Experimental & Behavioral Economics Lecture 6: Nonparametric tests and selection of sample size


 Edwin Shelton
 1 years ago
 Views:
Transcription
1 Experimental & Behavioral Economics Lecture 6: Nonparametric tests and selection of sample size Based on Siegel, Sidney, and N. J. Castellan (1988) Nonparametric statistics for the behavioral sciences, McGrawHill, New York, and teaching material by John Duffy (University of Pittsburgh) Bernd Rönz (HU Berlin) David Danz Summer term
2 Contents 1. Introduction (recap hypothesis testing) 2. Common (nonparametric) tests in experimental economics 3. Selection of sample size (power analysis) 2
3 Hypothesis testing The research hypothesis is the prediction derived from the theory under test. Null hypothesis (H 0 ) is an hypothesis of no effect (e.g., μ 1 = μ 2 ) usually formulated for the purpose of being rejected If rejected, the alternative hypothesis (H 1 ) is supported (not necessarily true) Alternative hypothesis (H 1 ) is the operational statement of the experimenter's research hypothesis. nature of the research hypothesis determines how H 1 should be stated (e.g., μ 1 μ 2, or μ 1 < μ 2, or μ 1 > μ 2 ) 3
4 Hypothesis testing The region of rejection is a region of the sampling distribution under H0 includes all possible values that a test statistic can take on. consists of a set of possible values which are so extreme that when H0 is true the probability of observing them is very small (α) Distribution of some test statistic under H0 4
5 NONPARAMETRIC TESTS 5
6 Nonparametric tests + If the sample size is very small, there may be no alternative to using a nonparametric statistical test (unless the nature of the population distribution is known exactly) Make usually fewer assumptions about the data Interpretation of nonparametric statistical tests is often more straightforward than the interpretation of parametric tests (easier to learn and to apply than are parametric tests) If assumptions of a parametric statistical model are met in the data, then parametric statistical tests are usually more efficient (lower powerefficiency with nonparametric tests) parametric statistical tests have been systematized: different tests are simply variations on a central theme (nonparametric tests less systematic) 6
7 Nonparametric tests Two independent samples (e.g., betweensubject design: same measure for each subject in two treatments) Fisher s Exact Test / ChiSquare Test of independence Median test WilcoxonMannWhitney Test / Robust Rank Order Test KolmogorovSmirnov Test Two dependent samples (e.g., withinsubject design: two measures or repeated measure for each subject) McNemar test Sign test / Wilcoxon Signed Ranks Tests 7
8 Scales 1. Nominal (or categorical) scale numbers or other symbols are used to classify an object, person, or characteristic (i.e., to identify the groups to which various objects belong) Example: Gender 2. Ordinal (or ranking) scale (1) + objects in one category of a scale stand in some kind of relation >R to objects in other categories ( higher, more preferred, more difficult, etc.) Example: Socioeconomic status, grades 3. Interval Scale (2) + distances or differences between any two numbers on the scale have can be interpreted in a meaningful way Example: Temperature 4. Ratio Scale (3) + has a true zero point as its origin, thus the ratio of any two scale points is independent of the unit of measurement Example: Weight, age 8
9 NONPARAMETRIC TESTS INDEPENDENT SAMPLES 9
10 Two independent samples Binary variables (nominal or ordinal) Fisher s Exact Test Two independent samples Binary variables 10
11 Two independent samples Binary variables (nominal or ordinal) Fisher s Exact Test H 0 : No relation between the variables (independence) under H0, the conditional probability of observing success for one variable is independent of the realization of the other variable. i.e., Pr(+ I) = Pr(+ II) = Pr(+) 11
12 Two independent samples Binary variables (nominal or ordinal) Fisher s Exact Test Hypergeometric distribution describes the probability of k successes in n draws without replacement from a finite population of size N containing exactly K successes. In our contingency table: K = (A+C) N = (A + B + C + D) k = A n = (A+B) 12
13 Two independent samples Binary variables (nominal or ordinal) Fisher s Exact Test Idea: Regard marginal totals as fixed A finite population of size N has (A+C) elements of group I and (B+D) elements of group II We draw a random sample of size (A+B) without replacement V is a random variable = number of observations sampled from group I In our sample, the realization of V is V = A Under H 0, the probability that V takes on the value A is given by the hypergeometric distribution 13
14 Two independent samples Binary variables (nominal or ordinal) Fisher s Exact Test With marginal totals being fixed, we can write down all possible contingency tables possible tables will be completely determined by alternative values for A (V) Pvalue is the probability (under H0) of sampling the observed or a more extreme contingency table Let A be the observed frequency in the cell where the row and column containing the smallest and second smallest marginal frequencies intersect. Observed or more extreme contingency tables (twosided): D D observed = A/(A+C) B/(B+D) Reject H0, if Pr( D D observed ) < α 14
15 Two independent samples Binary variables (nominal or ordinal) Fisher s Exact Test Example II = observed: D =
16 Two independent samples Binary variables (nominal or ordinal) Fisher s Exact Test Example II = observed: D = Pr( D D observed ) = 16
17 Two independent samples Nominal or ordinal scaling Chisquare test of independence Two variables, independent observations Generalization of Fisher s exact test to more than two discrete categories Expected frequencies in each discrete category should not be too small expected frequencies of each cell must exceed 1 at most 20% of the cells with expected frequencies less than 5 17
18 Two independent samples Nominal or ordinal scaling Chisquare test of independence H 0 : The variables are statistically independent = no relation = groups are sampled from the same population 18
19 Two independent samples Nominal or ordinal scaling Chisquare test of independence Idea: Test whether the deviations of observed cell proportions (conditional probabilities) from cell proportions expected under H 0 (independence) exceed what we can expect by chance (random deviations) 19
