(Foundation of Medical Statistics)
|
|
- Cathleen Terry
- 5 years ago
- Views:
Transcription
1 (Foundation of Medical Statistics) ( ) 4. ANOVA and the multiple comparisons 26/10/2018 Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
2 Analysis of variance (ANOVA) Consider more than 2 groups populations Ω 1, Ω 2,..., Ω m, m 3 whose means are µ 1, µ 2,..., µ m. Then 1 Null hypothesis (H 0 ) : µ 1 = µ 2 = = µ n. 2 Alternative hypothesis (H 1 ) : µ i µ j for some i and j. This test is called the (one-way) analysis of variance, ANOVA. Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
3 Analysis of variance (ANOVA) The two-way analysis of variance when there are two factors, and the multi-way analysis of variance when there are three or more factors. These are not treated here. Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
4 Assumption Suppose that the factor X is divided into m levels X 1,..., X m. Each X i follows a normal distribution. The variances are equal. Remark (1) When m = 2, the one-way ANOVA is equivalent to the t test (2) Equality of variances can be verified with the Bartlett test or the Levene test. Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
5 Table of data Table: Ex Gp size data total T mean X i var. V i level X 1 n 1 x 11 x 12 x 1n1 T 1 x 1 V 1 X 2 n 2 x 21 x 22 x 2n2 T 2 x 2 V X m n m x m1 x m2 x mnm T m x m V m tolal N T The mean of all data is x = T N. Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
6 Sum of squared deviation between groups Let S T be the sum of squared deviation with respect to the total mean x. S T = (x i j x) 2 i, j The sum of squared deviation between groups S A is defined by S A = m n i ( x i x) 2 i=1 Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
7 Sum of squared deviation within a group It is considered that if S A increases, then the difference between means of groups also increases. The sum of squared deviation within a group Sum of squares of errors S E is defined by S E = m n i (x i j x i ) 2 i=1 j=1 = (n 1 1)V (n m 1)V m Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
8 Degrees of freedom Theorem S T = S A + S E. Variances and the degrees of freedom the degrees of freedom of S A is ϕ A = m 1. the degrees of freedom of S E is ϕ E = N m. the degrees of freedom of S T is ϕ T = N 1. Variance V A = S A ϕ A, V E = S E ϕ E (variance of errors Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
9 Ratio of variances and F-distribution Let F 0 = V A V E. Fact F 0 follows F-distribution of degrees of freedom (ϕ A, ϕ E ) Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
10 Decision When F 0 F ϕ A ϕ E (α), p-value α (H 0 ) : µ 1 = = µ m is rejected. Hence µ i µ j for some i, j. When F 0 < F ϕ A ϕ E (α), p-value > α (H 0 ) : µ 1 = = µ m can not be rejected. Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
11 Using EZR Read ANOVA.csv into EZR. 1 Show the boxplots of groups Verify the equality of variances by the Bartlett test. 3 3 Perform the one-way ANOVA. 3 Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
12 Remark 1 When the normality or the equality of variances are not satisfied,, use Kruskal-Wallis test a nonparametric version of analysis of variance EZR: 3 R: kruskal.test ( list(data1, data2, data3,... )) Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
13 Remark 2 In the case of 2-way ANOVA (repeat), the effect of the two factors X, Y and the interaction X Y of X, Y can be tested. EZR: Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
14 Multiple comparison problem By the above test, the null hypothesis has been rejected. Thus it turns out that some population mean is different from the others. Question Which two population means differ? ANOVA does not answer this question. Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
15 Misuse of t test To see that, it seems to be necessary to repeat the t test for all pairs. But such a treatment should not be doing. Why? Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
16 Because... If there are 4 populations, we need to do 4 C 2 = 6 tests. Assuming that the reliability of a single t test is 95%, the total reliability of 6 times t tests is cb % = 73.5% Thus the total reliability is lower than 95%. Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
