Simultaneous Confidence Intervals and Multiple Contrast Tests
|
|
- Lionel Barber
- 6 years ago
- Views:
Transcription
1 Simultaneous Confidence Intervals and Multiple Contrast Tests Edgar Brunner Abteilung Medizinische Statistik Universität Göttingen 1
2 Contents Parametric Methods Motivating Example SCI Method Analysis of the Example Nonparametric Methods Motivating Example SCI Method Analysis of the Example Paricular Difficulties References 2
3 I Parametric Methods Motivating Example O 2 -Consumption of Leucocytes bars show min max O 2 -Consumption of Leucocytes D2 n 3 =7 D1 n 2 =8 PL n 1 =8 3,0 3,5 4,0 O 2 -Consumption [µl] Question Which dose is different from control? 3
4 Motivating Example Classical Analysis (1) ANOVA / H 0 : µ P = µ 1 = µ 2 (2) H 0 rejected multiple comparisons (FWE s = 0.05) (3) confidence intervals for µ 1 µ P and µ 2 µ P must be compatible to the decisions of the MCP i.e. confidence interval (CI) for µ i µ P may not contain 0 H 0 : µ i µ P = 0 is rejected, i = 1,2 Statistical Methods / Procedures ANOVA (F-test) multiple comparisons using closure principle (CTP) Bonferroni confidence intervals (1 α = 0.975) Results global hypothesis: F = 2.53 p-value (n.s.) MCP PL - D1: p = (n.s.) / PL - D2: p = (n.s.) 4
5 Motivating Example Shift of the D1 Data O 2 -Consumption of Leucocytes D2 n 3 =7 D1 n 2 =8 PL n 1 =8 3,0 3,5 4,0 O 2 -Consumption [µl] Results global hypothesis: F = 4.06 p-value (*) MCP (CTP) PL - D1: p = (*) / PL - D2: p = (*) 5
6 Conclusions from the Motivating Example Confidence Intervals (Bonferroni) PL - D1: [ 0.024, 0.557] - contains 0 / not compatible to CTP PL - D2: [ 0.063, 0.538] - contains 0 / not compatible to CTP Conclusions (undesirable properties decision on effect PL - D2 depends on effect PL - D1 confidence intervals are not compatible dependency of the statistics X 1 X P and X 2 X P not used (wasting information) different method needed 6
7 Different Method Idea statistical model is adapted and reduced to the particular questions of the experimenter take dependence of the statistics into account statistics completely dependent no α-adjusting necessary independence is the worst case example of O 2 -consumption ( ) C = = ( I 2 ) and X = (X P,X 1,X 2 ) ( ) ( ) X desired contrasts CX = 1 X P µ1 µ, µ X 2 X δ = P P µ 2 µ P consider the distribution of CX N(µ δ, Σ) [( ) ] n Σ = σ n 1 + n 1 P J 2 = (s i j ) i, j=1,2 2 7
8 Different Method Derivation of the Statistic s ii = σ 2 (n i + n P )/(n i n P ), i = 1,2 - diagonal elements of Σ ŝ ii : LS-estimator of s ii replacing σ 2 with the pooled estimator σ 2 N = 1 N 3 i=p,1,2 n i (X ik X i ) 2, N = n 1 + n 2 + n P k=1 ( ) X 1 X P studentize each component of CX = X 2 X P under H 0 : µ δ = 0, consider the statistics (i = 1,2) ni n P T i = (X i X / σ P ) N n i + n P. N(0,1), N, N/n i < N 0 < multivariate statistic T = (T 1,T 2 ). N(0,R), R: correlation matrix with ŝ ii 8
9 Different Method Derivation of the (1 α)-quantiles same quantile z 1 α,2,r for all components, such that z1 α,2,r z 1 α,2,r z1 α,2,r z 1 α,2,r dn(0,r) = 1 α better approximation: mulitvariate t-distribution: t 1 α,2,ν, R N R N : LS-estimator of R replacing σ 2 with σ 2 N diagonal elements = 1 off-diagonal elements depend only on sample sizes and σ 2 N T multivariate t-distribution references original paper: Bretz, Genz and Hothorn (2001) multivariate integration: Genz and Bretz (2009) heteroscedastic case: Hasler and Hothorn (2008) in general: C may be any appropriate contrast matrix 9
10 SCI-Method / Quantiles Korrelation = 0.99, Quantil= Korrelation = 0.5, Quantil= Korrelation = 0, Quantil= equi-coordinate quantiles of different bivariate normal distributions squares containing mass 1 α of the bivariate normal distributions computation by means of R-package mvtnorm SAS-macro: to be developed or input of R-code in SAS/IML Studio
