DOCSLIB.ORG

Methods for Meta–Analyses of Rare Events, Sparse Data, and Heterogeneity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Methods for Meta–Analyses of Rare Events, Sparse Data, and Heterogeneity Utah State University DigitalCommons@USU All Graduate Theses and Dissertations Graduate Studies 5-2019 Methods for Meta–Analyses of Rare Events, Sparse Data, and Heterogeneity Brinley Zabriskie Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/etd Part of the Mathematics Commons Recommended Citation Zabriskie, Brinley, "Methods for Meta–Analyses of Rare Events, Sparse Data, and Heterogeneity" (2019). All Graduate Theses and Dissertations. 7491. https://digitalcommons.usu.edu/etd/7491 This Dissertation is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. METHODS FOR META{ANALYSES OF RARE EVENTS, SPARSE DATA, AND HETEROGENEITY by Brinley Zabriskie A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mathematical Sciences (Statistics Specialization) Approved: Chris Corcoran, Sc.D. Adele Cutler, Ph.D. Major Professor Committee Member John Stevens, Ph.D. David Brown, Ph.D. Committee Member Committee Member Tyler Brough, Ph.D. Richard S. Inouye, Ph.D. Committee Member Vice Provost for Graduate Studies UTAH STATE UNIVERSITY Logan, Utah 2019 ii Copyright © Brinley Zabriskie 2019 All Rights Reserved iii ABSTRACT Methods for Meta{Analyses of Rare Events, Sparse Data, and Heterogeneity by Brinley Zabriskie, Doctor of Philosophy Utah State University, 2019 Major Professor: Dr. Chris Corcoran Department: Mathematics and Statistics The increasingly widespread use of meta{analysis has led to growing interest in meta{analytic methods for rare events and sparse data. Conventional approaches tend to perform very poorly in such settings. Recent work in this area has provided options for sparse data, but these are still often hampered when heterogeneity across the available studies differs based on treatment group. We propose several new exact methods that accommodate this common contingency, providing more reliable statistical tests when such patterns on heterogeneity are observed. First, we develop a permutation{based approach that can also be used as a basis for computing exact confidence intervals when estimating the effect size. Second, we extend the permutation{based approach to the network meta{analysis setting. Third, we develop a new exact confidence distribution approach for effect size estimation. We show these new methods perform markedly better than traditional methods when events are rare, and heterogeneity is present. (264 pages) iv PUBLIC ABSTRACT Methods for Meta{Analyses of Rare Events, Sparse Data, and Heterogeneity Brinley Zabriskie The vast and complex wealth of information available to researchers often leads to a systematic review, which involves a detailed and comprehensive plan and search strategy with the goal of identifying, appraising, and synthesizing all relevant studies on a particular topic. A meta{analysis, conducted ideally as part of a comprehensive systematic review, statistically synthesizes evidence from multiple independent studies to produce one overall conclusion. The increasingly widespread use of meta{analysis has led to growing interest in meta{analytic methods for rare events and sparse data. Conventional approaches tend to perform very poorly in such settings. Recent work in this area has provided options for sparse data, but these are still often hampered when heterogeneity across the available studies differs based on treatment group. Heterogeneity arises when participants in a study are more correlated than participants across studies, often stemming from differences in the administration of the treatment, study design, or measurement of the outcome. We propose several new exact methods that accommodate this common contingency, providing more reliable statistical tests when such patterns on heterogeneity are observed. First, we develop a permutation{based approach that can also be used as a basis for computing exact confidence intervals when estimating the effect size. Second, we extend the permutation{based approach to the network meta{analysis setting. Third, we develop a new exact confidence distribution approach for effect size estimation. We show these new methods perform markedly better than traditional methods when events are rare, and heterogeneity is present. v ACKNOWLEDGMENTS I would like to first thank Dr. Chris Corcoran for providing me the opportunity to pursue this degree and for his time, advice, and encouragement. I am also grateful for the teaching opportunities he gave me throughout my time at Utah State University. Dr. Corcoran has been very influential in my career development. I would also like to thank my committee