ENCYCLOPEDIA STATISTICS IN BEHAVIORAL SCIENCE potx

CO 80477USACATEGORICALDATAANALYSIS Alexander von Eye Department of Psychology Michigan State University East Lancing, MI USA CLASSICALTESTTHEORY/ ITEMRESPONSETHEORY/ KeeleUK DESIGN OFEXP

ENCYCLOPEDIA STATISTICS IN BEHAVIORAL

SCIENCE

OF

ENCYCLOPEDIA STATISTICS IN BEHAVIORAL

Copyright  2005 John Wiley & Sons Ltd,

This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the Publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought.

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Library of Congress Cataloguing-in-Publication Data

Encyclopedia of statistics in behavioral science / editors-in-chief, Brian S Everitt, David C Howell.

p cm.

Includes bibliographical references and index.

ISBN-13 978-0-470-86080-9 (hardcover : set)

ISBN-10 0-470-86080-4 (hardcover : set)

1 Psychometrics – Encyclopedias I Everitt, Brian II Howell, David C.

BF39.E498 2005

150
.1
5195 – dc22

2005004373

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN-13 978-0-470-86080-9 (HB)

ISBN-10 0-470-86080-4 (HB)

Typeset in 9 1/2/11 1/2 pt Times by Laserwords Private Limited, Chennai, India.

Printed and bound in Great Britain by Antony Rowe, Chippenham, UK.

This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which

at least two trees are planted for each one used for paper production.

Mary-Elizabeth and Donna

CO 80477USA

CATEGORICALDATAANALYSIS

Alexander von Eye

Department of Psychology

Michigan State University

East Lancing, MI

USA

CLASSICALTESTTHEORY/

ITEMRESPONSETHEORY/

KeeleUK

DESIGN OFEXPERIMENTS ANDSURVEYS

Roger Kirk

Department of Psychology andNeuroscience

Institute of StatisticsBaylor UniversityWaco, TXUSA

FACTORANALYSIS ANDSTRUCTURAL

EQUATIONMODELS

David Rindskopf

Educational PsychologyCUNY Graduate CenterNew York, NY

USA

viii Editorial Board

SCALING

Jan de Leeuw

Department of StatisticsUniversity of CaliforniaLos Angeles, CAUSA

STATISTICALMODELS

Jose Cortina

Department of PsychologyGeorge Mason UniversityFairfax, VA

USA

STATISTICALTHEORY

Ranald Macdonald

Department of PsychologyUniversity of StirlingStirling

Conway, ARUSA

Additive Constant Problem 16

Additive Genetic Variance 18

Analysis of Variance: Classification 66

Analysis of Variance: Multiple

Estimation 134Bayesian Methods for Categorical

Bayesian Statistics 146Bernoulli Family 150Binomial Confidence Interval 153Binomial Distribution: Estimating

and Testing Parameters 155Binomial Effect Size Display 157Binomial Test 158

Block Random Assignment 165

Bootstrap Inference 169Box Plots 176Bradley – Terry Model 178Breslow – Day Statistic 184Brown, William 186Bubble Plot 187Burt, Cyril Lodowic 187Bush, Robert R 189Calculating Covariance 191Campbell, Donald T 191Canonical Correlation Analysis 192Carroll – Arabie Taxonomy 196Carryover and Sequence Effects 197Case Studies 201Case – Cohort Studies 204Case – Control Studies 206Catalogue of Parametric Tests 207Catalogue of Probability Density

Functions 228Catastrophe Theory 234Categorizing Data 239

Central Limit Theory 249

Children of Twins Design 256

Classical Test Models 278

Classical Test Score Equating 282

Classification and Regression Trees 287

Common Pathway Model 330

Community Intervention Studies 331

Confidence Intervals: Nonparametric 375

Configural Frequency Analysis 381

Confounding in the Analysis of

Variance 389

Confounding Variable 391Contingency Tables 393Coombs, Clyde Hamilton 397Correlation 398Correlation and Covariance Matrices 400Correlation Issues in Genetics

Research 402Correlation Studies 403Correspondence Analysis 404Co-twin Control Methods 415Counterbalancing 418Counterfactual Reasoning 420Counternull Value of an Effect Size 422Covariance 423Covariance Matrices: Testing

Equality of 424Covariance Structure Models 426Covariance/variance/correlation 431Cox, Gertrude Mary 432Cram´er – von Mises Test 434Criterion-Referenced Assessment 435Critical Region 440Cross-classified and Multiple

Membership Models 441Cross-lagged Panel Design 450Crossover Design 451Cross-sectional Design 453Cross-validation 454Cultural Transmission 457

Data Mining 461

de Finetti, Bruno 465

de Moivre, Abraham 466Decision Making Strategies 466Deductive Reasoning and StatisticalInference 472DeFries – Fulker Analysis 475Demand Characteristics 477Deming, Edwards William 478Design Effects 479Development of Statistical Theory

in the 20th Century 483Differential Item Functioning 485Direct and Indirect Effects 490Direct Maximum Likelihood

Estimation 492Directed Alternatives in Testing 495Direction of Causation Models 496Discriminant Analysis 499

Dropouts in Longitudinal Data 515

Dropouts in Longitudinal Studies:

Methods of Analysis 518

Dummy Variables 522

Ecological Fallacy 525

Educational Psychology: Measuring

Change Over Time 527

Effect Size Measures 532

Event History Analysis 568

Exact Methods for Categorical Data 575

Expectancy Effect by Experimenters 581

Factor Analysis: Confirmatory 599

Factor Analysis: Exploratory 606

Factor Analysis: Multiple Groups 617

Finite Mixture Distributions 652

Fisher, Sir Ronald Aylmer 658

Fisherian Tradition in BehavioralGenetics 660Fixed and Random Effects 664Fixed Effect Models 665Focus Group Techniques 666Free Response Data Scoring 669Friedman’s Test 673Functional Data Analysis 675Fuzzy Cluster Analysis 678Galton, Francis 687Game Theory 688Gauss, Johann Carl Friedrich 694Gene-Environment Correlation 696Gene-Environment Interaction 698Generalizability 702Generalizability Theory: Basics 704Generalizability Theory: Estimation 711Generalizability Theory: Overview 717Generalized Additive Model 719Generalized Estimating Equations

Variables 749Gosset, William Sealy 753Graphical Chain Models 755Graphical Methods pre-20th Century 758Graphical Presentation of Longitudinal

Growth Curve Modeling 772Guttman, Louise (Eliyahu) 780Harmonic Mean 783Hawthorne Effect 784Heritability 786Heritability: Overview 787Heteroscedasticity and Complex

Variation 790Heuristics 795Heuristics: Fast and Frugal 795Hierarchical Clustering 799Hierarchical Item Response Theory

Modeling 805Hierarchical Models 810High-dimensional Regression 816

xii Contents

Hill’s Criteria of Causation 818

Histogram 820

Historical Controls 821

History of Analysis of Variance 823

History of Behavioral Statistics 826

History of the Control Group 829

History of Correlational Measurement 836

History of Discrimination and

History of Intelligence Measurement 858

History of Mathematical Learning

Incomplete Contingency Tables 899

Incompleteness of Probability Models 900

Independence: Chi-square and

Likelihood Ratio Tests 902

Independent Component Analysis 907

Independent Pathway Model 913

Classical Approaches 967Item Bias Detection:

Modern Approaches 970Item Exposure 974Item Response Theory (IRT): Cognitive

Item Response Theory (IRT) Modelsfor Dichotomous Data 982Item Response Theory (IRT) Modelsfor Polytomous Response Data 990Item Response Theory (IRT) Modelsfor Rating Scale Data 995Jackknife 1005Jonckheere – Terpstra Test 1007Kendall, Maurice George 1009Kendall’s Coefficient of

Concordance 1010

Kendall’s Tau – τ 1011Kernel Smoothing 1012

k-means Analysis 1017Kolmogorov, Andrey Nikolaevich 1022Kolmogorov – Smirnov Tests 1023Kruskal – Wallis Test 1026Kurtosis 1028Laplace, Pierre Simon (Marquis de) 1031Latent Class Analysis 1032Latent Transition Models 1033Latent Variable 1036Latin Squares Designs 1037Laws of Large Numbers 1040Least Squares Estimation 1041Leverage Plot 1045Liability Threshold Models 1046Linear Model 1048Linear Models: Permutation

Methods 1049Linear Multilevel Models 1054Linear Statistical Models for Causation:

A Critical Review 1061Linkage Analysis 1073Logistic Regression 1076Log-linear Models 1082

Contents xiii

Log-linear Rasch Models for Stability

and Change 1093

Longitudinal Data Analysis 1098

Longitudinal Designs in Genetic

Marginal Models for Clustered Data 1128

Markov Chain Monte Carlo and

Bayesian Statistics 1134

Markov Chain Monte Carlo Item

Response Theory Estimation 1143

Markov, Andrei Andreevich 1148

Markov Chains 1149

Martingales 1152

Matching 1154

Mathematical Psychology 1158

Maximum Likelihood Estimation 1164

Maximum Likelihood Item Response

Mendelian Genetics Rediscovered 1198

Mendelian Inheritance and Segregation

Theory Models 1272Multidimensional Scaling 1280Multidimensional Unfolding 1289Multigraph Modeling 1294Multilevel and SEM Approaches to

Growth Curve Modeling 1296Multiple Baseline Designs 1306Multiple Comparison Procedures 1309Multiple Comparison Tests:

Nonparametric and ResamplingApproaches 1325Multiple Imputation 1331Multiple Informants 1332Multiple Linear Regression 1333Multiple Testing 1338Multitrait – Multimethod Analyses 1343Multivariate Analysis: Bayesian 1348Multivariate Analysis: Overview 1352Multivariate Analysis of Variance 1359Multivariate Genetic Analysis 1363Multivariate Multiple Regression 1370Multivariate Normality Tests 1373Multivariate Outliers 1379Neural Networks 1387Neuropsychology 1393New Item Types and Scoring 1398Neyman, Jerzy 1401Neyman – Pearson Inference 1402Nightingale, Florence 1408Nonequivalent Group Design 1410Nonlinear Mixed Effects Models 1411Nonlinear Models 1416

