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1 ( ) IxJ ( 5 0% ) Pearson Fsher s exact test χ, LR Ch-square( G ) x, Odds Rato θ, Ch-square Ch-square (Goodness of ft) Pearson cross moment ( Mantel-Haenszel ), Ph-coeffcent, Gamma (γ ), Kendall τ (bnary) Cochran-Armtage Trend Ch-square sub (Ch-square ) Hermt Contrast 3 4 Generatng Herarchcal Structure 3 4 (assocaton) 3 (modelng) Generalzed Lnear Model (GLM: ) GLM (Regresson), (ANOVA: Analyss of Varance) Logstc, Log- Lnear Model Sehyug Kwon, Dept of Statstcs, Hannam Unv sprng, 00 58

2 3 Generalzed Lnear Model Neder & Wedderburn(97) GLM (component) 3 )random component: )systematc component: (predctor ) 3) lnk: systematc random 3 GLM (component) Random component natural exponental famly( ) Y = Y, Y, Y ) ( n f ( Y y; θ ) = a( θ ) b( y ) exp[ yq( θ )], θ (parameter) Posson ( ), Bnomal ( ), Standard Normal ( ) natural exponental famly θ ) Q θ ) ( ( Systematc component X ( data matrx desgn matr, β lnear predctor ( ) GLM systematc η = X β η = β 0 + βx + β x + + β p xp = β x for =,, n Lnk component Random systematc Y µ = E( Y ) µ η = g ( µ ) (monotonc dfferental functon) g( µ ) η lnk g = dentty lnk ( ) g( µ ) = µ E ( y ) = β x = β 0 + βx + β x = µ + + β β x Canoncal Lnk Canoncal Lnk g µ ) = Q( θ ) = β x = β 0 + β x + β x + + β p xp Canoncal Lnk ( p x p Sehyug Kwon, Dept of Statstcs, Hannam Unv sprng, 00 59

3 3 Logt model (, bnary ; /, 0, Pr(Y y = ) y = π Bernoull ) f ( y ; π ) = π ( π ) = ( π )[ π /( π )] = ( π ) exp[ y y ln( π π )] : NE Famly π Q( θ ) = ln( ) odds rato ln π Logt Logt π GLM Logt ln( 33 Log Lnear model π π ) = β x = β + β x + β x + + β 0 n Posson n E ( n ) = m n n exp( m )( m ) f n; m ) = = exp( m )( ) exp[ n ln( m )] : NE Famly n! n! ( Q θ ) = ln( m ) ( 34 GLM ln( m ) = β x = β 0 + β x + β x + + β p xp Random, Systematc, Lnk GLM Random ( ) Normal Systematc ( ) Identty ( ) ( ) Regresson Normal Identty ANOVA p x p Model ( ) Normal Identty Mxed ( + ) Regresson wth Indcator ANCOVA Bnomal Logt Mxed Logstc Regresson Posson Log Mxed Log-Lnear Multnomal Generalzed Logt Mxed Multnomal response (Least Square Method) ( ) GLM Sehyug Kwon, Dept of Statstcs, Hannam Unv sprng, 00 60

4 GLM Random log (lkelhood functon) strctly concave ML estmate( ) ML Fsher s scorng teraton algorthm Random 3 Logstc Regresson Y /, /, / 0, Pr(Y = ) = π Bernoull Y ( π π ( π ) Y ) Bnomal 3 Lnear Probablty Model ( ) ( Y ) = π ( = β + β x E 0 Identty Lnk ( ) x (lnear) GLM x, ( π ) V ( Y ) = π ( π ) 0 0 MVLUE 3 Logstc Regresson Model ( π ) x x π 0 S- π ( log t( π ( ) = ln( ) = β 0 + βx π ( exp( β 0 + β π ( = + exp( β + β 0 β > 0 π ( β < 0 0 x Sehyug Kwon, Dept of Statstcs, Hannam Unv sprng, 00 6

5 log odds Logt x β 0( β < 0 ) ( β 0 > ) β 0 π ( x ( ) = βπ( [ π ( ] x π ( = / ( ) β Inference ( ) Logt ( β, =,,, p ) MLE (Maxmum Lkelhood Estmate) Wald (943) β ± zα / ASE( β ) : ASE = Asymptotc Standard Error ( ) ' ' β * = ( β, β,, β q ) subset β* = ( β, β,, β q ) = 0 ( 0 H 0 : β = H0 : β = β 3 = 0 ' L Reduced- ( β* = ( β, β,, β q ) = 0,, 3 ) L Full-, ) GLM Devance( ) Reduced Full ( ) GLM Devance l ln( ) = [ln l ln l ] = [ L L] ~ χ ( q) : l Theorem ~ χ ( : 7 ) Wald(943) ˆ ' ˆ ˆ ˆ β * ( Cov( β* )) β* ~ χ ( q) : Wald Logt Alan Agrest (990), Wley publcaton- page -7 Sehyug Kwon, Dept of Statstcs, Hannam Unv sprng, 00 6

