THE CAMBRIDGE DICTIONARY OF Statistics Fourth Edition B.S. EVERITT InstituteofPsychiatry,King’sCollegeLondon A. SKRONDAL NorwegianInstituteofPublicHealth DepartmentofMathematics,UniversityofOslo CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521766999 © B. S. Everitt and A. Skrondal 2010 First, Second and Third Editions © Cambridge University Press 1998, 2002, 2006 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2010 ISBN-13 978-0-511-78827-7 eBook (EBL) ISBN-13 978-0-521-76699-9 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. A Aalen^Johansenestimator: Anestimator of thesurvivalfunctionfor asetofsurvivaltimes, when there are competing causes of death. Related to the Nelson–Aalen estimator. [ScandinavianJournalofStatistics,1978,5,141–50.] Aalen’slinearregressionmodel: A model for thehazardfunctionof a set ofsurvivaltimes givenby αðt;zðtÞÞ¼α ðtÞþα ðtÞz ðtÞþ(cid:2)(cid:2)(cid:2)þα ðtÞz ðtÞ 0 1 1 p p whereα(t)isthehazardfunctionattimetforanindividualwithcovariatesz(t)0=[z (t),...,z (t)]. 1 p The‘parameters’inthemodelarefunctionsoftimewithα (t)thebaselinehazardcorresponding 0 toz(t)=0 forallt,andα(t),theexcessrateattimetperunitincreaseinz (t).SeealsoCox’s q q proportionalhazardsmodel.[StatisticsinMedicine,1989,8,907–25.] Abbot’sformula: Aformulafortheproportionofanimals(usuallyinsects)dyinginatoxicitytrial thatrecognizesthatsomeinsectsmaydieduringtheexperimentevenwhentheyhavenot beenexposedtothetoxin,andamongthosewhohavebeensoexposed,somemaydieof naturalcauses.Explicitlytheformulais p(cid:3)¼pþð1(cid:4)pÞp i i wherep(cid:3)istheobservableresponseproportion,p istheexpectedproportiondyingatagiven i i doseandπistheproportionofinsectswhorespondnaturally.[ModellingBinaryData,2nd edition,2003,D.Collett,ChapmanandHall/CRCPress,London.] ABCmethod: Abbreviationforapproximatebootstrapconfidencemethod. Abilityparameter: SeeRaschmodel. Absolutedeviation: Synonymforaveragedeviation. Absoluterisk: Synonymforincidence. Absorbingbarrier: Seerandomwalk. AbsorbingMarkovchains: AstateofaMarkovchainisabsorbingifitisimpossibletoleaveit, i.e.theprobabilityofleavingthestateiszero,andaMarkovchainislabelled‘absorbing’ifit has at least one absorbing state. [International Journal of Mathematical Education in ScienceandTechnology,1996,27,197–205.] Absorptiondistributions: Probability distributions that represent the number of ‘individuals’ (e.g.particles)thatfailtocrossaspecifiedregioncontaininghazardsofvariouskinds.For example, the region may simply be a straight line containing a number of ‘absorption’ points.Whenaparticletravellingalongthelinemeetssuchapoint,thereisaprobabilityp thatitwillbeabsorbed.Ifitisabsorbeditfailstomakeanyfurtherprogress,butalsothe point is incapable of absorbing any more particles. When there are M active absorption 1 points, the probability of a particle being absorbed is [1 - (1 – p)M]. [Naval Research LogisticsQuarterly,1966,13,35–48.] Abundance matrices: Matrices that occur in ecological applications. They are essentially two-dimensional tables in which the classifications correspond to site and species. Thevalueintheijthcellgivesthenumberofspeciesjfoundatsitei.