DOCSLIB.ORG

Appendix: Contingency Table Analysis in Practice

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendix: Contingency Table Analysis in Practice Appendix A Appendix: Contingency Table Analysis in Practice A.1 Software for Categorical Data Analysis The free software R, for statistical computing and graphics, is of increasing popularity and usage (R web site: http://www.r-project.org/). Many researchers support their published papers with the related R code. This way, R software is continuously updated and one can find a variety of functions for basic or advanced analysis of categorical data and special types of them. R language and environment is similar to S and code written for S-Plus runs usually under R as well. Furthermore, standard statistical packages, such as SAS, SPSS, and Stata, are well supplied to treat categorical data. Especially in their updated versions, their features concerning categorical data analysis are enriched. They incorporate procedures for applying the recently developed methods and models in categorical data analysis following the new computing strategies. Briefly, one could say that their major new features concern mainly options for exact analysis and analysis of repeated categorical data. Thus, NLMIXED of SAS fits generalized linear mixed models while GEE analysis for marginal models can be performed in GENMOD. SPSS offers the “generalized estimating equations” sub-option under the “GLMs” option. The related R function is gee(). For categorical data analysis with SAS, we refer to Stokes et al. (2012) while a variety of SAS codes are presented and discussed in the Appendix of Agresti (2007, 2013). Advanced models are fitted in R using special functions, developed individually, and included in different libraries. Orientated toward categorical data analysis and models for ordinal data as well are the libraries MASS (Venables and Replay) and VGLM, VGAM developed by Yee (2008). For example, generalized linear mixed models can be fitted through the glmmPQL() function of the MASS library. Other software, as BMDP, Minitab, and SYSTAT, have also components for categorical data inference. M. Kateri, Contingency Table Analysis: Methods and Implementation Using R, 271 Statistics for Industry and Technology, DOI 10.1007/978-0-8176-4811-4, © Springer Science+Business Media New York 2014 272 A Appendix: Contingency Table Analysis in Practice Bayesian analysis of categorical data can be carried out through WINBUGS (http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/contents.shtml), which is a free soft- ware. Another option is to perform categorical data analysis through MATLAB, as Johnson and Albert (2000). The MATLAB functions they used are described in their Appendix. For categorical data analysis, there have been developed also some special pack- ages. Thus, exact analysis of categorical data is performed by StatXact while exact conditional logistic regression can be fitted by LogXact. SUDAAN is specialized for analysis of mixed data from stratified multistage cluster designs. It has also the feature of analyzing marginal models for nominal and ordinal responses by GEE. Software tool for estimating marginal regression models is also MAREG. Finally, some algorithms may be found in Fortran. For example, Haberman (1995) provided a Fortran program for fitting the association model RC(K)by the Newton–Raphson method while Ait-Sidi-Allal et al. (2004) implemented their algorithms for estimating parameters in association and correlation models also in Fortran. A.2 Contingency Table Analysis with R All procedures and models discussed in this book are worked out in R,inafashion aiming that even readers not familiar with R will be able to apply in practice all the models discussed here, even the nontrivial ones, fast and directly. A web companion of the book serves this goal. This section of the Appendix is basically the content description of the web companion of the book, to be found under http://cta.isw.rwth-aachen.de A.2.1 R Packages for Contingency Table Analysis An extensive list of special R packages, useful in the analysis of contingency tables, is provided in the web appendix. A.2.2 Data Input in R Alternative forms of defining contingency tables data in R are presented (matrix(), array(),anddata.frame()) and transformations from one type to another are illustrated. Ways of entering or reading data are discussed. A.3 R Functions Used 273 A.3 R Functions Used The R functions constructed for the descriptive and inferential needs of this book are given in the corresponding section of the web appendix, organized by