11 SCI-Method / Procedure Multiple Comparisons reject H (i) 0 : δ i = µ i µ P = 0 if T i z 1 α,2,r - or T i t 1 α,2,ν, R N Global Hypothesis reject H 0 : Cµ= µ δ = 0 if max{t 1,T 2 } z 1 α,2,r - or max{t 1,T 2 } t 1 α,2,ν, R N Simultaneous Confidence Intervals ( { [ P δ i X i X P ± z ]} ) 1 α,2, R N ni + n P. = σ N n i n 1 α P i I Error Control? FWE s (by Gabriel s Theorem, 1969) 11
12 Example: Analysis by SCI-Method Original Data Set (O 2 -Consumption of Leucocytes) O 2 -Consumption of Leucocytes D2 n 3 =7 D1 n 2 =8 PL n 1 =8 3,0 3,5 4,0 O 2 -Consumption [µl] SCI Classical PL - D1 t = 2.10 p-value n.s. n.s. PL - D2 t = 2.18 p-value n.s. n.s. 12
13 Example: Analysis by SCI-Method Shift of the D1 Data O 2 -Consumption of Leucocytes D2 n 3 =7 D1 n 2 =8 PL n 1 =8 3,0 3,5 4,0 O 2 -Consumption [µl] SCI Classical PL - D1 t = 2.53 p-value ( ) ( ) PL - D2 t = 2.18 p-value n.s. ( ) 13
14 Conclusions from the Analysis Confidence Intervals (D1 Shifted) PL - D1: [0.0049, ] - does not contain 0 / compatible PL - D2: [ , ] - contains 0 / compatible Conclusions decision on effect PL - D2 does not depend on effect PL - D1 confidence intervals are compatible dependency of the statistics X 1 X P and X 2 X P is used 14
15 Extensions / Generalizations Factorial Designs Biesheuvel and Hothorn (2002) / stratified samples general case under research: diploma thesis Large Number of Dimensions Σ N may become singular (breakdown?) Repeated Measures n d and n < d (breakdown?) high-dimensional data / Froemke, Hothorn and Kropf (2008) is there a limit distribution? Binomial Data Schaarschmidt, Sill and Hothorn (2008) Nonparametric effects non-normal data (Konietschke, 2009) ordinal data: ordinal effect size measure (Ryu and Agresti, 2008) 15
16 II Nonparametric Methods Motivating Example Toxicity Trial (60 Wistar Rats) damage by an inhalable substance on the mucosa of the nose 3 concentrations ( 2[ppm], 5[ppm], 10[ppm]) score (0 = no damage,..., 3 = severe damage ) ordinal data Concentration Number of Rats with Score [ppm] [ppm] [ppm]
17 Motivating Example Classical Analysis Strategy statistical model X ik F i (x), i = 1,2,3; k = 1,...,20 hypotheses H (1) 0 : F 1 = F 2 = F 3 H (2) 0 : F 1 = F 2 - relative effect: p 12 = F 1 df 2 H (3) 0 : F 1 = F 3 - relative effect: p 13 = F 1 df 3 H (4) 0 : F 2 = F 3 - relative effect: p 23 = F 2 df 3 relative effect p i j - interpretation p i j = F i df j = P(X i1 < X j1 )+ 1 2 P(X i1 = X j1 ) probability that the observations in group i tend to smaller values than in group j ordinal data: effect size measure (Ryu and Agresti, 2008) needed: confidence intervals for p i j = F i df j, i j = 1,2,3 error control: FWE s 17
18 SCI-Method Hypotheses of Interest H (1) 0 : p 12 = 1 2, H(2) 0 : p 13 = 1 2 Estimators of the Relative Effects p i j p i j = ( ) F i d F j = 1 (i j) n i R j n j+1 2 p = ( p12 asymptotic distribution of N( p p) N(0,V N ) depends on unknown parameters (elements of V N ) no pivotal quantity Statistics p 13 ) studentize each component (i, j) of p by v (i j) v (i j) : estimated variance of p i j (diagonal elements of V N ) (i j) j) estimation by means of ranks R ik, R(i jk, R(i) j) ik, and R( jk Reference: Brunner, Munzel und Puri (2002) 18
19 SCI-Method Asymptotic Distribution of the Statistics (i j) asymptotic distribution under H 0 : p i j = 1 2 of T i j = N ( p i j 1 2)/ v i j.. N(0,1) T = (T 12,T 13 ).. N(0,R), R: correlation matrix use the same procedure as in the parametric case error control: FWE s problem: confidence intervals may exceed the [0,1]-interval 19
20 SCI-Method / Properties Problem intervals are not range preserving lower and upper bound of a 95% confidence interval (n = 10) Solution multivariate δ-method 20