[Ecography,2006, 29,525–530.] Acceleratedfailuretimemodel: Ageneralmodelfordataconsistingofsurvivaltimes,inwhich explanatoryvariablesmeasuredonanindividualareassumedtoactmultiplicativelyonthe time-scale, and so affect the rate at which an individual proceeds along the time axis. Consequentlythemodelcanbeinterpretedintermsofthespeedofprogressionofadisease. In the simplest case of comparing two groups of patients, for example, those receiving treatmentAandthosereceivingtreatmentB,thismodelassumesthatthesurvivaltimeofan individualononetreatmentisamultipleofthesurvivaltimeontheothertreatment;asa result the probability that an individual on treatment A survives beyond time t is the probability that an individual on treatment B survives beyond time (cid:1)t, where (cid:1) is an unknownpositiveconstant.Whentheend-pointofinterestisthedeathofapatient,values of(cid:1)lessthanonecorrespondtoanaccelerationinthetimeofdeathofanindividualassigned totreatmentA,andvaluesof(cid:1)greaterthanoneindicatethereverse.Theparameter(cid:1)is known as the acceleration factor. [Modelling Survival Data in Medical Research, 2nd edition,2003,D.Collett,ChapmanandHall/CRCPress,London.] Acceleratedlifetesting: Asetofmethodsintendedtoensureproductreliabilityduringdesignand manufacture in which stress is applied to promote failure. The applied stresses might be temperature,vibration,shocketc.Inordertomakeavalidinferenceaboutthenormallifetime ofthesystemfromtheaccelerateddata(acceleratedinthesensethatashortenedtimetofailure isimplied),itisnecessarytoknowtherelationshipbetweentimetofailureandtheapplied stress.Oftenparametricstatisticalmodelsofthetimetofailureandofthemannerinwhich stressacceleratesagingareused.[AcceleratedTesting,2004,W.Nelson,Wiley,NewYork.] Accelerationfactor: Seeacceleratedfailuretimemodel. Acceptablequalitylevel: Seequalitycontrolprocedures. Acceptablerisk: Theriskforwhichthebenefitsofaparticularmedicalprocedureareconsideredto outweighthepotentialhazards.[AcceptableRisk,1984,B.Fischoff,CambridgeUniversity Press,Cambridge.] Acceptanceregion: Atermassociatedwithstatisticalsignificancetests,thatgivesthesetofvalues ofateststatisticforwhichthenullhypothesisisnotrejected.Suppose,forexample,az-test isbeingusedtotestthenullhypothesisthatthemeanbloodpressureofmenandwomenis equal against the alternative hypothesis that the two means are not equal. If the chosen significancelevelofthetestis0.05thentheacceptanceregionconsistsofvaluesofthetest statisticzbetween–1.96and1.96.[EncyclopediaofStatisticalSciences,2006,eds.S.Kotz, C.B.Read,N.BalakrishnanandB.Vidakovic,Wiley,NewYork.] Acceptance^rejectionalgorithm: Analgorithmforgeneratingrandomnumbersfromsome probabilitydistribution,f(x),byfirstgeneratingarandomnumberfromsomeotherdistri- bution,g(x),wherefandgarerelatedby fðxÞ(cid:5)kgðxÞ for all x withkaconstant.Thealgorithmworksasfollows: 2 * letrbearandomnumberfromg(x); * letsbearandomnumberfromauniformdistributionontheinterval(0,1); * calculatec=ksg(r); * ifc>f(r)rejectrandreturntothefirststep;ifc≤f(r)acceptrasarandomnumber from f. [Statistics in Civil Engineering, 1997, A.V. Metcalfe, Edward Arnold, London.] Acceptancesampling: A type ofqualitycontrolprocedurein which a sample is taken from a collectionorbatchofitems,andthedecisiontoacceptthebatchassatisfactory,orreject themasunsatisfactory,isbasedontheproportionofdefectiveitemsinthesample.[Quality ControlandIndustrialStatistics,4thedition,1974,A.J.Duncan,R.D.Irwin,Homewood, Illinois.] Accidentproneness: A personal psychological factor that affects an individual’s probability of sufferinganaccident.Theconcepthasbeenstudiedstatisticallyunderanumberofdifferent assumptionsforaccidents: * purechance,leadingtothePoissondistribution; * truecontagion,i.e.thehypothesisthatallindividualsinitiallyhavethesameprobability ofhavinganaccident,butthatthisprobabilitychangeseachtimeanaccidenthappens; * apparent contagion, i.e. the hypothesis that individuals have constant but unequal probabilitiesofhavinganaccident. The study of accident proneness has been valuable in the development of particular statisticalmethodologies,althoughinthepasttwodecadestheconcepthas,ingeneral, beenoutoffavour;attentionnowappearstohavemovedmoretowardsriskevaluation and analysis. [Accident Proneness, 1971, L. Shaw and H.S. Sichel, Pergamon Press, Oxford.] Accidentallyemptycells: Synonymforsamplingzeros. Accrualrate: Therateatwhicheligiblepatientsareenteredintoaclinicaltrial,measuredaspersons per unit of time. Often disappointingly low for reasons that may be both physician and patientrelated.