorded. Time, t (hr) 6.3 4.8 7.2 5.0 6.3 4.2 8.9 4.4 5.6 9.3 The microbiologist wishes to test whether the generation time for this micro- organism is significantly greater than a specific value of 5.0 hours. See following text for calculations: In performing the Sign Test, each data value, ti, is compared with the target value, mO, using the following conditions: ti < mO replace the data value with '-' ti = mO exclude the data value ti > mO replace the data value with '+' giving the results in Table 12.14 Difference + - + ----- + - + - + + Table 12.14 Signs of Differences in Example 12.14 12.4.p1 12.4 Sign test (Web) We now calculate the values of r + = number of '+' values: r + = 6 in Example 12.14 n = total number of data values not excluded: n = 9 in Example 12.14 Decision making in this test is based on the probability of observing particular values of r + for a given value of n. -
Nonparametric Statistics for Social and Behavioral Sciences
Statistics Incorporating a hands-on approach, Nonparametric Statistics for Social SOCIAL AND BEHAVIORAL SCIENCES and Behavioral Sciences presents the concepts, principles, and methods NONPARAMETRIC STATISTICS FOR used in performing many nonparametric procedures. It also demonstrates practical applications of the most common nonparametric procedures using IBM’s SPSS software. Emphasizing sound research designs, appropriate statistical analyses, and NONPARAMETRIC accurate interpretations of results, the text: • Explains a conceptual framework for each statistical procedure STATISTICS FOR • Presents examples of relevant research problems, associated research questions, and hypotheses that precede each procedure • Details SPSS paths for conducting various analyses SOCIAL AND • Discusses the interpretations of statistical results and conclusions of the research BEHAVIORAL With minimal coverage of formulas, the book takes a nonmathematical ap- proach to nonparametric data analysis procedures and shows you how they are used in research contexts. Each chapter includes examples, exercises, SCIENCES and SPSS screen shots illustrating steps of the statistical procedures and re- sulting output. About the Author Dr. M. Kraska-Miller is a Mildred Cheshire Fraley Distinguished Professor of Research and Statistics in the Department of Educational Foundations, Leadership, and Technology at Auburn University, where she is also the Interim Director of Research for the Center for Disability Research and Service. Dr. Kraska-Miller is the author of four books on teaching and communications. She has published numerous articles in national and international refereed journals. Her research interests include statistical modeling and applications KRASKA-MILLER of statistics to theoretical concepts, such as motivation; satisfaction in jobs, services, income, and other areas; and needs assessments particularly applicable to special populations. -
Nonparametric Estimation by Convex Programming
The Annals of Statistics 2009, Vol. 37, No. 5A, 2278–2300 DOI: 10.1214/08-AOS654 c Institute of Mathematical Statistics, 2009 NONPARAMETRIC ESTIMATION BY CONVEX PROGRAMMING By Anatoli B. Juditsky and Arkadi S. Nemirovski1 Universit´eGrenoble I and Georgia Institute of Technology The problem we concentrate on is as follows: given (1) a convex compact set X in Rn, an affine mapping x 7→ A(x), a parametric family {pµ(·)} of probability densities and (2) N i.i.d. observations of the random variable ω, distributed with the density pA(x)(·) for some (unknown) x ∈ X, estimate the value gT x of a given linear form at x. For several families {pµ(·)} with no additional assumptions on X and A, we develop computationally efficient estimation routines which are minimax optimal, within an absolute constant factor. We then apply these routines to recovering x itself in the Euclidean norm. 1. Introduction. The problem we are interested in is essentially as fol- lows: suppose that we are given a convex compact set X in Rn, an affine mapping x A(x) and a parametric family pµ( ) of probability densities. Suppose that7→N i.i.d. observations of the random{ variable· } ω, distributed with the density p ( ) for some (unknown) x X, are available. Our objective A(x) · ∈ is to estimate the value gT x of a given linear form at x. In nonparametric statistics, there exists an immense literature on various versions of this problem (see, e.g., [10, 11, 12, 13, 15, 17, 18, 21, 22, 23, 24, 25, 26, 27, 28] and the references therein). -
Repeated-Measures ANOVA & Friedman Test Using STATCAL