20 Two independent samples Nominal or ordinal scaling Chisquare test of independence Test statistic: n ij = observed number of cases categorized in the ith row of the jth column E ij = number of cases expected in the ith row of the jth column when H 0 is true 20
21 Two independent samples Nominal or ordinal scaling Chisquare test of independence Asymptotically (as N gets large), X 2 follows a chisquare distribution with df = (r 1)(c 1), where r is the number of rows and c is the number of columns in the contingency table 21
22 Two independent samples Nominal or ordinal scaling Chisquare test of independence Example 22
23 Two independent samples Nominal or ordinal scaling Chisquare test of independence Example 23
24 Two independent samples Nominal or ordinal scaling Chisquare test of independence Example df = (r 1)(c 1) = 2 Reject H 0 since value of X 2 is beyond the critical value with df = 2 and α =
25 Two independent samples Nominal or ordinal scaling Chisquare test of independence Remark for 2x2 tables: if N not too large, use Fisher s exact test If N large (say N > 30), use chisquare test, but employ test statistic with continuity correction (Yates): 25
26 Two independent samples At least ordinal scale The median test Two independent groups At least ordinal scale H 0 : Groups do not differ in central tendency = groups have been drawn from populations with the same median 26
27 Two independent samples At least ordinal scale The median test Idea: first determine the median score for the combined group (i.e., the median for all scores in both samples) if both groups are samples from populations whose medians are the same, we would expect about half of each group's scores to be above the combined median and about half to be below 27
28 Two independent samples At least ordinal scale The median test Under H 0, the sampling distribution of the number of the m cases in group I that fall above the combined median (A) and the number of the n cases in group II that fall above the combined median (B) is the hypergeometric distribution: 28
29 Two independent samples At least ordinal scale The median test Remarks When several scores may fall right at the combined median: i. The groups may be dichotomized as those scores that exceed the median and those that do not. ii. If m + n is large, and if only a few cases fall at the combined median, those few cases may be dropped from the analysis. Better do (i) and see whether it makes a difference when analysis based on greater than or equal to or greater than. There may be no alternative to the median test, even for intervalscale data, e.g., with censored data (some observations may be off the scale and therefore measured as the maximum (or minimum) previously assigned to the observations.) 29
30 Two independent samples At least ordinal scale Two independent groups At least ordinal scale Asymptotically equivalent to a ttest. H 0 : X and Y come from the same population, Pr(X>Y)= ½ =Pr(X<Y). the median is the same in both groups (assuming that variances of the distributions in both groups are equal) H 1 (onetail): Wilcoxon MannWhitney Test (a.k.a. Mann Whitney U test, Wilcoxon ranksum test, or Wilcoxon Mann Whitney test) X is stochastically larger than Y, Pr(X>Y) > ½ the bulk of the elements in X are larger than the bulk of the elements in Y H 1 (twotail): Pr(X>Y) ½ 30
31 Two independent samples At least ordinal scale Wilcoxon MannWhitney Test Idea: m = number of observations in the sample from group X n = number of observations in the sample from group Y combine the observations from both groups and rank them in order of increasing size lowest ranks are assigned to the largest negative values (if any) Note that the sum of the first N = (m+n) integers is N = N(N + 1)/2 W x is the sum of the ranks in group X W y is the sum of the ranks in group Y Thus, W x + W y = N(N + 1)/2 31
32 Two independent samples At least ordinal scale Wilcoxon MannWhitney Test Idea: If H0 is true, we would expect the average ranks in each of the two groups to be about equal. If W x is very large (or very small), then we may have reason to suspect that the samples were not drawn from the same population. The sampling distribution of W x (together with m and n) when H0 is true is known Hence, we can determine the probability associated with the occurrence under H0 of any W x as extreme as the observed value. 32
33 Two independent samples At least ordinal scale Wilcoxon MannWhitney Test Example W x = = 15 W Y = = 13 Pr(W x 15, n = 4, m = 3 ) =.20 (Pr(W x 15, n = 4, m = 3 ) =.8857) Do not reject H0 33
34 Two independent samples At least ordinal scale Wilcoxon MannWhitney Test Remarks Use normal approximation for large samples (m > 10 or n > 10) Then, is asymptotically normally distributed with zero mean and unit variance. Wilcoxon test has greater power than the median test The Wilcoxon test considers the rank value of each observation rather than simply its location with respect to the combined median, and, thus, uses more of the information in the data. 34
35 Two independent samples At least ordinal scale Wilcoxon MannWhitney Test Remarks When ties occur each of the tied observations the average of the ranks they would have had if no ties had occurred Correction of test statistic may be necessary (see Siegel & Castellan, 1988) Wilcoxon MannWhitney Test may be regarded as a permutation test applied to the ranks of the observations and, thus, constitutes a good approximation to the permutation test. 35
36 Two independent samples At least ordinal scale Robust Rank Order Test In order to interpret Wilcoxon tests as a test for equality of medians, we have to assume equal variances in both groups The robust Rank Order Test relaxes the assumption of the same variances, i.e., the underlying distributions may be different when testing equality of medians As before: Two independent groups, at least ordinal scale m = number of observations in the sample from group X n = number of observations in the sample from group Y combine the observations from both groups and rank them in order of increasing size, were lowest ranks are assigned to the largest negative values (if any) 36
37 Two independent samples At least ordinal scale Robust Rank Order Test Procedure For each observation in X [Y] we count the number of observations of Y [X] with a lower rank ( placement of Xi [Yj] ) =: U(YXi) [=: U(XYj)] Calculate the mean of the placements in X [and Y]: Calculate the index of variability of U(YXi) and U(XYj): Test statistic with known distribution: 37