chapter of their first use. A.3.1 R Functions of Chap.1 • Binomial–Normal Distribution Graph: bin_norm( ) A.3.2 R Functions of Chap.2 • Likelihood Ratio Statistic for Testing Independence in Two-way Contingency Tables: G2( ) • Odds Ratio for a 2 × 2Table:odds.ratio( ) • Local Odds Ratios for an I × J Table: local.odds.DM( ) • Global Odds Ratios for an I × J Table: global.odds.DM( ) • Cumulative Odds Ratios for an I × J Table: cum.odds.DM( ) • Continuation Odds Ratios for an I × J Table: cont.odds.DM( ) • Linear Trend Test: linear.trend( ) • Midrank Scores Computation: midrank( ) • Fourfold Plots for the Local Odds Ratios of an I × J Table: ffold.local( ) A.3.3 R Functions of Chap.3 • Breslow–Day–Tarone Test of Homogeneous Association: BDT( ) • Woolf’s Test of Homogeneous Association: woolf( ) A.3.4 R Functions of Chap.5 • Independence (I) Model for Two-way Contingency Tables: fit.I( ) • Quasi-Independence (QI) Model for Two-way Contingency Tables: fit.QI( ) 274 A Appendix: Contingency Table Analysis in Practice A.3.5 R Functions of Chap.6 • Scores’ Rescaling to Obey the Weighted Constraints (6.17): rescale( ) • Uniform (U) Association Model: fit.U( ) • Row Effect (R) Association Model: fit.R( ) • Column Effect (C) Association Model: fit.C( ) • Row–Column (RC) Association Model: fit.RC( ) • RC(M) Association Model: fit.RCm( ) • Plotting the Row and Column Scores in Two Dimensions: plot_2dim( ) A.3.6 R Functions of Chap.9 • (1 − a)100% Asymptotic Confidence Interval for the Difference of Correlated Proportions: McNemar.CI( ) • Factors Needed to Fit Symmetry Models on an I × I Table in glm: SYMV( ) • Scores’ Rescaling to Satisfy Constraints (9.38): rescale.square( ) A.4 Contingency Table Analysis with SPSS The association and symmetry models cannot be fitted directly in SPSS through the options of the windows commands. Association models that are GLM can be fitted through the GLM option by defining the appropriate vectors, as explained in Sect. 6.6. For all two-way association models (RC(M) included, which is nonlinear and thus cannot be fitted in GLM) and the symmetry models, we provide appropriate syntax codes to be fitted in SPSS MATRIX. In particular, we provide MATRIX codes for: • Independence for two-way tables using SPSS MATRIX • Association models for two-way tables (uniform (U), row effect (R), column effect (C), and RC (M) association models) • Symmetry models References Agresti, A.: Generalized odds ratios for ordinal data. Biometrics 36, 59–67 (1980) Agresti, A.: A survey of strategies for modeling cross–classifications having ordinal variables. J. Am. Stat. Assoc. 78, 184–198 (1983a) Agresti, A.: A simple diagonals–parameters symmetry and quasi-symmetry model. Stat. Probab. Lett. 1, 313–316 (1983b) Agresti, A.: Testing marginal homogeneity for ordinal categorical variables. Biometrics 39, 505–510 (1983c) Agresti, A.: Applying R2-type measures to ordered categorical data. Technometrics 28, 133–138 (1986) Agresti, A.: A model for agreement between ratings on an ordinal scale. Biometrics 44, 539–548 (1988) Agresti, A.: A survey of exact inference for contingency tables (with discussion). Stat. Sci. 7, 131–177 (1992) Agresti, A.: Computing conditional maximum likelihood estimates for generalized Rasch models using simple loglinear models with diagonal parameters. Scand. J. Stat. 20, 63–71 (1993) Agresti, A.: On logit confidence intervals for the odds ratio with small samples. Biometrics 55, 597–602 (1999) Agresti, A.: Exact inference for categorical data: recent advances and continuing controversies. Stat. Med. 20, 2709–2722 (2001) Agresti, A.: Dealing with discreteness: making ‘exact’ confidence intervals for proportions, differences of proportions, and odds ratios more exact. Stat. Meth. Med. Res. 12, 3–21 (2003) Agresti, A:. An Introduction to Categorical Data Analysis, 2nd edn. Wiley, New York (2007) Agresti, A.: Analysis of Ordinal Categorical Data, 2nd edn. Wiley, New York (2010) Agresti, A.: Categorical Data Analysis, 3rd edn. Wiley, Hoboken (2013) Agresti, A., Chuang, C.: Model-based Bayesian methods for estimating cell proportions in cross- classification tables having ordered categories. Comput. Stat. Data Anal. 7, 245–258 (1989) Agresti, A., Coull, B.A.: Approximate is better than ‘exact’ for interval estimation of binomial proportions. Am. Stat. 52, 119–126 (1998) Agresti, A., Coull, B.A.: The analysis of contingency tables under inequality constraints. J. Stat. Plann. Infer. 