21 Range Preserving Intervals Procedure continuous transformation of G( p i j ) (, ) G : (G 1,...,G q ) : (0,1) q R q strictly monotone, i.e. G l (p i j) 0 differentiable, bijective, G l ( 1 2 ) = 0, l = 1,...,q in the example: q = 2 asymptotic distribution of G: Cramer s δ-theorem transformed estimators are also multivariat normal elements v i j of the covariance matrix of G multivariate δ-theorem: v i j = [G (p i j )] 2 v i j back transformation of the limits [0,1] - range preserving 21
22 Example: Analysis by SCI-Method Toxicity Trial (60 Wistar Rats) Results (Probit) Concentration Number of Rats with Score [ppm] [ppm] [ppm] Comparison Effect Interval p-value 2 vs / [0.501; 0.787] vs / [0.753; 0.970] <
23 Nonparametric Methods / Difficulties Non-Transitivity pairwise relative effects are not transitive e.g.: p 1 < p 2 < p 3 < p 1 counter-example: Efron s paradox dice (Rump, 2001) Brown and Hettmansperger (2002) - one-way layout Thangavelu and Brunner (2007) - stratified Wilcoxon tests New Definition of Relative Effects for a > 2 e.g. p i = HdF i, H = mean of the F i all distributions are compared to H or all distributions are compared to the same reference to be worked out covariance matrix of N( p 1,..., p d ) is quite involved first results: Konietschke (2009) Factorial Designs consider each factor separately? or combine all comparisons in one vector? to be worked out 23
24 Discussion aund Outlook SCI-Method unifies 3 steps of the classical analysis strategy ANOVA multiple comparisons (controlling FWE s ) confidence intervals for the effects - compatible to the multiple comparisons in one procedure further research detailed results regarding power extension to factorial designs extension to repeated measures designs for parametric as well as nonparametric models Software so far only for independent samples (one-factorial design) parametric models: R-package: SimComp in CRAN nonparametric models: R-package: nparcomp in CRAN 24
25 Cooperation / Credits Ludwig Hothorn and assistants (Biostatistik, LU Hannover) Frank Konietschke (Medizinische Statistik, University of Göttingen) 25
26 References BIESHEUVEL, E. and HOTHORN, L.A. (2002). Many-to-one comparisons in stratified designs. BIOMETRICAL JOURNAL 44, BRETZ, F., GENZ, A., and HOTHORN, L.A. (2001). On the numerically availibilty of multiple comparison procedures, Biometrical Journal 43, BROWN, B. M. and HETTMANSPERGER, T. P. (2002). Kruskal-Wallis, Multiple Comparisons and Efron Dice. Australian and New Zealand Journal of Statistics 44, BRUNNER,E., MUNZEL, U., and PURI, M., (2002). The multivariate nonparametric Behrens-Fisher problem. Journal of Statistical Planning and Inference 108, FROEMKE C., HOTHORN L.A. and KROPF S. (2008). Nonparametric relevance-shifted multiple testing procedures for the analysis of high-dimensional multivariate data with small sample sizes. BMC Bioinformatics, 9:54 doi: /
27 References GABRIEL, K.R. (1969). Simultaneous Test Procedures - Some Theory of Multiple Comparisons. The Annals of Mathematical Statistics 40, GENZ, A. and BRETZ F. (2009). Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics 195. Springer, Heidelberg, New York. HASLER M. and HOTHORN L.A. (2008). Multiple Contrast Tests in the Presence of Heteroscedasticity. Biometrical Journal 50, KONIETSCHKE, F. (2009). Simultane Konfidenzintervalle für nichtparametrische relative Kontrasteffekte. Dissertation, Georg-August-Universität Göttingen RUMP, C. M. (2001). Strategies for Rolling the Efron dice. Mathematics Magazine 74,
28 References RYU, E. and AGRESTI, A. (2008). Modeling and inference for an ordinal effect size measure. Statistics in Medicine 27, SCHAARSCHMIDT, F., SILL, M. and HOTHORN, L.A. (2008). Approximate Simultaneous Confidence Intervals for Multiple Contrasts of Binomial Proportions. Biometrical Journal 50, THANGAVELU, K. and BRUNNER, E. (2007). Wilcoxon Mann-Whitney Test for Stratified Samples and Efron s Paradox Dice. Journal of Statistical Planning and Inference 137,