38 Two independent samples At least ordinal scale Robust Rank Order Test Example U(YX) = 3 U(XY) =.75 V x = 2 V y = 2.75 Ù = 1.13 Pr(Ù > 1.13) > 0.1 do not reject H 0 (same conclusion with Wilcoxon MannWhitney test) 38
39 More than two independent samples At least ordinal scale KruksalWallis Test Do k > 2 independent samples (ordinal/ordered data) come from the same or different populations? Extension of MannWhitney to three or more samples. Analogue to the Ftest used in analysis of variance, but without the assumption that all populations under comparison are normally distributed. H0: All k samples have the same distribution functions. H1: At least two of the samples have different distribution functions. 39
40 Two independent samples At least interval scale KolmogorovSmirnov Test Prerequisites Here: Two independent samples/groups At least interval scale H0: samples have been drawn from the same population (i.e., from populations with the same distribution) sensitive to any kind of difference in the distributions from which the two samples were drawn differences in location (central tendency), in dispersion, in skewness, etc. 40
41 Two independent samples At least interval scale KolmogorovSmirnov Test Idea If the two samples have been drawn from the same population distribution, then the cumulative distribution functions (CDF) of both samples are expected to be close to each other If the two sample CDFs are "too far apart" at any point, this suggests that the samples come from different populations large deviations between the two sample CDFs is evidence against H0 41
42 Two independent samples At least interval scale KolmogorovSmirnov Test Procedure determine the empirical CDF for each sample by using the same intervals for both distributions for each interval we subtract one step function from the other test focuses on the largest of these observed deviations S m (X) := empirical CDF for sample A (of size m), i.e., S m (X) =K/m, where K is the number of observations equal to or less than X S n (X) := empirical CDF for sample B (of size n) KolmogorovSmirnov twosample test statistic onesided: D m,n = max[s m (X) S n (X)] twosided: D m,n = max[ S m (X) S n (X) ] Reject H0 if D m,n is too large (sampling distributions of D m,n are known, depend on nature of H1) 42
43 Two independent samples At least interval scale KolmogorovSmirnov Test Example 43
44 Two independent samples At least interval scale KolmogorovSmirnov Test Example D m,n = 0.70, m = 9, n = 10 Value of test statistic greater than critical value > reject H0. 44
45 Two independent samples At least interval scale KolmogorovSmirnov Test Remarks Can also be used to test an empirical distribution from one sample against some theoretical distribution (as the corresponding chisquare test); then the theoretical distribution must be continuous 45
46 NONPARAMETRIC TESTS DEPENDENT SAMPLES 46
47 Two dependent samples Binary variable (nominal or ordinal scale) McNemar test Two related (dependent) samples Binary variable Test for the significance of changes in some binary response (e.g., by treatment manipulation) Often used in the context of before and after designs 47
48 Two dependent samples Binary variable (nominal or ordinal scale) McNemar test Idea B, C: # individuals who responded the same on each treatment (+ and, respectively) A, D: # individuals whose responses changed between treatments (from + to, and from to +, respectively) 48
49 Two dependent samples Binary variable (nominal or ordinal scale) McNemar test Idea Thus, (A + D) is the total number of people whose responses changed. Focus on cells in which changes may occur: Without any treatment effect, the number of changes in each direction would be equally likely. H0: Expected number of observations in each cell is (A + D)/2 49
50 Two dependent samples Binary variable (nominal or ordinal scale) McNemar test Remember: Test statistic for the Chisquare test of independence O i = number of cases observed in category i E i = number of cases expected in category i (under H0) Applied to cells counting changes, we yield McNemar s statistic: which (approximately) follows a chisquare distribution with df = 1 50
51 Two dependent samples Binary variable (nominal or ordinal scale) McNemar test Remarks Correction for continuity (Yates) gives better approximation (correction is necessary because a continuous distribution (chisquare) is used to approximate a discrete distribution): If the total number of changes (A+D) is less than 10, use the binomial test rather than the McNemar test. 51
52 Two dependent samples Binary variable (nominal or ordinal scale) McNemar test Example Under H 0, Pr(X 2 > 1.25) > 0.05 Do not reject H 0 52
53 Two dependent samples At least ordinal scale Sign test Two related samples Variable under consideration has a continuous distribution Xi: score of subject i in treatment X Yi: score of subject i in treatment Y H0: Pr(Xi > Yi) = Pr(Xi < Yi) = ½ = median difference between X and Y is zero 53
54 Two dependent samples At least ordinal scale Sign test Idea focus on the direction of the difference between every Xi and Yi, noting whether the sign of the difference is positive or negative When H0 is true, we would expect the number of pairs which have (Xi > Yi) to be equal to the number of pairs which have (Xi < Yi). H0 is rejected if too few differences of one sign occur. 54
55 Two dependent samples At least ordinal scale Sign test The probability associated with the occurrence of a particular number of positive (and negative) differences can be determined by the binomial distribution with p = 1/2, N = the number of pairs. If a matched pair shows no difference (i.e., the difference is zero and has no sign), it is dropped from the analysis and N is reduced accordingly. 55
56 Two dependent samples At least ordinal scale Sign test Example 56
57 Two dependent samples At least ordinal scale Sign test Example Probability of observing k of n ranks being negative: k pdf cdf With n = 12, p = ½, the probability of observing 2 or less negative (or positive) signs = 2*Pr(X 2) = 2* = Reject H0 57
58 Two dependent samples At least ordinal scale Sign test Remark For large samples (say, N > 35), normal approximation to the binomial distribution is used 58
59 Two dependent samples At least ordinal scale Wilcoxon SignedRank Test (a.k.a. Wilcoxon T test) Sign test uses only information about the direction of the differences within pairs Wilcoxon signedrank test uses also the relative magnitude gives more weight to a pair which shows a large difference between the two conditions than to a pair which shows a small difference. 59