Nonparametric Correlation (r s) 1419Nonparametric Correlation (tau) 1420Nonparametric Item Response Theory

Nonparametric Regression 1426Nonrandom Samples 1430Nonresponse in Sample Surveys 1433Nonshared Environment 1436Normal Scores and Expected Order

Statistics 1439

Odds and Odds Ratios 1462

One Way Designs: Nonparametric and

Page’s Ordered Alternatives Test 1503

Paired Observations, Distribution Free

Methods 1505

Panel Study 1510

Paradoxes 1511

Parsimony/Occham’s Razor 1517

Partial Correlation Coefficients 1518

Partial Least Squares 1523

Path Analysis and Path Diagrams 1529

Extensions 1584Probability: Foundations of 1594Probability: An Introduction 1600Probability Plots 1605

Procrustes Analysis 1610Projection Pursuit 1614Propensity Score 1617Prospective and Retrospective

Studies 1619Proximity Measures 1621Psychophysical Scaling 1628Qualitative Research 1633Quantiles 1636Quantitative Methods in PersonalityResearch 1637Quartiles 1641Quasi-experimental Designs 1641Quasi-independence 1644Quasi-symmetry in Contingency

Nonparametric Analyses 1681Randomized Block Designs 1686Randomized Response Technique 1687

Rank Based Inference 1688Rasch Modeling 1691Rasch Models for Ordered ResponseCategories 1698Rater Agreement 1707Rater Agreement – Kappa 1712Rater Agreement – Weighted Kappa 1714

Regression Discontinuity Design 1725

Regression Model Coding for the

Residuals in Structural Equation,

Factor Analysis, and Path Analysis

Robust Testing Procedures 1768

Robustness of Standard Tests 1769

Scaling Asymmetric Matrices 1787

Scaling of Preferential Choice 1790

Issues, Methods 1820Sex-Limitation Models 1823Shannon, Claude E 1827Shared Environment 1828Shepard Diagram 1830Sibling Interaction Effects 1831Sign Test 1832Signal Detection Theory 1833Signed Ranks Test 1837Simple Random Assignment 1838Simple Random Sampling 1840Simple V Composite Tests 1841Simulation Methods for CategoricalVariables 1843Simultaneous Confidence Interval 1849Single-Case Designs 1850Single and Double-blind Procedures 1854Skewness 1855Slicing Inverse Regression 1856Snedecor, George Waddell 1863Social Interaction Models 1864Social Networks 1866Social Psychology 1871Social Validity 1875Software for Behavioral Genetics 1876Software for Statistical Analyses 1880Spearman, Charles Edward 1886Spearman’s Rho 1887Sphericity Test 1888Standard Deviation 1891Standard Error 1891Standardized Regression Coefficients 1892Stanine Scores 1893Star and Profile Plots 1893State Dependence 1895Statistical Models 1895Stem and Leaf Plot 1897Stephenson, William 1898Stevens, S S 1900Stratification 1902Structural Equation Modeling:

Categorical Variables 1905Structural Equation Modeling:

Checking SubstantivePlausibility 1910

xvi Contents

Structural Equation Modeling:

Latent Growth Curve Analysis 1912

Structural Equation Modeling:

Survey Questionnaire Design 1969

Survey Sampling Procedures 1977

Teaching Statistics to Psychologists 1994

Teaching Statistics: Sources 1997

Telephone Surveys 1999

Test Bias Detection 2001

Test Construction 2007

Test Construction: Automated 2011

Test Dimensionality: Assessment of 2014

Test Translation 2021

Tetrachoric Correlation 2027

Theil Slope Estimate 2028

Thomson, Godfrey Hilton 2030

Three Dimensional (3D) Scatterplots 2031

Three-mode Component and Scaling

Methods 2032

Thurstone, Louis Leon 2045

Time Series Analysis 2046Tolerance and Variance Inflation

Transformation 2056Tree Models 2059Trellis Graphics 2060Trend Tests for Counts and

Proportions 2063Trimmed Means 2066T-Scores 2067Tukey, John Wilder 2067Tukey Quick Test 2069Tversky, Amos 2070Twin Designs 2071Twins Reared Apart Design 2074Two by Two Contingency Tables 2076Two-mode Clustering 2081Two-way Factorial: Distribution-FreeMethods 2086Type I, Type II and Type III Sums ofSquares 2089Ultrametric Inequality 2093Ultrametric Trees 2094Unidimensional Scaling 2095Urban, F M 2097Utility Theory 2098Validity Theory and Applications 2103Variable Selection 2107Variance 2110Variance Components 2110Walsh Averages 2115Wiener, Norbert 2116Wilcoxon, Frank 2117Wilcoxon – Mann – Whitney Test 2118Winsorized Robust Measures 2121Within Case Designs: Distribution FreeMethods 2122Yates’ Correction 2127Yates, Frank 2129Yule, George Udny 2130