Vienna Medical University 11/2014. Quality ranking. or... Comparisons against the grand mean
1 / 30 Vienna Medical University 11/2014 Quality ranking or... Comparisons against the grand mean Ludwig A. Hothorn hothorn@biostat.uni-hannover.de Institute of Biostatistics, Leibniz University Hannover,
More informationUser-defined contrasts within multiple contrast tests- case studies using R
1 / 37 Vienna Section ROES User-defined contrasts within multiple contrast tests- case studies using R Ludwig A. Hothorn hothorn@biostat.uni-hannover.de Institute of Biostatistics, Leibniz University Hannover,
More informationPSY 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 informationCHAPTER 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 informationNonparametric 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 informationIntroduction 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 informationarxiv: v1 [stat.me] 20 Feb 2018
How to analyze data in a factorial design? An extensive simulation study Maria Umlauft arxiv:1802.06995v1 [stat.me] 20 Feb 2018 Institute of Statistics, Ulm University, Germany Helmholtzstr. 20, 89081
More informationDETAILED 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 informationarxiv: v1 [math.st] 12 Oct 2017
Wild Bootstrapping Rank-Based Procedures: Multiple Testing in Nonparametric Split-Plot Designs Maria Umlauft,*, Marius Placzek 2, Frank Konietschke 3, and Markus Pauly arxiv:70.04532v [math.st] 2 Oct 207
More informationReports of the Institute of Biostatistics
Reports of the Institute of Biostatistics No 01 / 2010 Leibniz University of Hannover Natural Sciences Faculty Titel: Multiple contrast tests for multiple endpoints Author: Mario Hasler 1 1 Lehrfach Variationsstatistik,
More informationNonparametric 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 informationCHI 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 informationNon-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 informationReports of the Institute of Biostatistics
Reports of the Institute of Biostatistics No 02 / 2008 Leibniz University of Hannover Natural Sciences Faculty Title: Properties of confidence intervals for the comparison of small binomial proportions
More information3 Joint Distributions 71
2.2.3 The Normal Distribution 54 2.2.4 The Beta Density 58 2.3 Functions of a Random Variable 58 2.4 Concluding Remarks 64 2.5 Problems 64 3 Joint Distributions 71 3.1 Introduction 71 3.2 Discrete Random
More information4/6/16. 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 informationNon-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 informationBIO 682 Nonparametric Statistics Spring 2010
BIO 682 Nonparametric Statistics Spring 2010 Steve Shuster http://www4.nau.edu/shustercourses/bio682/index.htm Lecture 8 Example: Sign Test 1. The number of warning cries delivered against intruders by
More informationModeling and inference for an ordinal effect size measure
STATISTICS IN MEDICINE Statist Med 2007; 00:1 15 Modeling and inference for an ordinal effect size measure Euijung Ryu, and Alan Agresti Department of Statistics, University of Florida, Gainesville, FL
More informationData 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 informationLecture 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 informationNon-parametric confidence intervals for shift effects based on paired ranks
Journal of Statistical Computation and Simulation Vol. 76, No. 9, September 2006, 765 772 Non-parametric confidence intervals for shift effects based on paired ranks ULLRICH MUNZEL* Viatris GmbH & Co.