60 Two dependent samples At least ordinal scale Wilcoxon SignedRank Test Idea Calculate the difference di = Xi Yi for each matched pair of observations Rank di's without respect to sign Assign to each rank the sign (+ or ) of the di which it represents. If H0 is true, the sum of ranks having plus signs and summed those ranks having minus signs, are expected to be equal Reject H0 if the sum of the positive ranks is too different from the sum of the negative ranks, (suggesting that treatment X differs from treatment Y) 60
61 Two dependent samples At least ordinal scale Wilcoxon SignedRank Test N = number of nonzero di s. T + = sum of the ranks which have a positive sign T = sum of the ranks which have a negative sign Note: the sum of all of the ranks is N(N + 1)/2 = T + + T Distribution of T + under H0 is known (Wilcoxon SignedRank Test corresponds to permutation test (for paired observations) based on ranks rather than scores di) 61
62 Two dependent samples At least ordinal scale Wilcoxon SignedRank Test Example T + = = 73 2*Pr(T + 73, N=12) = Reject H0 (as with sign test, but note lower pvalue here) 62
63 Two dependent samples At least ordinal scale Wilcoxon SignedRank Test Remarks Ties pairs with di = 0 are dropped from the analysis and the sample size is reduced accordingly. When two or more d's have the same magnitude, their rank is the average of the ranks which would have been assigned if the d's had differed slightly Large Samples T + is approximately normally distributed with 63
64 SELECTION OF SAMPLE SIZE  POWER ANALYSIS 64
65 Power analysis True state of the world (population) H 0 is true H 1 is true Test result (based on sample) Do not reject H 0 Reject H 0 Correct (1 α) Type I Error α Type II Error β Correct (1 β) power 65
66 Power analysis Type I error: rejecting H0 when it is, in fact, true. Pr(Type I error) =: α In experimental economics, common values of α are.05 and.01 Type II error: failing to reject H0 when, in fact, it is false. Pr(Type II error) =: β Distribution of test statistic under H0 True distribution of the test statistic: 66
67 Power analysis Type I error: rejecting H0 when it is, in fact, true. Pr(Type I error) =: α In experimental economics, common values of α are.05 and.01 Type II error: failing to reject H0 when, in fact, it is false. Pr(Type II error) =: β Power of a test: Probability of correctly concluding a significant effect when it really exist in the population = 1  Pr(Type II error) = 1  β Usually desired to be
68 Power analysis Distribution of test statistic under H0: True distribution of the test statistic: 68
69 Power analysis Power depends on Level of significance α (+) True effect size in the population (+) Sample size N (+) Variance in the data ( ) The kind of test (e.g., Sign test versus Wilcoxon signed rank test) The nature of H1 (onesided > twosided) and other variables, depending upon the test being done. 69
70 Power analysis Given The kind of test (and nature of H1) Probability of TypeI error Power (1 β) Presumed size of effect/parameter Variance in the data (and further assumptions, depending on the test) we can determine the lowest sample size we need in order to detect the presumed effect (with probability (1 β)) 70
71 Power analysis Ways to determine power If the (approximate) distribution of the test statistic is known, we may calculate the power for given parameters directly If the distribution of the test statistic is not known or if an analytical solution is to tedious (e.g. some parameter of a structural model), we may determine power by simulation Example: Twosample test of proportions See script 71
Review of the Topics for Midterm II
Review of the Topics for Midterm II STA 100 Lecture 18 I. Confidence Interval 1. Point Estimation a. A point estimator of a parameter is a statistic used to estimate that parameter. b. Properties of a
More informationBy Hui Bian. Office for Faculty Excellence
By Hui Bian Office for Faculty Excellence A parametric statistical test is a test whose model specifies certain conditions about the parameters of the population from which the research sample was drawn.
More informationUnit 14: Nonparametric Statistical Methods
Unit 14: Nonparametric Statistical Methods Statistics 571: Statistical Methods Ramón V. León 7/26/2004 Unit 14  Stat 571  Ramón V. León 1 Introductory Remarks Most methods studied so far have been based
More informationChapter 16: Nonparametric Tests
Chapter 16: Nonparametric Tests In Chapter 113 we discussed tests of hypotheses in a parametric statistics framework: which assumes that the functional form of the (population) probability distribution
More informationIntrinsically Ties Adjusted NonParametric Method for the Analysis of Two Sampled Data
Journal of Modern Applied Statistical Methods Volume Issue Article 8 03 Intrinsically Ties Adjusted NonParametric Method for the Analysis of Two Sampled Data G. U. Ebuh Nnamdi Azikiwe University, Awka,
More informationLecture 7: Binomial Test, Chisquare
Lecture 7: Binomial Test, Chisquare Test, and ANOVA May, 01 GENOME 560, Spring 01 Goals ANOVA Binomial test Chi square test Fisher s exact test Su In Lee, CSE & GS suinlee@uw.edu 1 Whirlwind Tour of One/Two
More informationThe Gaussian distribution (normal distribution)
The Gaussian distribution (normal distribution) When the distribution of the observations is normal, then 95% of all observation are located in the interval: mean1.96 SD to mean+1.96 SD represents a descriptive
More informationOutcome 1 Outcome 2 Treatment 1 A B Treatment 2 C D
Chapter 6 Categorical data So far, we ve focused on analyzing numerical data This section focuses on data that s categorical (eg, with values like red or blue that don t have any ordering) We ll start
More informationCHAPTER 5 NONPARAMETRIC TESTS
CHAPTER 5 NONPARAMETRIC TESTS The methods described in chapter 4 are all based on the following assumptions: 1. Simple random sample. 2. The data is numeric. 3. The data is normally distributed. 4. For
More informationNonparametric Methods Testing with Ordinal Data or Nonnormal Distributions Chapter 16
Nonparametric Methods Testing with Ordinal Data or Nonnormal Distributions Chapter 16 Rational The classical inference methods require a distribution to describe the population of a variable Only the parameters
More informationStatistical 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 informationNonparametric tests. Nonparametric tests and ANOVAs: What you need to know. Quick Reference Summary: Sign Test. ( ) n(x P = 2 * Pr[x!