z Scores 2131

Ackerman, Terry A University of North

Carolina, Greensboro, NC, USA

Aiken, Leona S Arizona State University,

Tempe, AZ, USA

Airoldi, Mara London School of

Economics and Political Science,

London, UK

Andrich, David Murdoch University,

Murdoch, Western Australia, Australia

Ayton, Peter City University, London,

UK

Banks, Adrian University of Surrey,

Guildford, UK

Barrett, Paul University of Auckland,

Auckland, New Zealand

Bartholomew, David J London School

of Economics and Political Science,

Beller, Michal Educational Testing

Service, Princeton, NJ, USA

Berge, Jos M.F Ten University of

Groningen, Groningen, The Netherlands

Berger, Martijn P.F University of

Maastricht, Maastricht, The Netherlands

Berger, Vance W University of Maryland,

Baltimore, MD, USA

Besag, Julian University of Washington,

Seattle, WA, USA

Billiet, Jaak Katholieke Universiteit,

Leuven, Belgium

Bock, Hans-Hermann RWTH Aachen

University, Aachen, Germany

B¨ockenholt, Ulf McGill University,

Quebec, Montreal, Canada

Bogartz, Richard S University of

Massachusetts, Amherst, MA, USA

Bogat, G Anne Michigan State University,

East Lansing, MI, USA

Boik, Robert J Montana State University,

Bozeman, MT, USA

Boker, Steven M University of Notre

Dame, Notre Dame, IN, USA

Bolt, Daniel University of Wisconsin,

Madison, WI, USA

Bookstein, Fred L University of

Michigan, Ann Arbor, MI, USA

Boomsma, Dorret I Vrije Universiteit,

Amsterdam, The Netherlands

Borg, Ingwer Center for Survey Research

and Methodology, Mannheim, Germany

& University of Giessen, Giessen, Germany

Boulet, John R Educational Commission

for Foreign Medical Graduates, Philadelphia, PA, USA

Brockwell, P.J Colorado State University,

Fort Collins, CO, USA

Brown, Bruce L Brigham Young

University, Provo, UT, USA

Bub, Kristen L Harvard University,

Cambridge, MA, USA

Buchanan, Roderick D University of

Melbourne, Parkville, Victoria, Australia

Buchner, Axel Heinrich-Heine-Universit¨at

D¨usseldorf, D¨usseldorf, Germany

Burke, Ailbhe MRC SGDP Centre, King’s

College, London, UK

Busk, Patricia L University of San

Francisco, San Francisco, CA, USA

Byrne, Barbara M University of Ottawa,

Punta Gorda, FL, USA

xviii Contributors

Cade, Brian S Fort Collins Science

Center, U S Geological Survey, Fort

Collins, CO, USA

Canter, David University of Liverpool,

Liverpool, UK

Canty, A.J McMaster University,

Hamilton, Ontario, Canada

Cardon, Lon R University of Oxford,

Oxford, UK

Chaplin, William F St John’s University,

New York, NY, USA

Chechile, Richard A Tufts University,

Medford, MA, USA

Chen, Chun-Houh Institute of Statistical

Science, Academia Sinica, Taiwan

Chen, Gilad Texas A&M University,

College Station, TX, USA

Cherny, Stacey S University of Oxford,

Oxford, UK

Christensen, Ronald University of New

Mexico, Albuquerque, NM, USA

Clark-Carter, David Staffordshire

University, Stoke-on-Trent, UK

Clauser, Brian E National Board of

Medical Examiners, Philadelphia, PA,

Cohen, Patricia New York State

Psychiatric Institute, New York, NY, USA

Colman, Andrew M University of

Leicester, Leicester, UK

Congdon, Peter Queen Mary University of

London, London, UK

Conrad, Karen M University of Illinois,

Chicago, IL, USA

Conrad, Kendon J University of Illinois,

Chicago, IL, USA

Corcoran, Chris Utah State University,

Logan, UT, USA

Corley, R.P University of Colorado,

Boulder, CO, USA

Corter, James E Columbia University,

New York, NY, USA

Cortina, Jose George Mason University,

Fairfax, VA, USA

Cortina, Lilia M University of Michigan,

Ann Arbor, MI, USA

Cotton, John W University of California,

Santa Barbara, CA, USA

Cowles, Michael York University, Toronto,

Ontario, Canada

Cribbie, Robert A York University,

Toronto, Ontario, Canada

Cussens, James University of York, York,

UK

Cutler, Adele Utah State University,

Logan, UT, USA

D’Agostino Jr, Ralph B Wake Forest

University, Winston-Salem, NC, USA

D’Onofrio, Brian M University of

Virginia, Charlottesville, VA, USA

Daniels, Richard University of Georgia,

Athens, GA, USA

Darlington, Richard B Cornell

University, Ithaca, NY, USA

Davey, Tim Educational Testing Service,

Princeton, NJ, USA

Davison, Anthony C ´ Ecole Polytechnique F´ed´erale de Lausanne (EPFL),

Lausanne, Switzerland

de Boeck, Paul Katholieke Universiteit

Leuven, Leuven, Belgium

de Champlain, Andre F National Board

of Medical Examiners, Philadelphia, PA, USA

Dehue, Trudy University of Groningen,

Groningen, The Netherlands

Delaney, Harold D University of New

Mexico, Albuquerque, NM, USA

de Leeuw, Edith D Utrecht University,

Utrecht, The Netherlands, and MethodikA, Amsterdam, The Netherlands

de Leeuw, Jan University of California,

Los Angeles, CA, USA

de Munter, Agnes Katholieke Universiteit,

Leuven, Belgium

DePuy, Venita Duke Clinical Research

Institute, Durham, NC, USA

Dick, Danielle M Washington University

School of Medicine, St Louis, MO, USA

Dickins, David University of Liverpool,

Liverpool, UK

Donner, Allan University of Western

Ontario, London, Ontario, Canada

Doyle, James K Worcester Polytechnic

Institute, Worcester, MA, USA

Contributors xix

Dracup, Chris Northumbria University,

Newcastle, UK

Drasgow, Fritz University of Illinois,

Champaign, IL, USA

Driskell, Robyn Bateman Baylor

University, Waco, TX, USA

Durkalski, Valerie Medical University of

South Carolina, Columbia, SC, USA

Dykacz, Janice Marie Johns Hopkins

University, Baltimore, MD, USA

Eley, Thalia C King’s College, London,

London, UK

Elston, Robert C Case Western Reserve

University, Cleveland, OH, USA

Enders, Craig K University of Nebraska,

Lincoln, NE, USA

Engelhard, George Emory University,

Atlanta, GA, USA

Erdfelder, Edgar Universit¨at Mannheim,

Mannheim, Germany

Evans, David M University of Oxford,

Oxford, UK

Everitt, Brian S Institute of Psychiatry,

King’s College, London, UK

Faber, Diana University of Liverpool,

Liverpool, UK

Fabrigar, Leandre R Queen’s University,

Kingston, Ontario, Canada

Fahoome, Gail Wayne State University,

Detroit, MI, USA

Faul, Franz

Christian-Albrechts-Universit¨at, Kiel, Germany

Faux, Robert B University of Pittsburgh,

Pittsburgh, PA, USA

Ferron, John University of South Florida,

Fischer, Mary University of Washington,

Seattle, WA, USA

Fisher, Dennis G California State

University, Long Beach, CA, USA

Fitzmaurice, Garrett M Harvard School

of Public Health, Boston, MA, USA

Fox, John McMaster University, Hamilton,

Ontario, Canada

Fraley, R Chris University of Illinois,

Champaign, IL, USA

Freedman, D.A University of California,

Berkeley, CA, USA

Gable, Philip A Ouachita Baptist

University, Arkadelphia, AR, USA

Garthwaite, Paul H The Open University,

Milton Keynes, UK

Gentle, James E George Mason

University, Fairfax, VA, USA

Gessaroli, Marc E National Board of

Medical Examiners, Philadelphia, PA, USA

Gierl, Mark J University of Alberta,

Edmonton, Alberta, Canada

Gillespie, Nathan A Virginia Institute of

Psychiatric and Behavioral Genetics, Richmond, VA, USA

Glas, Cees A.W University of Twente,

Enschede, The Netherlands

Goldberg, Jack University of Washington,

Seattle, WA, USA

Goldsmith, L Jane University of

Louisville, Louisville, KY, USA

Goldstein, Harvey University of London,

Gottfredson, Gary D University of

Maryland, College Park, MD, USA

Gower, John The Open University, Milton

Keynes, UK

Greenacre, Michael Universitat Pompeu

Fabra, Barcelona, Spain

Grietens, Hans Katholieke Universiteit

Leuven, Leuven, Belgium

Groenen, Patrick J.F Erasmus University,

Rotterdam, The Netherlands

Guti´errez-Pe ˜na, Eduardo National

University of Mexico, Mexico City, Mexico

xx Contributors

Hagger-Johnson, Gareth University of

Edinburgh, Edinburgh, UK

Hambleton, Ronald K University of

Massachusetts, Amherst, MA, USA

Hand, David J Imperial College London,

London, UK

Hardin, James W University of South

Carolina, Columbia, SC, USA

Harold, Crystal M George Mason

University, Mason, VA, USA

Harris, Richard J American Society of

Radiologic Technologists, Albuquerque,

NM, USA

Hartung, Joachim University of

Dortmund, Dortmund, Germany

Hatch, John P University of Texas, San

Antonio, TX, USA

Hayes, Andrew F Ohio State University,

Columbus, OH, USA

Hazelton, Martin L University of Western

Australia, Crawley, WA, Australia

Hedeker, Donald University of Illinois at

Chicago, Chicago, IL, USA

Heerwegh, Dirk Katholieke Universiteit

Leuven, Leuven, Belgium

Henderson, Norman Oberlin College,

Oberlin, OH, USA

Hendrickson, Amy B University of

Maryland, College Park, MD, USA

Hendrix, Kent A NPS Pharmaceuticals,

Salt Lake City, UT, USA

Hendrix, Suzanne B Myriad Genetics

Inc., Provo, UT, USA

Herrera, Gina Coffee University of

Wisconsin, Madison, WI, USA

Hershberger, Scott L California State

University, Long Beach, CA, USA

Hettmansperger, T.P Pennsylvania State

University, University Park, PA, USA

Hewitt, John K University of Colorado,

Boulder, CO, USA

Hoffrage, Ulrich University of Lausanne,

Lausanne, Switzerland

Holford, Theodore Yale University, New

Haven, CT, USA

Holland, Burt Temple University,

Philadelphia, PA, USA

Holland, Paul W Educational Testing

Service, Princeton, NJ, USA

Hopper, John L University of Melbourne,

Melbourne, Victoria, Australia

Horsten, Leon Katholieke Universiteit

Leuven, Leuven, Belgium

Howell, David C University of Vermont,

Burlington, VT, USA

Hox, Joop Utrecht University, Utrecht, The

Netherlands

Huang, Guan-Hua National Chiao Tung

University, Hsinchu, Taiwan

Huberty, Carl J University of Georgia,

Athens, GA, USA

Huitema, Bradley E Western Michigan

University, Kalamazoo, MI, USA

Indurkhya, Alka Harvard University,

Boston, MA, USA

Irtel, Hans University of Mannheim,

Mannheim, Germany

Jackson, J Edward Retired, Rochester,

NY, USA

Jacobson, Kristen C Virginia

Commonwealth University, Richmond,

VA, USA

Jaffe, Adi California State University,

Long Beach, CA, USA

Johnson, Matthew S City University of

New York, NY, USA

Jolliffe, Ian University of Aberdeen,

Aberdeen, UK

Judd, Charles M University of Colorado,

Boulder, CO, USA

Junker, Brian W Carnegie Mellon

University, Pittsburgh, PA, USA

Kaplan, David University of Delaware,

Newark, DE, USA

Kaplan, Seth A Tulane University, New

Orleans, LA, USA

Karabatsos, George University of Illinois,

Chicago, IL, USA

Kay, Jim W University of Glasgow,

Glasgow, UK

Kazdin, Alan E Yale University School of

Medicine, New Haven, CT, USA

Keller, Lisa A University of

Massachusetts, Amherst, MA, USA

Kenny, David A University of

Connecticut, Storrs, CT, USA

Contributors xxi

Kenward, Michael G London School of

Hygiene and Tropical Medicine, London,

UK

Keselman, H.J University of Manitoba,

Winnipeg, Manitoba, Canada

Khamis, Harry J Wright State University,

Dayton, OH, USA

Kim, Jee-Seon University of Wisconsin,

Madison, WI, USA

Kirk, Roger E Baylor University, Waco,

Kolen, Michael J University of Iowa,

Iowa City, IA, USA

Kornbrot, Diana University of

Hertfordshire, Hatfield, UK

Kovtun, Mikhail Duke University,

Durham, NC, USA

Kratochwill, Thomas R University of

Wisconsin, Madison, WI, USA

Kreft, Ita G.G California State

University, Los Angeles, CA, USA

Kroonenberg, Pieter M Leiden

University, Leiden, The Netherlands

Krzanowski, Wojtek J University of

Exeter, Exeter, UK

Kulikowich, Jonna M Pennsylvania State

University, University Park, PA, USA

Kuncheva, Ludmila I University of

Wales, Bangor, UK

Laming, Donald University of Cambridge,

Cambridge, UK

Lance, Charles E University of Georgia,

Athens, GA, USA

Landau, Sabine Institute of Psychiatry,

King’s College, London, UK

Landis, Ronald S Tulane University, New

Orleans, LA, USA

Langeheine, Rolf Bordesholm, Germany

Langholz, Bryan University of Southern

California, Los Angeles, CA, USA

Lee, Peter M University of York, York,

UK

Leese, Morven Institute of Psychiatry,

King’s College, London, UK

Lemeshow, Stanley Ohio State University,

Columbus, OH, USA

Lesaffre, Emmanuel Katholieke

Universiteit Leuven, Leuven, Belgium

Levy, Paul S RTI International, Research

Triangle Park, NC, USA

Lewis, Robert Michael College of William

& Mary, Williamsburg, VA, USA

Lewis-Beck, Michael S University of

Iowa, Iowa City, IA, USA

Lindstrom, Mary J University of

Wisconsin, Madison, WI, USA

Lipsey, Mark W Vanderbilt University,

Nashville, TN, USA

Little, Roderick J University of Michigan,

Ann Arbor, MI, USA

Little, Todd D University of Kansas,

Lawrence, KS, USA

Li, Shuhong University of Massachusetts,

Amherst, MA, USA

Li, Zhen University of Maryland,

Baltimore, MD, USA

Liu, Li Aventis Pharmaceuticals,

Bridgewater, NJ, USA

Lix, Lisa M University of Manitoba,

Winnipeg, Manitoba, Canada

Longford, Nicholas T SNTL, Leicester,

Luce, R Duncan University of California,

Irvine, CA, USA

Ludbrook, John University of Melbourne,

Melbourne, Victoria, and University of Adelaide, Adelaide, South Australia, Australia