More informationAnalysis of Variance (ANOVA) Cancer Research UK 10 th of May 2018 D.-L. Couturier / R. Nicholls / M. Fernandes
Analysis of Variance (ANOVA) Cancer Research UK 10 th of May 2018 D.-L. Couturier / R. Nicholls / M. Fernandes 2 Quick review: Normal distribution Y N(µ, σ 2 ), f Y (y) = 1 2πσ 2 (y µ)2 e 2σ 2 E[Y ] =
More informationSmall n, σ known or unknown, underlying nongaussian
READY GUIDE Summary Tables SUMMARY-1: Methods to compute some confidence intervals Parameter of Interest Conditions 95% CI Proportion (π) Large n, p 0 and p 1 Equation 12.11 Small n, any p Figure 12-4
More informationOctober 1, Keywords: Conditional Testing Procedures, Non-normal Data, Nonparametric Statistics, Simulation study
A comparison of efficient permutation tests for unbalanced ANOVA in two by two designs and their behavior under heteroscedasticity arxiv:1309.7781v1 [stat.me] 30 Sep 2013 Sonja Hahn Department of Psychology,
More informationApplied Multivariate and Longitudinal Data Analysis
Applied Multivariate and Longitudinal Data Analysis Chapter 2: Inference about the mean vector(s) Ana-Maria Staicu SAS Hall 5220; 919-515-0644; astaicu@ncsu.edu 1 In this chapter we will discuss inference
More informationNonparametric Statistics Notes
Nonparametric Statistics Notes Chapter 5: Some Methods Based on Ranks Jesse Crawford Department of Mathematics Tarleton State University (Tarleton State University) Ch 5: Some Methods Based on Ranks 1
More informationIntroduction to Statistical Inference Lecture 10: ANOVA, Kruskal-Wallis Test
Introduction to Statistical Inference Lecture 10: ANOVA, Kruskal-Wallis Test la Contents The two sample t-test generalizes into Analysis of Variance. In analysis of variance ANOVA the population consists
More informationExam 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 informationSEVERAL 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 informationContents. Acknowledgments. xix
Table of Preface Acknowledgments page xv xix 1 Introduction 1 The Role of the Computer in Data Analysis 1 Statistics: Descriptive and Inferential 2 Variables and Constants 3 The Measurement of Variables
More informationNon-parametric methods
Eastern Mediterranean University Faculty of Medicine Biostatistics course Non-parametric methods March 4&7, 2016 Instructor: Dr. Nimet İlke Akçay (ilke.cetin@emu.edu.tr) Learning Objectives 1. Distinguish
More informationR-functions for the analysis of variance
1 R-functions for the analysis of variance The following R functions may be downloaded from the directory http://www.uni-koeln.de/~luepsen/r/ Usage advices: Variables used as factors have to declared as
More informationModule 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 informationRelative Potency Estimations in Multiple Bioassay Problems
Relative Potency Estimations in Multiple Bioassay Problems Gemechis Dilba Institute of Biostatistics, Leibniz University of Hannover, Germany 5 th International Conference on Multiple Comparison Procedures
More informationTransition Passage to Descriptive Statistics 28
viii Preface xiv chapter 1 Introduction 1 Disciplines That Use Quantitative Data 5 What Do You Mean, Statistics? 6 Statistics: A Dynamic Discipline 8 Some Terminology 9 Problems and Answers 12 Scales of
More informationBiostatistics 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 informationOnline publication date: 22 March 2010
This article was downloaded by: [South Dakota State University] On: 25 March 2010 Access details: Access Details: [subscription number 919556249] Publisher Taylor & Francis Informa Ltd Registered in England
More informationHYPOTHESIS 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 informationNon-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 informationMATH 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 informationKruskal-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 informationComparison of Two Samples