Nonparametric tests and ANOVAs: What you need to know Nonparametric tests Nonparametric tests are usually based on ranks There are nonparametric versions of most parametric tests arametric Onesample and
More informationOn Consistency and Limitation of paired ttest, Sign and Wilcoxon Sign Rank Test
IOSR Journal of Mathematics (IOSRJM) eissn: 22785728, pissn:2319765x. Volume 10, Issue 1 Ver. IV. (Feb. 2014), PP 0106 On Consistency and Limitation of paired ttest, Sign and Wilcoxon Sign Rank
More informationCHISQUARE (Χ 2 ) TESTS FOR FREQUENCIES Notes from: Hinkle, et al., (2003)
KEY CONCEPTS: ( ) TESTS FOR FREQUENCIES Notes from: Hinkle, et al., (003) Nonparametric Tests: Statistical tests of significance that require fewer assumptions than parametric tests. ChiSquare ( ) Distribution:
More informationCourse Notes  Statistics
EPI546: Fundamentals of Epidemiology and Biostatistics Course Notes  Statistics MSc (Credit to Roger J. Lewis, MD, PhD) Outline: I. Classical Hypothesis (significance) testing A. Type I (alpha) error
More informationParametric versus Nonparametric Statisticswhen to use them and which is more powerful? Dr Mahmoud Alhussami
Parametric versus Nonparametric Statisticswhen to use them and which is more powerful? Dr Mahmoud Alhussami Parametric Assumptions The observations must be independent. Dependent variable should be continuous
More informationSimple example. Simple example. Simple example. Example
Simple example Simple example A box contains 1000 balls some white some black It is believed (hypothesized) that there are 990 W and 10 B 5 balls are drawn at random ( with replacement). They were all
More informationStatistics can be classified
Statistics in a nutshell Statistics can be classified Descriptive: Describes a population sample Able to apply measures of central tendency & dispersion Inferential: Try to infer something that applies
More informationChapter 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 informationStatistics in medicine
Statistics in medicine Lecture 3: Bivariate association : continuous variables Fatma Shebl, MD, MS, MPH, PhD Assistant Professor Chronic Disease Epidemiology Department Yale School of Public Health Fatma.shebl@yale.edu
More informationModule 9: Nonparametric Tests. The Applied Research Center
Module 9: Nonparametric Tests The Applied Research Center Module 9 Overview } Nonparametric Tests } Parametric vs. Nonparametric Tests } Restrictions of Nonparametric Tests } OneSample ChiSquare Test
More informationCOMP6053 lecture: The chisquared test and distribution.
COMP6053 lecture: The chisquared test and distribution jn2@ecs.soton.ac.uk Assumption of normal error term In all of the techniques we've covered so far, there has been a background assumption that the
More informationCRITICAL VALUES FOR THE MOOD TEST OF EQUALITY OF DISPERSION. Justice I. Odiase and Sunday M. Ogbonmwan
CRITICAL VALUES FOR THE OOD TEST OF EQUALITY OF DISPERSION Justice I. Odiase and Sunday. Ogbonmwan Abstract. An exhaustive unconditional permutation distribution of a test statistic is necessary in the
More informationLearning objectives. Nonparametric tests. Inhomogeneity of variance. Why nonparametric test? Nonparametric tests. NOT normally distributed
Learning objectives Criteria for choosing a nonparametric test Part Why nonparametric test? Inhomogeneity of variance Violation of the assumption of normally distributed data Group Group Group Inhomogeneity
More informationThe purpose of Statistics is to ANSWER QUESTIONS USING DATA Know more about your data and you can choose what statistical method...
The purpose of Statistics is to ANSWER QUESTIONS USING DATA Know the type of question and you can choose what type of statistics... Aim: DESCRIBE Type of question: What's going on? Examples: How many chapters
More information1 of 1 7/9/009 8:3 PM Virtual Laboratories > 9. Hy pothesis Testing > 1 3 4 5 6 7 6. ChiSquare Tests In this section, we will study a number of important hypothesis tests that fall under the general term
More informationLecture 8 Descriptive and Quantitative Analysis
Lecture 8 Descriptive and Quantitative Analysis Descriptive Statistics (error checking, data cleaning, assumptions of independence) Descriptive (SAMPLE) statistics help us understand and summarize the
More informationSTATISTIKA 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 information3. 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 informationChapter 3: Nonparametric Tests
B. Weaver (15Feb00) Nonparametric Tests... 1 Chapter 3: Nonparametric Tests 3.1 Introduction Nonparametric, or distribution free tests are socalled because the assumptions underlying their use are fewer
More informationNonparametric Statistics
Nonparametric Statistics J. Lozano University of Goettingen Department of Genetic Epidemiology Interdisciplinary PhD Program in Applied Statistics & Empirical Methods Graduate Seminar in Applied Statistics
More informationCHAPTER 3 COMMONLY USED STATISTICAL TERMS
CHAPTER 3 COMMONLY USED STATISTICAL TERMS There are many statistics used in social science research and evaluation. The two main areas of statistics are descriptive and inferential. The third class of
More informationInference Procedures for One Sample and PairedData Location Problems
Inference Procedures for One Sample and PairedData Location Problems Robert J. Serfling October, 006 In 1 4 we examine and compare three types of procedures for inference about location parameters, in
More informationChi Square & Correlation
Chi Square & Correlation Nonparametric Test of Chi 2 Used when too many assumptions are violated in TTests: Sample size too small to reflect population Data are not continuous and thus not appropriate
More informationLecture Notes H: More Hypothesis Testing  ANOVA, the Chi Square Test and the KolmogorovSmirnov Test for Normality
