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+10,MONOVOLUME,SP,65,F,Solteiro,33423,"0,81",0 +4,MONOVOLUME,SP,73,F,Viuvo,25956,"0,79",0 +10,SEDAN,SP,69,F,Casado,59251,"0,02",0 +10,PICAPE LEVE,SP,70,M,Casado,34950,"0,79",0 +3,CAMIONETA LEVE,SP,65,M,Casado,10018,"0,83",0 +10,PICAPE PESADA,RS,77,M,Viuvo,42372,"0,80",0 +10,SEDAN,SP,65,M,Casado,84677,"0,59",0 +6,SEDAN,SP,70,M,Casado,25111,"0,77",0 +10,PICAPE LEVE,SP,47,M,Solteiro,21808,"0,81",0 +8,SEDAN,PR,66,M,Casado,25066,"0,80",0 +2,HATCH,SP,36,F,Casado,31301,"0,51",0 +1,HATCH,SP,55,M,Casado,17613,"0,79",0 +3,STATION WAGON,SC,40,M,Casado,31129,"0,25",2 +1,SEDAN,BA,45,M,Casado,25692,"0,81",2 +1,HATCH,SP,26,F,Casado,51205,"0,47",0 +10,HATCH,SP,35,M,Casado,16003,"0,76",0 +2,HATCH,RS,61,F,Casado,27511,"0,77",0 +2,SEDAN,PR,57,M,Casado,40726,"0,44",0 +7,SUV,PA,63,M,Casado,71918,"0,77",0 +10,SUV,SP,52,F,Casado,59598,"0,34",0 +10,HATCH,SP,20,F,Solteiro,43050,"0,49",0 +6,HATCH,SC,49,M,Casado,34332,"0,79",0 +3,SUV,SP,38,M,Solteiro,28028,"0,80",0 +1,HATCH,SC,28,F,Casado,15952,"0,48",0 +8,PICAPE LEVE,SP,35,M,Casado,34477,"0,83",0 +4,CAMIONETA LEVE,SP,59,F,Casado,46260,"0,35",0 +1,SUV,RJ,30,M,Solteiro,62798,"0,80",0 +10,HATCH,SP,29,M,Solteiro,33293,"0,08",0 +1,SEDAN,SP,50,F,Solteiro,15742,"0,80",0 +10,MONOVOLUME,SP,53,M,Casado,40533,"0,57",0 +10,SEDAN,SP,50,F,Casado,35820,"0,76",0 +10,PICAPE LEVE,MG,50,M,Casado,30275,"0,83",0 +10,SEDAN,SP,38,M,Solteiro,72544,"0,80",0 +10,PICAPE PESADA,MG,76,M,Casado,45320,"0,76",0 +10,MONOVOLUME,SP,53,F,Divorciado,31102,"0,81",0 +5,SEDAN,SC,40,M,Solteiro,23195,"0,82",0 +10,SEDAN,SP,62,M,Casado,32375,"0,77",0 +1,SUV,SP,37,F,Casado,196650,"0,04",0 +10,HATCH,SP,52,F,Solteiro,14178,"0,81",0 +2,HATCH,GO,68,F,Casado,36860,"0,78",0 +10,SUV,RS,54,M,Casado,66519,"0,12",0 +10,MONOVOLUME,SP,47,F,Solteiro,44410,"0,48",0 +2,HATCH,SP,36,M,Casado,30453,"0,82",0 +2,HATCH,MG,45,M,Solteiro,28776,"0,59",0 +10,HATCH,SP,53,F,Viuvo,15644,"0,78",0 +3,SEDAN,DF,36,M,Casado,14745,"0,80",0 +9,MONOVOLUME,SP,46,F,Casado,26695,"0,42",0 +10,HATCH,SP,62,F,Solteiro,22319,"0,76",0 +2,MONOVOLUME,SP,40,M,Casado,19922,"0,83",0 +10,SEDAN,SP,49,M,Casado,22428,"0,76",0 +10,STATION WAGON,SP,50,F,Casado,50639,"0,79",0 +5,PICAPE LEVE,SP,53,F,Casado,33856,"0,78",0 +7,MONOVOLUME,PR,70,M,Casado,33423,"0,80",0 +10,MONOVOLUME,SP,43,M,Casado,30358,"0,82",0 +10,SUV,SP,48,F,Casado,52374,"0,78",1 +10,HATCH,SP,67,F,Solteiro,13358,"0,76",0 +10,SEDAN,SP,73,M,Casado,35959,"0,81",0 +8,HATCH,SP,60,F,Casado,27488,"0,76",0 +10,SUV,SP,65,M,Casado,51100,"0,83",0 +2,SEDAN,CE,37,M,Casado,23734,"0,78",0 +1,SUV,MG,41,M,Casado,25002,"0,82",0 +1,HATCH,SP,68,F,Casado,16238,"0,08",0 +0,HATCH,SC,31,M,Casado,19591,"0,67",0 +10,SEDAN,RJ,54,M,Casado,43588,"0,53",0 +10,PICAPE LEVE,SP,44,M,Casado,20742,"0,17",0 +0,SUV,SP,69,F,Divorciado,43257,"0,77",0 +9,SUV,SP,59,F,Solteiro,37113,"0,82",0 +6,SUV,RS,61,M,Casado,58813,"0,55",0 +10,SEDAN,SP,67,M,Casado,44243,"0,82",1 +0,SEDAN,PE,63,M,Casado,28748,"0,79",0 +9,SUV,RS,38,F,Casado,62167,"0,79",0 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+5,MONOVOLUME,SP,58,F,Casado,33423,"0,82",0 +9,SUV,RS,59,F,Divorciado,48888,"0,83",0 +10,SUV,SP,69,M,Casado,91608,"0,42",0 +10,HATCH,SP,62,M,Casado,16484,"0,76",0 +10,SEDAN,SP,63,F,Casado,53380,"0,76",0 +2,HATCH,SP,76,M,Casado,27839,"0,78",0 +9,HATCH,SP,28,F,Solteiro,22245,"0,13",0 +10,MONOVOLUME,SP,34,M,Casado,38903,"0,76",0 +2,SEDAN,SP,67,M,Divorciado,15997,"0,76",0 +1,HATCH,DF,56,F,Casado,42200,"0,76",0 +4,SEDAN,SP,36,M,Casado,19187,"0,78",0 +2,SEDAN,SP,70,M,Casado,84677,"0,76",0 +10,SUV,SP,47,M,Casado,42713,"0,83",0 +10,MONOVOLUME,SP,73,M,Casado,33946,"0,83",0 +3,HATCH,RS,47,F,Casado,37612,"0,36",0 +10,SUV,SP,74,M,Casado,18055,"0,78",0 +3,SEDAN,SP,53,F,Casado,25844,"0,76",0 +10,PICAPE LEVE,SP,66,M,Casado,15590,"0,83",0 +7,SUV,PR,51,M,Casado,41971,"0,76",0 +10,MONOVOLUME,SP,33,M,Casado,56546,"0,68",1 +0,HATCH,PE,46,M,Solteiro,17683,"0,15",0 +0,SUV,RS,39,M,Casado,32478,"0,79",0 +3,PICAPE LEVE,PR,54,M,Casado,28002,"0,80",0 +9,SEDAN,SP,44,F,Divorciado,22203,"0,78",0 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+10,SEDAN,PR,69,M,Casado,21548,"0,77",0 +3,HATCH,RS,24,F,Solteiro,27839,"0,82",0 +0,HATCH,SC,28,F,Solteiro,32092,"0,76",0 +10,HATCH,SP,27,M,Casado,35780,"0,81",0 +7,SUV,RS,58,M,Casado,36520,"0,77",0 +4,HATCH,SP,37,M,Casado,19812,"0,80",0 +10,SEDAN,SP,55,M,Casado,35820,"0,61",1 +10,HATCH,SP,50,F,Solteiro,50104,"0,77",0 +1,HATCH,SP,55,F,Casado,34211,"0,78",0 +7,SEDAN,SP,34,M,Casado,59894,"0,21",0 +10,PICAPE PESADA,MG,55,F,Casado,67773,"0,77",0 +10,HATCH,SP,33,M,Solteiro,49105,"0,10",0 +10,PICAPE LEVE,SP,60,M,Casado,15201,"0,80",0 +4,SEDAN,SP,28,F,Casado,35820,"0,22",0 +10,MONOVOLUME,SP,52,M,Casado,17573,"0,81",1 +4,STATION WAGON,SP,55,M,Casado,36647,"0,77",1 +10,SUV,SP,36,F,Casado,37008,"0,81",1 +10,SEDAN,SP,66,F,Divorciado,17316,"0,10",0 +10,MONOVOLUME,SP,58,M,Casado,21963,"0,77",0 +5,HATCH,SP,28,M,Casado,42510,"0,07",0 +10,SEDAN,SP,37,F,Casado,61685,"0,76",0 +10,HATCH,SP,59,F,Viuvo,22425,"0,80",0 +10,HATCH,SP,50,M,Casado,10530,"0,83",0 +1,SUV,MG,79,M,Viuvo,21810,"0,77",0 +9,HATCH,SP,56,M,Divorciado,30719,"0,79",0 +9,HATCH,SP,56,M,Casado,10421,"0,20",0 +6,HATCH,SP,70,F,Divorciado,12265,"0,82",0 +10,HATCH,SP,63,F,Divorciado,34337,"0,82",0 +9,SEDAN,SP,51,M,Casado,24522,"0,81",0 +10,HATCH,SP,44,F,Casado,34325,"0,01",0 +10,HATCH,SP,67,M,Casado,36929,"0,05",0 +8,PICAPE PESADA,SP,45,F,Casado,67773,"0,77",0 +10,HATCH,SP,79,M,Casado,7919,"0,82",0 +10,MONOVOLUME,SP,68,F,Casado,32075,"0,79",0 +9,SEDAN,SP,57,F,Casado,73521,"0,80",0 +7,PICAPE LEVE,SP,45,M,Casado,18695,"0,82",0 +9,HATCH,SP,54,M,Casado,21423,"0,81",0 +5,MONOVOLUME,PI,26,M,Solteiro,25956,"0,82",0 +3,HATCH,SP,36,F,Casado,33800,"0,13",0 +0,HATCH,PR,64,M,Casado,26995,"0,77",0 +9,SUV,SP,41,F,Casado,48676,"0,12",0 +10,SEDAN,SP,38,M,Casado,51929,"0,37",0 +10,SEDAN,SP,49,F,Casado,60278,"0,83",0 +0,SEDAN,RS,45,M,Casado,39511,"0,03",0 +1,SUV,SP,44,F,Casado,46710,"0,77",0 +0,SEDAN,SP,71,M,Casado,34474,"0,79",0 +10,HATCH,SP,61,M,Casado,23829,"0,79",0 +3,HATCH,SP,37,M,Casado,20556,"0,67",0 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+7,HATCH,SP,49,F,Casado,29711,"0,81",0 +9,SUV,SP,63,F,Casado,86778,"0,61",0 +10,SEDAN,SP,57,M,Casado,14148,"0,79",0 +4,HATCH,MG,38,F,Casado,16448,"0,81",0 +10,SEDAN,SP,51,F,Casado,26970,"0,82",0 +6,SEDAN,SP,30,F,Casado,70633,"0,18",0 +4,HATCH,PR,37,M,Casado,37994,"0,82",0 +0,SUV,SP,52,F,Casado,47685,"0,79",1 +5,HATCH,SP,37,F,Casado,37994,"0,03",0 +10,HATCH,SP,69,F,Casado,23085,"0,80",0 +0,HATCH,MG,39,M,Casado,24886,"0,80",0 +1,SUV,RS,42,M,Casado,58759,"0,23",0 +10,PICAPE LEVE,SP,65,M,Casado,9781,"0,78",0 +2,HATCH,SP,66,M,Casado,34431,"0,80",0 +3,PICAPE LEVE,PR,45,M,Casado,27794,"0,76",0 +10,MONOVOLUME,SP,39,F,Casado,18192,"0,44",0 +1,SEDAN,SP,53,F,Casado,22428,"0,81",0 +10,HATCH,RS,64,F,Casado,37090,"0,12",0 +10,SEDAN,SP,39,F,Casado,64016,"0,40",0 +10,SEDAN,SP,36,M,Casado,27779,"0,47",0 +0,HATCH,SP,69,M,Casado,37372,"0,77",0 +10,HATCH,SP,69,M,Casado,27568,"0,81",0 +10,HATCH,SP,52,F,Casado,17048,"0,79",1 +0,SEDAN,SP,28,F,Casado,18645,"0,45",0 +10,HATCH,SP,62,M,Casado,21423,"0,82",0 +2,PICAPE LEVE,PR,69,M,Casado,23662,"0,83",0 +10,SEDAN,SP,55,M,Casado,51929,"0,38",0 +6,HATCH,MG,27,M,Solteiro,17455,"0,81",0 +10,HATCH,SP,26,M,Casado,30719,"0,12",0 +5,HATCH,SP,77,F,Divorciado,31937,"0,79",0 +10,SEDAN,SP,66,F,Solteiro,24273,"0,76",0 +0,HATCH,PR,32,M,Casado,39164,"0,81",0 +1,SEDAN,SP,38,M,Casado,34913,"0,31",0 +7,HATCH,RS,56,F,Casado,32092,"0,82",0 +4,PICAPE LEVE,SP,30,M,Casado,49142,"0,81",0 +10,HATCH,SP,33,F,Casado,26255,"0,52",0 +7,MONOVOLUME,SP,48,M,Casado,25956,"0,77",0 +0,PICAPE LEVE,SE,60,M,Casado,47750,"0,78",0 +3,SUV,PR,65,F,Casado,36520,"0,78",0 +10,MONOVOLUME,SP,35,M,Casado,44538,"0,01",0 +1,HATCH,SP,60,F,Viuvo,28810,"0,77",0 +3,HATCH,CE,65,M,Casado,34679,"0,83",0 +10,SEDAN,PR,33,F,Casado,44799,"0,79",0 +1,HATCH,SP,50,M,Casado,34325,"0,50",0 +3,SEDAN,SP,57,M,Casado,37929,"0,32",0 +0,HATCH,SP,48,F,Casado,15952,"0,36",1 +8,SEDAN,MG,80,M,Casado,29415,"0,50",0 +8,STATION WAGON,RS,57,F,Casado,25685,"0,81",0 +10,SEDAN,SP,62,M,Casado,12043,"0,82",0 +7,PICAPE LEVE,MG,63,M,Casado,34677,"0,75",0 +10,HATCH,SP,68,F,Viuvo,8593,"0,53",1 +10,SEDAN,SP,88,M,Casado,23815,"0,78",0 +10,SEDAN,SP,60,M,Divorciado,22428,"0,78",0 +10,SEDAN,SP,56,F,Divorciado,17316,"0,42",0 +10,SEDAN,SP,57,M,Casado,71278,"0,49",0 +10,SUV,CE,39,F,Casado,64452,"0,76",0 +10,HATCH,SP,59,M,Casado,15560,"0,78",0 +10,SEDAN,SP,40,F,Solteiro,32287,"0,78",0 +10,SEDAN,SP,58,M,Casado,34253,"0,39",0 +0,SUV,SC,56,M,Casado,101020,"0,83",0 +10,SUV,RS,36,M,Casado,64785,"0,79",0 +10,MONOVOLUME,SP,80,M,Casado,25335,"0,78",0 +10,HATCH,MG,33,F,Solteiro,26100,"0,81",0 +10,MONOVOLUME,SP,64,M,Casado,62820,"0,76",0 +7,HATCH,SP,33,M,Casado,16238,"0,31",0 +0,STATION WAGON,SP,65,M,Casado,41644,"0,77",0 +0,MONOVOLUME,BA,58,F,Casado,45876,"0,79",0 +3,SEDAN,SP,61,F,Casado,21352,"0,82",0 +10,SEDAN,SP,41,M,Casado,59894,"0,82",0 +1,PICAPE PESADA,PR,62,M,Casado,41921,"0,03",0 +5,SEDAN,SP,33,M,Solteiro,75025,"0,76",0 +0,MONOVOLUME,SP,53,F,Casado,24823,"0,78",0 +10,HATCH,SP,51,F,Solteiro,13657,"0,83",0 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+10,MONOVOLUME,SP,79,M,Viuvo,15996,"0,80",0 +0,HATCH,SP,40,F,Casado,16356,"0,05",0 +10,MONOVOLUME,SP,58,M,Casado,19922,"0,82",0 +0,HATCH,SP,23,F,Casado,15952,"0,77",0 +10,SEDAN,MG,56,M,Casado,26278,"0,80",0 +10,SEDAN,SP,48,F,Casado,33844,"0,20",0 +6,SUV,PE,36,M,Casado,65410,"0,84",0 +4,MONOVOLUME,SP,59,M,Casado,21783,"0,81",0 +8,SEDAN,CE,77,F,Viuvo,77545,"0,81",0 +3,CAMIONETA LEVE,PR,45,M,Casado,18376,"0,83",0 +6,HATCH,SP,71,M,Casado,60941,"0,79",0 +2,HATCH,SP,55,F,Casado,18691,"0,71",0 +2,HATCH,SP,34,F,Casado,28689,"0,12",0 +2,HATCH,SP,64,M,Casado,18104,"0,82",0 +8,HATCH,SP,57,F,Casado,34325,"0,77",0 +10,SUV,SP,72,F,Viuvo,47534,"0,82",0 +10,SUV,SP,52,M,Casado,52478,"0,50",0 +3,SEDAN,SP,54,M,Casado,34854,"0,75",0 +10,MONOVOLUME,MG,61,M,Solteiro,36848,"0,38",0 +10,SUV,SP,65,M,Casado,80429,"0,29",0 +3,SEDAN,DF,50,F,Casado,24755,"0,81",0 +6,SEDAN,SP,63,F,Casado,19608,"0,80",0 +3,HATCH,SP,46,M,Solteiro,39703,"0,12",0 +10,HATCH,PR,63,M,Casado,27713,"0,83",0 +10,MONOVOLUME,SP,71,M,Casado,17884,"0,79",0 +10,SUV,SC,64,M,Casado,35069,"0,80",0 +7,SUV,SP,44,M,Casado,54510,"0,53",0 +10,HATCH,SP,69,F,Solteiro,3884,"0,04",0 +10,HATCH,SP,49,F,Casado,20073,"0,13",0 +8,SUV,SC,46,F,Casado,34816,"0,76",0 +4,CAMIONETA LEVE,PR,46,F,Casado,36982,"0,83",0 +10,HATCH,SP,51,M,Casado,17818,"0,80",0 +6,MONOVOLUME,SP,29,F,Solteiro,17222,"0,79",0 +0,SEDAN,RJ,62,M,Casado,60278,"0,31",1 +10,HATCH,PE,44,F,Casado,23723,"0,77",0 +5,SEDAN,SP,55,F,Divorciado,35493,"0,59",0 +10,HATCH,SP,64,F,Casado,42200,"0,79",0 +10,SEDAN,SP,54,M,Casado,42532,"0,81",0 +10,HATCH,SP,64,F,Casado,39377,"0,24",0 +10,SEDAN,SP,65,M,Casado,55200,"0,80",0 +10,SUV,SP,65,F,Casado,59839,"0,84",0 +1,SEDAN,SP,46,M,Casado,44904,"0,27",0 +1,MONOVOLUME,SP,61,M,Casado,21963,"0,04",0 +1,HATCH,SP,52,F,Casado,23329,"0,80",0 +10,HATCH,PR,68,M,Casado,34325,"0,81",0 +1,SEDAN,DF,29,M,Casado,43752,"0,82",0 +10,HATCH,SP,45,F,Solteiro,11740,"0,83",0 +10,SUV,SP,70,M,Casado,52478,"0,30",0 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+9,HATCH,SP,47,F,Casado,32585,"0,81",0 +10,SEDAN,SP,39,F,Casado,34624,"0,82",0 +10,SUV,SP,50,M,Casado,45846,"0,79",0 +10,HATCH,SP,53,F,Casado,34325,"0,83",0 +10,HATCH,SP,64,F,Casado,16448,"0,76",0 +10,SEDAN,MG,79,M,Casado,74088,"0,05",0 +2,HATCH,SP,65,F,Casado,27726,"0,82",0 +3,SUV,SP,41,F,Casado,66560,"0,29",0 +2,HATCH,SP,57,F,Casado,45387,"0,79",0 +2,HATCH,SP,41,F,Casado,31287,"0,82",0 +0,SUV,MG,29,M,Casado,36093,"0,81",0 +3,HATCH,MG,38,M,Casado,20806,"0,79",0 +10,HATCH,SP,60,F,Casado,6268,"0,78",0 +10,SEDAN,SP,63,F,Casado,35959,"0,79",0 +2,HATCH,SP,48,M,Casado,26170,"0,68",0 +10,HATCH,SP,52,F,Divorciado,27839,"0,77",0 +2,HATCH,SP,35,F,Casado,32546,"0,11",0 +6,MONOVOLUME,SC,41,F,Casado,16114,"0,16",0 +1,HATCH,SP,84,M,Casado,14378,"0,58",1 +0,HATCH,SP,45,F,Casado,21211,"0,79",0 +10,SEDAN,SP,70,M,Casado,84525,"0,79",0 +10,SEDAN,SP,42,M,Casado,25826,"0,81",0 +6,HATCH,SP,38,M,Casado,34225,"0,76",1 +10,SEDAN,SC,63,M,Casado,53185,"0,01",0 +10,SEDAN,MG,83,M,Casado,28346,"0,78",0 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+ +%% ====================================================================== +%% Formatando slides para seções e subseções +\AtBeginSection[]{ + \begin{frame}[c, allowframebreaks] + \begin{center} + \textcolor{teal}{\thesection} \\ \vspace{0.3cm} + \parbox{0.6\textwidth}{ + \centering \textcolor{teal}{\LARGE \bf \insertsection}}\\ + \end{center} + \end{frame} +} + +\AtBeginSubsection{ + \begin{frame}[c, allowframebreaks] + \begin{center} + \textcolor{teal}{\thesection.