Repeated-Measures ANOVA & Friedman Test Using STATCAL (R) & SPSS Prana Ugiana Gio Download STATCAL in www.statcal.com i CONTENT 1.1 Example of Case 1.2 Explanation of Some Book About Repeated-Measures ANOVA 1.3 Repeated-Measures ANOVA & Friedman Test 1.4 Normality Assumption and Assumption of Equality of Variances (Sphericity) 1.5 Descriptive Statistics Based On SPSS dan STATCAL (R) 1.6 Normality Assumption Test Using Kolmogorov-Smirnov Test Based on SPSS & STATCAL (R) 1.7 Assumption Test of Equality of Variances Using Mauchly Test Based on SPSS & STATCAL (R) 1.8 Repeated-Measures ANOVA Based on SPSS & STATCAL (R) 1.9 Multiple Comparison Test Using Boferroni Test Based on SPSS & STATCAL (R) 1.10 Friedman Test Based on SPSS & STATCAL (R) 1.11 Multiple Comparison Test Using Wilcoxon Test Based on SPSS & STATCAL (R) ii 1.1 Example of Case For example given data of weight of 11 persons before and after consuming medicine of diet for one week, two weeks, three weeks and four week (Table 1.1.1). Tabel 1.1.1 Data of Weight of 11 Persons Weight Name Before One Week Two Weeks Three Weeks Four Weeks A 89.43 85.54 80.45 78.65 75.45 B 85.33 82.34 79.43 76.55 71.35 C 90.86 87.54 85.45 80.54 76.53 D 91.53 87.43 83.43 80.44 77.64 E 90.43 84.45 81.34 78.64 75.43 F 90.52 86.54 85.47 81.44 78.64 G 87.44 83.34 80.54 78.43 77.43 H 89.53 86.45 84.54 81.35 78.43 I 91.34 88.78 85.47 82.43 78.76 J 88.64 84.36 80.66 78.65 77.43 K 89.51 85.68 82.68 79.71 76.5 Average 89.51 85.68 82.68 79.71 76.69 Based on Table 1.1.1: The person whose name is A has initial weight 89,43, after consuming medicine of diet for one week 85,54, two weeks 80,45, three weeks 78,65 and four weeks 75,45. -
1 Sample Sign Test 1
Non-Parametric Univariate Tests: 1 Sample Sign Test 1 1 SAMPLE SIGN TEST A non-parametric equivalent of the 1 SAMPLE T-TEST. ASSUMPTIONS: Data is non-normally distributed, even after log transforming. This test makes no assumption about the shape of the population distribution, therefore this test can handle a data set that is non-symmetric, that is skewed either to the left or the right. THE POINT: The SIGN TEST simply computes a significance test of a hypothesized median value for a single data set. Like the 1 SAMPLE T-TEST you can choose whether you want to use a one-tailed or two-tailed distribution based on your hypothesis; basically, do you want to test whether the median value of the data set is equal to some hypothesized value (H0: η = ηo), or do you want to test whether it is greater (or lesser) than that value (H0: η > or < ηo). Likewise, you can choose to set confidence intervals around η at some α level and see whether ηo falls within the range. This is equivalent to the significance test, just as in any T-TEST. In English, the kind of question you would be interested in asking is whether some observation is likely to belong to some data set you already have defined. That is to say, whether the observation of interest is a typical value of that data set. Formally, you are either testing to see whether the observation is a good estimate of the median, or whether it falls within confidence levels set around that median. -
Signed-Rank Tests for ARMA Models
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MULTIVARIATE ANALYSIS 39, l-29 ( 1991) Time Series Analysis via Rank Order Theory: Signed-Rank Tests for ARMA Models MARC HALLIN* Universith Libre de Bruxelles, Brussels, Belgium AND MADAN L. PURI+ Indiana University Communicated by C. R. Rao An asymptotic distribution theory is developed for a general class of signed-rank serial statistics, and is then used to derive asymptotically locally optimal tests (in the maximin sense) for testing an ARMA model against other ARMA models. Special cases yield Fisher-Yates, van der Waerden, and Wilcoxon type tests. The asymptotic relative efficiencies of the proposed procedures with respect to each other, and with respect to their normal theory counterparts, are provided. Cl 1991 Academic Press. Inc. 1. INTRODUCTION 1 .l. Invariance, Ranks and Signed-Ranks Invariance arguments constitute the theoretical cornerstone of rank- based methods: whenever weaker unbiasedness or similarity properties can be considered satisfying or adequate, permutation tests, which generally follow from the usual Neyman structure arguments, are theoretically preferable to rank tests, since they are less restrictive and thus allow for more powerful results (though, of course, their practical implementation may be more problematic). Which type of ranks (signed or unsigned) should be adopted-if Received May 3, 1989; revised October 23, 1990. AMS 1980 subject cIassifications: 62Gl0, 62MlO. Key words and phrases: signed ranks, serial rank statistics, ARMA models, asymptotic relative effhziency, locally asymptotically optimal tests. *Research supported by the the Fonds National de la Recherche Scientifique and the Minist&e de la Communaute Franpaise de Belgique. -
7 Nonparametric Methods