17 Multiple comparisons Bonferroni method Various multiple comparison methods have been posed to avoid such difficulties. Bonferroni correction Taking the significance level to be smaller, in order to guarantee the reliability of 95% even when repeating the t test. Since (1 α) n 1 nα, if we take the siginificance level to be α/n, after n times t tests, the total significance level is less than α. Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
18 Ex If n = 6 and α = 0.05, we may perform 6 times t tests under the significance level = Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
19 Multiple comparisons Holm method Boferroni s method is conservative, i.e., if n is larger, power is lower since α/n is very small. There is the Holm s method improved the Bonferroni method. Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
20 Multiple comparisons Holm metho Repeat the t test at the significance level α and n times, and arrange the resulting p values (which the software will output) in ascending order p 1 < p 2 < < p n. Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
21 Procedure 1 If p 1 < α/n, p 1 is significant. 2 If p 2 < α/(n 1), p 2 is significant 3 If p 3 < α/(n 2), p 3 is significant and so on. 4 If p k α/(n k + 1) for the first time, p k,..., p n are not significant. Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
22 Multiple comparisons Tukey-Kramer method A method to compare all pairs of m groups in one test. Assumption Each group follows a normal distribution. The variances are equal. Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
23 Using EZR Read ANOVA.csv into EZR. Perform the Tukey-Kramer method 3 Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
24 Using EZR Result: The simultaneous confidence intervals are displayed. As a result, there is a difference between group 1 and group 3, also group 1 and group 4 Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
25 Multiple comparisons Dunnet method A method of comparison between the control group X 1 and each of the other groups X 2,..., X m (there are m 1 combinations.) Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
26 Multiple comparisons nonparametric methods When the normality is not satisfied, use nonparametric methods. Assumption It is assumed that the distributions of all groups are the same shape. The sample size of each group is large (10 or more in each group). Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
27 Nonparametric methods all pair comparisons Steel-Dwass method pair comparisons between the control group and the other groups Steel method These methods are found in 3 Reference (Japanese) Math and Stat in Medical Sciences Basic Statistics 26/10/ / 27
13: Additional ANOVA Topics
13: Additional ANOVA Topics Post hoc comparisons Least squared difference The multiple comparisons problem Bonferroni ANOVA assumptions Assessing equal variance When assumptions are severely violated Kruskal-Wallis
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 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 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 Seven: Multi-Sample Methods 1/52
Chapter Seven: Multi-Sample Methods 1/52 7.1 Introduction 2/52 Introduction The independent samples t test and the independent samples Z test for a difference between proportions are designed to analyze
More informationy ˆ i = ˆ " T u i ( i th fitted value or i th fit)
1 2 INFERENCE FOR MULTIPLE LINEAR REGRESSION Recall Terminology: p predictors x 1, x 2,, x p Some might be indicator variables for categorical variables) k-1 non-constant terms u 1, u 2,, u k-1 Each u
More informationMore about Single Factor Experiments
More about Single Factor Experiments 1 2 3 0 / 23 1 2 3 1 / 23 Parameter estimation Effect Model (1): Y ij = µ + A i + ɛ ij, Ji A i = 0 Estimation: µ + A i = y i. ˆµ = y..  i = y i. y.. Effect Modell
More informationLec 3: Model Adequacy Checking
November 16, 2011 Model validation Model validation is a very important step in the model building procedure. (one of the most overlooked) A high R 2 value does not guarantee that the model fits the data
More informationTable 1: Fish Biomass data set on 26 streams
Math 221: Multiple Regression S. K. Hyde Chapter 27 (Moore, 5th Ed.) The following data set contains observations on the fish biomass of 26 streams. The potential regressors from which we wish to explain
More informationI i=1 1 I(J 1) j=1 (Y ij Ȳi ) 2. j=1 (Y j Ȳ )2 ] = 2n( is the two-sample t-test statistic.