members, Drs. Adele Cutler, John Stevens, Dave Brown, and Tyler Brough for their time, input, and assistance. Additionally, the faculty, staff, and students in the Department of Mathematics and Statistics have made my time here enjoyable and memorable. I would also like to thank the co{founders of Cytel Inc., Drs. Cyrus Mehta and Nitin Patel, whose pioneering advancements made this work possible. I greatly appreciate Dr. Pralay Senchaudhuri, Senior Vice President for Research & Development at Cytel Inc., who has taken the time to provide invaluable advice throughout this process. I would also like to thank software specialist Sumit Singh who provided technical support. This work would also not have been possible without the Center for High Performance Computing at the University of Utah. I am grateful for their resources and technical assistance. Lastly, I am grateful for my family for their assistance, support, and encouragement. Brinley Zabriskie vi CONTENTS Page ABSTRACT......................................... iii PUBLIC ABSTRACT.................................... iv ACKNOWLEDGMENTS.................................. v LIST OF TABLES...................................... ix LIST OF FIGURES..................................... x 1 INTRODUCTION TO META{ANALYSIS METHODS FOR RARE EVENTS AND HETEROGENEITY................................. 1 1.1 Background....................................1 1.2 Common Statistical Methods for Meta{Analyses...............4 1.3 Common Statistical Methods for Meta{Analyses of Rare Events......8 1.4 Novel Statistical Methods for Meta{Analyses of Rare Events........ 13 1.5 Summary of the Remaining Chapters..................... 16 2 EXACT META{ANALYSIS FOR RARE EVENTS AND HETEROGENEITY . 18 2.1 Background.................................... 18 2.2 Methodology................................... 19 2.2.1 Exact Conditional Logistic Regression................. 19 2.2.2 Exact Conditional Logistic Regression for Correlated Data..... 22 2.2.3 Exact Estimation Method........................ 26 2.2.4 Network Algorithm............................ 29 2.3 Innovation.................................... 34 2.3.1 Exact Test: Variation of Treatment Level Within Study....... 34 2.3.2 Exact Test: Variation of Correlation Structure Among Treatment Levels 36 2.3.3 Network Algorithm: Variation of Treatment Level Within Study.. 38 2.3.4 Network Algorithm: Variation of Correlation Structure Among Treat- ment Levels.............................. 40 2.4 Application.................................... 41 2.4.1 Simulation Study............................. 42 vii 2.4.1.1 Type I Error........................... 44 2.4.1.2 Bias................................ 46 2.4.1.3 Confidence Interval Coverage.................. 49 2.4.2 Illustrative Examples........................... 51 2.4.2.1 Stomach Ulcers......................... 51 2.4.2.2 Antibiotics............................ 54 2.4.3 Network Algorithm............................ 57 2.4.3.1 Original Exact Test for Correlated Data........... 57 2.4.3.2 Variation of Treatment Level Within Study.......... 68 2.4.3.3 Variation of Correlation Structure Among Treatment Levels 68 2.5 Conclusion.................................... 72 3 EXACT NETWORK META{ANALYSIS....................... 76 3.1 Background.................................... 76 3.1.1 Introduction and Terminology...................... 76 3.1.2 Basic Methods.............................. 80 3.1.3 Assumptions and Validity Considerations............... 82 3.2 Methodology and Innovation.......................... 83 3.3 Application.................................... 87 3.4 Conclusion.................................... 92 4 COMBINING CONFIDENCE DISTRIBUTIONS FOR RARE EVENT META{ ANALYSIS...................................... 93 4.1 Background.................................... 93 4.2 Methodology................................... 100 4.2.1 Confidence Distribution Definition................... 100 4.2.2 Combining Confidence Intervals as in Tian et al.(2009)....... 101 4.2.3 Combining p{value Functions as in Liu et al.(2014)......... 103 4.2.4 Combining p{values as in Fisher(1932)................ 105 4.3 Innovation.................................... 106 4.3.1 Modification 1: Use the Logit Function Under the General CD Frame- work without Weights........................ 107 4.3.2 Modification 2: Use the Logit Function Under the General CD Frame- work with Weights Defined by Liu et al.(2014).......... 108 viii 4.3.3 Modification 3: Use the Tian et al.(2009) Framework with Weights Defined by Liu et al.(2014)..................... 109 4.3.4 Summary................................. 109 4.4 Application.................................... 109 4.4.1 Simulation Results...........................