\thesubsection} \\ \vspace{0.3cm} + \parbox{0.6\textwidth}{ + \centering \textcolor{teal!80}{\large \insertsection}\\ + \centering \textcolor{teal}{\Large \bf \insertsubsection}}\\ + \end{center} + \end{frame} +} + +%% ====================================================================== +%% Metadados do documento +\title{Modelos de Regressão para Dados de Contagem com R} +\author[Walmes Zeviani, Eduardo Jr \& Cesar Taconelli]{ + Prof. Dr. Walmes M. Zeviani\\ + Eduardo E. Ribeiro Jr\\ + Prof. Dr. Cesar A. Taconelli +} +\institute[UFPR]{ + Laboratório de EstatÃstica e Geoinformação \\ + Departamento de EstatÃstica \\ + Universidade Federal do Paraná} +\date{\today \\[0.1cm] \url{edujrrib@gmail.com}} +%\titlegraphic{\includegraphics[width=2cm]{images/MRDCr_logo}} + +%% ====================================================================== + +%% Inicia o documento + +\begin{document} + +\begin{frame} + \titlepage +\end{frame} + + + +\begin{frame}{Conteúdo} + +\begin{multicols}{2} + +\begin{enumerate} + \item Introdução + \item Modelos Lineares Generalizados + \item Modelo de Regressão Poisson + \item Modelo de Quase-Verossimilhança + \item Modelos Paramétricos Alternativos + + \begin{enumerate} + \item Modelo Binomial Negativo + \item Modelo Poisson Generalizado + \item Modelo COM-Poisson + \item Modelo Gamma-Count + \end{enumerate} + \item Modelos para Excesso de Zeros + \begin{enumerate} + \item Modelos de Barreira (Hurdle) + \item Modelos de Mistura (Zero Inflated) + \end{enumerate} + \item Modelos com Efeitos Aleatórios +\end{enumerate} + +\end{multicols} + +\end{frame} + + + + +%%%----------------------------------------------------------------------------------------- + + +%%% Slide 2 + +\begin{frame}\frametitle{Dados de contagens} +Alguns exemplos de problemas envolvendo contagens: + +\vspace{0,2cm} +\begin{itemize} +\item Número de acidentes em uma rodovia por semana; +\item Número de automóveis vendidos por dia; +\item Número de gols marcados por times de futebol por partida; +\item Número de falhas por metro de fio de cobre produzido; +\item Número de colônias de bactérias por $0,01mm^{2}$ de uma dada cultura... +\end{itemize} +\end{frame} + + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 3 + +\begin{frame}\frametitle{Modelos probabilÃsticos para dados de contagens} + +\begin{itemize} + \item Modelos probabilÃsticos para variáveis aleatórias discretas, com suporte no conjunto de números inteiros não-negativos, são potenciais candidatos para a análise de dados de contagens. + +\vspace{0.5cm} + + \item Algumas alternativas: Distribuição Binomial, Poisson e generalizações; distribuições geradas por misturas, como a beta-binomial, binomial negativa; distribuições fundamentadas na modelagem do tempo entre eventos, na razão de probabilidades sucessivas... + + +\end{itemize} +\end{frame} + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 4 + +\begin{frame}\frametitle{Regressão para dados de contagens} + +\begin{itemize} + +\item Modelos de regressão são utilizados para modelar a distribuição de uma variável aleatória $Y$ condicional aos valores de um conjunto de variáveis explicativas $x_{1},x_{2},...,x_{p}$. + +\vspace{0,5cm} + +\item Métodos para inferência e modelos de regressão para dados de contagem estão bem +aquém, em quantidade e diversidade, em relação ao verificado para dados contÃnuos. + +\vspace{0,5cm} + +\item A aplicação de modelos de regressão com erros normais na análise de contagens, embora frequente, em geral é desaconselhável. + +\end{itemize} +\end{frame} + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 5 + +\begin{frame}{Regressão com erros normais na análise de dados de contagens} + + \vspace{0,5cm} + + \begin{itemize} + \item O modelo linear com erros normais não considera a natureza discreta dos dados; + + \vspace{0,5cm} + + \item Associa probabilidade nula a qualquer possÃvel contagem; + + \vspace{0,5cm} + + \item Admite probabilidades não nulas a valores negativos da variável; + + \end{itemize} + +\end{frame} + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 6 + + +\begin{frame}{Regressão com erros normais na análise de dados de contagens} + + + \vspace{0,2cm} + + \begin{itemize} + + \item O uso de transformações dificulta a interpretação dos resultados; + + \vspace{0,5cm} + + \item O uso da transformação logarÃtmica apresenta problemas para contagens iguais a zero; + + \vspace{0,5cm} + + \item Não se contempla a relação não constante entre variância e média, caracterÃstica de dados de contagens. + + + \end{itemize} + +\end{frame} + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 7 + +\begin{frame}{A distribuição de Poisson} + + \begin{itemize} + \item A distribuição de Poisson é a principal referência para a análise de dados de contagens. + \vspace{0,3cm} + \item Função de probabilidades: + + +$$ + P\left ( Y=k \right )=\frac{e^{-\lambda}\lambda^{k}}{k!},\hspace{5mm}k=0,1,2,...; \lambda>0. +$$ + + \vspace{0,2cm} + + \item Se os eventos sob contagem ocorrem independentemente e sujeitos a uma taxa constante $\lambda >0$, sob o modelo Poisson, para um intervalo de exposição de tamanho $t$ tem-se: + +$$ + P\left ( Y_t=k \right )=\frac{e^{-\lambda t}(\lambda t)^{k}}{k!