Serik Sagitov, Chalmers and GU, February, 08 Solutions chapter Matlab commands: x = data matrix boxplot(x) anova(x) anova(x) Problem.3 Consider one-way ANOVA test statistic For I = and = n, put F = MS
More information13: Additional ANOVA Topics. Post hoc Comparisons
13: Additional ANOVA Topics Post hoc Comparisons ANOVA Assumptions Assessing Group Variances When Distributional Assumptions are Severely Violated Post hoc Comparisons In the prior chapter we used ANOVA
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 informationComparing the means of more than two groups
Comparing the means of more than two groups Chapter 15 Analysis of variance (ANOVA) Like a t-test, but can compare more than two groups Asks whether any of two or more means is different from any other.
More informationAnalysis of variance (ANOVA) ANOVA. Null hypothesis for simple ANOVA. H 0 : Variance among groups = 0
Analysis of variance (ANOVA) ANOVA Comparing the means of more than two groups Like a t-test, but can compare more than two groups Asks whether any of two or more means is different from any other. In
More informationLinear Combinations of Group Means
Linear Combinations of Group Means Look at the handicap example on p. 150 of the text. proc means data=mth567.disability; class handicap; var score; proc sort data=mth567.disability; by handicap; proc
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 informationReview: General Approach to Hypothesis Testing. 1. Define the research question and formulate the appropriate null and alternative hypotheses.
1 Review: Let X 1, X,..., X n denote n independent random variables sampled from some distribution might not be normal!) with mean µ) and standard deviation σ). Then X µ σ n In other words, X is approximately
More informationWhat is a Hypothesis?
What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population mean Example: The mean monthly cell phone bill in this city is μ = $42 population proportion Example:
More informationSTAT 135 Lab 9 Multiple Testing, One-Way ANOVA and Kruskal-Wallis
STAT 135 Lab 9 Multiple Testing, One-Way ANOVA and Kruskal-Wallis Rebecca Barter April 6, 2015 Multiple Testing Multiple Testing Recall that when we were doing two sample t-tests, we were testing the equality
More 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 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 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 informationLecture Slides for INTRODUCTION TO. Machine Learning. ETHEM ALPAYDIN The MIT Press,
Lecture Slides for INTRODUCTION TO Machine Learning ETHEM ALPAYDIN The MIT Press, 2004 alpaydin@boun.edu.tr http://www.cmpe.boun.edu.tr/~ethem/i2ml CHAPTER 14: Assessing and Comparing Classification Algorithms
More informationCh 2: Simple Linear Regression
Ch 2: Simple Linear Regression 1. Simple Linear Regression Model A simple regression model with a single regressor x is y = β 0 + β 1 x + ɛ, where we assume that the error ɛ is independent random component
More informationIEOR165 Discussion Week 12
IEOR165 Discussion Week 12 Sheng Liu University of California, Berkeley Apr 15, 2016 Outline 1 Type I errors & Type II errors 2 Multiple Testing 3 ANOVA IEOR165 Discussion Sheng Liu 2 Type I errors & Type
More informationWeek 14 Comparing k(> 2) Populations
Week 14 Comparing k(> 2) Populations Week 14 Objectives Methods associated with testing for the equality of k(> 2) means or proportions are presented. Post-testing concepts and analysis are introduced.