2 Comparison of Two Samples 2.1 Introduction Problems of comparing two samples arise frequently in medicine, sociology, agriculture, engineering, and marketing. The data may have been generated by observation
More informationTextbook Examples of. SPSS Procedure
Textbook s of IBM SPSS Procedures Each SPSS procedure listed below has its own section in the textbook. These sections include a purpose statement that describes the statistical test, identification of
More informationChapter 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 informationA Regression Framework for Rank Tests Based on the Probabilistic Index Model
A Regression Framework for Rank Tests Based on the Probabilistic Index Model Jan De Neve and Olivier Thas We demonstrate how many classical rank tests, such as the Wilcoxon Mann Whitney, Kruskal Wallis
More informationNonparametric Location Tests: k-sample
Nonparametric Location Tests: k-sample Nathaniel E. Helwig Assistant Professor of Psychology and Statistics University of Minnesota (Twin Cities) Updated 04-Jan-2017 Nathaniel E. Helwig (U of Minnesota)
More informationON MULTIVARIATE t AND GAUSS PROBABILITIES IN R. Introduction
ON MULTIVARIATE t AND GAUSS PROBABILITIES IN R TORSTEN HOTHORN, FRANK BRETZ, AND ALAN GENZ Introduction The numerical computation of a multivariate normal or t probability is often a difficult problem.
More informationsphericity, 5-29, 5-32 residuals, 7-1 spread and level, 2-17 t test, 1-13 transformations, 2-15 violations, 1-19
additive tree structure, 10-28 ADDTREE, 10-51, 10-53 EXTREE, 10-31 four point condition, 10-29 ADDTREE, 10-28, 10-51, 10-53 adjusted R 2, 8-7 ALSCAL, 10-49 ANCOVA, 9-1 assumptions, 9-5 example, 9-7 MANOVA
More informationST4241 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= 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 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 informationThe One-Way Independent-Samples ANOVA. (For Between-Subjects Designs)
The One-Way Independent-Samples ANOVA (For Between-Subjects Designs) Computations for the ANOVA In computing the terms required for the F-statistic, we won t explicitly compute any sample variances or
More informationIntroduction 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 informationparameter space Θ, depending only on X, such that Note: it is not θ that is random, but the set C(X).
4. Interval estimation The goal for interval estimation is to specify the accurary of an estimate. A 1 α confidence set for a parameter θ is a set C(X) in the parameter space Θ, depending only on X, such
More informationUnit 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 informationNonparametric tests. Timothy Hanson. Department of Statistics, University of South Carolina. Stat 704: Data Analysis I
1 / 16 Nonparametric tests Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I Nonparametric one and two-sample tests 2 / 16 If data do not come from a normal
More informationDr. 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 informationWorkshop Research Methods and Statistical Analysis
Workshop Research Methods and Statistical Analysis Session 2 Data Analysis Sandra Poeschl 08.04.2013 Page 1 Research process Research Question State of Research / Theoretical Background Design Data Collection
More informationProbabilistic Index Models
Probabilistic Index Models Jan De Neve Department of Data Analysis Ghent University M3 Storrs, Conneticut, USA May 23, 2017 Jan.DeNeve@UGent.be 1 / 37 Introduction 2 / 37 Introduction to Probabilistic
More informationStatistical 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 informationParametric 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 informationBasic Statistical Analysis
indexerrt.qxd 8/21/2002 9:47 AM Page 1 Corrected index pages for Sprinthall Basic Statistical Analysis Seventh Edition indexerrt.qxd 8/21/2002 9:47 AM Page 656 Index Abscissa, 24 AB-STAT, vii ADD-OR rule,
More informationAdaptive Treatment Selection with Survival Endpoints
Adaptive Treatment Selection with Survival Endpoints Gernot Wassmer Institut für Medizinische Statisti, Informati und Epidemiologie Universität zu Köln Joint wor with Marus Roters, Omnicare Clinical Research,