Luecht, Richard M University of North

Carolina, Greensboro, NC, USA

Luellen, Jason K University of Memphis,

Memphis, TN, USA

Lunneborg, Clifford E University of

Washington, Seattle, WA, USA

Macdonald, Ranald R University of

Stirling, Stirling, UK

xxii Contributors

Maes, Hermine H Virginia

Commonwealth University, Virginia, VA,

USA and Katholieke Universiteit, Leuven,

Belgium

Magidson, Jay Statistical Innovations Inc.,

Belmont, MA, USA

Mair, Patrick University of Vienna,

Vienna, Austria

Mangione, Thomas W John Snow, Inc.,

Boston, MA, USA

Manton, Kenneth G Duke University,

Durham, NC, USA

Marcoulides, George A California State

University, Fullerton, CA, USA

Marden, John I University of Illinois,

Champaign, IL, USA

Margolis, Melissa J National Board of

Medical Examiners, Philadelphia, PA,

USA

Marks, Michael J University of Illinois,

Champaign, IL, USA

Martin, Nicholas G Queensland

Institute of Medical Research, Herston,

Queensland, Australia

Massaro, Joseph M Quintiles Inc.,

Cambridge, MA, USA

Mattern, Krista University of Illinois,

Champaign, IL, USA

Maxwell, Scott E University of Notre

Dame, Notre Dame, IN, USA

McArdle, John J The University of

Virginia, Virginia, VA, USA

McGuffin, Peter Institute of Psychiatry,

King’s College, London, UK

McKean, J.W Western Michigan

University, Kalamazoo, MI, USA

McKenzie, Dean P Monash University,

Melbourne, Victoria, Australia

Mead, Alan D Economic Research

Institute, Lockport, IL, USA

Mechelen, Iven Van Katholieke

Universiteit Leuven, Leuven, Belgium

Meijer, Rob R University of Twente,

Enschede, The Netherlands

Meiser, Thorsten University of Jena, Jena,

Germany

Mellenbergh, Gideon J University of

Amsterdam, Amsterdam, The Netherlands

Michailidis, George University of

Michigan, Ann Arbor, MI, USA

Michell, Joel University of Sydney, Sydney,

New South Wales, Australia

Miles, Jeremy University of York, York,

UK

Miller, Richard University of Nebraska,

Kearney, NE, USA

Mislevy, Robert J University of

Maryland, College Park, MD, USA

Mitchell, Dayton C University of

Maryland, College Park, MD, USA

Molenaar, P.C.M University of

Amsterdam, Amsterdam, The Netherlands

Molenberghs, Geert Limburgs

Universitair Centrum, Diepenbeek, Belgium

Moore, Kris K Baylor University, Waco,

TX, USA

Moors, Guy Tilburg University, Tilburg,

The Netherlands

Mulaik, Stanley A Georgia Institute of

Technology, Atlanta, GA, USA

Muller, Dominique University of Paris,

Paris, France

Mun, Eun Young University of Alabama,

Birmingham, AL, USA

Mu ˜niz, Jos´e University of Oviedo, Oviedo,

Spain

Murphy, Kevin R Pennsylvania State

University, University Park, PA, USA

Murray, David M University of Memphis,

Memphis, TN, USA

Murtagh, Fionn Royal Holloway College,

University of London, Egham, UK

Myung, Jay I Ohio State University,

Columbus, OH, USA

Navarro, Daniel J Ohio State University,

Columbus, OH, USA

Neale, Ben Institute of Psychiatry, King’s

College, London, UK

Neale, Michael C Virginia Commonwealth

University, Richmond, VA, USA

Neiderhiser, Jenae M George Washington

University, Washington, DC, USA

Nickerson, Raymond S Tufts University,

Medford, MA, USA

Onghena, Patrick Katholieke Universiteit

Leuven, Leuven, Belgium

Contributors xxiii

Pardy, Susan A Queen’s University,

Kingston, Ontario, Canada

Pattison, Philippa University of

Melbourne, Melbourne, Victoria,

Australia

Patton, Michael Quinn Union Institute

and University, Saint Paul, MN, USA

Pettit, Lawrence Queen Mary, University

of London, London, UK

Phillips, Carl V University of Texas,

Houston, TX, USA

Phillips, Lawrence D London School

of Economics and Political Science,

London, UK

Plake, Barbara S University of Nebraska,

Lincoln, NE, USA

Ployhart, Robert E University of South

Carolina, Columbia, SC, USA

Posthuma, Danielle Vrije Universiteit,

Amsterdam, The Netherlands

Prentice, Ross L Fred Hutchinson Cancer

Research Center, Seattle, WA, USA

Press, Jim University of California,

Riverside, CA, USA

Price, Thomas S University of Oxford,

Oxford, UK

Prinzie, Peter Leiden University, Leiden,

The Netherlands

Pruzek, Robert State University of New

York at Albany, NY, USA

Putt, Mary E University of Pennsylvania,

Philadelphia, PA, USA

Rabe-Hesketh, Sophia University of

California, Berkeley, CA, USA

Ramsay, James McGill University,

Montreal, Quebec, Canada

Ramsey, Philip H Queens College of

CUNY, Flushing, NY, USA

Ranyard, Rob University of Bolton,

Bolton, UK

Rasbash, Jon University of London,

London, UK

Raykov, Tenko Fordham University,

Bronx, NY, USA

Read, Daniel University of Durham,

Durham, UK

Reichardt, Charles S University of

Denver, Denver, CO, USA

Rende, Richard Brown Medical School,

Providence, NY, USA

Rendina-Gobioff, Gianna University of

South Florida, Tampa, FL, USA

Rhee, S.H University of Colorado,

Boulder, CO, USA

Rigdon, Edward E Georgia State

University, Atlanta, GA, USA

Rijsdijk, Fr ¨uhling Institute of Psychiatry,

King’s College, London, UK

Rindskopf, David CUNY Graduate

Center, New York, NY, USA

Robin, Frederic Educational Testing

Service, Princeton, NJ, USA

Rodabough, Tillman Baylor University,

Waco, TX, USA

Rodgers, Joseph Lee University of

Oklahoma, Norman, OK, USA

Rogers, H Jane University of Connecticut,

Storrs, CT, USA

Rose, Richard J Indiana University,

Bloomington, IN, USA

Rosenbaum, Paul R University of

Pennsylvania, Philadelphia, PA, USA

Rosenthal, Robert University of

California, Riverside, CA, USA

Ross, Helen University of Stirling, Stirling,

UK

Rovine, Michael J Pennsylvania State

University, University Park, PA, USA

Rudas, Tam´as E¨otv¨os Lor´and University,

Budapest, Hungary

Rupp, Andr´e A University of Ottawa,

Ottawa, Ontario, Canada

Sampson, Paul D University of

Washington, Seattle, WA, USA

Sanders, Piet F Cito (Dutch National

Institute for Test Development), Arnhem, The Netherlands

Saudino, Kimberly J Boston University,

Boston, MA, USA

Sawilowsky, Shlomo Wayne State

University, Detroit, MI, USA

Schalkwyk, Leonard C Institute of

Psychiatry, King’s College, London, UK

Schmitt, Neal Michigan State University,

East Lansing, MI, USA

Schumacker, Randall E University of

North Texas, Denton, TX, USA

xxiv Contributors

Schuster, Christof

Justus-Liebig-Universit¨at Giessen, Giessen,

Germany

Scott, Heather University of South

Florida, Tampa, FL, USA

Segal, Nancy L California State

University, Fullerton, CA, USA

Senn, Stephen University of Glasgow,

Glasgow, UK

Serlin, Ronald C University of Wisconsin,

Madison, WI, USA

Shadish, William R University of

California, Merced, CA, USA

Shafran, Randi University of Minnesota,

Duluth, MN, USA

Sham, P Institute of Psychiatry, King’s

College, London, UK

Shavelson, Richard J Stanford University,

Stanford, CA, USA

Sheather, S.J Texas A&M University,

College Station, TX, USA

Shi, Dailun Fudan University, Shanghai,

China

Sijtsma, Klaas Tilburg University, Tilburg,

The Netherlands

Silver, N Clayton University of Nevada,

Las Vegas, NV, USA

Singer, Judith D Harvard Graduate

School of Education, Cambridge, MA,

USA

Sireci, Stephen G University of

Massachusetts, Amherst, MA, USA

Skrondal, Anders London School of

Economics and Political Science,

London, UK

Slegers, David W University of Kansas,

Lawrence, KS, USA

Slomkowski, Cheryl Brown Medical

School, Providence, NY, USA

Smith, David University of Notre Dame,

Notre Dame, IN, USA

Smith, Laurence D University of Maine,

Orono, ME, USA

Smith, Philip T University of Reading,

Reading, UK

Smith, Randolph A Kennesaw State

University, Kennesaw, GA, USA

Snijders, Tom A.B University of

Groningen, Groningen, The Netherlands

Spence, Ian University of Toronto,

Toronto, Ontario, Canada

Spilich, George Washington College,

Chestertown, MD, USA

Spinath, Frank M Saarland University,

Saarbruecken, Germany

Stallings, M.C University of Colorado,

Boulder, CO, USA

Steele, Fiona University of London,

London, UK

Stefanescu, Catalina London Business

School, London, UK

Steinley, Douglas University of Missouri,

Columbia, MO, USA

Stewart, David W University of Southern

California, Los Angeles, CA, USA

Stoel, Reinoud D University of

Amsterdam, Amsterdam, The Netherlands

Stone, James V Sheffield University,

Sheffield, UK

Stuetzle, Werner University of

Washington, Seattle, WA, USA

Suchy, Yana University of Utah, Salt Lake

City, UT, USA

Swaminathan, Hariharan University of

Connecticut, Storrs, CT, USA

Szabat, Kathryn A LaSalle University,

Philadelphia, PA, USA

Takane, Yoshio McGill University,

Montreal, Quebec, Canada

Thach, Chau Merck & Co., Inc., Rahway,

NJ, USA

Thomas, Roger University of Georgia,

Athens, GA, USA

Thompson, Bruce Texas A&M University,

College Station, TX, USA

Thorndike, Robert M Western

Washington University, Bellingham, WA, USA

Thum, Y.M University of California, Los

Angeles, CA, USA

Tidd, Simon T Vanderbilt University,

Nashville, TN, USA

Tisak, John Bowling Green State

University, Bowling Green, OH, USA

Tisak, Marie S Bowling Green State

University, Bowling Green, OH, USA

Tong, Ye University of Iowa, Iowa City,

IA, USA

Contributors xxv

Toone, Leslie Utah State University,

Logan, Utah, USA

Toothaker, Larry E University of

Oklahoma, Oklahoma City, OK, USA

Tourangeau, Roger University of

Maryland, College Park, MD, USA and

University of Michigan, Ann Arbor, MI,

USA

Towers, Hilary George Washington

University, Washington, DC, USA

Treat, Teresa A Yale University, New