ECON 497 Lecture H Page 1 of 1 Metropolitan State University ECON 497: Research and Forecasting Lecture Notes H: More Hypothesis Testing  ANOVA, the Chi Square Test and the KolmogorovSmirnov Test for
More informationALTERNATIVES TO t AND F
CHAPTER 15 ALTERNATIVES TO t AND F OBJECTIVES After completing this chapter, you should be able to compute and use nonparametric alternatives to parametric tests, such as the t test for independent samples
More informationNONPARAMETRIC TESTS. LALMOHAN BHAR Indian Agricultural Statistics Research Institute Library Avenue, New Delhi12
NONPARAMETRIC TESTS LALMOHAN BHAR Indian Agricultural Statistics Research Institute Library Avenue, New Delhi1 lmb@iasri.res.in 1. Introduction Testing (usually called hypothesis testing ) play a major
More informationThe Wilcoxon test: Graham Hole Research Skills, version 1.0
The Wilcoxon test: Use this when the same participants perform both conditions of your study: i.e., it is appropriate for analysing the data from a repeatedmeasures design with two conditions. Use it
More informationChapter 10. Chisquare Test of Independence
Lecture notes, Lang Wu, UBC 1 Chapter 10. Chisquare Test of Independence 10.1. Association between Two Discrete Variables To study the relationship or association between two continuous variables, we
More informationChapter 21 Statistical Tests for Ordinal Data. Table 21.1
Chapter 21 Statistical Tests for Ordinal Data The MannWhitney (RankSum) Test In this example, we deal with a dependent variable that is measured on a ratio scale (time in seconds), but its distribution
More informationSTATISTICAL POWER COMPARISONS FOR EQUAL SKEWNESS DIFFERENT KURTOSIS AND EQUAL KURTOSIS DIFFERENT SKEWNESS COEFFICIENTS IN NONPARAMETRIC TESTS
Ekonometri ve İstatistik Sayı:18 2013 81115 İSTANBUL ÜNİVERSİTESİ İKTİSAT FAKÜLTESİ EKONOMETRİ VE İSTATİSTİK DERGİSİ STATISTICAL POWER COMPARISONS FOR EQUAL SKEWNESS DIFFERENT KURTOSIS AND EQUAL KURTOSIS
More informationNonParametric Tests
NonParametric Tests Non Parametric Tests Do not make as many assumptions about the distribution of the data as the t test. Do not require data to be Normal Good for data with outliers Nonparametric tests
More informationStatistics Advanced Placement G/T Essential Curriculum
Statistics Advanced Placement G/T Essential Curriculum UNIT I: Exploring Data employing graphical and numerical techniques to study patterns and departures from patterns. The student will interpret and
More informationNonparametric test One sample tests Two sample tests Testing for three or more samples
Nonparametric test One sample tests Two sample tests Testing for three or more samples 1 Background So far we have stressed that in order to carry out hypothesis tests we need to make certain assumptions
More informationRandom Numbers. Quasi Random Numbers. Midsquare method Linear Congruential Generator Linear Shift Feedback Register
Randomness Applications in scientific computing Random variables and Pseudo Properties of a good random number generator (RNG) RNGs Midsquare method Linear Congruential Generator Linear Shift Feedback
More informationNon Parametric Statistics
Non Parametric Statistics Διατμηματικό ΠΜΣ Επαγγελματική και Περιβαλλοντική ΥγείαΔιαχείριση και Οικονομική Αποτίμηση Δημήτρης Φουσκάκης Introduction So far in the course we ve assumed that the data come
More information3.6: 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 informationBIOSTATISTICAL ANALYSIS OF RESEARCH DATA
BIOSTATISTICAL ANALYSIS OF RESEARCH DATA March 27 th and April 3 rd, 2015 Kris Attwood, PhD Department of Biostatistics & Bioinformatics Roswell Park Cancer Institute Outline Biostatistics in Research
More informationTo test an average or pair of averages when σ is known, we use ztests
ttests To test an average or pair of averages when σ is known, we use ztests But often σ is unknown, e.g., in specially constructed psycholinguistics tests, in tests of reactions of readers or software
More informationIntroduction to Quantitative Research Analysis and SPSS. SW242 Session 6 Slides
Introduction to Quantitative Research Analysis and SPSS SW242 Session 6 Slides 2 Creation & Description of a Data Set Ø Four Levels of Measurement Nominal, ordinal, interval, ratio Ø Variable Types Independent
More informationInferential Statistics
Inferential Statistics Sampling and the normal distribution Zscores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are
More informationData 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 informationThe ChiSquare Distributions
MATH 183 The ChiSquare Distributions Dr. Neal, WKU The chisquare distributions can be used in statistics to analyze the standard deviation " of a normally distributed measurement and to test the goodness
More informationPermutation Tests for Univariate and Multivariate Ordered Categorical Data
AUSTRIAN JOURNAL OF STATISTICS Volume 35 (2006), Number 2&3, 315 324 Permutation Tests for Univariate and Multivariate Ordered Categorical Data Fortunato Pesarin and Luigi Salmaso University of Padova,
More informationCHI SQUARE ANALYSIS 8/18/2011 HYPOTHESIS TESTS SO FAR PARAMETRIC VS. NONPARAMETRIC
CHI SQUARE ANALYSIS I NTRODUCTION TO NON PARAMETRI C ANALYSES HYPOTHESIS TESTS SO FAR We ve discussed Onesample ttest Dependent Sample ttests Independent Samples ttests OneWay Between Groups ANOVA
More informationNAG C Library Chapter Introduction. g08 Nonparametric Statistics
g08 Nonparametric Statistics Introduction g08 NAG C Library Chapter Introduction g08 Nonparametric Statistics Contents 1 Scope of the Chapter... 2 2 Background to the Problems... 2 2.1 Parametric and Nonparametric
More informationDidacticiel Études de cas. Non parametric tests for differences in location. (K 2) independent samples.