[JournalofClinicalOncology,2001,19,3554–61.] Accuracy: Thedegreeofconformitytosomerecognizedstandardvalue.Seealsobias. ACE: Abbreviationforalternatingconditionalexpectation. ACEmodel: Abiometricalgeneticmodelthatpostulatesadditivegeneticfactors,commonenviron- mental factors, and specific environmental factors in a phenotype. The model is used to quantify the contributions of genetic and environmental influences to variation. [Encyclopedia of Behavioral Statistics, Volume 1, 2005, eds. B.S. Everitt and D.C. Howell,Wiley,Chichester.] ACES: Abbreviationforactivecontrolequivalencestudies. ACF: Abbreviationforautocorrelationfunction. ACORN: An acronym for ‘A Classification of Residential Neighbourhoods’. It is a system for classifying households according to the demographic, employment and housing charac- teristics of their immediate neighbourhood. Derived by applying cluster analysis to 40 variables describing each neighbourhood including age, class, tenure, dwelling type and carownership. [Statistics in Society, 1999, eds. D. Dorling andS. Simpson, Arnold, London.] 3 Acquiescencebias: Thebiasproducedbyrespondentsinasurveywhohavethetendencytogive positiveresponses,suchas‘true’,‘like’,‘often’or‘yes’toaquestion.Atitsmostextreme, thepersonrespondsinthiswayirrespectiveofthecontentoftheitem.Thusapersonmay respond‘true’totwoitemslike‘Ialwaystakemymedicationontime’and‘Ioftenforgetto take my pills’. See also end-aversion bias. [Journal of Intellectual Disability Research, 1995,39,331–40.] Actionlines: Seequalitycontrolprocedures. Activecontrolequivalencestudies(ACES): Clinicaltrialsinwhichtheobjectissimplyto showthatthenewtreatmentisatleastasgoodastheexistingtreatment.Suchstudiesare becomingmorewidespreadduetocurrenttherapiesthatreflectprevioussuccessesinthe developmentofnewtreatments.Thestudiesrelyonanimplicithistoricalcontrolassump- tion, since to conclude that a new drug is efficacious on the basis of this type of study requiresafundamentalassumptionthattheactivecontroldrugwouldhaveperformedbetter thanaplacebo,hadaplacebobeenusedinthetrial.[StatisticalIssuesinDrugDevelopment, 2ndedition,2008,S.Senn,Wiley-Blackwell,Chichester.] Activecontroltrials: Clinicaltrialsin which the trial drug is compared with some other active compoundratherthanaplacebo.[AnnalsofInternalMedicine,2000,135,62–4.] Activelifeexpectancy(ALE): Definedforagivenageastheexpectedremainingyearsfreeof disability.Ausefulindexofpublichealthandqualityoflifeforpopulations.Aquestionof greatinterestiswhetherrecenttrendstowardslongerlifeexpectancyhavebeenaccompanied byacomparableincreaseinALE.[NewEnglandJournalofMedicine,1983,309,1218–24.] Actuarialestimator: Anestimatorofthesurvivalfunction,S(t),oftenusedwhenthedataarein groupedform.Givenexplicitlyby " # Y d SðtÞ¼ 1(cid:4) j N (cid:4)1w j(cid:6)0 j 2 j tðjþ1Þ(cid:5)t wheretheorderedsurvivaltimesare0<t <···<t ,N isthenumberofpeopleatriskat (1) (n) i thestartoftheintervalt ,t ,d istheobservednumberofdeathsintheintervalandw the (i) (i+1) i i numberofcensoredobservationsintheinterval.