},\hspace{5mm}k=0,1,2,.... +$$ + \end{itemize} + +\end{frame} + +%%%----------------------------------------------------------------------------------------- + + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 11 + +\begin{frame}{Propriedades da distribuição de Poisson} + +Dentre as principais propriedades da distribuição de Poisson, tem-se: + +\vspace{0,3cm} + +\begin{itemize} + + \item Média: $E(Y)= \lambda$; + \vspace{0,5cm} + + \item Variância: $Var(Y)=\lambda$ (equidispersão); + \vspace{0,5cm} + + \item Razão de probabilidades sucessivas: $\frac{P\left ( X=k \right )}{P\left ( X=k-1 \right )}=\frac{\lambda}{k},$ gerando a relação de recorrência: + + $$ + P(Y=k)k=P(Y=k-1)\lambda; + $$ + + \item Se $Y_{1},Y_{2},...,Y_{n}$ são v.a.s independentes com $Y_{i}\sim Poisson(\lambda_{i})$, e $\sum\lambda_{i}<\infty$, então $\sum Y_{i}\sim Poisson(\sum\lambda_{i})$. + + \end{itemize} +\end{frame} + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 13 + +\begin{frame}{Distribuição Poisson para diferentes valores de $\lambda$} + + \begin{figure}[h] + \includegraphics[height=6cm,width=9cm]{images/Graf_Poisson.pdf} + \caption{Distribuição de Poisson para diferentes valores de $\lambda$} + \label{Fig1} + \centering + +\end{figure} +\end{frame} + +%%%----------------------------------------------------------------------------------------- + + +%%% Slide 8 + +\begin{frame}{Motivações para a distribuição de Poisson} + + + + \vspace{0,5cm} + + \begin{itemize} + \item Caso limite da distribuição binomial$(n, \pi)$ quando $n\rightarrow \infty$ e $\pi\rightarrow 0$, fixado $\lambda=n\pi$, ou seja: + +\vspace{0.4cm} + +$$ + \underset{n \to \infty \pi \to 0 }{lim} +\left [ \begin{pmatrix} + n\\k +\end{pmatrix} \left ( \frac{\lambda}{n} \right )^{k}\left ( 1-\frac{\lambda}{n} \right )^{n-k}\right ]=\frac{e^{-\lambda}\lambda^{k}}{k!}. +$$ + + \vspace{0,5cm} + + \item Resultado do processo estocástico de Poisson, em que os eventos contados ocorrem \textbf{aleatoriamente} ao longo do tempo, espaço,... + + \end{itemize} + +\end{frame} + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 9 + +\begin{frame}{Motivações para a distribuição de Poisson} + +\begin{itemize} + + + \item Se o tempo decorrido entre sucessivos eventos é uma variável aleatória com distribuição exponencial de média $\mu=1/\lambda$, então o número de eventos ocorridos em um intervalo t de tempo tem distribuição de Poisson com média $\lambda t$. + + \vspace{0,3cm} + +\begin{itemize} + + \item A dualidade entre as distribuições Poisson e exponencial implica que a taxa de ocorrência do evento, definida por: + + +\end{itemize} + + +$$ + \lambda (t) =\lim_{\Delta t\rightarrow 0}\frac{P\left \{ \text{evento ocorrer em} \left ( t,t+\Delta t \right ) \right \}}{\Delta t}, +$$ + + \vspace{0,3cm} + +dado que o evento não ocorreu até o tempo $t$, \textbf{é constante} para todo $t>0$. + + + \end{itemize} + +\end{frame} + + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 13 + +\begin{frame}{Diferentes comportamentos para $\lambda (t)$} + \vspace{-1.5cm} + \begin{figure}[h] + \includegraphics[height=8cm,width=10cm]{images/Graf_Risco.pdf} + \vspace{-0.8cm} + \caption{Diferentes comportamentos para $\lambda (t)$} + \label{Fig1} + \centering + +\end{figure} +\end{frame} + +%%%----------------------------------------------------------------------------------------- + + +%%% Slide 10 + +\begin{frame}{O processo de Poisson} + + O Processo de Poisson configura um processo de contagem em que $Y(t),t\geqslant 0$, representa o número de eventos que ocorrem até $t$, satisfazendo: + + \vspace{0,5cm} + + \begin{enumerate} + \item $Y(t)$ é inteiro e não negativo; + \item Para $s<t$, $Y(s)\leq Y(t)$; + \item $Y(t)-Y(s)$ é o número de eventos que ocorrem no intervalo $(s,t]$; + \item O processo é estacionário: + + $$ + Y(t_{2}+s)-Y(t_{1}+s) \overset{i.d. }{\sim}Y(t_{2})-Y(t_{1}), \forall s>0 + $$ + + \item O processo tem incrementos independentes, ou seja, os números de eventos verificados em intervalos disjuntos são independentes. + \end{enumerate} + +\end{frame} + +%%%----------------------------------------------------------------------------------------- + + +%%% Slide 14 + +\begin{frame}{Diferentes padrões em processos de contagens} + +\begin{figure}[h] + \includegraphics[scale=0.6]{images/processos14.pdf} + \caption{Ilustração de diferentes tipos de processos pontuais} + \label{Fig2} + \centering + +\end{figure} +\end{frame} + + + +%%% Slide 15 + +\begin{frame}{Regressão Poisson} + +\begin{itemize} + \item O modelo de regressão Poisson (ou modelo log linear de Poisson) é o mais usado para a análise de dados de contagens. + + \vspace{0,5cm} + + \item A regressão Poisson baseia-se nos pressupostos inerentes ao processo e à distribuição de Poisson. + + \vspace{0,5cm} + + \item Caso tais pressupostos não sejam atendidos, a regressão Poisson ainda pode ser uma alternativa apropriada, desde que usada com os cuidados necessários. + +\end{itemize} + +\end{frame} + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 16 + +\begin{frame}{Regressão Poisson - Especificação do modelo} + +\begin{itemize} + \item Sejam $Y_{1},Y_{2},...,Y_{n}$ variáveis aleatórias condicionalmente independentes, dado o vetor de covariáveis ${\boldsymbol{x_{i}}}'=(x_{i1},x_{i2},...,x_{ip}), i=1,2,...,n$. A regressão Poisson é definida pela distribuição de Poisson: + + +$$ + f(y_{i}|\boldsymbol{x_{i}})=\frac{e^{-\mu_{i}}(\mu_{i})^{y_{i}}}{y_{i}!},\hspace{5mm}y=0,1,2,..., +$$ + +\vspace{0,2cm} + +sendo as covariáveis inseridas ao modelo por: + +$$ +\ln(\mu _{i})=\boldsymbol{x_{i}'} \boldsymbol{\beta}, +$$ + +\vspace{0,2cm} + +em que ${\boldsymbol{\beta }}$ é o vetor de parâmetros de regressão. + + +\end{itemize} + +\end{frame} + +%%%----------------------------------------------------------------------------------------- + + +%%% Slide 17 + +\begin{frame}{Regressão Poisson - Propriedades} + + +\vspace{0,8cm} + + \begin{itemize} + + \item $f(y_{i}|\boldsymbol{x_{i}})=\frac{e^{-exp(\boldsymbol{x'_i\beta})}{exp({\boldsymbol{x'_i\beta}})}^{y_i}}{y_i!}$ + + + + \vspace{0,8cm} + + + \item $E\left [ y_{i}|\boldsymbol{x_{i}} \right ]= \mu_{i}=exp\left ( \boldsymbol{x'_{i}\beta} \right );$ + +\vspace{0,8cm} + + \item $Var\left [ y_{i}|\mathbf{x_{i}} \right ]= \mu_{i}=exp\left ( \boldsymbol{x'_{i}\beta} \right ).$ + + + + +\end{itemize} + +\end{frame} + +%%%----------------------------------------------------------------------------------------- + + + + + +%%% Slide 18 + +\begin{frame}{Regressão Poisson - Estimação por máxima verossimilhança} + +Para a regressão Poisson: + +\vspace{0.5cm} + +\begin{itemize} + +\item Log-verossimilhança: $l(\boldsymbol{\beta})=\sum_{i=1}^{n} \{ y_{i}\boldsymbol{x_{i}'\beta}-\exp{(\boldsymbol{x_{i}'\beta}}\}-\ln{y_{i}!