7 NONPARAMETRIC METHODS 7 Nonparametric Methods SW Section 7.11 and 9.4-9.5 Nonparametric methods do not require the normality assumption of classical techniques. I will describe and illustrate selected non-parametric methods, and compare them with classical methods. Some motivation and discussion of the strengths and weaknesses of non-parametric methods is given. The Sign Test and CI for a Population Median The sign test assumes that you have a random sample from a population, but makes no assumption about the population shape. The standard t−test provides inferences on a population mean. The sign test, in contrast, provides inferences about a population median. If the population frequency curve is symmetric (see below), then the population median, iden- tified by η, and the population mean µ are identical. In this case the sign procedures provide inferences for the population mean. The idea behind the sign test is straightforward. Suppose you have a sample of size m from the population, and you wish to test H0 : η = η0 (a given value). Let S be the number of sampled observations above η0. If H0 is true, you expect S to be approximately one-half the sample size, .5m. If S is much greater than .5m, the data suggests that η > η0. If S is much less than .5m, the data suggests that η < η0. Mean and Median differ with skewed distributions Mean and Median are the same with symmetric distributions 50% Median = η Mean = µ Mean = Median S has a Binomial distribution when H0 is true. The Binomial distribution is used to construct a test with size α (approximately). -
6 Single Sample Methods for a Location Parameter
6 Single Sample Methods for a Location Parameter If there are serious departures from parametric test assumptions (e.g., normality or symmetry), nonparametric tests on a measure of central tendency (usually the median) are used. Recall: M is a median of a random variable X if P (X M) = P (X M) = :5. • ≤ ≥ The distribution of X is symmetric about c if P (X c x) = P (X c + x) for all x. • ≤ − ≥ For symmetric continuous distributions, the median M = the mean µ. Thus, all conclusions • about the median can also be applied to the mean. If X be a binomial random variable with parameters n and p (denoted X B(n; p)) then • n ∼ P (X = x) = px(1 p)n−x for x = 0; 1; : : : ; n x − n n! where = and k! = k(k 1)(k 2) 2 1: • x x!(n x)! − − ··· · − Tables exist for the cdf P (X x) for various choices of n and p. The probabilities and • cdf values are also easy to produce≤ using SAS or R. Thus, if X B(n; :5), we have • ∼ n P (X = x) = (:5)n x x n P (X x) = (:5)n ≤ k k X=0 P (X x) = P (X n x) because the B(n; :5) distribution is symmetric. ≤ ≥ − For sample sizes n > 20 and p = :5, a normal approximation (with continuity correction) • to the binomial probabilities is often used instead of binomial tables. (x :5) :5n { Calculate z = ± − : Use x+:5 when x < :5n and use x :5 when x > :5n. :5pn − { The value of z is compared to N(0; 1), the standard normal distribution. -
Tests of Hypotheses Using Statistics
Tests of Hypotheses Using Statistics Adam Massey¤and Steven J. Millery Mathematics Department Brown University Providence, RI 02912 Abstract We present the various methods of hypothesis testing that one typically encounters in a mathematical statistics course. The focus will be on conditions for using each test, the hypothesis tested by each test, and the appropriate (and inappropriate) ways of using each test. We conclude by summarizing the di®erent tests (what conditions must be met to use them, what the test statistic is, and what the critical region is). Contents 1 Types of Hypotheses and Test Statistics 2 1.1 Introduction . 2 1.2 Types of Hypotheses . 3 1.3 Types of Statistics . 3 2 z-Tests and t-Tests 5 2.1 Testing Means I: Large Sample Size or Known Variance . 5 2.2 Testing Means II: Small Sample Size and Unknown Variance . 9 3 Testing the Variance 12 4 Testing Proportions 13 4.1 Testing Proportions I: One Proportion . 13 4.2 Testing Proportions II: K Proportions . 15 4.3 Testing r £ c Contingency Tables . 17 4.4 Incomplete r £ c Contingency Tables Tables . 18 5 Normal Regression Analysis 19 6 Non-parametric Tests 21 6.1 Tests of Signs . 21 6.2 Tests of Ranked Signs . 22 6.3 Tests Based on Runs . 23 ¤E-mail: [email protected] yE-mail: [email protected] 1 7 Summary 26 7.1 z-tests . 26 7.2 t-tests . 27 7.3 Tests comparing means . 27 7.4 Variance Test . 28 7.5 Proportions . 28 7.6 Contingency Tables . -
9 Blocked Designs