More informationA posteriori multiple comparison tests
A posteriori multiple comparison tests 11/15/16 1 Recall the Lakes experiment Source of variation SS DF MS F P Lakes 58.000 2 29.400 8.243 0.006 Error 42.800 12 3.567 Total 101.600 14 The ANOVA tells us
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 information9 One-Way Analysis of Variance
9 One-Way Analysis of Variance SW Chapter 11 - all sections except 6. The one-way analysis of variance (ANOVA) is a generalization of the two sample t test to k 2 groups. Assume that the populations of
More informationMathematical statistics
November 15 th, 2018 Lecture 21: The two-sample t-test Overview Week 1 Week 2 Week 4 Week 7 Week 10 Week 14 Probability reviews Chapter 6: Statistics and Sampling Distributions Chapter 7: Point Estimation
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 informationStatistics for Managers Using Microsoft Excel Chapter 10 ANOVA and Other C-Sample Tests With Numerical Data
Statistics for Managers Using Microsoft Excel Chapter 10 ANOVA and Other C-Sample Tests With Numerical Data 1999 Prentice-Hall, Inc. Chap. 10-1 Chapter Topics The Completely Randomized Model: One-Factor
More informationOutline. Topic 19 - Inference. The Cell Means Model. Estimates. Inference for Means Differences in cell means Contrasts. STAT Fall 2013
Topic 19 - Inference - Fall 2013 Outline Inference for Means Differences in cell means Contrasts Multiplicity Topic 19 2 The Cell Means Model Expressed numerically Y ij = µ i + ε ij where µ i is the theoretical
More informationDegrees of freedom df=1. Limitations OR in SPSS LIM: Knowing σ and µ is unlikely in large
Z Test Comparing a group mean to a hypothesis T test (about 1 mean) T test (about 2 means) Comparing mean to sample mean. Similar means = will have same response to treatment Two unknown means are different
More 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 informationz and t tests for the mean of a normal distribution Confidence intervals for the mean Binomial tests
z and t tests for the mean of a normal distribution Confidence intervals for the mean Binomial tests Chapters 3.5.1 3.5.2, 3.3.2 Prof. Tesler Math 283 Fall 2018 Prof. Tesler z and t tests for mean Math
More informationChapter 24. Comparing Means
Chapter 4 Comparing Means!1 /34 Homework p579, 5, 7, 8, 10, 11, 17, 31, 3! /34 !3 /34 Objective Students test null and alternate hypothesis about two!4 /34 Plot the Data The intuitive display for comparing
More informationThe entire data set consists of n = 32 widgets, 8 of which were made from each of q = 4 different materials.
One-Way ANOVA Summary The One-Way ANOVA procedure is designed to construct a statistical model describing the impact of a single categorical factor X on a dependent variable Y. Tests are run to determine
More informationLAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES- Part 2
LAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES- Part 2 Data Analysis: The mean egg masses (g) of the two different types of eggs may be exactly the same, in which case you may be tempted to accept
More information2 Hand-out 2. Dr. M. P. M. M. M c Loughlin Revised 2018
Math 403 - P. & S. III - Dr. McLoughlin - 1 2018 2 Hand-out 2 Dr. M. P. M. M. M c Loughlin Revised 2018 3. Fundamentals 3.1. Preliminaries. Suppose we can produce a random sample of weights of 10 year-olds
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 informationMultiple Pairwise Comparison Procedures in One-Way ANOVA with Fixed Effects Model
Biostatistics 250 ANOVA Multiple Comparisons 1 ORIGIN 1 Multiple Pairwise Comparison Procedures in One-Way ANOVA with Fixed Effects Model When the omnibus F-Test for ANOVA rejects the null hypothesis that
More informationIntroduction. Chapter 8
Chapter 8 Introduction In general, a researcher wants to compare one treatment against another. The analysis of variance (ANOVA) is a general test for comparing treatment means. When the null hypothesis
More informationSTAT 263/363: Experimental Design Winter 2016/17. Lecture 1 January 9. Why perform Design of Experiments (DOE)? There are at least two reasons:
STAT 263/363: Experimental Design Winter 206/7 Lecture January 9 Lecturer: Minyong Lee Scribe: Zachary del Rosario. Design of Experiments Why perform Design of Experiments (DOE)? There are at least two
More informationANOVA Situation The F Statistic Multiple Comparisons. 1-Way ANOVA MATH 143. Department of Mathematics and Statistics Calvin College
1-Way ANOVA MATH 143 Department of Mathematics and Statistics Calvin College An example ANOVA situation Example (Treating Blisters) Subjects: 25 patients with blisters Treatments: Treatment A, Treatment
More informationDesign of Experiments. Factorial experiments require a lot of resources
Design of Experiments Factorial experiments require a lot of resources Sometimes real-world practical considerations require us to design experiments in specialized ways. The design of an experiment is
More informationStatistics For Economics & Business