});$ + +\vspace{0.5cm} + +\item Vetir escore: $\boldsymbol{S}(\boldsymbol{\beta})=\frac{\partial l(\boldsymbol{\beta};\boldsymbol{y})}{\partial \boldsymbol{\beta}}= + \sum_{i=1}^{n}(y_{i}-\exp(\boldsymbol{x_{i}'\beta}))\boldsymbol{x_{i}};$ +\vspace{0.5cm} + +\item Matriz Informação: $ +\boldsymbol{I({\beta})} = \sum_{i=1}^n \mu_i \boldsymbol{x_i x'_i} = \exp{(\boldsymbol{x'_i \beta})\boldsymbol{x_i x'_i}}; +$ + +\vspace{0.5cm} + +\item Distribuição assintótica: $ +\boldsymbol{\hat{\beta}} \overset{a}{\sim} N \left ( \boldsymbol{\beta}, \left [ \sum_{i=1}^n \mu_i \boldsymbol{x_i x'_i} \right ]^{-1} \right ). +$ + +\end{itemize} + +\end{frame} + + +%%%----------------------------------------------------------------------------------------- + + +%%% Slide 34 + +\begin{frame}{Regressão Poisson - Modelo Linear Generalizado} + + +A Regressão Poisson é um caso particular dos Modelos Lineares Generalizados (MLG). Algumas propriedades dessa classe de modelos: + +\begin{itemize} + +\item Os estimadores são consistentes ainda que a distribuição especificada seja incorreta, mas desde que a média condicional de $Y$ seja declarada corretamente; + +\item Os erros padrões, intervalos de confiança e testes de hipóteses, no entanto, ficam comprometidos; + +\item O ajuste de um MLG requer apenas a especificação: + +\begin{itemize} +\item Da esperança de $Y$ condicional à s covariáveis, mediante especificação do preditor linear e da função de ligação; + +\item Da variância condicional, mediante especificação da função de variância $V(\mu)$, possÃvel inclusão do parâmetro de dispersão $(\phi)$, ou sua estimação por métodos robustos (abordagem de Quase-Verossimilhança). + +\end{itemize} + +\end{itemize} + +\end{frame} + + + +%%%----------------------------------------------------------------------------------------- + + +\begin{frame}{Regressão Poisson - Quase-Verossimilhança} + +\begin{itemize} + \item Para o ajuste de um modelo alternativo via Quase-Verossimilhança, definimos: + +\vspace{0.4cm} + +$$g(E(y_i|\boldsymbol{x_i}))=\boldsymbol{x'_i \beta};$$ + +$$Var(y_i|\boldsymbol{x_i})=\phi V(\mu_i).$$ + +\item A obtenção dos estimadores se dá pela maximização da função de quase-verossimilhança: + +$$ Q\left ( \mu \right )=\int _y^\mu \frac{y-t}{\phi V(t)}dt $$. + + +\item As funções de quase-verossimilhança, quase-escore e quase-informação compartilham propriedades comuns à s correspondentes funções no caso paramétrico, para MLGs. + +\end{itemize} + + + +\end{frame} + + + + + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 39 + +\begin{frame}{Estimação via Quase-Verossimilhança} + +\begin{itemize} + +\item Distribuição assintótica: + +\vspace{0.3cm} + +$$ \boldsymbol{\hat{\beta}_{QL}} \overset{a}{\sim} N(\boldsymbol{\beta}, Var(\boldsymbol{\hat{\beta}_{QL}} ))$$ + +\vspace{0.8cm} + +\item Para o modelo Quase-Poisson, assume-se: + +$$\ln(E(y_i|\boldsymbol{x_i}))=\boldsymbol{x'_i \beta};$$ + + + + + +\end{itemize} + +\end{frame} + + + +%%%----------------------------------------------------------------------------------------- + +%%% Slide45 + +\begin{frame}{Estimação via Quase Verossimilhança} + +\begin{itemize} + +\item A matriz de covariâncias assintótica para $\boldsymbol{\hat{\beta}_{QL}}$ fica dada por: + +$$ +Var(\boldsymbol{\hat{\beta}_{QL}})=\left [ \sum_{i=1}^{n} \boldsymbol{x_{i}x'_{i}}\mu_{i} \right ]^{-1} +\sum_{i=1}^{n} \boldsymbol{x_{i}x'_{i}} \omega_{i} \left [ \sum_{i=1}^{n} \boldsymbol{x_{i}x'_{i}}\mu_{i} \right ]^{-1}, +$$ + +com $\mu_{i}=exp({\boldsymbol{x'_{i}\beta}})$ e $\omega_{i}=Var(y_{i}|\boldsymbol{x_{i}})$. + +\vspace{0.5cm} + + +\item Podemos considerar $\omega_i=V(\mu_i;\phi)$, como $\omega_i =\phi \mu_i$, $\omega_i =\phi \mu_i^2$ ou, simplesmente, o estimador robusto, baseado em $\omega_i=(y_i-\mu_i)^2$. + +\item Um estimador para $\phi$: + +$$ \hat{\phi} = \frac{1}{n-p}\sum_{i=1}^n \frac{y_i-\hat{\mu}_i}{V(\hat{\mu}_i)}. $$ + + +\end{itemize} + +\end{frame} + + + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 47 + +\begin{frame}{Distribuição binomial negativa} + +\begin{itemize} + +\item Função de probabilidades: + +$$ +P(Y=k)=\frac{\Gamma(\alpha+k)}{\Gamma(k+1)\Gamma(\alpha)}\left ( \frac{\lambda}{\lambda+\alpha} \right )^{k} \left( \frac{\alpha}{\lambda+\alpha} \right )^{\alpha}, k=0,1,2,...; \alpha > 0, \lambda>0 +$$ + +\item Propriedades: + +$$ +E(Y)=\lambda ; \quad Var(Y)= \lambda+ \alpha^{-1} \lambda^2 +$$ + +\item Assim, para qualquer $\alpha>0$, temos $Var(Y)>\lambda$. + +\item A distribuição binomial negativa tem como caso limite distribuição Poisson, quando $\alpha \rightarrow \infty$. + + +\end{itemize} + +\end{frame} + + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 48 + +\begin{frame}{Distribuição binomial negativa} + +\begin{itemize} + +\item Uma parametrização alternativa: + +\vspace{0.5cm} + +$$ +P(Y=k) = \left ( \begin{matrix} +r+k-1\\ +r-1 +\end{matrix} \right ) (1-p)^rp^k, \hspace{0,5cm} k=0,1,2,..., +$$ + +\vspace{0.5cm} + +sendo $r=\alpha$ e $p=\lambda/(\lambda+\alpha),$ com $0<p<1$ e $r>0$. +\vspace{0.5cm} + +\item Modelagem do número de "sucessos" até o r-ésimo "fracasso" ($r = 1,2,3,...$), configurando uma generalização da distribuição geométrica (para $r=1$). + +\vspace{0.5cm} + +\item Modelagem de alguns tipos de processos pontuais envolvendo contágio. + +\end{itemize} + +\end{frame} + + + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 48 + +\begin{frame}{Distribuição binomial negativa} + +\begin{itemize} + +\item A principal motivação para a distribuição binomial negativa baseia-se num processo de contagem heterogêneo, em que $Y \sim Poisson( \theta)$ e $\theta$ tem distribuição $Gama(\alpha, \beta):$ + +$$ +g\left ( \theta;\alpha,\beta \right )=\frac{\beta^{\alpha}}{\Gamma\left ( \alpha \right )}\theta^{\alpha-1}e^{-\beta \theta},\quad \alpha, \beta, \nu>0, +$$ + +com $E(\theta)=\theta=\alpha /\beta$ e variância $Var(\theta)=\alpha /\beta^2.$ + +\vspace{0.5cm} + +\item Como resultado, temos uma mistura Poisson-Gamma, resultando, marginalmente (em relação a $\theta$), na distribuição binomial negativa. + + +\end{itemize} + +\end{frame} + + +%%%----------------------------------------------------------------------------------------- + + +%%% Slide 50 + +\begin{frame}{Distribuição binomial negativa para diferentes parâmetros} + + \begin{figure}[h] + \includegraphics[height=5.5cm,width=9cm]{images/Graf_BN.pdf} + \caption{Distribuição binomial negativa para $\lambda=2$ e diferentes valores de $\alpha.