[SurvivalModelsandDataAnalysis,1999, R.G.Elandt–JohnsonandN.L.Johnson,Wiley,NewYork.] Actuarialstatistics: The statistics used by actuaries to evaluate risks, calculate liabilities and planthefinancialcourseofinsurance,pensions,etc.Anexampleislifeexpectancyfor people of various ages, occupations, etc. See also life table. [Financial and Actuarial Statistics: An Introduction, 2003, D.S. Borowiak and A.F. Shapiro, CRC Press, Boca Raton.] Adaptiveclustersampling: Aprocedureinwhichaninitialsetofsubjectsisselectedbysome samplingprocedureand,wheneverthevariableofinterestofaselectedsubjectsatisfiesa given criterion, additional subjects in the neighbourhood of that subject areadded to the sample.[Biometrika,1996,84,209–19.] Adaptivedesigns: Clinicaltrialsthataremodifiedinsomewayasthedataarecollectedwithinthe trial.Forexample,theallocationoftreatmentmaybealteredasafunctionoftheresponseto protect patients from ineffective or toxic doses. [Controlled Clinical Trials, 1999, 20, 172–86.] 4 Adaptiveestimator: Seeadaptivemethods. Adaptivelasso: Seelasso. Adaptivemethods: Procedures that use various aspects of the sample data to select the most appropriatetypeofstatisticalmethodforanalysis.Anadaptiveestimator,T,forthecentre ofadistribution,forexample,mightbe T ¼mid-range when k (cid:5)2 ¼arithmetic mean when25k55 ¼median when k (cid:6)5 where k is the samplekurtosis. So if the sample looks as if it arises from a short-tailed distribution,theaverageofthelargestandsmallestobservationsisused;ifitlookslikea long-tailed situation the median is used, otherwise the mean of the sample is calculated. [JournaloftheAmericanStatisticalAssociation,1967,62,1179–86.] Adaptivemethods of treatmentassignment: Any method of treatment allocation in a clinicaltrialthat uses accumulating outcome data to affect the treatment selection, for example,theO’Brien-Flemingmethod.[Biometrika,1977,64,191–199.] Adaptivesamplingdesign: Asamplingdesignin which the procedure for selectingsampling unitson which to make observations may depend on observed values of the variable of interest. In a survey for estimating the abundance of a natural resource, for example, additionalsites(thesamplingunitsinthiscase)inthevicinityofhighobservedabundance maybeaddedtothesampleduringthesurvey.Themainaiminsuchadesignistoachieve gainsinprecisionorefficiencycomparedtoconventionaldesignsofequivalentsamplesize bytakingadvantageofobservedcharacteristicsofthepopulation.Forthistypeofsampling designtheprobabilityofagivensampleofunitsisconditionedonthesetofvaluesofthe variable of interest in the population. [Adaptive Sampling, 1996, S.K. Thompson and G.A.F.Seber,Wiley,NewYork.] Addedvariableplot: Agraphicalprocedureusedinalltypesofregressionanalysisforidentifying whether or not a particular explanatory variable should be included in a model, in the presenceofotherexplanatoryvariables.Thevariablethatisthecandidateforinclusioninthe model may be new or it may simply be a higher power of one currently included. If thecandidatevariableisdenotedx,thentheresidualsfromtheregressionoftheresponse i variableonalltheexplanatoryvariables,savex,areplottedagainsttheresidualsfromthe i regressionofx ontheremainingexplanatoryvariables.Astronglinearrelationshipinthe i plot indicates the need for x in the regression equation (Fig. 1). [Regression Analysis, i Volume2,1993,editedbyM.S.Lewis-Beck,SagePublications,London.] Addition rule for probabilities: For two events, A and B that are mutuallyexclusive, the probabilityofeithereventoccurringisthesumoftheindividualprobabilities,i.e. PrðAor BÞ¼PrðAÞþPrðBÞ where Pr(A) denotes the probability of event A etc. For k mutually exclusive events A , 1 A ,...,A,themoregeneralruleis 2 k PrðA or A ... or A Þ¼PrðA ÞþPrðA Þþ(cid:2)(cid:2)(cid:2)þPrðA Þ 1 2 k 1 2 k SeealsomultiplicationruleforprobabilitiesandBoole’sinequality.