Haven, CT, USA

Trosset, Michael W College of William &

Mary, Williamsburg, VA, USA

Tweney, Ryan D Bowling Green State

University, Bowling Green, OH, USA

van de Geer, Sara A University of

Leiden, Leiden, The Netherlands

van de Pol, Frank Heerlen, The

Netherlands

van de Velden, Michel Erasmus

University Rotterdam, Rotterdam, The

Netherlands and Groningen University,

Groningen, The Netherlands

van den Noortgate, Wim Katholieke

Universiteit Leuven, Leuven, Belgium

van den Oord, Edwin J.C.G Virginia

Commonwealth University, Richmond,

VA, USA

van der Maas, H.L.J University of

Amsterdam, Amsterdam, The Netherlands

Vandenberg, Robert J University of

Georgia, Athens, GA, USA

Veldkamp, Bernard P University of

Twente, Enschede, The Netherlands

Verbeke, Geert Katholieke Universiteit

Leuven, Leuven, Belgium

Vermunt, Jeroen K Tilburg University,

Tilburg, The Netherlands

Victoria-Feser, Maria-Pia HEC –

University of Geneva, Geneva,

Switzerland

Visser, Penny S University of Chicago,

Chicago, IL, USA

von Eye, Alexander Michigan State

University, East Lansing, MI, USA

von Weber, Stefan Fachhochschule

Furtwangen, Furtwangen, Germany

Vorperian, Houri K University of

Wisconsin, Madison, WI, USA

Vos, Hans J University of Twente,

Enschede, The Netherlands

Wade, Claire Whitehead Institute for

Biomedical Research, Cambridge, MA, USA

Wagenmakers, E.-J University of

Amsterdam, Amsterdam, The Netherlands

Wagner-Menghin, Michaela M.

University of Vienna, Vienna, Austria

Wainer, Howard National Board of

Medical Examiners, Philadelphia, PA, USA

Wakefield, Jon University of Washington,

Seattle, WA, USA

Walwyn, Rebecca Institute of Psychiatry,

King’s College, London, UK

Wang, Molin Harvard School of Public

Health and Dana-Farber Cancer Institute, Boston, MA, USA

Wasserman, Stanley Indiana University,

Bloomington, Indiana, USA

Webb, Noreen M University of

California, Los Angeles, CA, USA

Weber, Erik Ghent University, Ghent,

Belgium

Weersing, V Robin Yale Child Study

Center, New Haven, CT, USA

Weidman, Nadine Harvard University,

Cambridge, MA, USA

Welkenhuysen-Gybels, Jerry Katholieke

Universiteit Leuven, Leuven, Belgium

Well, Arnold D University of

Massachusetts, Amherst, MA, USA

Wermuth, Nanny Chalmers/Gothenburg

University, Gothenburg, Sweden

West, Stephen G Arizona State

University, Tempe, AZ, USA

Whitaker, Christopher J University of

Wales, Bangor, UK

Wiggins, Lacey Baylor University, Waco,

TX, USA

Wight, Randall D Ouachita Baptist

University, Arkadelphia, AR, USA

Wilcox, Rand R University of Southern

California, Los Angeles, CA, USA

xxvi Contributors

Willett, John B Harvard University

Graduate School of Education,

Cambridge, MA, USA

Williams, Siˆan E University of Sussex,

Brighton, UK

Wishart, David University of St Andrews,

St Andrews, UK

Wixted, John T University of California,

San Diego, CA, USA

Wothke, Werner Research Department,

CTB/McGraw-Hill, Monterey, CA, USA

Wright, Daniel B University of Sussex,

Brighton, UK

Wuensch, Karl L East Carolina

University, Greenville, NC, USA

Yashin, Anatoli Duke University, Durham,

NC, USA

Zenisky, April L University of

Massachusetts, Amherst, MA, USA

Zhang, Jialu University of Maryland,

Baltimore, MD, USA

Zhao, Yue University of Massachusetts,

Amherst, MA, USA

Zhou, YanYan Florida International

University, Miami, FL, USA

Zumbo, Bruno D University of British

Columbia, Vancouver, British Columbia, Canada

Forty years ago there was hardly a field called ‘behavioral sciences’ In fact, psychology largely

was the behavioral sciences, with some help from group theory in sociology and

decision-making in economics Now, of course, psychology has expanded and developed in a myriad

of ways, to the point where ‘behavioral sciences’ is often the more useful term Physiologicalpsychology has become neuroscience, covering areas not previously part of psychology.Decision-making has become decision science, involving people from economics, marketing,and other disciplines Learning theory has become cognitive science, again exploring problemsthat were not even considered 40 years ago And developments in computing have broughtforth a host of new techniques that were not possible in the days of manual or electroniccalculators And with all of these changes, there have been corresponding changes in theappropriate statistical methodologies

Originally many of the statistical methods within psychology were held in common acrossthe majority of sub-disciplines Those working in learning theory, perception, developmentalpsychology, and social psychology largely relied on the same basic techniques Most researchinvolved some variation on the analysis of variance, though factor analysis was central tothe study of individual differences, and specialized techniques were developed in scaling andpsychometrics Clinical psychology largely relied on correlation and regression techniques,with a substantial dose of single-subject designs, which had only a rudimentary statisticalcomponent At that time, one could point to a few classic texts that covered the importantmethods: namely Winer (1962) [6] for the analysis of variance, Draper and Smith (1966) [1]for regression, Lawley and Maxwell (1963) [3] for factor analysis, Nunnally (1967) [4] forpsychometric theory, and Siegel (1956) [5] for nonparametric methods And then, of course,there was Hays’ (1963) classic text [2], through which many of our contemporaries had theirfirst taste of statistics

The past forty years have seen a huge change in statistical methods in the behavioralsciences New fields, with their own techniques, have developed, and the traditional fields havebroadened their approach to the point that the classic methods of the past have been supplanted

by more refined and useful techniques Traditional repeated measures analysis of variance,for example, has now been largely replaced by generalized linear mixed effects models thatcan deal appropriately with both normally and nonnormally distributed responses Clinicalpsychology has profited enormously from the development of concepts centered on latentvariables and structural equation modeling There are statistical techniques for dealing withbehavioral genetics that had not even been thought of 40 years ago The work of biostatisticians

xxviii Preface

went largely untouched in the behavioral sciences, but in recent years, techniques such aslogistic regression and survival analysis, and concepts like risk, odds, incidence, and prevalencehave made their way into the field, and not just among clinical researchers And this list could

go on for a great deal longer In compiling this encyclopedia, one of our major goals was

to address this huge expansion of statistical methods in the behavioral sciences Much of therelevant literature is scattered in professional journals and books, and is not readily accessible

to those who would profit from it We have aimed to bring together material on each of themany disciplines in behavioral science, and make it available in one source, not to replace theprofessional literature, but to offer a summary of it so that readers have the opportunity to see

it in its broader context

There are, of course, other large-scale reference works that are at least partially relevant

Wiley’s Encyclopedia of Statistical Science (EoSS ) (1982 – 1999) is the classic example, and

is used by many statisticians The more recent Encyclopedia of Biostatistics, Second Edition (EOB ) (2005) is a further excellent source of material, particularly for medical statisticians However, in the Encyclopedia of Statistics in Behavioral Science (ESBS ), we focus on statistics

as developed and applied in the behavioral sciences The result is that we give more emphasis

to, for example, topics such as structural equation modeling, factor analysis, scaling, and

mea-surement than either the EoSS or the EOB We also thought it essential to include solid age of the statistical theory that underlies the methods described in the ESBS, although in some- what less depth than both the EoSS and the EOB The current work does inevitably overlap with the EoSS and the EOB, but each of the encyclopedias contains much material outside the

cover-scope of the other two The three works are essentially complementary rather than competitive

We began this project by breaking the field into 15 broad categories For each category,

we recruited a section editor with known expertise in his or her field These categories, andthe section editors, are:

1 Statistical models (Jose Cortina)

2 Scaling (Jan de Leeuw)

3 Classical test theory/Item response theory/Psychological measurement

(Ronald K Hambleton)

4 Design of experiments and surveys (Roger Kirk)

5 Multivariate methods (Sabine Landau)

6 Historical/Institutional (Sandy Lovie)

7 Descriptive statistics/Graphical procedures (Pat Lovie)

8 Nonparametric and computationally intensive methods (Cliff Lunneborg)

9 Statistical theory (Ranald Macdonald)

10 Intervention/Observational studies (Patrick Onghena)

11 Behavioral genetics (Shaun Purcell)

12 Longitudinal/multilevel methods (Sophia Rabe-Hesketh and Anders Skrondal)

13 Factor analysis and structural equation modeling (David Rindskopf)

14 Teaching statistics, software, and overviews (Chris Spatz)

15 Categorical data analysis (Alexander von Eye)

in technical level and mathematical content among the entries will, with luck, largely reflectthe type of reader likely to turn to a particular entry.