} + \label{Fig1} + \centering + +\end{figure} +\end{frame} + + + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 51 + +\begin{frame}{Distribuição binomial negativa} + +\begin{itemize} + +\item O modelo de regressão com resposta binomial negativa pode ser especificado fazendo $E(y|\boldsymbol{x})=exp(\boldsymbol{x'\beta}).$ + +\vspace{0.5cm} + +\item Para valores fixados de $\alpha$, a distribuição binomial negativa fica expressa na forma da famÃlia exponencial linear, contemplada pela teoria de MLG. + +\vspace{0.5cm} + +\item A estimação dos parâmetros do modelo se dá numericamente, segundo um algoritmo em duas etapas, em que $\alpha$ e $\boldsymbol{\beta}$ são estimados condicionalmente até convergência. + +\end{itemize} + +\end{frame} + +\end{document} \ No newline at end of file diff --git a/inst/slides/images/Graf_BN.pdf b/inst/slides/images/Graf_BN.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4135b78e2bf1a20026cfaedae397421d7e5ad6d1 Binary files /dev/null and b/inst/slides/images/Graf_BN.pdf differ diff --git a/inst/slides/images/Graf_Poisson.pdf b/inst/slides/images/Graf_Poisson.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d19722fe69d2604b592fe8658ac5ab15142613f4 Binary files /dev/null and b/inst/slides/images/Graf_Poisson.pdf differ diff --git a/inst/slides/images/Graf_Risco.pdf b/inst/slides/images/Graf_Risco.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d81d406389c5171526cc29972395498de70e947d Binary files /dev/null and b/inst/slides/images/Graf_Risco.pdf differ diff --git a/inst/slides/images/processos14.pdf b/inst/slides/images/processos14.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fb02e99a3893f869fcbdd8d52ede6b4a2439ebf0 Binary files /dev/null and b/inst/slides/images/processos14.pdf differ diff --git a/vignettes/Ovelhas.Rmd b/vignettes/Ovelhas.Rmd new file mode 100644 index 0000000000000000000000000000000000000000..dc2d6bfc3ffb40493bac604262a8c46869e0a79b --- /dev/null +++ b/vignettes/Ovelhas.Rmd @@ -0,0 +1,264 @@ +--- +title: "Análise de Contagens - Quase-Verossimilhança" +author: > + Walmes M. Zeviani, + Eduardo E. Ribeiro Jr & + Cesar A. Taconeli +vignette: > + %\VignetteIndexEntry{Análise de Contagens - Quase-Verossimilhança} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteEncoding{UTF-8} +--- + +```{r setup, include=FALSE} +source("_setup.R") +``` + +Dados referentes a um experimento realizado com o objetivo de +investigar o efeito de uma intervenção, por parte do cuidador, no +comportamento de ovelhas. + +Para isso, foram consideradas ovelhas de duas +linhagens distintas (pouco ou muito reativas), submetidas a dois tipos +diferentes de intervenção (observação ou observação + intervenção). + +A variável resposta aqui considerada é o número de mudanças na postura +corporal do animal ao longo do perÃodo de observação (3 minutos). + + +```{r, echo = FALSE, include=FALSE} +##### Carregamento e tratamento inicial dos dados + +dados <- read.csv('Dadoscomp.csv',sep=',') +dados$tratamento <- factor(dados$tratamento) +levels(dados$tratamento) <- c('Observ', 'Observ + Interv') +dados$linhagem <- factor(dados$linhagem) +levels(dados$linhagem) <- c('Pouco reativo', 'Muito reativo') +dados2 <- dados[1:38,c(1:5,30:53)] +dados3 <- dados[1:38,30:53] ### Somente as mudanças de postura durante a intervenção. +r1 <- rowSums(dados3[,1:3])-1 +table(r1) +r2 <- rowSums(dados3[,4:6])-1 +table(r2) +r4 <- rowSums(dados3[,12:16])-1 +table(r4) +d2 <- rowSums(dados[1:38,57:59])-1 +table(d2) + +datapost1 <- data.frame(r1, dados2[ ,c('tratamento', 'linhagem')]) +names(datapost1)[1] <- 'Nposturas' +datapost2 <- data.frame(r2, dados2[ ,c('tratamento', 'linhagem')]) +names(datapost2)[1] <- 'Nposturas' +datapost4 <- data.frame(r4, dados2[ ,c('tratamento', 'linhagem')]) +names(datapost4)[1] <- 'Nposturas' + +##### Pacotes requeridos +require('lmtest') +require('boot') +require('car') +require('RColorBrewer') +require('sandwich') +require('hnp') +require('knitr') +``` + + +### Verificação do conteúdo e a estrutura dos dados. + +```{r} +str(datapost4) +summary(datapost4) +``` + + + +### Análise descritiva + +```{r} +tab <- data.frame(with(datapost4, table(tratamento, Nposturas))) + +myColours <- brewer.pal(2,"Blues") + +my.settings <- list( + superpose.polygon=list(col=myColours[2:5], border="transparent"), + strip.background=list(col=myColours[6]), + strip.border=list(col="black") +) + +bwplot(Nposturas ~ linhagem | tratamento, + data=datapost4, + main="Mudanças de postura vs tratamento e linhagem", + xlab="Linhagem", ylab="Frequência") + +### Variância e média amostrais por tratamento e linhagem. + +mdp <- aggregate(Nposturas ~ tratamento:linhagem, datapost4, function(x) c(mean = mean(x), var = var(x))) +mdp + +``` + + +### Regressão poisson com estimação por máxima verossimilhança. + +```{r} +ajusteps <- glm(Nposturas ~ tratamento + linhagem, data=datapost4, family=poisson) +summary(ajusteps) + +### Avaliando possÃvel efeito de interação. +ajustepsint <- glm(Nposturas ~ tratamento * linhagem, data=datapost4, family=poisson) +anova(ajusteps,ajustepsint, test = 'Chisq') + +exp(coef(ajusteps)[2]) +### Estima-se que a taxa de variação de postura no grupo com intervenção seja aproximadamente +### a metade em relação ao grupo sem intervenção. + +##### Estimação do parâmetro de dispersão. +X2 <- sum(resid(ajusteps,type='pearson')**2) +phichap <- X2/ajusteps$df.residual +phichap +``` + +### Diagnóstico do ajuste (gráficos). + +```{r} +##### Diagnóstico do modelo - gráficos padrão do R. +par(mfrow=c(2,2)) +plot(ajusteps, pch = 20, cex = 1.25) +``` + +```{r, echo = FALSE} +envelope=function(modelo){ + dados=na.omit(modelo$data) + nsim=100 + n=modelo$df.null+1 + r1=sort(rstandard(modelo,type='deviance')) + m1=matrix(0,nrow=n,ncol=nsim) + a2=simulate(modelo,nsim=nsim) + + for (i in 1:nsim){ + dados$y=a2[,i] + aj=update(modelo,y~.,data=dados) + m1[,i]=sort(rstandard(aj,type='deviance'))} + + li=apply(m1,1,quantile,0.025) + m=apply(m1,1,quantile,0.5) + ls=apply(m1,1,quantile,0.975) + + quantis=qnorm((1:n-0.5)/n) + + plot(rep(quantis,2),c(li,ls),type='n',xlab='Percentil da N(0,1)',ylab='ResÃduos') + title('Gráfico Normal de