9 Blocked Designs 9.1 Friedman's Test 9.1.1 Application 1: Randomized Complete Block Designs • Assume there are k treatments of interest in an experiment. In Section 8, we considered the k-sample Extension of the Median Test and the Kruskal-Wallis Test to test for any differences in the k treatment medians. • Suppose the experimenter is still concerned with studying the effects of a single factor on a response of interest, but variability from another factor that is not of interest is expected. { Suppose a researcher wants to study the effect of 4 fertilizers on the yield of cotton. The researcher also knows that the soil conditions at the 8 areas for performing an experiment are highly variable. Thus, the researcher wants to design an experiment to detect any differences among the 4 fertilizers on the cotton yield in the presence a \nuisance variable" not of interest (the 8 areas). • Because experimental units can vary greatly with respect to physical characteristics that can also influence the response, the responses from experimental units that receive the same treatment can also vary greatly. • If it is not controlled or accounted for in the data analysis, it can can greatly inflate the experimental variability making it difficult to detect real differences among the k treatments of interest (large Type II error). • If this source of variability can be separated from the treatment effects and the random experimental error, then the sensitivity of the experiment to detect real differences between treatments in increased (i.e., lower the Type II error). • Therefore, the goal is to choose an experimental design in which it is possible to control the effects of a variable not of interest by bringing experimental units that are similar into a group called a \block". -
1 Exercise 8: an Introduction to Descriptive and Nonparametric
1 Exercise 8: An Introduction to Descriptive and Nonparametric Statistics Elizabeth M. Jakob and Marta J. Hersek Goals of this lab 1. To understand why statistics are used 2. To become familiar with descriptive statistics 1. To become familiar with nonparametric statistical tests, how to conduct them, and how to choose among them 2. To apply this knowledge to sample research questions Background If we are very fortunate, our experiments yield perfect data: all the animals in one treatment group behave one way, and all the animals in another treatment group behave another way. Usually our results are not so clear. In addition, even if we do get results that seem definitive, there is always a possibility that they are a result of chance. Statistics enable us to objectively evaluate our results. Descriptive statistics are useful for exploring, summarizing, and presenting data. Inferential statistics are used for interpreting data and drawing conclusions about our hypotheses. Descriptive statistics include the mean (average of all of the observations; see Table 8.1), mode (most frequent data class), and median (middle value in an ordered set of data). The variance, standard deviation, and standard error are measures of deviation from the mean (see Table 8.1). These statistics can be used to explore your data before going on to inferential statistics, when appropriate. In hypothesis testing, a variety of statistical tests can be used to determine if the data best fit our null hypothesis (a statement of no difference) or an alternative hypothesis. More specifically, we attempt to reject one of these hypotheses. -
Nonparametric Analogues of Analysis of Variance
*\ NONPARAMETRIC ANALOGUES OF ANALYSIS OF VARIANCE by °\\4 RONALD K. LOHRDING B. A., Southwestern College, 1963 A MASTER'S REPORT submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE Department of Statistics and Statistical Laboratory KANSAS STATE UNIVERSITY Manhattan, Kansas 1966 Approved by: ' UKC )x 0-i Major PrProfessor L IVoJL CONTENTS 1. INTRODUCTION 1 2. COMPLETELY RANDOMIZED DESIGN 4 2. 1. The Kruskal-Wallis One-way Analysis of Variance by Ranks 4 2. 2. Mosteller's k-Sample Slippage Test 7 2.3. Conover's k-Sample Slippage Test 8 2.4. A k-Sample Kolmogorov-Smirnov Test 11 2 2. 5. The Contingency x Test 13 3. RANDOMIZED COMPLETE BLOCK 14 3. 1. Cochran Q Test 15 3.2. Friedman's Two-way Analysis of Variance by Ranks . 16 4. PROCEDURES WHICH GENERALIZE TO SEVERAL DESIGNS . 17 4.1. Mood's Median Test 18 4.2. Wilson's Median Test 21 4.3. The Randomized Rank-Sum Test 27 4.4. Rank Test for Paired-Comparison Experiments ... 30 5. BALANCED INCOMPLETE BLOCK 33 5. 1. Bradley's Rank Analysis of Incomplete Block Designs 33 5.2. Durbin's Rank Analysis of Incomplete Block Designs 40 6. A 2x2 FACTORIAL EXPERIMENT 44 7. PARTIALLY BALANCED INCOMPLETE BLOCK 47 OTHER RELATED TESTS 50 ACKNOWLEDGEMENTS 53 REFERENCES 54 NONPARAMETRIC ANALOGUES OF ANALYSIS OF VARIANCE 1. Introduction J. V. Bradley (I960) proposes that the history of statistics can be divided into four major stages. The first or one parameter stage was when statistics were merely thought of as averages or in some instances ratios.