[KA1Chapter8.] 5 e bl a ari v se 40 n o p s e of r 0 n 2 o si s e gr e m r 0 o s fr al u d 0 si 2 e− R Fig.1 Addedvariable 0 500 1000 plotindicatingavariable Residuals from regression of the candidate variable thatshouldbeincludedin on other explanatory variables themodel. Additiveclusteringmodel: Amodelforclusteranalysiswhichattemptstofindthestructureofa similaritymatrixwithelementss byfittingamodeloftheform ij XK s ¼ w p p þ(cid:3) ij k ik jk ij k¼1 whereKisthenumberofclustersandw isaweightrepresentingthesalienceoftheproperty k correspondingtoclusterk.Ifobjectihasthepropertyofclusterk,thenp =1,otherwiseitis ik zero.[PsychologicalReview,1979,86,87–123.] Additiveeffect: Atermusedwhentheeffectofadministeringtwotreatmentstogetheristhesumof theirseparateeffects.Seealsoadditivemodel.[JournalofBoneMineralResearch,1995, 10,1303–11.] Additivegeneticvariance: The variance of a trait due to the main effects of genes. Usually obtainedbyafactorialanalysisofvarianceoftraitvaluesonthegenespresentatoneormore loci.[StatisticsinHumanGenetics,1998,P.Sham,Arnold,London.] Additivemodel: Amodelinwhichtheexplanatoryvariableshaveanadditiveeffectontheresponse variable.So,forexample,ifvariableAhasaneffectofsizeaonsomeresponsemeasureand variableBoneofsizebonthesameresponse,theninanassumedadditivemodelforAandB theircombinedeffectwouldbea+b. Additiveoutlier: A term applied to an observation in a timeserieswhich is affected by a non- repetitive intervention such as a strike, a war, etc. Only the level of the particular observation is considered affected. In contrast an innovational outlier is one which corresponds to an extraordinary shock at some time point Twhich also influences sub- sequentobservationsintheseries.[JournaloftheAmericanStatisticalAssociation,1996, 91,123–31.] Additivetree: Aconnected,undirectedgraphwhereeverypairofnodesisconnectedbyaunique pathandwherethedistancesbetweenthenodesaresuchthat 6 a. Dissimiliaritles A B C D E Worker A Worker B 15 Worker C 20 25 Worker D 18 23 6 Worker E 20 25 20 18 b. Additive Tree 10 E 2 D 6 4 C Fig.2 Anexampleofanadditivetree. 10 B (ReproducedbypermissionofSagePublications 5 fromTreeModelsofSimilarityandAssociation, 5 A 1996,J.E.Corter.) d þd (cid:5)max½d þd ;d þd (cid:7) for allx;y;u; andv xy uv xu yv xv yu AnexampleofsuchatreeisshowninFig.2.Seealsoultrametrictree.[TreeModelsof Similarity and Association, 1996, J.E. Corter, Sage University Papers 112, Sage Publications,ThousandOaks.] Adelstein,Abe(1916^1993): BorninSouthAfrica,AdelsteinstudiedmedicineattheUniversity of the Witwatersrand. In the 1960s he emigrated to Manchester where he worked in the Department of Social Medicine. Later he was appointed Chief Medical Statistician for EnglandandWales.Adelsteinmadesignificantcontributionstotheclassificationofmental illnessandtotheepidemiologyofsuicideandalcoholism. Adequatesubset: Atermusedinregressionanalysisforasubsetoftheexplanatoryvariablesthatis thoughttocontainasmuchinformationabouttheresponsevariableasthecompleteset.See alsoselectionmethodsinregression. Adjacencymatrix: Amatrixwithelements,x ,usedtoindicatetheconnectionsinadirectedgraph. ij Ifnodeirelatestonodej,x =1,otherwisex =0.Forasimplegraphwithnoself-loops,the ij ij adjacencymatrixmusthavezerosonthediagonal.Foranundirectedgraphtheadjacency matrixissymmetric.