Probabilidades') + lines(quantis,li,type='l') + lines(quantis,m,type='l',lty=2) + lines(quantis,ls,type='l') + points(quantis,r1,pch=16,cex=0.75) +} + +``` + +```{r} +##### Gráfico quantil-quantil com envelopes simulados. +par(mfrow=c(1,1)) +envelope(ajusteps) +``` + + +### Ajustando modelos por quase-verossimilhança. + +```{r} + +### Modelo quasi poisson (V(mu) = mu). +ajuste12 <- glm(r4 ~ tratamento+linhagem, data=datapost4, family = 'quasipoisson') +# summary(ajuste12) + +### Forma alternativa de declarar o Modelo quase-poisson (V(mu) = mu). +ajuste13 <- glm(r4 ~ tratamento+linhagem, data=datapost4, family = quasi(variance = 'mu', link='log')) +# summary(ajuste13) + +### Modelo de quase-verossimilhança (V(mu) = mu^2). +ajuste14 <- glm(r4 ~ tratamento+linhagem, data=datapost4, family = quasi(variance = 'mu^2', link='log')) +summary(ajuste14) + +### Gráficos de diagnóstico para o modelo de quase-verossimilhança. +par(mfrow = c(2,2)) +plot(ajuste14, pch = 20, cex = 1.25) + +### Gráficos quantil-quantil para os resÃduos dos modelos Poisson e de Quase-Verossimilhança. +par(mfrow=c(1,2)) +qqnorm(resid(ajusteps,type='deviance'), pch = 20, main = 'Poisson', las = 1) +qqline(resid(ajusteps,type='deviance')) +qqnorm(resid(ajuste14,type='deviance'), pch = 20, main = 'QL', las = 1) +qqline(resid(ajuste14,type='deviance')) +``` + + +### Usando estimação robusta e bootstrap. + +```{r} + +estrb <- coeftest(ajusteps, vcov=sandwich) ### Estimador sanduÃche +estrb ### Estimação robusta. + +### Usando bootstrap (R=999 simulações) +ajusteboot <- Boot(ajusteps) +summary(ajusteboot) ### Resultados obtidos via bootstrap. + + +``` + + +### Resumo geral dos resultados. + +```{r} + +erroz <- rbind(summary(ajusteps)$coefficients[2,2:3], summary(ajuste13)$coefficients[2,2:3], + summary(ajuste14)$coefficients[2,2:3], estrb[2,2:3], c(summary(ajusteboot)[2,4], + mean(ajusteboot$t[,2]/summary(ajusteboot)[2,4]))) + +ics <- rbind(confint.default(ajusteps)[2,],confint.default(ajuste13)[2,], confint.default(ajuste14)[2,], +estrb[2,1] + c(-1.96,1.96) * estrb[2,2], mean(ajusteboot$t[,2])+c(-1.96,1.96)*summary(ajusteboot)[2,4]) + +quadres <- cbind(erroz, ics) + +rownames(quadres) <- c('Poisson', 'Quasi(mu)', 'Quasi(mu^2)', 'Robusto (sanduiche)', 'Bootstrap') + +### Quadro resumo para as estimativas produzidas pelos quatro modelos para o efeito de intervenção. +kable(quadres, format = "markdown", caption = "Comparativo dos modelos ajustados") + +``` + +### Verificando efeito das observações com maiores resÃduos na análise. + +```{r} +dadosexclud <- datapost4[-c(8,18,28),] + +### Ajustando o modelo Poisson sem as três observações. +ajusteexcludpoi <- glm(Nposturas ~ tratamento+linhagem, data=dadosexclud, family = poisson) + +### Estimativa do parâmetro de dispersão. +sum(resid(ajusteexcludpoi, type = 'pearson')**2)/ajusteexcludpoi$df.residual + +### Gráficos de diagnóstico. +par(mfrow=c(2,2)) +plot(ajusteexcludpoi) + +### Agora, o modelo quase-poisson +ajusteexcludquasi <- glm(Nposturas ~ tratamento+linhagem, data=dadosexclud, family = quasi(variance = 'mu', link='log')) + +### Estimativas produzidas pelo modelo quasipoisson com e sem as três observações. + +c1 <- compareCoefs(ajusteps, ajuste13, ajusteexcludpoi, ajusteexcludquasi, print = FALSE) +colnames(c1) <- c('Est. Poisson c/out','E.P. Poisson c/out', + 'Est. Quasi c/out','E.P. Quasi c/out', + 'Est. Poisson s/out','E.P. Poisson s/out', + 'Est. Quasi s/out','E.P. Quasi s/out') +kable(round(c1,3), align = 'c') + +### Efeito da intervenção desconsiderando as três observações. +exp(coef(ajusteexcludpoi)[2]) + +### O efeito de intervenção aumenta, e torna-se mais significativo, mediante exclusão dos outliers. + +``` + + diff --git a/vignettes/Sinistros.Rmd b/vignettes/Sinistros.Rmd new file mode 100644 index 0000000000000000000000000000000000000000..88fd2dca863072c3555f7b41043596db3a156cad --- /dev/null +++ b/vignettes/Sinistros.Rmd @@ -0,0 +1,329 @@ +--- +title: "Análise de Contagens - Poisson e Binomial Negativa" +author: > + Walmes M. Zeviani, + Eduardo E. Ribeiro Jr & + Cesar A. Taconeli +vignette: > + %\VignetteIndexEntry{Análise de Contagens - Poisson e Binomial Negativa} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteEncoding{UTF-8} +--- + +```{r setup, include=FALSE} +source("_setup.R") +``` + + +Dados referentes ao número de sinistros registrados por 16483 clientes de +uma seguradora de automóveis ao longo de um ano, contemplando as +seguintes variáveis: + +* **Sinistros**: Número de sinistros registrados; +* **Exposicao**: PerÃodo de cobertura do cliente durante o ano sob análise; +* **Idade**: Idade do cliente (em anos); +* **Sexo**: M para masculino e F para feminino; +* **Valor**: Valor do veÃculo segurado (em reais). + + + +```{r, echo = FALSE, include=FALSE} + +require(lattice) +require(MASS) +require(effects) +require(knitr) + +dados <- read.csv('Baseauto2.csv', header=TRUE, sep=',', dec = ',') + +options(scipen = 3) ### Era zero. + +##### Preparando os dados. +names(dados) <- c('Bonus','Tipo','Estado','Idade','Sexo','Civil','Valor','Exposicao','Sinistros') +dados <- dados[,c('Idade','Sexo','Valor','Exposicao','Sinistros')] +dados <- dados[which(dados$Valor != 0),] +dados$lexpo <- log(dados$Exposicao) +dados$Valor <- dados$Valor/1000 + +``` + +### Verificação do conteúdo e a estrutura dos dados. + +```{r} +head(dados, 10) ### Dez primeiras linhas da base. +str(dados) +``` + +### Análise descritiva da distribuição do número de sinistros. + +```{r} +table(dados$Sinistros) ### Distribuição do números de sinistros. +taxageral <- sum(dados$Sinistros)/sum(dados$Exposicao); taxageral ### Taxa de sinistros na amostra. + +tab <- aggregate(cbind(Exposicao, Sinistros) ~ Sexo, data = dados, sum) +taxa <- with(tab, Sinistros/Exposicao) +tab <- cbind(tab, taxa) +### Distribuição do número de sinistros por sexo. +kable(tab, align = 'c', caption = '**Taxa de sinistros segundo Sexo**') + +dados$idadecat <- cut(dados$Idade, breaks=c(18,30,60, 92), include.lowest = T) +tab <- aggregate(cbind(Exposicao, Sinistros) ~ idadecat, data = dados, sum) +taxa <- with(tab, Sinistros/Exposicao) +tab <- cbind(tab, taxa) +### Distribuição do número de sinistros por sexo. +kable(tab, align = 'c', caption = '**Taxa de sinistros