the r book generalized linear models

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1、13GeneralizedLinearModelsWecanusegeneralizedlinearmodels(GLMs)–pronounced‘glims’–whenthevarianceisnotconstant,and/orwhentheerrorsarenotnormallydistributed.Certainkindsofresponsevariablesinvariablysufferfromthesetwoimportantcontraventionsofthestandardassu

2、mptions,andGLMsareexcellentatdealingwiththem.Specifically,wemightconsiderusingGLMswhentheresponsevariableis:
countdataexpressedasproportions(e.g.logisticregressions);
countdatathatarenotproportions(e.g.log-linearmodelsofcounts);
binaryresponsevariables(e.

3、g.deadoralive);
dataontimetodeathwherethevarianceincreasesfasterthanlinearlywiththemean(e.g.timedatawithgammaerrors).VarianceVariance41302002468102468100246810MeanMeanVarianceVariance0.01.01.52.02.53.00.502040608010002468100246810MeanMeanTheRBook,SecondE

4、dition.MichaelJ.Crawley.©2013JohnWiley&Sons,Ltd.Published2013byJohnWiley&Sons,Ltd.558THERBOOKThecentralassumptionthatwehavemadeuptothispointisthatvariancewasconstant(topleft-handgraph).Incountdata,however,wheretheresponsevariableisanintegerandthereareoft

5、enlotsofzerosinthedataframe,thevariancemayincreaselinearlywiththemean(toptight).Withproportiondata,wherewehaveacountofthenumberoffailuresofaneventaswellasthenumberofsuccesses,thevariancewillbeaninvertedU-shapedfunctionofthemean(bottomleft).Wheretherespon

6、sevariablefollowsagammadistribution(asintime-to-deathdata)thevarianceincreasesfasterthanlinearlywiththemean(bottomright).ManyofthebasicstatisticalmethodssuchasregressionandStudent’sttestassumethatvarianceisconstant,butinmanyapplicationsthisassumptionisun

7、tenable.HencethegreatutilityofGLMs.AGLMhasthreeimportantproperties:
theerrorstructure;
thelinearpredictor;
thelinkfunction.Thesearealllikelytobeunfamiliarconcepts.Theideasbehindthemarestraightforward,however,anditisworthlearningwhateachoftheconceptsinvol

8、ves.13.1ErrorstructureUptothispoint,wehavedealtwiththestatisticalanalysisofdatawithnormalerrors.Inpractice,however,manykindsofdatahavenon-normalerrors:forexample:
errorsthatarestronglyskewed;
errorsthatarekurtotic;
errorst