Statistics For Economics & Business Analysis of Variance In this chapter, you learn: Learning Objectives The basic concepts of experimental design How to use one-way analysis of variance to test for differences
More informationSTA2601. Tutorial letter 203/2/2017. Applied Statistics II. Semester 2. Department of Statistics STA2601/203/2/2017. Solutions to Assignment 03
STA60/03//07 Tutorial letter 03//07 Applied Statistics II STA60 Semester Department of Statistics Solutions to Assignment 03 Define tomorrow. university of south africa QUESTION (a) (i) The normal quantile
More informationMultiple comparisons - subsequent inferences for two-way ANOVA
1 Multiple comparisons - subsequent inferences for two-way ANOVA the kinds of inferences to be made after the F tests of a two-way ANOVA depend on the results if none of the F tests lead to rejection of
More informationAnalysis of variance
Analysis of variance 1 Method If the null hypothesis is true, then the populations are the same: they are normal, and they have the same mean and the same variance. We will estimate the numerical value
More information12.10 (STUDENT CD-ROM TOPIC) CHI-SQUARE GOODNESS- OF-FIT TESTS
CDR4_BERE601_11_SE_C1QXD 1//08 1:0 PM Page 1 110: (Student CD-ROM Topic) Chi-Square Goodness-of-Fit Tests CD1-1 110 (STUDENT CD-ROM TOPIC) CHI-SQUARE GOODNESS- OF-FIT TESTS In this section, χ goodness-of-fit
More informationLinear models and their mathematical foundations: Simple linear regression
Linear models and their mathematical foundations: Simple linear regression Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/21 Introduction
More informationLec 1: An Introduction to ANOVA
Ying Li Stockholm University October 31, 2011 Three end-aisle displays Which is the best? Design of the Experiment Identify the stores of the similar size and type. The displays are randomly assigned to
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 informationReview for Final. Chapter 1 Type of studies: anecdotal, observational, experimental Random sampling
Review for Final For a detailed review of Chapters 1 7, please see the review sheets for exam 1 and. The following only briefly covers these sections. The final exam could contain problems that are included
More informationPerformance Evaluation and Comparison
Outline Hong Chang Institute of Computing Technology, Chinese Academy of Sciences Machine Learning Methods (Fall 2012) Outline Outline I 1 Introduction 2 Cross Validation and Resampling 3 Interval Estimation
More information1. What does the alternate hypothesis ask for a one-way between-subjects analysis of variance?
1. What does the alternate hypothesis ask for a one-way between-subjects analysis of variance? 2. What is the difference between between-group variability and within-group variability? 3. What does between-group
More informationCHAPTER 13: F PROBABILITY DISTRIBUTION
CHAPTER 13: F PROBABILITY DISTRIBUTION continuous probability distribution skewed to the right variable values on horizontal axis are 0 area under the curve represents probability horizontal asymptote
More information7.2 One-Sample Correlation ( = a) Introduction. Correlation analysis measures the strength and direction of association between
7.2 One-Sample Correlation ( = a) Introduction Correlation analysis measures the strength and direction of association between variables. In this chapter we will test whether the population correlation
More informationDifference between means - t-test /25
Difference between means - t-test 1 Discussion Question p492 Ex 9-4 p492 1-3, 6-8, 12 Assume all variances are not equal. Ignore the test for variance. 2 Students will perform hypothesis tests for two
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 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 informationIntroduction to Analysis of Variance (ANOVA) Part 2
Introduction to Analysis of Variance (ANOVA) Part 2 Single factor Serpulid recruitment and biofilms Effect of biofilm type on number of recruiting serpulid worms in Port Phillip Bay Response variable:
More information22s:152 Applied Linear Regression. Chapter 8: 1-Way Analysis of Variance (ANOVA) 2-Way Analysis of Variance (ANOVA)
22s:152 Applied Linear Regression Chapter 8: 1-Way Analysis of Variance (ANOVA) 2-Way Analysis of Variance (ANOVA) We now consider an analysis with only categorical predictors (i.e. all predictors are
More informationCorrelation Analysis
Simple Regression Correlation Analysis Correlation analysis is used to measure strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the
More informationAssignment #7. Chapter 12: 18, 24 Chapter 13: 28. Due next Friday Nov. 20 th by 2pm in your TA s homework box
Assignment #7 Chapter 12: 18, 24 Chapter 13: 28 Due next Friday Nov. 20 th by 2pm in your TA s homework box Lab Report Posted on web-site Dates Rough draft due to TAs homework box on Monday Nov. 16 th
More informationTukey Complete Pairwise Post-Hoc Comparison
Tukey Complete Pairwise Post-Hoc Comparison Engineering Statistics II Section 10.2 Josh Engwer TTU 2018 Josh Engwer (TTU) Tukey Complete Pairwise Post-Hoc Comparison 2018 1 / 23 PART I PART I: Gosset s
More informationChap The McGraw-Hill Companies, Inc. All rights reserved.