More informationCDA Chapter 3 part II
CDA Chapter 3 part II Two-way tables with ordered classfications Let u 1 u 2... u I denote scores for the row variable X, and let ν 1 ν 2... ν J denote column Y scores. Consider the hypothesis H 0 : X
More informationST4241 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 informationStatistics for EES Factorial analysis of variance
Statistics for EES Factorial analysis of variance Dirk Metzler June 12, 2015 Contents 1 ANOVA and F -Test 1 2 Pairwise comparisons and multiple testing 6 3 Non-parametric: The Kruskal-Wallis Test 9 1 ANOVA
More informationOne-way ANOVA Model Assumptions
One-way ANOVA Model Assumptions STAT:5201 Week 4: Lecture 1 1 / 31 One-way ANOVA: Model Assumptions Consider the single factor model: Y ij = µ + α }{{} i ij iid with ɛ ij N(0, σ 2 ) mean structure random
More informationRank-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 informationExtending the Robust Means Modeling Framework. Alyssa Counsell, Phil Chalmers, Matt Sigal, Rob Cribbie
Extending the Robust Means Modeling Framework Alyssa Counsell, Phil Chalmers, Matt Sigal, Rob Cribbie One-way Independent Subjects Design Model: Y ij = µ + τ j + ε ij, j = 1,, J Y ij = score of the ith
More information1.0 Hypothesis Testing
.0 Hypothesis Testing The Six Steps of Hypothesis Testing Rejecting or Accepting the Null Samples and Data. Parametric Tests for Comparing Two Populations Testing Equality of Population Means (T Test!
More informationLecture 10: Non- parametric Comparison of Loca6on. GENOME 560, Spring 2015 Doug Fowler, GS
Lecture 10: Non- parametric Comparison of Loca6on GENOME 560, Spring 2015 Doug Fowler, GS (dfowler@uw.edu) 1 Review What do we mean by nonparametric? What is a desirable localon stalslc for ordinal data?
More informationNonparametric 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 informationNONPARAMETRICS. Statistical Methods Based on Ranks E. L. LEHMANN HOLDEN-DAY, INC. McGRAW-HILL INTERNATIONAL BOOK COMPANY
NONPARAMETRICS Statistical Methods Based on Ranks E. L. LEHMANN University of California, Berkeley With the special assistance of H. J. M. D'ABRERA University of California, Berkeley HOLDEN-DAY, INC. San
More informationLecture 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 informationAn Application of the Closed Testing Principle to Enhance One-Sided Confidence Regions for a Multivariate Location Parameter
An Application of the Closed Testing Principle to Enhance One-Sided Confidence Regions for a Multivariate Location Parameter Universität Bern Institut für mathematische Statistik und Versicherungslehre
More informationLecture 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 informationHypothesis Testing One Sample Tests
STATISTICS Lecture no. 13 Department of Econometrics FEM UO Brno office 69a, tel. 973 442029 email:jiri.neubauer@unob.cz 12. 1. 2010 Tests on Mean of a Normal distribution Tests on Variance of a Normal
More informationData 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 informationSAS/STAT 14.1 User s Guide. Introduction to Nonparametric Analysis
SAS/STAT 14.1 User s Guide Introduction to Nonparametric Analysis This document is an individual chapter from SAS/STAT 14.1 User s Guide. The correct bibliographic citation for this manual is as follows:
More informationNonparametric Methods
Nonparametric Methods Marc H. Mehlman marcmehlman@yahoo.com University of New Haven Nonparametric Methods, or Distribution Free Methods is for testing from a population without knowing anything about the
More informationStat 5101 Lecture Notes
Stat 5101 Lecture Notes Charles J. Geyer Copyright 1998, 1999, 2000, 2001 by Charles J. Geyer May 7, 2001 ii Stat 5101 (Geyer) Course Notes Contents 1 Random Variables and Change of Variables 1 1.1 Random
More informationHypothesis Testing. Hypothesis: conjecture, proposition or statement based on published literature, data, or a theory that may or may not be true