[IntroductoryGraphTheory,1985,G.Chartrand,Dover,NewYork.] Adjustedcorrelationmatrix: Acorrelationmatrixinwhichthediagonalelementsarereplaced bycommunalities.Thebasisofprincipalfactoranalysis. Adjustedtreatmentmeans: Usuallyusedforestimatesofthetreatmentmeansinananalysisof covariance,afteradjustingalltreatmentstothesamemeanlevelforthecovariate(s),using theestimatedrelationshipbetweenthecovariate(s)andtheresponsevariable.[Biostatistics: A Methodology for the Health Sciences, 2nd edn, 2004, G. Van Belle, L.D. Fisher, P.J. HeagertyandT.S.Lumley,Wiley,NewYork.] Adjustingforbaseline: The process of allowing for the effectofbaselinecharacteristicson the responsevariableusuallyinthecontextofalongitudinalstudy. SeealsoLord’sparadox 7 andbaselinebalance.[StatisticalIssuesinDrugDevelopment,2ndedition,2008,S.Senn, Wiley-Blackwell,Chichester.] Administrativedatabases: Databasesstoring information routinely collected for purposes of managing a health-care system. Used by hospitals and insurers to examine admissions, proceduresandlengthsofstay.[HealthcareManagementForum,1995,8,5–13.] Admissibility: Averygeneralconceptthatisapplicabletoanyprocedureofstatisticalinference.The underlyingnotionisthataprocedureisadmissibleifandonlyiftheredoesnotexistwithin that class of procedures another one which performs uniformly at least as well as the procedure in question and performs better than it in at least one case. Here ‘uniformly’ means for all values of the parameters that determine the probability distribution of the randomvariablesunderinvestigation.[KA2Chapter31.] Admixtureinhumanpopulations: Theinter-breeding betweentwoormorepopulationsthat werepreviouslyisolatedfromeachotherforgeographicalorculturalreasons.Population admixturecanbeasourceofspuriousassociationsbetweendiseasesandallelesthatareboth morecommoninoneancestralpopulationthantheothers.However,populationsthathave been admixed for several generations may beuseful for mapping diseasegenes,because spuriousassociationstendtobedissipatedmorerapidlythantrueassociationsinsuccessive generations of random mating. [Statistics in Human Genetics, 1998, P. Sham, Arnold, London.] Adoptionstudies: Studies of the rearing of anonbiological child in afamily. Such studies have playedanimportantroleintheassessmentofgeneticvariationinhumanandanimaltraits. [Foundations of Behavior Genetics, 1978, J.L. Fulker and W.R. Thompson, Mosby, St.Louis.] Adverseselection: Atermusedininsurancewhentheinsurercannotdistinguishbetweenmembers of good- and poor-risk categories for a certain hazard and the poor-risks are the only purchasers of coverage with the consequence that the insurer expects to lose money on eachpolicysold.[QuarterlyJournalofEconomics,1976,90,629–650.] Aetiologicalfraction: Synonymforattributablerisk. Affineinvariance: Atermappliedtostatisticalprocedureswhichgiveidenticalresultsafterthedata hasbeensubjectedtoanaffinetransformation.AnexampleisHotelling’sT2test.[Canadian JournalofStatistics,2003,31,437–55.] Affinetransformation: Thetransformation,Y=AX+bwhereAisanonsingularmatrixandbis anyvectorofrealnumbers.Importantinmanyareasofstatisticsparticularlymultivariate analysis. Age-dependentbirthanddeathprocess: Abirthanddeathprocesswherethebirthrateand deathratearenotconstantovertime,butchangeinamannerwhichisdependentontheage oftheindividual.[StochasticModellingofScientificData,1995,P.Guttorp,Chapmanand Hall/CRCPress,London.] Ageheaping: Atermappliedtothecollectionofdataonageswhentheseareaccurateonlytothe nearestyear,halfyearormonth.Occursbecausemanypeople(particularlyolderpeople) tendnottogivetheirexactageinasurvey.Insteadtheyroundtheirageupordowntothe nearestnumberthatendsin0or5.SeealsocoarsedataandWhippleindex.[Population Studies,1991,45,497–518.] 8