Load more

Recommended publications
  • Some General Points on the -Measure of Heterogeneity in Meta-Analysis
    Metrika DOI 10.1007/s00184-017-0622-3 Some general points on the I2-measure of heterogeneity in meta-analysis Dankmar Böhning1 · Rattana Lerdsuwansri2 · Heinz Holling3 Received: 27 January 2017 © The Author(s) 2017. This article is an open access publication Abstract Meta-analysis has developed to be a most important tool in evaluation research. Heterogeneity is an issue that is present in almost any meta-analysis. How- ever, the magnitude of heterogeneity differs across meta-analyses. In this respect, Higgins’ I 2 has emerged to be one of the most used and, potentially, one of the most useful measures as it provides quantification of the amount of heterogeneity involved in a given meta-analysis. Higgins’ I 2 is conventionally interpreted, in the sense of a variance component analysis, as the proportion of total variance due to heterogeneity. However, this interpretation is not entirely justified as the second part involved in defin- ing the total variation, usually denoted as s2, is not an average of the study-specific variances, but in fact some other function of the study-specific variances. We show that s2 is asymptotically identical to the harmonic mean of the study-specific variances and, for any number of studies, is at least as large as the harmonic mean with the inequality being sharp if all study-specific variances agree. This justifies, from our point of view, the interpretation of explained variance, at least for meta-analyses with larger number B Dankmar Böhning [email protected] Rattana Lerdsuwansri [email protected] Heinz Holling [email protected] 1 Southampton Statistical Sciences Research Institute and Mathematical Sciences, University of Southampton, Southampton, UK 2 Department of Mathematics and Statistics, Faculty of Science, Thammasat University, Rangsit Campus, Pathumthani, Thailand 3 Statistics and Quantitative Methods, Faculty of Psychology and Sports Science, University of Münster, Münster, Germany 123 D.
    [Show full text]
  • Contingency Tables Are Eaten by Large Birds of Prey
    Case Study Case Study Example 9.3 beginning on page 213 of the text describes an experiment in which fish are placed in a large tank for a period of time and some Contingency Tables are eaten by large birds of prey. The fish are categorized by their level of parasitic infection, either uninfected, lightly infected, or highly infected. It is to the parasites' advantage to be in a fish that is eaten, as this provides Bret Hanlon and Bret Larget an opportunity to infect the bird in the parasites' next stage of life. The observed proportions of fish eaten are quite different among the categories. Department of Statistics University of Wisconsin|Madison Uninfected Lightly Infected Highly Infected Total October 4{6, 2011 Eaten 1 10 37 48 Not eaten 49 35 9 93 Total 50 45 46 141 The proportions of eaten fish are, respectively, 1=50 = 0:02, 10=45 = 0:222, and 37=46 = 0:804. Contingency Tables 1 / 56 Contingency Tables Case Study Infected Fish and Predation 2 / 56 Stacked Bar Graph Graphing Tabled Counts Eaten Not eaten 50 40 A stacked bar graph shows: I the sample sizes in each sample; and I the number of observations of each type within each sample. 30 This plot makes it easy to compare sample sizes among samples and 20 counts within samples, but the comparison of estimates of conditional Frequency probabilities among samples is less clear. 10 0 Uninfected Lightly Infected Highly Infected Contingency Tables Case Study Graphics 3 / 56 Contingency Tables Case Study Graphics 4 / 56 Mosaic Plot Mosaic Plot Eaten Not eaten 1.0 0.8 A mosaic plot replaces absolute frequencies (counts) with relative frequencies within each sample.