The section editors did an outstanding job of soliciting entries, and out of approximately

700 potential entries, they were able to recruit all but a handful of authors The section editorsthen worked with the authors as supporters, editors, and, occasionally, cajolers The editors-in-chief collected the finished products from the section editors, sometimes going back to theauthors with queries or suggestions, set up the cross-referencing, and then compiled all of theentries into the encyclopedia But any success this encyclopedia achieves is largely due to theoften heroic efforts of the section editors

We also owe a debt to the nearly 600 authors, who worked so hard, often throughmultiple drafts, to produce this work With the editors-in-chief on different sides of theAtlantic, and section editors and authors spread all over the world, this work could nothave been carried out without the Internet We want to thank all who responded promptly

to messages sent in the middle of the night (their time) or just after they went home for theweekend

This encyclopedia uses a system of cross-referencing to make the material more accessible

to the reader Terms appearing as head words, such as incomplete contingency tables, appear

in bold face when first used in another head word entry, although frequently appearing terms

such as mean, P value and so on are not emboldened in this way In some cases, the wording of

the text and the wording of the corresponding head word will not be identical, but the reader

should have no trouble finding the relevant entry So, for example, ‘maximum likelihood

was used ’ will refer to the entry entitled ‘Maximum likelihood estimation’ When the

wording in the text does not allow a direct reference of this sort, the intended article isgiven in parentheses, for example, ‘the parameters were estimated by a process involving the

maximization of the likelihood (see Maximum Likelihood Estimation)’.

Some head words simply reference another entry where the particular topic is discussed;

for example, ‘Structural Zeros: See Incomplete Contingency Tables,’ or ‘Linear

Regres-sion: See Multiple Linear Regression.’ If such a term appears in an entry it is not bolded.

Instead the entry where the topic is described is again given explicitly in parentheses, for

exam-ple, ‘in some cross-classified tables, structural zeros (see Incomplete Contingency Tables),

.’ In this way, readers are led directly to the relevant entry

Helen Ramsey, who was our Publishing Editor at Wiley, did a superb job leading usalong, providing encouragement where it was needed, and somehow keeping complete track

of over 600 entries in various stages of completion We could not have done this withouther We are also grateful for the assistance offered by her colleagues at Wiley, including SamCrowe, Louise Rush, Geoff Reynolds, and Helen Green Harriett Meteyard did sterling work

Brian S EverittDavid C Howell

April 2005

References

[1] Draper, N.R & Smith, H (1966) Applied Regression Analysis, Wiley, New York.

[2] Hays, W.L (1963) Statistics for Psychologists, 1st Edition, Holt, Rinehart, and Winston, New York.

[3] Lawley, D.N & Maxwell, A.E (1963) Factor Analysis as a Statistical Method, Butterworth, London.

[4] Nunnally, J.C (1967) Psychometric Theory, McGraw-Hill, New York.

[5] Siegel, S (1956) Nonparametric Statistics for the Behavioral Sciences, McGraw-Hill, New York.

[6] Winer, B.J (1962) Statistical Principles in Experimental Design, McGraw-Hill, New York.

Further Reading

Armitage, P & Colton, T (2005) Encyclopedia of Biostatistics, 2nd Edition, Wiley, Chichester.

Kotz, S (2003) Encyclopedia of Statistical Science, Wiley, Chichester.

Abbreviations and Acronyms

AAPOR American Association for Public Opinion ResearchABP Antibiotic Prophylaxis

AC Available-case

ACE American Council on Education

ACE Average Causal Effect

ACT American College Testing

AD Average Deviation

ADF Asymptotic Distribution Free

ADHD Attention Deficit Hyperactivity Disorder

ADL Activity of Daily Living

AERA American Educational Research AssociationAGFI Adjusted Goodness-of-fit Index

AIC Akaike’s Information Criterion

AICPA American Institute of Certified Public Accountants

AJ Analytical Judgment

ALPH Alpha Factor Arrest

ANCOVA Analysis of Covariance

ANOVA Analysis of Variance

AoA Age of Acquisition

APA American Psychological Association

AR Absolute Risk Reduction

AR Appearance-reality

AR Autoregressive

ARE Asymptotic Relative Efficiency

ART Appearance-reality Distinction Task

ASB Antisocial Behavior

AS-CAT A-Stratified Computerized Adaptive TestingASE Asymptotic Standard Error

ASP Affected Sib-pair

ATA Automated Test Assembly

ATD Alternating Treatments Design

BBN Bayesian Belief Networks

BDI Beck Depression Inventory

BDS Behavioral Dyscontrol Scale

BESD Binomial Effect Size Display

BF Bayes Factor

BFM Best Fitting Model

BHP Benign Prostate Hypertrophy

BIBD Balanced Incomplete Block Designs

xxxii Abbreviations and Acronyms

BIC Bayesian Information Criterion

BLUE Best Linear Unbiased Estimator

BLUP Best Linear Unbiased Predictor

BMDP Biomedical Statistical Software Package

CAIC Consistent Akaike Information Criterion

CA-MST Computer-adaptive Multistage Testing

CART Classification and Regression Trees

CASK Canonical Analysis of SKew symmetric dataCASRO Council of American Survey Research OrganizationsCAT Computer Adaptive Test

CATI Computer-Assisted Telephone-Interviewing

CBT Cognitive Behavioral Therapy

CBT Computer Based Testing/Tests

CC Chain Components

CC Chest Circumference

CC Complete-case

CCA Canonical Correlation Analysis

CCA Covariance Component Analysis

CCAT-UST Constrained Cat Using Shadow Tests

CCC Category Characteristic Curve

CCT Computerized Classification Test

CDA Confirmatory Data Analysis

cdfs Cumulative Distribution Functions

CDP Composite Direct Product

CERN Centre Europ´een De Recherche Nucl´eaire

CES-D Center for Epidemiologic Studies DepressionCFA Configural Frequency Analysis

CFA Confirmatory Factor Analysis

CFI Comparative Fit Index

CFT Computerized Fixed Test

CI Confidence Interval

CKR Common Knowledge and Rationality

CLP Category-level-point

CLPC Cross-lagged Panel Correlation

CLR Conditional Likelihood Ratio

cM centiMorgan

CMH Cochran – Mantel – Haenszel

CML Conditional Maximum Likelihood

COG Cognitive Performance

CoT Children of Twins

CPO Conditional Predictive Ordinates

CR Conditioned Response

CR Constructed-response

CRD Completely Randomized Design

Abbreviations and Acronyms xxxiii

CRF Category Response Function

CRT Criterion-referenced Testing

CS Conditioned Stimulus

CSA Covariance Structure Analysis

CSI Classical Statistical Inference

CSR Covariance Structure Residual

CTA Confirmatory Tetrad Analysis

CTT Classical Test Theory

CV Coefficient of Variation

CV Cross-validation

CVM Categorical Variable Matrix

CWE Comparison-wise Error

DA Decision Accuracy

DAG Directed Acyclic Graph

DASL Data and Story Library

DC Decision Consistency

DC Dichorionic

DDA Descriptive Discriminant Analysis

DEDICOM DEcomposing DIrectional COMponents

DFA Discriminant Function Analysis

DHHS Department of Health and Human Services

DIF Differential Item Functioning

DMM Double Monotonicity Model

DOC Direction of Causation

EAP Expected a Posteriori

ECD Evidenced-centered Assessment Design

ECLS Early Childhood Longitudinal Study

ECMO Extra Corporeal Membrane Oxygenation

ECVI Expected Cross Validation Index

EDA Exploratory Data Analysis

EEA Equal Environments Assumption

EEG Electroencephalogram

EFA Exploratory Factor Analysis

EGEE Extended Generalized Estimating Equation

EI Essentially Independent

EICR Extended Individual Case Residual

eips Elementary Information Processing Operators

ELU Centrifugal Elutriation

EM Expectation Maximization

EMEA European Medicine Agency

EPC Expected Parameter Change

EPQ Eysenck Personality Questionnaire

xxxiv Abbreviations and Acronyms

EQQ Empirical Quantile – Quantile

ES Effect Size

ESS Evolutionarily Stable Strategy

ETS Educational Testing Service

EU Expected Utility

FA Factor Analysis

FAR False Alarm Rate

FBI Federal Bureau of Investigation

FDA Food and Drug Administration

FDA Functional Data Analysis

FDR False Discovery Rate

FEM Fixed Effects Model

FGA Fractile Graphical Analysis

FIA Fisher-information Approximation

FIFA Full Information Factor Analysis

FIML Full Information Maximum Likelihood

FML Full Maximum Likelihood

fMRI Functional Magnetic Resonance Imaging

FSE Factor Score Estimation

FT Facet Theory

FWE Family-wise or Experiment-wise

GAM General Additive Model

GCT Gender Constancy Test

GIGO Garbage in – Garbage out

GLM/GLIM Generalized Linear Model

GLTM General Component Latent Trait Model

GPA Undergraduate Grade Point Average

GT Generalizability Theory

GxE Gene X Environment Interaction

HLM Hierarchical Linear Models

ICC Item Characteristic Curve

ICR Individual Case Residual

IFA Image Factor Analysis

IPS Intergenerational Panel Study of Parents and ChildrenIQR The Interquartile Range