1 Topic Non parametric tests for differences in location. (K 2) independent samples. The tests for comparison of population try to determine if K (K 2) samples come from the same underlying population
More informationNonparametric 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 informationFundamentals of Data Analysis: Selecting a DA Strategy
MBACATÓLICA JAN/APRIL 2006 Marketing Research Fernando S. Machado Week 7 Fundamentals of Data Analysis: Selecting a Data Analysis Strategy Frequency Distribution and CrossTabulation Hypothesis Testing:
More informationChiSquare. The goodnessoffit test involves a single (1) independent variable. The test for independence involves 2 or more independent variables.
ChiSquare Parametric statistics, such as r and t, rest on estimates of population parameters (x for μ and s for σ ) and require assumptions about population distributions (in most cases normality) for
More informationStatistics for Fraud Detection
Statistics for Fraud Detection Statistics are useful for summarizing large amounts of data into a form that can be more easily interpreted. Accountants and managers can use statistical theory and statistics
More informationANSWERS TO TEST NUMBER 6
Question 1: (15 points) A point estimate of µ x  µ y is ANSWERS TO TEST NUMBER 6 Ȳ X 7.0 3.0 4. Given the small samples, we are required to assume that the populations are normally distributed with the
More information1/1/2014. Scales of Measurement GETTING TO THE STANDARD NORMAL DISTRIBUTION. Scales of Measurement. Scales of Measurement
MBA 605 Business Analytics Don Conant, PhD. GETTING TO THE STANDARD NORMAL DISTRIBUTION Measurement is a process of assigning numbers to characteristics according to a defined rule. Not all measurement
More informationChisquare test Fisher s Exact test
Lesson 1 Chisquare test Fisher s Exact test McNemar s Test Lesson 1 Overview Lesson 11 covered two inference methods for categorical data from groups Confidence Intervals for the difference of two proportions
More informationSAS Program Notes Biostatistics: A Guide to Design, Analysis, and Discovery Chapter 9: Nonparametric Tests
SAS Program Notes Biostatistics: A Guide to Design, Analysis, and Discovery Chapter 9: Nonparametric Tests In the program notes for chapter 9, we discuss nonparametric or distributionfree methods. Note
More informationChi Square for Contingency Tables
2 x 2 Case Chi Square for Contingency Tables A test for p 1 = p 2 We have learned a confidence interval for p 1 p 2, the difference in the population proportions. We want a hypothesis testing procedure
More informationBIOSTATISTICS FOR THE CLINICIAN
BIOSTATISTICS FOR THE CLINICIAN Mary Lea Harper, Pharm.D. Learning Objectives 1. Understand when to use and how to calculate and interpret different measures of central tendency (mean, median, and mode)
More informationCOMPARING DATA ANALYSIS TECHNIQUES FOR EVALUATION DESIGNS WITH NON NORMAL POFULP_TIOKS Elaine S. Jeffers, University of Maryland, Eastern Shore*
COMPARING DATA ANALYSIS TECHNIQUES FOR EVALUATION DESIGNS WITH NON NORMAL POFULP_TIOKS Elaine S. Jeffers, University of Maryland, Eastern Shore* The data collection phases for evaluation designs may involve
More informationRankBased NonParametric Tests
RankBased NonParametric 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 informationSuperiority by a Margin Tests for the Odds Ratio of Two Proportions
Chapter 07 Superiority by a Margin Tests for the Odds Ratio of Two Proportions Introduction This module computes power and sample size for hypothesis tests for superiority of the odds ratio of two independent
More informationRandom Number By Prof. S.Shakya 1
Random Number By Prof. S.Shakya 1 Random Number Random numbers are samples drawn from a uniformly distributed random variable between some satisfied intervals, they have equal probability of occurrence.
More informationSome Critical Information about SOME Statistical Tests and Measures of Correlation/Association
Some Critical Information about SOME Statistical Tests and Measures of Correlation/Association This information is adapted from and draws heavily on: Sheskin, David J. 2000. Handbook of Parametric and
More informationAnalysis of numerical data S4
Basic medical statistics for clinical and experimental research Analysis of numerical data S4 Katarzyna Jóźwiak k.jozwiak@nki.nl 22nd November 2016 1/44 Hypothesis tests: numerical and ordinal data 1 group:
More informationParametric Statistics 1 Nonparametric Statistics
Parametric Statistics 1 Nonparametric Statistics Timothy C. Bates tim.bates@ed.ac.uk Assume data are drawn from samples with a certain distribution (usually normal) Compute the likelihood that groups are
More informationMILLER AND PRASNIKAR: STRATEGIC PLAY,draft 1
MILLER AND PRASNIKAR: STRATEGIC PLAY,draft 1 Introduction The solution to the game predicts the outcomes of experimental subjects playing the game in a classroom laboratory. The conduct of the experiment
More informationSTA3123: Statistics for Behavioral and Social Sciences II. Text Book: McClave and Sincich, 12 th edition. Contents and Objectives
STA3123: Statistics for Behavioral and Social Sciences II Text Book: McClave and Sincich, 12 th edition Contents and Objectives Initial Review and Chapters 8 14 (Revised: Aug. 2014) Initial Review on
More informationNONPARAMETRIC 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 pvalues classical significance testing depend on assumptions
More informationX 2 has mean E [ S 2 ]= 2. X i. n 1 S 2 2 / 2. 2 n 1 S 2
Week 11 notes Inferences concerning variances (Chapter 8), WEEK 11 page 1 inferences concerning proportions (Chapter 9) We recall that the sample variance S = 1 n 1 X i X has mean E [ S ]= i =1 n and is
More informationChiSquare Tests. In This Chapter BONUS CHAPTER
BONUS CHAPTER ChiSquare Tests In the previous chapters, we explored the wonderful world of hypothesis testing as we compared means and proportions of one, two, three, and more populations, making an educated
More informationTests of relationships between variables Chisquare Test Binomial Test Run Test for Randomness OneSample KolmogorovSmirnov 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 informationBiostatistics: Pretest Primer. Larry Liang MD University of Texas Southwestern Medical Center
Biostatistics: Pretest Primer Larry Liang MD University of Texas Southwestern Medical Center Question 1 Which of the following will increase the sensitivity of a test? A. Decrease Type I Error B. Increase
More informationPermutation Tests for Comparing Two Populations
Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. JaeWan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of
More informationSTA 4504/5503 Sample Exam 1 Spring 2011 Categorical Data Analysis. 1. Indicate whether each of the following is true (T) or false (F).