    [Show full text]
  • Use of Chi-Square Statistics
    This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this site. Copyright 2008, The Johns Hopkins University and Marie Diener-West. All rights reserved. Use of these materials permitted only in accordance with license rights granted. Materials provided “AS IS”; no representations or warranties provided. User assumes all responsibility for use, and all liability related thereto, and must independently review all materials for accuracy and efficacy. May contain materials owned by others. User is responsible for obtaining permissions for use from third parties as needed. Use of the Chi-Square Statistic Marie Diener-West, PhD Johns Hopkins University Section A Use of the Chi-Square Statistic in a Test of Association Between a Risk Factor and a Disease The Chi-Square ( X2) Statistic Categorical data may be displayed in contingency tables The chi-square statistic compares the observed count in each table cell to the count which would be expected under the assumption of no association between the row and column classifications The chi-square statistic may be used to test the hypothesis of no association between two or more groups, populations, or criteria Observed counts are compared to expected counts 4 Displaying Data in a Contingency Table Criterion 2 Criterion 1 1 2 3 . C Total . 1 n11 n12 n13 n1c r1 2 n21 n22 n23 . n2c r2 . 3 n31 . r nr1 nrc rr Total c1 c2 cc n 5 Chi-Square Test Statistic
    [Show full text]
  • Knowledge and Human Capital As Sustainable Competitive Advantage in Human Resource Management
    sustainability Article Knowledge and Human Capital as Sustainable Competitive Advantage in Human Resource Management Miloš Hitka 1 , Alžbeta Kucharˇcíková 2, Peter Štarcho ˇn 3 , Žaneta Balážová 1,*, Michal Lukáˇc 4 and Zdenko Stacho 5 1 Faculty of Wood Sciences and Technology, Technical University in Zvolen, T. G. Masaryka 24, 960 01 Zvolen, Slovakia; [email protected] 2 Faculty of Management Science and Informatics, University of Žilina, Univerzitná 8215/1, 010 26 Žilina, Slovakia; [email protected] 3 Faculty of Management, Comenius University in Bratislava, Odbojárov 10, P.O. BOX 95, 82005 Bratislava, Slovakia; [email protected] 4 Faculty of Social Sciences, University of SS. Cyril and Methodius in Trnava, Buˇcianska4/A, 917 01 Trnava, Slovakia; [email protected] 5 Institut of Civil Society, University of SS. Cyril and Methodius in Trnava, Buˇcianska4/A, 917 01 Trnava, Slovakia; [email protected] * Correspondence: [email protected]; Tel.: +421-45-520-6189 Received: 2 August 2019; Accepted: 9 September 2019; Published: 12 September 2019 Abstract: The ability to do business successfully and to stay on the market is a unique feature of each company ensured by highly engaged and high-quality employees. Therefore, innovative leaders able to manage, motivate, and encourage other employees can be a great competitive advantage of an enterprise. Knowledge of important personality factors regarding leadership, incentives and stimulus, systematic assessment, and subsequent motivation factors are parts of human capital and essential conditions for effective development of its potential. Familiarity with various ways to motivate leaders and their implementation in practice are important for improving the work performance and reaching business goals.