IRB Institutional Review Board

IRF Item Response Function

IRT Item Response Theory

ITT Intention-to-treat

JAP Journal of Applied Psychology

JCA Joint Correspondence Analysis

JML Joint Maximum Likelihood

k-nn K-nearest Neighbor classifier

Abbreviations and Acronyms xxxv

LCF Linear Classification Function

LDA Linear Discriminant Analysis

LDF Linear Discriminant Functions

LD Linkage Disequilibrium

LGC Latent Growth Curve

LR Likelihood Ratio

MAICE Minimum AIC Estimate

MANOVA Multivariate Analysis of Variance

MAP Maximum a Posteriori

MAR Missing at Random

MASS Modern Applied Statistics with S

MCAR Missing Completely at Random

MCC Minimally Competent Candidate

MGV Minimum Generalized Variance

ML Maximum Likelihood

MLE Maximum Likelihood Estimate

MLFA Maximum Likelihood Factor Analysis

MML Marginal Maximum Likelihood

MMPI Minnesota Multiphasic Personality Inventory

MNAR Missing Not at Random

MPT Multinomial Processing Tree

MR Multiple Linear Regression

MSA Multidimensional Scalogram Analysis

MSE Mean-squared Error

MSR Mean Structure Residual

MST Multistage Test

MTMM Multitrait – Multimethod Matrix Method

MVE Minimum Volume Ellipsoid

MZ Monozygotic

NFI Bentler – Bonett Normed fit index

NNFI Bentler – Bonett Non-Normed fit index

OLS Ordinary Least Squares

2PLM Two-parameter Logistic Model

PCA Principal Component Analysis

PDA Predictive Discriminant Analysis

PDF Probability Density Function

PP Personnel Psychology

PRE Proportionate Reduction in Error

RAFT R Econstruction AFter F Eedback with TAke the Best

RMSD Root Mean – Squared Deviation

ROC Receiver Operating Characteristic

RP Response Probability

xxxvi Abbreviations and Acronyms

SBS Sequential Backward Selection

SE Standard Errors

SED Seriously Emotionally Disturbed

SEM Structural Equation Modeling

SEM Standard Error of Measurement

SFS Sequential Forward Selection

SME Subject Matter Expert

SP ‘Scree’ Plot

SPRT Sequential Probability Ratio Test

SSE Sum of Squared Errors

SSRC UK Social Science Research Council

SVD Singular Value Decomposition

TAAS Texas Assessment of Academic Skills

TAU Treatment As Usual

TB-CAT Testlet-based Cat

TCF Test Characteristic Function

TEDS Twins Early Development Study

TL Traditional Linear

TLI Tucker – Lewis Index

TS Tensile Strength

UCI Unobserved Conditional Invariance

UCL University College, London

UCPAE Uniform Certified Public Accountant Examination

UD Unidimensionality

ULS or OLS Unweighted or Ordinary Least Squares

UPGA Unweighted Pair Group Average

UR Unconditioned Biological Or Behavioral Response

US Unconditioned Stimulus

UTI Urinary Tract Infections

VAP Ventilator Associated Pneumonia

VIF Variance Inflation Factor

VLCD Very-low-calorie Diet

VRBM Verbal Meaning

VTSABD Virginia Twin Study of Adolescent Behavior DevelopmentWDM Weighted Deviation Model

WHI Women’s Health Initiative

WLI Weak Local Independence

WLS Weighted Least Squares

WMW Wilcoxon – Mann – Whitney Test

ZIP Zero-inflated Poisson

A Priori Power Analysis see

Power

A Priori v Post Hoc

Testing

Macdonald [11] points out some of the problems

with post hoc analyses, and offers as an example the

P value one would ascribe to drawing a particular

card from a standard deck of 52 playing cards If

the null hypothesis is that all 52 cards have the

same chance (1/52) to be selected, and the alternative

hypothesis is that the ace of spades will be selected

with probability one, then observing the ace of spades

would yield a P value of 1/52 For a Bayesian

perspective (see Bayesian Statistics) on a similar

situation involving the order in which songs are

played on a CD, see Sections 4.2 and 4.4 of [13]

Now then, with either cards or songs on a CD, if

no alternative hypothesis is specified, then there is

the problem of inherent multiplicity Consider that

regardless of what card is selected, or what song is

played first, one could call it the target (alternative

hypothesis) after-the-fact (post hoc), and then draw

the proverbial bull’s eye around it, quoting a P value

of 1/52 (or 1/12 if there are 12 songs on the CD) We

would have, then, a guarantee of a low P value (at

least in the case of cards, or more so for a lottery),

thereby violating the probabilistic interpretation that

under the null hypothesis a P value should, in the

continuous case, have a uniform distribution on the

unit interval [0,1] In any case, the P value should

be less than any number k in the unit interval [0,1], with probability no greater than k [8].

The same problem occurs when somebody findsthat a given baseball team always wins on Tues-days when they have a left-handed starting pitcher.What is the probability of such an occurrence? Thisquestion cannot even be properly formulated, letalone answered, without first specifying an appro-priate probability model within which to embed thisevent [6] Again, we have inherent multiplicity Howmany other outcomes should we take to be as statis-tically significant as or more statistically significant

than this one? To compute a valid P value, we need

the null probability of all of these outcomes in theextreme region, and so we need both an enumeration

of all of these outcomes and their ranking, based onthe extent to which they contradict the null hypothe-sis [3, 10]

Inherent multiplicity is also at the heart of a tial controversy when an interim analysis is used, thenull hypothesis is not rejected, the study continues

poten-to the final analysis, and the final P value is greater

than the adjusted alpha level yet less than the overall

alpha level (see Sequential Testing) For example,

suppose that a maximum of five analyses are planned,and the overall alpha level is 0.05 two-sided, sothat 1.96 would be used as the critical value for asingle analysis But with five analyses, the criticalvalues might instead be{2.41, 2.41, 2.41, 2.41, 2.41}

if the Pocock sequential boundaries are used or

2 A Priori v Post Hoc Testing

{4.56, 3.23, 2.63, 2.28, 2.04} if the O’Brien–Fleming

sequential boundaries are used [9] Now suppose that

none of the first four tests result in early stopping, and

the test statistic for the fifth analysis is 2.01 In fact,

the test statistic might even assume the value 2.01

for each of the five analyses, and there would be no

early stopping

In such a case, one can lament that if only no

penalty had been applied for the interim analysis,

then the final results, or, indeed, the results of any

of the other four analyses, would have attained

statistical significance And this is true, of course,

but it represents a shift in the ranking of all possible

outcomes Prior to the study, it was decided that a

highly significant early difference would have been

treated as more important than a small difference at

the end of the study That is, an initial test statistic

greater than 2.41 if the Pocock sequential boundaries

are used, or an initial test statistic greater than 4.56 if

the O’Brien-Fleming sequential boundaries are used,

would carry more weight than a final test statistic of

1.96 Hence, the bet (for statistical significance) was

placed on the large early difference, in the form of the

interim analysis, but it turned out to be a losing bet,

and, to make matters worse, the standard bet of 1.96

with one analysis would have been a winning bet

Yet, lamenting this regret is tantamount to requesting

a refund on a losing lottery ticket In fact, almost any

time there is a choice of analyses, or test statistics,

the P value will depend on this choice [4] It is

clear that again inherent multiplicity is at the heart

of this issue

Clearly, rejecting a prespecified hypotheses is

more convincing than rejecting a post hoc hypotheses,

even at the same alpha level This suggests that

the timing of the statement of the hypothesis could

have implications for how much alpha is applied

to the resulting analysis In fact, it is difficult to

answer the questions ‘Where does alpha come from?’

and ‘How much alpha should be applied?’, but

in trying to answer these questions, one may well

suggest that the process of generating alpha requires

a prespecified hypothesis [5] Yet, this is not very

satisfying because sometimes unexpected findings

need to be explored In fact, discarding these findings

may be quite problematic itself [1] For example, a

confounder may present itself only after the data are

in, or a key assumption underlying the validity of

the planned analysis may be found to be violated

In theory, it would always be better to test the

hypothesis on new data, rather than on the samedata that suggested the hypothesis, but this is notalways feasible, or always possible [1] Fortunately,there are a variety of approaches to controlling theoverall Type I error rate while allowing for flexibility

in testing hypotheses that were suggested by the data.Two such approaches have already been mentioned,specifically the Pocock sequential boundaries andthe O’Brien – Fleming sequential boundaries, whichallow one to avoid having to select just one analysistime [9]

In the context of the analysis of variance, Fisher’s

least significant difference (LSD) can be used tocontrol the overall Type I error rate when arbi-

trary pairwise comparisons are desired (see Multiple

Comparison Procedures) The approach is based on

operating in protected mode, so that these pairwisecomparisons occur only if an overall equality null

hypothesis is first rejected (see Multiple Testing).

Of course, the overall Type I error rate that is beingprotected is the one that applies to the global nullhypothesis that all means are the same This mayoffer little consolation if one mean is very large,another is very small, and, because of these two,all other means can be compared without adjustment

(see Multiple Testing) The Scheffe method offers

simultaneous inference, as in any linear tion of means can be tested Clearly, this generalizesthe pairwise comparisons that correspond to pairwisecomparisons of means

combina-Another area in which post hoc issues arise is theselection of the primary outcome measure Some-times, there are various outcome measures, or endpoints, to be considered For example, an interven-tion may be used in hopes of reducing childhoodsmoking, as well as drug use and crime It maynot be clear at the beginning of the study which ofthese outcome measures will give the best chance todemonstrate statistical significance In such a case,

it can be difficult to select one outcome measure toserve as the primary outcome measure Sometimes,however, the outcome measures are fusible [4], and,

in this case, this decision becomes much easier Toclarify, suppose that there are two candidate outcomemeasures, say response and complete response (how-ever these are defined in the context in question).Furthermore, suppose that a complete response alsoimplies a response, so that each subject can be clas-sified as a nonresponder, a partial responder, or acomplete responder

A Priori v Post Hoc Testing 3

In this case, the two outcome measures are

fusible, and actually represent different cut points

of the same underlying ordinal outcome measure [4]

By specifying neither component outcome measure,

but rather the information-preserving composite end

point (IPCE), as the primary outcome measure, one

avoids having to select one or the other, and can

find legitimate significance if either outcome

mea-sure shows significance The IPCE is simply the

underlying ordinal outcome measure that contains

each component outcome measure as a binary

sub-endpoint Clearly, using the IPCE can be cast as a

method for allowing post hoc testing, because it

obvi-ates the need to prospectively select one outcome

measure or the other as the primary one Suppose,

for example, that two key outcome measures are

response (defined as a certain magnitude of

bene-fit) and complete response (defined as a somewhat

higher magnitude of benefit, but on the same scale)