11 pter11 Chap Analysis of Variance Overview of ANOVA Multiple Comparisons Tests for Homogeneity of Variances Two-Factor ANOVA Without Replication General Linear Model Experimental Design: An Overview
More information9-6. Testing the difference between proportions /20
9-6 Testing the difference between proportions 1 Homework Discussion Question p514 Ex 9-6 p514 2, 3, 4, 7, 9, 11 (use both the critical value and p-value for all problems. 2 Objective Perform hypothesis
More informationChapter 12 - Lecture 2 Inferences about regression coefficient
Chapter 12 - Lecture 2 Inferences about regression coefficient April 19th, 2010 Facts about slope Test Statistic Confidence interval Hypothesis testing Test using ANOVA Table Facts about slope In previous
More informationM A N O V A. Multivariate ANOVA. Data
M A N O V A Multivariate ANOVA V. Čekanavičius, G. Murauskas 1 Data k groups; Each respondent has m measurements; Observations are from the multivariate normal distribution. No outliers. Covariance matrices
More informationPROBLEM TWO (ALKALOID CONCENTRATIONS IN TEA) 1. Statistical Design
PROBLEM TWO (ALKALOID CONCENTRATIONS IN TEA) 1. Statistical Design The purpose of this experiment was to determine differences in alkaloid concentration of tea leaves, based on herb variety (Factor A)
More informationT.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS
ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS In our work on hypothesis testing, we used the value of a sample statistic to challenge an accepted value of a population parameter. We focused only
More informationThe Chi-Square Distributions
MATH 03 The Chi-Square Distributions Dr. Neal, Spring 009 The chi-square distributions can be used in statistics to analyze the standard deviation of a normally distributed measurement and to test the
More informationSTAT22200 Spring 2014 Chapter 5
STAT22200 Spring 2014 Chapter 5 Yibi Huang April 29, 2014 Chapter 5 Multiple Comparisons Chapter 5-1 Chapter 5 Multiple Comparisons Note the t-tests and C.I. s are constructed assuming we only do one test,
More informationMachine Learning: Evaluation
Machine Learning: Evaluation Information Systems and Machine Learning Lab (ISMLL) University of Hildesheim Wintersemester 2007 / 2008 Comparison of Algorithms Comparison of Algorithms Is algorithm A better
More informationData Analysis and Statistical Methods Statistics 651
Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Suhasini Subba Rao Motivations for the ANOVA We defined the F-distribution, this is mainly used in
More informationApplication of Variance Homogeneity Tests Under Violation of Normality Assumption
Application of Variance Homogeneity Tests Under Violation of Normality Assumption Alisa A. Gorbunova, Boris Yu. Lemeshko Novosibirsk State Technical University Novosibirsk, Russia e-mail: gorbunova.alisa@gmail.com
More informationChapter 7 Comparison of two independent samples
Chapter 7 Comparison of two independent samples 7.1 Introduction Population 1 µ σ 1 1 N 1 Sample 1 y s 1 1 n 1 Population µ σ N Sample y s n 1, : population means 1, : population standard deviations N
More 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 informationThe Chi-Square Distributions
MATH 183 The Chi-Square Distributions Dr. Neal, WKU The chi-square distributions can be used in statistics to analyze the standard deviation σ of a normally distributed measurement and to test the goodness
More informationUsing SPSS for One Way Analysis of Variance
Using SPSS for One Way Analysis of Variance This tutorial will show you how to use SPSS version 12 to perform a one-way, between- subjects analysis of variance and related post-hoc tests. This tutorial