    [Show full text]
  • Pearson-Fisher Chi-Square Statistic Revisited
    Information 2011 , 2, 528-545; doi:10.3390/info2030528 OPEN ACCESS information ISSN 2078-2489 www.mdpi.com/journal/information Communication Pearson-Fisher Chi-Square Statistic Revisited Sorana D. Bolboac ă 1, Lorentz Jäntschi 2,*, Adriana F. Sestra ş 2,3 , Radu E. Sestra ş 2 and Doru C. Pamfil 2 1 “Iuliu Ha ţieganu” University of Medicine and Pharmacy Cluj-Napoca, 6 Louis Pasteur, Cluj-Napoca 400349, Romania; E-Mail: [email protected] 2 University of Agricultural Sciences and Veterinary Medicine Cluj-Napoca, 3-5 M ănăş tur, Cluj-Napoca 400372, Romania; E-Mails: [email protected] (A.F.S.); [email protected] (R.E.S.); [email protected] (D.C.P.) 3 Fruit Research Station, 3-5 Horticultorilor, Cluj-Napoca 400454, Romania * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel: +4-0264-401-775; Fax: +4-0264-401-768. Received: 22 July 2011; in revised form: 20 August 2011 / Accepted: 8 September 2011 / Published: 15 September 2011 Abstract: The Chi-Square test (χ2 test) is a family of tests based on a series of assumptions and is frequently used in the statistical analysis of experimental data. The aim of our paper was to present solutions to common problems when applying the Chi-square tests for testing goodness-of-fit, homogeneity and independence. The main characteristics of these three tests are presented along with various problems related to their application. The main problems identified in the application of the goodness-of-fit test were as follows: defining the frequency classes, calculating the X2 statistic, and applying the χ2 test.
    [Show full text]
  • Measures of Association for Contingency Tables
    Newsom Psy 525/625 Categorical Data Analysis, Spring 2021 1 Measures of Association for Contingency Tables The Pearson chi-squared statistic and related significance tests provide only part of the story of contingency table results. Much more can be gleaned from contingency tables than just whether the results are different from what would be expected due to chance (Kline, 2013). For many data sets, the sample size will be large enough that even small departures from expected frequencies will be significant. And, for other data sets, we may have low power to detect significance. We therefore need to know more about the strength of the magnitude of the difference between the groups or the strength of the relationship between the two variables. Phi The most common measure of magnitude of effect for two binary variables is the phi coefficient. Phi can take on values between -1.0 and 1.0, with 0.0 representing complete independence and -1.0 or 1.0 representing a perfect association. In probability distribution terms, the joint probabilities for the cells will be equal to the product of their respective marginal probabilities, Pn( ij ) = Pn( i++) Pn( j ) , only if the two variables are independent. The formula for phi is often given in terms of a shortcut notation for the frequencies in the four cells, called the fourfold table. Azen and Walker Notation Fourfold table notation n11 n12 A B n21 n22 C D The equation for computing phi is a fairly simple function of the cell frequencies, with a cross- 1 multiplication and subtraction of the two sets of diagonal cells in the numerator.
    [Show full text]
  • Chi-Square Tests
    Chi-Square Tests Nathaniel E. Helwig Associate Professor of Psychology and Statistics University of Minnesota October 17, 2020 Copyright c 2020 by Nathaniel E. Helwig Nathaniel E. Helwig (Minnesota) Chi-Square Tests c October 17, 2020 1 / 32 Table of Contents 1. Goodness of Fit 2. Tests of Association (for 2-way Tables) 3. Conditional Association Tests (for 3-way Tables) Nathaniel E. Helwig (Minnesota) Chi-Square Tests c October 17, 2020 2 / 32 Goodness of Fit Table of Contents 1. Goodness of Fit 2. Tests of Association (for 2-way Tables) 3. Conditional Association Tests (for 3-way Tables) Nathaniel E. Helwig (Minnesota) Chi-Square Tests c October 17, 2020 3 / 32 Goodness of Fit A Primer on Categorical Data Analysis In the previous chapter, we looked at inferential methods for a single proportion or for the difference between two proportions. In this chapter, we will extend these ideas to look more generally at contingency table analysis. All of these methods are a form of \categorical data analysis", which involves statistical inference for nominal (or categorial) variables. Nathaniel E. Helwig (Minnesota) Chi-Square Tests c October 17, 2020 4 / 32 Goodness of Fit Categorical Data with J > 2 Levels Suppose that X is a categorical (i.e., nominal) variable that has J possible realizations: X 2 f0;:::;J − 1g. Furthermore, suppose that P (X = j) = πj where πj is the probability that X is equal to j for j = 0;:::;J − 1. PJ−1 J−1 Assume that the probabilities satisfy j=0 πj = 1, so that fπjgj=0 defines a valid probability mass function for the random variable X.