If one outcome measure needs to be selected as the

primary one, then it may be unclear which one to

select Yet, because both outcome measures are

mea-sured on the same scale, this decision need not be

addressed, because one could fuse the two outcome

measures together into a single trichotomous outcome

measure, as in Table 1

Even when one recognizes that an outcome

mea-sure is ordinal, and not binary, there may still be

a desire to analyze this outcome measure as if it

were binary by dichotomizing it Of course, there is

a different binary sub-endpoint for each cut point of

the original ordinal outcome measure In the

previ-ous paragraph, for example, one could analyze the

binary response outcome measure (20/30 in the

con-trol group vs 20/30 in the active group in the fictitious

data in Table 1), or one could analyze the binary

com-plete response outcome measure (10/30 in the control

group vs 20/30 in the active group in the fictitious

data in Table 1) With k ordered categories, there are

k− 1 binary sub-endpoints, together comprising the

Lancaster decomposition [12]

In Table 1, the overall response rate would not

differentiate the two treatment groups, whereas the

No

response

Partialresponse

Completeresponse

Noresponse

Partialresponse

Completeresponse

complete response rate would If one knew this ahead

of time, then one might select the overall responserate But the data could also turn out as in Table 2.Now the situation is reversed, and it is the over-all response rate that distinguishes the two treat-ment groups (30/30 or 100% in the active group

vs 20/30 or 67% in the control group), whereas thecomplete response rate does not (10/30 or 33% inthe active group vs 10/30 or 33% in the controlgroup) If either pattern is possible, then it might not

be clear, prior to collecting the data, which of thetwo outcome measures, complete response or over-all response, would be preferred The Smirnov test

(see Kolmogorov–Smirnov Tests) can help, as it

allows one to avoid having to prespecify the ticular sub-endpoint to analyze That is, it allows forthe simultaneous testing of both outcome measures

par-in the cases presented above, or of all k− 1 outcomemeasures more generally, while still preserving theoverall Type I error rate This is achieved by lettingthe data dictate the outcome measure (i.e., selectingthat outcome measure that maximizes the test statis-tic), and then comparing the resulting test statisticnot to its own null sampling distribution, but rather

to the null sampling distribution of the maximallychosen test statistic

Adaptive tests are more general than the Smirnovtest, as they allow for an optimally chosen set ofscores for use with a linear rank test, with the scoresessentially being selected by the data [7] That is, theSmirnov test allows for a data-dependent choice ofthe cut point for a subsequent application on of an

analogue of Fisher’s exact test (see Exact Methods

for Categorical Data), whereas adaptive tests allow

the data to determine the numerical scores to beassigned to the columns for a subsequent linear ranktest Only if those scores are zero to the left of a givencolumn and one to the right of it will the linear ranktest reduce to Fisher’s exact test For the fictitiousdata in Tables 1 and 2, for example, the Smirnovtest would allow for the data-dependent selection ofthe analysis of either the overall response rate or thecomplete response rate, but the Smirnov test would

4 A Priori v Post Hoc Testing

No

response

Partialresponse

Completeresponse

not allow for an analysis that exploits reinforcing

effects To see why this can be a problem, consider

Table 3

Now both of the aforementioned measures can

distinguish the two treatment groups, and in the same

direction, as the complete response rates are 50%

and 33%, whereas the overall response rates are 83%

and 67% The problem is that neither one of these

measures by itself is as large as the effect seen in

Table 1 or Table 2 Yet, overall, the effect in Table 3

is as large as that seen in the previous two tables,

but only if the reinforcing effects of both measures

are considered After seeing the data, one might wish

to use a linear rank test by which numerical scores

are assigned to the three columns and then the mean

scores across treatment groups are compared One

might wish to use equally spaced scores, such as 1,

2, and 3, for the three columns Adaptive tests would

allow for this choice of scores to be used for Table 3

while preserving the Type I error rate by making the

appropriate adjustment for the inherent multiplicity

The basic idea behind adaptive tests is to subject

the data to every conceivable set of scores for use

with a linear rank test, and then compute the

min-imum of all the resulting P values This minmin-imum

P value is artificially small because the data were

allowed to select the test statistic (that is, the scores

for use with the linear rank test) However, this

min-imum P value can be used not as a (valid) P value,

but rather as a test statistic to be compared to the

null sampling distribution of the minimal P value so

computed As a result, the sample space can be

parti-tioned into regions on which a common test statistic

is used, and it is in this sense that the adaptive test

allows the data to determine the test statistic, in a

post hoc fashion Yet, because of the manner in which

the reference distribution is computed (on the basis

of the exact design-based permutation null

distribu-tion of the test statistic [8] factoring in how it was

selected on the basis of the data), the resulting test is

exact This adaptive testing approach was first

pro-posed by Berger [2], but later generalized by Berger

and Ivanova [7] to accommodate preferred alternative

hypotheses and to allow for greater or lesser belief inthese preferred alternatives

Post hoc comparisons can and should be explored,but with some caveats First, the criteria for selectingsuch comparisons to be made should be specifiedprospectively [1], when this is possible Of course,

it may not always be possible Second, plausibilityand subject area knowledge should be considered(as opposed to being based exclusively on statisticalconsiderations) [1] Third, if at all possible, thesecomparisons should be considered as hypothesis-generating, and should lead to additional studies toproduce new data to test these hypotheses, whichwould have been post hoc for the initial experiments,but are now prespecified for the additional ones

References

[1] Adams, K.F (1998) Post hoc subgroup analysis and the

truth of a clinical trial, American Heart Journal 136,

753–758.

[2] Berger, V.W (1998) Admissibility of exact conditional

tests of stochastic order, Journal of Statistical Planning

and Inference 66, 39–50.

[3] Berger, V.W (2001) The p-value interval as an

infer-ential tool, The Statistician 50(1), 79–85.

[4] Berger, V.W (2002) Improving the information content

of categorical clinical trial endpoints, Controlled Clinical

Trials 23, 502–514.

[5] Berger, V.W (2004) On the generation and ownership

of alpha in medical studies, Controlled Clinical Trials

25, 613–619.

[6] Berger, V.W & Bears, J (2003) When can a clinical

trial be called ‘randomized’? Vaccine 21, 468–472.

[7] Berger, V.W & Ivanova, A (2002) Adaptive tests for

ordered categorical data, Journal of Modern Applied

[9] Demets, D.L & Lan, K.K.G (1994) Interim

analy-sis: the alpha spending function approach, Statistics in

Medicine 13, 1341–1352.

[10] Hacking, I (1965) The Logic of Statistical Inference,

Cambridge University Press, Cambridge.

[11] Macdonald, R.R (2002) The incompleteness of bility models and the resultant implications for theories

proba-of statistical inference, Understanding Statistics 1(3),

167–189.

[12] Permutt, T & Berger, V.W (2000) A new look

at rank tests in ordered 2 × k contingency tables,

Communications in Statistics – Theory and Methods 29,

989–1003.

The ACE model refers to a genetic

epidemiologi-cal model that postulates that additive genetic factors

(A) (see Additive Genetic Variance), common

envi-ronmental factors (C), and specific envienvi-ronmental

factors (E) account for individual differences in a

phenotype (P) (see Genotype) of interest This model

is used to quantify the contributions of genetic and

environmental influences to variation and is one of

the fundamental models of basic genetic

epidemiol-ogy [6] Its name is therefore a simple acronym that

allows researchers to communicate the fundamentals

of a genetic model quickly, which makes it a useful

piece of jargon for the genetic epidemiologist The

focus is thus the causes of variation between

individu-als In mathematical terms, the total variance of a trait

(VP)is predicted to be the sum of the variance

compo-nents: VP= VA+ VC+ VE, where VAis the additive

genetic variance, VCthe shared environmental

vari-ance (see Shared Environment), and VEthe specific

environmental variance The aim of fitting the ACE

model is to answer questions about the importance of

nature and nurture on individual differences such as

‘How much of the variation in a trait is accounted

for by genetic factors?’ and ‘Do shared

environ-mental factors contribute significantly to the trait

variation?’ The first of these questions addresses

her-itability, defined as the proportion of the total

vari-ance explained by genetic factors (h2= VA/VP) The

nature-nurture question is quite old It was Sir

Fran-cis Galton [5] who first recognized that comparing

the similarity of identical and fraternal twins yields

information about the relative importance of heredity

versus environment on individual differences At the

time, these observations seemed to conflict with

Gre-gor Mendel’s classical experiments that demonstrated

that the inheritance of model traits in carefully bred

material agreed with a simple theory of particulate

inheritance Ronald Fisher [4] synthesized the views

of Galton and Mendel by providing the first coherentaccount of how the ‘correlations between relatives’could be explained ‘on the supposition of Mendelianinheritance’ In this chapter, we will first explain each

of the sources of variation in quantitative traits inmore detail Second, we briefly discuss the utility of

the classical twin design and the tool of path

analy-sis to represent the twin model Finally, we introduce

the concepts of model fitting and apply them by ting models to actual data We end by discussing thelimitations and assumptions, as well as extensions ofthe ACE model

fit-Quantitative Genetics

Fisher assumed that the variation observed for a traitwas caused by a large number of individual genes,each of which was inherited in a strict conformity

to Mendel’s laws, the so-called polygenic model Ifthe model includes many environmental factors also

of small and equal effect, it is known as the tifactorial model When the effects of many smallfactors are combined, the distribution of trait val-ues approximates the normal (Gaussian) distribution,

mul-according to the central limit theorem Such a

dis-tribution is often observed for quantitative traits thatare measured on a continuous scale and show indi-vidual variation around a mean trait value, but mayalso be assumed for qualitative or categorical traits,which represent an imprecise measurement of an

underlying continuum of liability to a trait (see

Lia-bility Threshold Models), with superimposed

thresh-olds [3] The factors contributing to this variation canthus be broken down in two broad categories, geneticand environmental factors Genetic factors refer toeffects of loci on the genome that contain variants(or alleles) Using quantitative genetic theory, we candistinguish between additive and nonadditive geneticfactors Additive genetic factors (A) are the sum of allthe effects of individual loci Nonadditive genetic fac-tors are the result of interactions between alleles onthe same locus (dominance, D) or between alleles on

different loci (epistasis) Environmental factors are

those contributions that are nongenetic in origin andcan be divided into shared and nonshared environ-mental factors Shared environmental factors (C) areaspects of the environment that are shared by mem-bers of the same family or people who live together,