( ) 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 http://wolfpackhannamackr sprng, 00 58
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 http://wolfpackhannamackr sprng, 00 59
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 http://wolfpackhannamackr sprng, 00 60
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 http://wolfpackhannamackr sprng, 00 6
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 http://wolfpackhannamackr sprng, 00 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 http://wolfpackhannamackr sprng, 00 63
Example Thymdne (LI) : LI 4 7 ( ) LI π ( 8 0 0 0 0 0 3 0 0 4 3 0 0 6 3 0 0 8 0 3 /3 / 4 0 0 6 8 3 0 0 34 38 3 /3 DATA CANCER; INPUT LI CASE GOOD @@; CARDS; 8 0 0 0 3 0 4 3 0 6 3 0 8 0 3 4 0 6 8 3 0 34 38 3 ; 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 http://wolfpackhannamackr sprng, 00 64
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 ) = 0507 + 0088 * LI ( L L) LI (0088) LI Sehyug Kwon, Dept of Statstcs, Hannam Unv http://wolfpackhannamackr sprng, 00 65
Logt Model πˆ( ln( ) = 377 + 0449 * LI πˆ( πˆ( = + exp ( 377 + 0449 * LI ) LI (0449) LI Probt Model Φ ( π ˆ( x )) = 378 + 00878 *LI π ˆ( x ) = Φ( 378 + 00878 * LI ) LI (0038) LI Sehyug Kwon, Dept of Statstcs, Hannam Unv http://wolfpackhannamackr sprng, 00 66
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 http://wolfpackhannamackr sprng, 00 67
HOMEWORK #7 Due 5 6 (X) (Y) 5,5 35 45 55 65 765 95 ) Logt Model (ft) ) Probt Model (ft) 3), Logt Model, Probt Model Sehyug Kwon, Dept of Statstcs, Hannam Unv http://wolfpackhannamackr sprng, 00 68