107, 45–73 (2002) Agresti, A., Hartzel, J.: Strategies for comparing treatments on a binary response with multi-centre data. Stat. Med. 19, 1115–1139 (2000) Agresti, A., Hitchcock, D.B.: Bayesian inference for categorical data analysis. Stat. Methods Appl. 14, 297–330 (2005) Agresti, A., Kateri, M.: Some remarks on latent variable models in categorical data analysis. Commun. Stat. Theory Meth. 43, 1–14 (2014) M. Kateri, Contingency Table Analysis: Methods and Implementation Using R, 275 Statistics for Industry and Technology, DOI 10.1007/978-0-8176-4811-4, © Springer Science+Business Media New York 2014 276 References Agresti, A., Kezouh, A.: Association models for multidimensional cross-classifications of ordinal variables. Commun. Stat. Ser. A 12, 1261–1276 (1983) Agresti, A., Lang, J.B.: Quasi-symmetric latent class models, with application to rater agreement. Biometrics 49, 131–139 (1993) Agresti, A., Min, Y.: On small-sample confidence intervals for parameters in discrete distributions. Biometrics 57, 963–971 (2001) Agresti, A., Min, Y.: Unconditional small-sample confidence intervals for the odds ratio. Biostatis- tics 3, 379–386 (2002) Agresti, A., Yang, M.: An empirical investigation of some effects of sparseness in contingency tables.
Load more

Recommended publications
  • 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]
  • 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]
  • Linear Information Models: an Introduction
    Journal of Data Science 5(2007), 297-313 Linear Information Models: An Introduction Philip E. Cheng, Jiun W. Liou, Michelle Liou and John A. D. Aston Academia Sinica Abstract: Relative entropy identities yield basic decompositions of cat- egorical data log-likelihoods. These naturally lead to the development of information models in contrast to the hierarchical log-linear models. A recent study by the authors clarified the principal difference in the data likelihood analysis between the two model types. The proposed scheme of log-likelihood decomposition introduces a prototype of linear information models, with which a basic scheme of model selection can be formulated accordingly. Empirical studies with high-way contingency tables are exem- plified to illustrate the natural selections of information models in contrast to hierarchical log-linear models. Key words: Contingency tables, log-linear models, information models, model selection, mutual information. 1. Introduction Analysis of contingency tables with multi-way classifications has been a fun- damental area of research in the history of statistics. From testing hypothesis of independence in a 2 × 2 table (Pearson, 1904; Yule, 1911; Fisher, 1934; Yates, 1934) to testing interaction across a strata of 2 × 2 tables (Bartlett, 1935), many discussions had emerged in the literature to build up the foundation of statisti- cal inference of categorical data analysis. In this vital field of applied statistics, three closely related topics have gradually developed and are still theoretically incomplete even after half a century of investigation. The first and primary topic is born out of the initial hypothesis testing for independence in a 2×2 table.
    [Show full text]
  • Three-Way Contingency Tables
    Newsom Psy 525/625 Categorical Data Analysis, Spring 2021 1 Three-Way Contingency Tables Three-way contingency tables involve three binary or categorical variables. I will stick mostly to the binary case to keep things simple, but we can have three-way tables with any number of categories with each variable. Of course, higher dimensions are also possible, but they are uncommon in practice and there are few commonly available statistical tests for them. Notation We described two-way contingency tables earlier as having I rows and J columns, forming an I × J matrix. If a third variable is included, the three-way contingency table is described as I × J × K, where I is the number of rows, J is the number of columns, and K is the number of superordinate columns each level containing a 2 × 2 matrix. The indexes i, j, and k are used to enumerate the rows or columns for each dimension. Each within-cell count or joint proportion has three subscripts now, nijk and pijk. There are two general types of marginal counts (or proportions), with three specific marginals for each type. The first type of marginal count combines one of the three dimensions. Using the + notation from two- way tables, the marginals combining counts across either I, J, or K dimensions are n+jk, ni+k, or nij+, respectively. The second general type of marginal collapses two dimensions and is given as ni++, n+j+, or n++k. The total count (sample size) is n+++ (or sometimes just n). The simplest case is a 2 × 2 × 2 shown below to illustrate the notation.
    [Show full text]