6 33 Inverse CDF( ) Lnks 6 ( π ( ) ( β > 0 ) (cumulatve probablty densty functon) β < 0 x x ( π ( = F ( β 0 + β : F ) GLM F ( π ( ) = β 0 + β x β > 0 logstc π ( x ) = exp( β 0 + β x ) /[ + exp( β 0 + β ] Logstc exp( ( x µ ) / β ) π β Logstc (pdf) f ( x µ, β ) = = µ, = β [ + exp( ( x µ ) / β )] 3 Logstc (cdf) F ( x µ, β ) = [ + exp( ( x µ ) / β )] β > 0 logstc π ( x ) = exp( β 0 + β x ) /[ + exp( β 0 + β ] Logstc Logstc regresson F µ = 0, τ = CDF π ( = F ( β 0 + β π β β α / 3 CDF Logt logstc CDF Logstc Probt model F CDF Φ π = Φ( β + β ) Probt ( 0 x Logstc π ( 0 Pr obt( π ( ) = Φ ( π ( ) = β 0 + βx 34 Lnear probablty model: Logt model: Probt model: π ˆ( = β 0 + βx πˆ( ln( ) = β0 + β x πˆ( Φ ( π ˆ( ) = β 0 + βx Sehyug Kwon, Dept of Statstcs, Hannam Unv sprng, 00 63

7 Example Thymdne (LI) : LI 4 7 ( ) LI π ( /3 / /3 DATA CANCER; INPUT LI CASE CARDS; ; TITLE 'Lnear Lnk Functon'; PROC GENMOD DATA=CANCER; MODEL GOOD/CASE=LI /LINK=IDENTITY DIST=NORMAL; OUTPUT OUT=OUT PRED=YHAT_LI; TITLE 'Logt Lnk Functon'; PROC GENMOD DATA=CANCER; MODEL GOOD/CASE=LI /LINK=LOGIT DIST=BIN; OUTPUT OUT=OUT PRED=YHAT_LO; TITLE 'Probt Lnk Functon'; PROC GENMOD DATA=CANCER; Sehyug Kwon, Dept of Statstcs, Hannam Unv sprng, 00 64

8 MODEL GOOD/CASE=LI /LINK=PROBIT; OUTPUT OUT=OUT3 PRED=YHAT_PR; DATA FIN; MERGE OUT OUT OUT3; PROC PRINT DATA=FIN; GENMOD GENeralzed lnear MODel Model = LINK Lnear Probablty Model (DIST=Normal) Logt model Probt DIST OUTPUT (statement) OUT SAS data PRED=YHAT (predcted value) YHAT P= RES= / U= / L= Lnear Prob Model πˆ ( x ) = * LI ( L L) LI (0088) LI Sehyug Kwon, Dept of Statstcs, Hannam Unv sprng, 00 65

9 Logt Model πˆ( ln( ) = * LI πˆ( πˆ( = + exp ( * LI ) LI (0449) LI Probt Model Φ ( π ˆ( x )) = *LI π ˆ( x ) = Φ( * LI ) LI (0038) LI Sehyug Kwon, Dept of Statstcs, Hannam Unv sprng, 00 66

10 πˆ ( SYMBOL I=L3 V=NONE C=BLACK; SYMBOL I=L3 V=NONE C=RED; SYMBOL3 I=L3 V=NONE C=BLUE; AXIS ORDER=0 TO BY 05 LABEL=('PHI_HAT'); AXIS ORDER=8 TO 38 BY 0 LABEL=('LI LEVEL'); TITLE 'PHI HAT BY MODELs'; PROC GPLOT DATA=FIN; Symbol: V= value C=color I=nterpolate Axs ORDER=, LABEL= PLOT (YHAT_LI YHAT_LO YHAT_PR)*LI /OVERLAY VAXIS=AXIS HAXIS=AXIS; Sehyug Kwon, Dept of Statstcs, Hannam Unv sprng, 00 67

11 HOMEWORK #7 Due 5 6 (X) (Y) 5, ) Logt Model (ft) ) Probt Model (ft) 3), Logt Model, Probt Model Sehyug Kwon, Dept of Statstcs, Hannam Unv sprng, 00 68

Categorical Data Analysis Ch. 1 Introduction 1.1. 범주형자료분석이란자연과학, 사회과학은물론의학관련분야에까지범주형자료분석은널리활용되고있다. 기업의부실여부판단, 새로운의학치료법에대한가치평가, 사람들의의견에영향을주는요인들에대한평가등범주

Categorical Data Analysis Ch. 1 Introduction 1.1. 범주형자료분석이란자연과학, 사회과학은물론의학관련분야에까지범주형자료분석은널리활용되고있다. 기업의부실여부판단, 새로운의학치료법에대한가치평가, 사람들의의견에영향을주는요인들에대한평가등범주 Ch. 1 Introducton 1.1. 범주형자료분석이란자연과학, 사회과학은물론의학관련분야에까지범주형자료분석은널리활용되고있다. 기업의부실여부판단, 새로운의학치료법에대한가치평가, 사람들의의견에영향을주는요인들에대한평가등범주형자료에대한분석의필요성은증가하고있다. 그러나범주형자료분석은다른일반적인분석과는달리모형이다소복잡하고결과해석이용이하지않아 1.1.1. 변수 (varable)