segundo Idade**') + +tabidsex <- aggregate(cbind(Exposicao, Sinistros) ~ Sexo + idadecat, data = dados, sum) +taxa <- with(tabidsex, Sinistros/Exposicao) +tabidsex <- cbind(tabidsex, taxa) +### Distribuição do número de sinistros por idade e sexo. +kable(tabidsex, align = 'c', caption = '**Taxa de sinistros segundo Sexo e Idade**') + +``` + +## Regressão usando o modelo log-linear poisson. + +```{r} +dados <- na.omit(dados) +ajusteps <- glm(Sinistros ~ Sexo+Idade+I(Idade**2)+Valor+offset(log(Exposicao)), data = dados, family=poisson) +summary(ajusteps) + +### Estimação do parâmetro de dispersão. + +X2 <- sum(resid(ajusteps,type='pearson')**2) +phichap <- X2/ajusteps$df.residual +phichap ### Indicador de superdispersão. +``` + +```{r, echo = FALSE} +envelope=function(modelo){ + dados=na.omit(modelo$data) + nsim=100 + n=modelo$df.null+1 + r1=sort(rstandard(modelo,type='deviance')) + m1=matrix(0,nrow=n,ncol=nsim) + a2=simulate(modelo,nsim=nsim) + + for (i in 1:nsim){ + dados$y=a2[,i] + aj=update(modelo,y~.,data=dados) + m1[,i]=sort(rstandard(aj,type='deviance'))} + + li=apply(m1,1,quantile,0.025) + m=apply(m1,1,quantile,0.5) + ls=apply(m1,1,quantile,0.975) + + quantis=qnorm((1:n-0.5)/n) + + plot(rep(quantis,2),c(li,ls),type='n',xlab='Percentil da N(0,1)',ylab='ResÃduos') + title('Gráfico Normal de Probabilidades') + lines(quantis,li,type='l') + lines(quantis,m,type='l',lty=2) + lines(quantis,ls,type='l') + points(quantis,r1,pch=16,cex=0.75) +} +``` + + +### Diagnóstico do ajuste (gráficos). + +```{r} +##### Diagnóstico do modelo - gráficos. +par(mfrow=c(2,2)) +plot(ajusteps) + +par(mfrow=c(1,1)) +envelope(ajusteps) +``` + + +### Ajuste do modelo associando um parâmetro ao termo offset (log-exposicao) + +```{r} +ajusteps2 <- glm(Sinistros ~ Sexo + Idade +I(Idade**2) + Valor + log(Exposicao), data = dados, family=poisson) +summary(ajusteps2) +anova(ajusteps, ajusteps2, test = 'Chisq') +``` + + +## Regressão usando a distribuição binomial negativa. + +```{r} +ajustenb2 <- glm.nb(Sinistros ~ Sexo+Idade+I(Idade**2)+Valor+log(Exposicao),data= dados) +summary(ajustenb2) + +``` + + +### Diagnóstico do ajuste (gráficos). + +```{r} +##### Diagnóstico do modelo - gráficos. +par(mfrow=c(2,2)) +plot(ajustenb2) +``` + + +```{r, echo = FALSE} +dadosnb3 <- dados[,c('Sexo','Idade','Valor','Exposicao','Sinistros')] +dadosnb3$lexpo <- log(dados$Exposicao) +ajustenb2 <- glm.nb(Sinistros ~ Sexo+Idade+I(Idade**2)+Valor+lexpo,data= dadosnb3) +envelope=function(modelo){ + dados=na.omit(dadosnb3) + nsim=100 + n=modelo$df.null+1 + r1=sort(rstandard(modelo,type='deviance')) + m1=matrix(0,nrow=n,ncol=nsim) + a2=simulate(modelo,nsim=nsim) + + for (i in 1:nsim){ + dados$y=a2[,i] + aj=update(modelo,y~.,data=dados) + m1[,i]=sort(rstandard(aj,type='deviance'))} + + li=apply(m1,1,quantile,0.025) + m=apply(m1,1,quantile,0.5) + ls=apply(m1,1,quantile,0.975) + + quantis=qnorm((1:n-0.5)/n) + + plot(rep(quantis,2),c(li,ls),type='n',xlab='Percentil da N(0,1)',ylab='ResÃduos') + title('Gráfico Normal de Probabilidades') + lines(quantis,li,type='l') + lines(quantis,m,type='l',lty=2) + lines(quantis,ls,type='l') + points(quantis,r1,pch=16,cex=0.75) +} +``` + + +```{r} +par(mfrow=c(1,1)) +envelope(ajustenb2) +``` + + +### Explorando os efeitos das covariáveis. Estimativas pontuais e ICs (95%) + +```{r} +intervalos <- confint(ajustenb2) +estimat <- cbind(ajustenb2$coefficients, intervalos) +colnames(estimat)[1] <- 'Estimativa pontual' + +### Quadro de estimativas +kable(round(estimat, 5), align = 'c', caption = '**Estimativas pontuais e intervalos de confiança - Modelo Binomial Negativo**') +``` + + +### Gráficos de efeitos + +```{r, echo = FALSE} + +``` + +```{r} + +efeitos <- allEffects(ajustenb2, given.values=c(lexpo=0)) +trellis.par.set(list(axis.text = list(cex = 1.2))) + +plot(efeitos[[2]], type='response',main=list( + label="Taxa de sinistros vs. Idade", + cex=1.5), + xlab=list( + label="Idade (anos)", + cex=1.5), + ylab=list( + label="Taxa de sinistros", + cex=1.5)) + +plot(efeitos[[1]], type='response',main=list( + label="Taxa de sinistros vs. Sexo", + cex=1.5), + xlab=list( + label="Sexo", + cex=1.5), + ylab=list( + label="Taxa de sinistros", + cex=1.5)) + +plot(efeitos[[4]], type='response',main=list( + label="Taxa de sinistros vs. Valor do automóvel", + cex=1.5), + xlab=list( + label="Valor (x1000 reais)", + cex=1.5), + ylab=list( + label="Taxa de sinistros", + cex=1.5)) + +``` + + + + +```{r, echo = FALSE, include=F} + +## Frequências ajustadas pelas duas distribuições, com e sem covariaveis. + +### Poisson sem ajuste de covariáveis. +n <- nrow(dados) +mediasin <- mean(dados$Sinistros) +freqps <- round(n*dpois(0:10,mediasin)) + + +### Poisson com covariaveis +pred1 <- predict(ajusteps2,type='response') +intervalo <- 0:10 +matprob <- matrix(0, nrow=nrow(dados), ncol=length(intervalo)) +probpois <- function(interv, taxa) + dpois(intervalo,taxa) +for(i in 1:nrow(dados)) + matprob[i,] <- probpois(interv = intervalo, taxa = pred1[i]) +pbarra <- colMeans(matprob) +freqpsaj <- round(n*pbarra) + + + +### Binomial negativa sem covariaveis. +ajustenb <- glm.nb(Sinistros ~ 1,data=dados) + +media <- exp(coefficients(ajustenb)) +shape <- ajustenb$theta +freqbn <- round(n*dnbinom(0:10, mu = media, size = shape)) + + +### Binomial negativa com covariaveis +pred2 <- predict(ajustenb2,type='response') + +intervalo <- 0:10 +matprob <- matrix(0,nrow=nrow(dados),ncol=length(intervalo)) +probnb <- function(interv, media, shape) + dnbinom(intervalo, mu = media, size = shape) +for(i in 1:nrow(dados)) + matprob[i,] <- probnb(interv = intervalo, media = pred2[i], shape = ajustenb2$theta) +pbarra <- colMeans(matprob) +frebnaj <- round(n*pbarra) +ams <- c(table(dados$Sinistros), rep(0,5)) +matfreq <- rbind(ams, freqps, freqpsaj, freqbn, frebnaj) +colnames(matfreq) <- 0:10 +rownames(matfreq) <- c('Amostra', 'Poisson não ajustada por covariáveis', 'Poisson ajustada por covariáveis', + 'BN não ajustada por covariáveis', 'BN ajustada por covariáveis') +``` + + +## Comparação dos ajustes + +```{r, results = 'markup'} +kable(matfreq, format = "markdown", caption = "Frequências amostrais e frequências ajustadas para o número de sinistros") +``` + + + + + + + + + + +