    [Show full text]
  • Good Statistical Practices for Contemporary Meta-Analysis: Examples Based on a Systematic Review on COVID-19 in Pregnancy
    Review Good Statistical Practices for Contemporary Meta-Analysis: Examples Based on a Systematic Review on COVID-19 in Pregnancy Yuxi Zhao and Lifeng Lin * Department of Statistics, Florida State University, Tallahassee, FL 32306, USA; [email protected] * Correspondence: [email protected] Abstract: Systematic reviews and meta-analyses have been increasingly used to pool research find- ings from multiple studies in medical sciences. The reliability of the synthesized evidence depends highly on the methodological quality of a systematic review and meta-analysis. In recent years, several tools have been developed to guide the reporting and evidence appraisal of systematic reviews and meta-analyses, and much statistical effort has been paid to improve their methodological quality. Nevertheless, many contemporary meta-analyses continue to employ conventional statis- tical methods, which may be suboptimal compared with several alternative methods available in the evidence synthesis literature. Based on a recent systematic review on COVID-19 in pregnancy, this article provides an overview of select good practices for performing meta-analyses from sta- tistical perspectives. Specifically, we suggest meta-analysts (1) providing sufficient information of included studies, (2) providing information for reproducibility of meta-analyses, (3) using appro- priate terminologies, (4) double-checking presented results, (5) considering alternative estimators of between-study variance, (6) considering alternative confidence intervals, (7) reporting predic- Citation: Zhao, Y.; Lin, L. Good tion intervals, (8) assessing small-study effects whenever possible, and (9) considering one-stage Statistical Practices for Contemporary methods. We use worked examples to illustrate these good practices. Relevant statistical code Meta-Analysis: Examples Based on a is also provided.
    [Show full text]
  • 11 Basic Concepts of Testing for Independence
    11-3 Contingency Tables Basic Concepts of Testing for Independence In this section we consider contingency tables (or two-way frequency tables), which include frequency counts for A contingency table (or two-way frequency table) is a table in categorical data arranged in a table with a least two rows and which frequencies correspond to two variables. at least two columns. (One variable is used to categorize rows, and a second We present a method for testing the claim that the row and variable is used to categorize columns.) column variables are independent of each other. Contingency tables have at least two rows and at least two columns. 11 Example Definition Below is a contingency table summarizing the results of foot procedures as a success or failure based different treatments. Test of Independence A test of independence tests the null hypothesis that in a contingency table, the row and column variables are independent. Notation Hypotheses and Test Statistic O represents the observed frequency in a cell of a H0 : The row and column variables are independent. contingency table. H1 : The row and column variables are dependent. E represents the expected frequency in a cell, found by assuming that the row and column variables are ()OE− 2 independent χ 2 = ∑ r represents the number of rows in a contingency table (not E including labels). (row total)(column total) c represents the number of columns in a contingency table E = (not including labels). (grand total) O is the observed frequency in a cell and E is the expected frequency in a cell.