STA 4504/5503 Sample Exam 1 Spring 2011 Categorical Data Analysis 1. Indicate whether each of the following is true (T) or false (F). (a) T In 2 2 tables, statistical independence is equivalent to a population
More informationThe Logic of Statistical Inference  Testing Hypotheses
The Logic of Statistical Inference  1 Testing Hypotheses Confirming your research hypothesis (relationship between 2 variables) is dependent on ruling out: Rival hypotheses Research design problems (e.g.
More informationInferential Statistics for Social and Behavioural Research
Research Journal of Mathematics and Statistics 1(2): 4754, 2009 ISSN: 20407505 Maxwell Scientific Organization, 2009 Submitted Date: August 26, 2009 Accepted Date: September 04, 2009 Published Date:
More informationAppendix A: Background
12 Appendix A: Background The purpose of this Appendix is to review background material on the normal distribution and its relatives, and an outline of the basics of estimation and hypothesis testing as
More informationChapter 13 Statistical Foundations: Ordinal Data Analysis 406. Chapter 13 Statistical Foundations: Ordinal Data Analysis
Chapter 13 Statistical Foundations: Ordinal Data Analysis 406 Chapter 13 Statistical Foundations: Ordinal Data Analysis I. Introduction A. The Nature of Ordinal Data 1. Ordinal data are ordered categories,
More informationOne tailed vs two tailed tests: a normal distribution filled with a rainbow of colours. = height (14) = height (10) = D
One tailed ttest using SPSS: Divide the probability that a twotailed SPSS test produces. SPSS always produces twotailed significance level. One tailed vs two tailed tests: a normal distribution filled
More informationTwoSample TTests Allowing Unequal Variance (Enter Means)
Chapter 44 TwoSample TTests Allowing Unequal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one or twosided twosample ttests when no assumption
More informationTwo sample hypothesis tests
Two sample hypothesis tests As we noted when discussing regression models, in many situations we have two natural subgroups in the data, and we re interested in how they compare to each other (there s
More informationVariables and Data A variable contains data about anything we measure. For example; age or gender of the participants or their score on a test.
The Analysis of Research Data The design of any project will determine what sort of statistical tests you should perform on your data and how successful the data analysis will be. For example if you decide
More informationHow to choose a statistical test. Francisco J. Candido dos Reis DGOFMRP University of São Paulo
How to choose a statistical test Francisco J. Candido dos Reis DGOFMRP 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 informationThe Math Part of the Course
The Math Part of the Course Measures of Central Tendency Mode: The number with the highest frequency in a dataset Median: The middle number in a dataset Mean: The average of the dataset When to use each:
More informationIOP 201Q (Industrial Psychological Research) Tutorial 6
IOP 201Q (Industrial Psychological Research) Tutorial 6 TRUE/FALSE [1 point each] Indicate whether the sentence or statement is true or false. 1. Probability values are always greater than or equal to
More informationHITCHHIKER'S GUIDE TO ELEMENTARY STATISTICS
\user\jobb\stat.98 980504 Torbjörn Ledin, Dept ENT, University Hospital, Linköping email Torbjorn.Ledin@inr.liu.se Adopted for the Basic Statistics Course for the PhD students. The examples shall be seen
More informationChapter 10: USING BETWEENSUBJECTS AND WITHINSUBJECTS EXPERIMENTAL DESIGNS
Chapter 10: USING BETWEENSUBJECTS AND WITHINSUBJECTS EXPERIMENTAL DESIGNS An experimental design is used when you want to establish a causal relationship and you can manipulate a variable (IV). Manipulate
More informationGuide to the Summary Statistics Output in Excel
How to read the Descriptive Statistics results in Excel PIZZA BAKERY SHOES GIFTS PETS Mean 83.00 92.09 72.30 87.00 51.63 Standard Error 9.47 11.73 9.92 11.35 6.77 Median 80.00 87.00 70.00 97.50 49.00 Mode
More informationNonParametric Tests of Significance, Second Set of Exercises Vartanian: SW 540
NonParametric Tests of Significance, Second Set of Exercises Vartanian: SW 540 1. You are examining the rankings of players on two rugby teams. Is there a significant difference in the two teams? Use
More informationChapter 12: Inferring From Data Estimation And Significance Testing
Chapter Outlines for: Frey, L., Botan, C., & Kreps, G. (1999). Investigating communication: An introduction to research methods. (2nd ed.) Boston: Allyn & Bacon. Chapter 12: Inferring From Data Estimation
More informationn x Nonparametric Statistics
Parametric Procedures 1. Involve Population Parameters Example: Population Mean 2. Require Interval Scale Ratio Scale Whole Numbers Fractions Example: Height in Inches (72, 60.5, 54.7) 3. Have Stringent
More information