More information1 One-way Analysis of Variance
1 One-way Analysis of Variance Suppose that a random sample of q individuals receives treatment T i, i = 1,,... p. Let Y ij be the response from the jth individual to be treated with the ith treatment
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 information5 Inferences about a Mean Vector
5 Inferences about a Mean Vector In this chapter we use the results from Chapter 2 through Chapter 4 to develop techniques for analyzing data. A large part of any analysis is concerned with inference that
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 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 informationNonparametric tests, Bootstrapping
Nonparametric tests, Bootstrapping http://www.isrec.isb-sib.ch/~darlene/embnet/ Hypothesis testing review 2 competing theories regarding a population parameter: NULL hypothesis H ( straw man ) ALTERNATIVEhypothesis
More informationChapter 11 - Lecture 1 Single Factor ANOVA
April 5, 2013 Chapter 9 : hypothesis testing for one population mean. Chapter 10: hypothesis testing for two population means. What comes next? Chapter 9 : hypothesis testing for one population mean. Chapter
More informationInstitute of Actuaries of India
Institute of Actuaries of India Subject CT3 Probability & Mathematical Statistics May 2011 Examinations INDICATIVE SOLUTION Introduction The indicative solution has been written by the Examiners with the
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 informationAquatic Toxicology Lab 10 Pimephales promelas Larval Survival and Growth Test Data Analysis 1. Complete test initiated last week 1.
Aquatic Toxicology Lab 10 Pimephales promelas Larval Survival and Growth Test Data Analysis 1. Complete test initiated last week 1. make day 7 observations 2. prepare fish for drying 3. weigh aluminum
More informationNotes for Week 13 Analysis of Variance (ANOVA) continued WEEK 13 page 1
Notes for Wee 13 Analysis of Variance (ANOVA) continued WEEK 13 page 1 Exam 3 is on Friday May 1. A part of one of the exam problems is on Predictiontervals : When randomly sampling from a normal population
More informationYou can compute the maximum likelihood estimate for the correlation
Stat 50 Solutions Comments on Assignment Spring 005. (a) _ 37.6 X = 6.5 5.8 97.84 Σ = 9.70 4.9 9.70 75.05 7.80 4.9 7.80 4.96 (b) 08.7 0 S = Σ = 03 9 6.58 03 305.6 30.89 6.58 30.89 5.5 (c) You can compute
More informationAssumptions of classical multiple regression model
ESD: Recitation #7 Assumptions of classical multiple regression model Linearity Full rank Exogeneity of independent variables Homoscedasticity and non autocorrellation Exogenously generated data Normal
More informationA discussion on multiple regression models
A discussion on multiple regression models In our previous discussion of simple linear regression, we focused on a model in which one independent or explanatory variable X was used to predict the value
More informationANOVA: Analysis of Variance
ANOVA: Analysis of Variance Marc H. Mehlman marcmehlman@yahoo.com University of New Haven The analysis of variance is (not a mathematical theorem but) a simple method of arranging arithmetical facts so
More informationStat 710: Mathematical Statistics Lecture 41
Stat 710: Mathematical Statistics Lecture 41 Jun Shao Department of Statistics University of Wisconsin Madison, WI 53706, USA Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, 2009 1 / 10 Lecture 41: One-way
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 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 information