    [Show full text]
  • On Median Tests for Linear Hypotheses G
    ON MEDIAN TESTS FOR LINEAR HYPOTHESES G. W. BROWN AND A. M. MOOD THE RAND CORPORATION 1. Introduction and summary This paper discusses the rationale of certain median tests developed by us at Iowa State College on a project supported by the Office of Naval Research. The basic idea on which the tests rest was first presented in connection with a problem in [1], and later we extended that idea to consider a variety of analysis of variance situations and linear regression problems in general. Many of the tests were pre- sented in [2] but without much justification. And in fact the tests are not based on any solid foundation at all but merely on what appeared to us to be reasonable com- promises with the various conflicting practical interests involved. Throughout the paper we shall be mainly concerned with analysis of variance hypotheses. The methods could be discussed just as well in terms of the more gen- eral linear regression problem, but the general ideas and difficulties are more easily described in the simpler case. In fact a simple two by two table is sufficient to illus- trate most of the fundamental problems. After presenting a test devised by Friedman for row and column effects in a two way classification, some median tests for the same problem will be described. Then we shall examine a few other situations, in particular the question of testing for interactions, and it is here that some real troubles arise. The asymptotic character of the tests is described in the final section of the paper.
    [Show full text]
  • Learn to Use the Phi Coefficient Measure and Test in R with Data from the Welsh Health Survey (Teaching Dataset) (2009)
    Learn to Use the Phi Coefficient Measure and Test in R With Data From the Welsh Health Survey (Teaching Dataset) (2009) © 2019 SAGE Publications, Ltd. All Rights Reserved. This PDF has been generated from SAGE Research Methods Datasets. SAGE SAGE Research Methods Datasets Part 2019 SAGE Publications, Ltd. All Rights Reserved. 2 Learn to Use the Phi Coefficient Measure and Test in R With Data From the Welsh Health Survey (Teaching Dataset) (2009) Student Guide Introduction This example dataset introduces the Phi Coefficient, which allows researchers to measure and test the strength of association between two categorical variables, each of which has only two groups. This example describes the Phi Coefficient, discusses the assumptions underlying its validity, and shows how to compute and interpret it. We illustrate the Phi Coefficient measure and test using a subset of data from the 2009 Welsh Health Survey. Specifically, we measure and test the strength of association between sex and whether the respondent has visited the dentist in the last twelve months. The Phi Coefficient can be used in its own right as a means to assess the strength of association between two categorical variables, each with only two groups. However, typically, the Phi Coefficient is used in conjunction with the Pearson’s Chi-Squared test of association in tabular analysis. Pearson’s Chi-Squared test tells us whether there is an association between two categorical variables, but it does not tell us how important, or how strong, this association is. The Phi Coefficient provides a measure of the strength of association, which can also be used to test the statistical significance (with which that association can be distinguished from zero, or no-association).
    [Show full text]
  • A Test of Independence in Two-Way Contingency Tables Based on Maximal Correlation
    A TEST OF INDEPENDENCE IN TWO-WAY CONTINGENCY TABLES BASED ON MAXIMAL CORRELATION Deniz C. YenigÄun A Dissertation Submitted to the Graduate College of Bowling Green State University in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2007 Committee: G¶abor Sz¶ekely, Advisor Maria L. Rizzo, Co-Advisor Louisa Ha, Graduate Faculty Representative James Albert Craig L. Zirbel ii ABSTRACT G¶abor Sz¶ekely, Advisor Maximal correlation has several desirable properties as a measure of dependence, includ- ing the fact that it vanishes if and only if the variables are independent. Except for a few special cases, it is hard to evaluate maximal correlation explicitly. In this dissertation, we focus on two-dimensional contingency tables and discuss a procedure for estimating maxi- mal correlation, which we use for constructing a test of independence. For large samples, we present the asymptotic null distribution of the test statistic. For small samples or tables with sparseness, we use exact inferential methods, where we employ maximal correlation as the ordering criterion. We compare the maximal correlation test with other tests of independence by Monte Carlo simulations. When the underlying continuous variables are dependent but uncorre- lated, we point out some cases for which the new test is more powerful. iii ACKNOWLEDGEMENTS I would like to express my sincere appreciation to my advisor, G¶abor Sz¶ekely, and my co-advisor, Maria Rizzo, for their advice and help throughout this research. I thank to all the members of my committee, Craig Zirbel, Jim Albert, and Louisa Ha, for their time and advice.
    [Show full text]