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1、17BinaryResponseVariablesManystatisticalproblemsinvolvebinaryresponsevariables.Forexample,weoftenclassifyindividualsasdeadoralive,occupiedorempty,healthyordiseased,wiltedorturgid,maleorfemale,literateorilliterate,matureorimmature,solventorinsolvent,oremployedorunemployed.Itisintere
2、stingtounderstandthefactorsthatareassociatedwithanindividualbeinginoneclassortheother.Binaryanalysiswillbeausefuloptionwhenatleastoneofyourexplanatoryvariablesiscontinuous(ratherthancategorical).Inastudyofcompanyinsolvency,forinstance,thedatawouldconsistofalistofmeasurementsmadeontheinsolve
3、ntcompanies(theirage,size,turnover,location,managementexperience,workforcetraining,andsoon)andasimilarlistforthesolventcompanies.Thequestionthenbecomeswhich,ifany,oftheexplanatoryvariablesincreasetheprobabilityofanindividualcompanybeinginsolvent.Theresponsevariablecontainsonly0sand1s;forexa
4、mple,0torepresentdeadindividualsand1torepresentliveones.Thus,thereisonlyasinglecolumnofnumbersfortheresponse,incontrasttoproportiondatawheretwovectors(successesandfailures)wereboundtogethertoformtheresponse(seeChapter16).ThewaythatRtreatsbinarydataistoassumethatthe0sand1scomefromabinomialtr
5、ialwithsamplesize1.Iftheprobabilitythatanindividualisdeadisp,thentheprobabilityofobtainingy(whereyiseitherdeadoralive,0or1)isgivenbyanabbreviatedformofthebinomialdistributionwithn=1,knownastheBernoullidistribution:P(y)=py(1−p)(1−y).TheRBook,SecondEdition.MichaelJ.Crawley.©2013JohnWiley&Sons
6、,Ltd.Published2013byJohnWiley&Sons,Ltd.BINARYRESPONSEVARIABLES651Therandomvariableyhasameanofpandavarianceofp(1–p),andtheobjectiveistodeterminehowtheexplanatoryvariablesinfluencethevalueofp.Thetricktousingbinaryresponsevariableseffectivelyistoknowwhenitisworthusingthem,andwhenitisbettertolum
7、pthesuccessesandfailurestogetherandanalysethetotalcountsofdeadindividuals,occupiedpatches,insolventfirmsorwhatever.Thequestionyouneedtoaskyourselfis:doIhaveuniquevaluesofoneormoreexplanatoryvariablesforeachandeveryindividualcase?Iftheansweris‘yes’,thenana