Hypothesis esting Hypothesis: conjecture, proposition or statement based on published literature, data, or a theory that may or may not be true Statistical Hypothesis: conjecture about a population parameter
More informationAnalysis 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 informationN Utilization of Nursing Research in Advanced Practice, Summer 2008
University of Michigan Deep Blue deepblue.lib.umich.edu 2008-07 536 - Utilization of ursing Research in Advanced Practice, Summer 2008 Tzeng, Huey-Ming Tzeng, H. (2008, ctober 1). Utilization of ursing
More informationAN IMPROVEMENT TO THE ALIGNED RANK STATISTIC
Journal of Applied Statistical Science ISSN 1067-5817 Volume 14, Number 3/4, pp. 225-235 2005 Nova Science Publishers, Inc. AN IMPROVEMENT TO THE ALIGNED RANK STATISTIC FOR TWO-FACTOR ANALYSIS OF VARIANCE
More informationHANDBOOK OF APPLICABLE MATHEMATICS
HANDBOOK OF APPLICABLE MATHEMATICS Chief Editor: Walter Ledermann Volume VI: Statistics PART A Edited by Emlyn Lloyd University of Lancaster A Wiley-Interscience Publication JOHN WILEY & SONS Chichester
More informationComparison of two samples
Comparison of two samples Pierre Legendre, Université de Montréal August 009 - Introduction This lecture will describe how to compare two groups of observations (samples) to determine if they may possibly
More informationANOVA - 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 informationSimultaneous identifications of the minimum effective dose in each of several groups
Journal of Statistical Computation and Simulation Vol. 77, No. 2, February 2007, 149 161 Simultaneous identifications of the minimum effective dose in each of several groups SHOW-LI JAN*, YUH-ING CHEN
More informationTwo-stage k-sample designs for the ordered alternative problem
Two-stage k-sample designs for the ordered alternative problem Guogen Shan, Alan D. Hutson, and Gregory E. Wilding Department of Biostatistics,University at Buffalo, Buffalo, NY 14214, USA July 18, 2011
More informationMultiple Endpoints: A Review and New. Developments. Ajit C. Tamhane. (Joint work with Brent R. Logan) Department of IE/MS and Statistics
1 Multiple Endpoints: A Review and New Developments Ajit C. Tamhane (Joint work with Brent R. Logan) Department of IE/MS and Statistics Northwestern University Evanston, IL 60208 ajit@iems.northwestern.edu
More informationChapter 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 informationTABLE OF CONTENTS CHAPTER 1 COMBINATORIAL PROBABILITY 1
TABLE OF CONTENTS CHAPTER 1 COMBINATORIAL PROBABILITY 1 1.1 The Probability Model...1 1.2 Finite Discrete Models with Equally Likely Outcomes...5 1.2.1 Tree Diagrams...6 1.2.2 The Multiplication Principle...8
More informationIntroduction to Statistical Analysis
Introduction to Statistical Analysis Changyu Shen Richard A. and Susan F. Smith Center for Outcomes Research in Cardiology Beth Israel Deaconess Medical Center Harvard Medical School Objectives Descriptive
More informationThis paper has been submitted for consideration for publication in Biometrics
BIOMETRICS, 1 10 Supplementary material for Control with Pseudo-Gatekeeping Based on a Possibly Data Driven er of the Hypotheses A. Farcomeni Department of Public Health and Infectious Diseases Sapienza
More informationAnalysis of Variance (ANOVA)
Analysis of Variance (ANOVA) Much of statistical inference centers around the ability to distinguish between two or more groups in terms of some underlying response variable y. Sometimes, there are but
More informationEmpirical Power of Four Statistical Tests in One Way Layout
International Mathematical Forum, Vol. 9, 2014, no. 28, 1347-1356 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.47128 Empirical Power of Four Statistical Tests in One Way Layout Lorenzo
More information1 Statistical inference for a population mean
1 Statistical inference for a population mean 1. Inference for a large sample, known variance Suppose X 1,..., X n represents a large random sample of data from a population with unknown mean µ and known
More information