The corresponding link function is the complementary log-log link The logistic model is comparable with the probit model if

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1 SK300 and SK400 Lnk funtons for bnomal GLMs Autumn 08 We motvate the dsusson by the beetle eample GLMs for bnomal and multnomal data Covers the followng materal from hapters 5 and 6: Seton 5.6., 5.6.3, and 5.7. Setons 5.., 5..4, and 5..5 Setons 6.., 6.., 6..3, 6..4, and 6.3. Let be the probablty that a beetle des at dose Logst regresson β0+ β β0+ β e / ( + e and the probt model Φ ( β + β 0 Ørnulf Borgan Department of Mathemats Unversty of Oslo gve smlar model ft, but qute dfferent parameter estmates (f. R ode on web page How an ths be eplaned? Assume the lethal dose for a beetle s gven as where Z ~ F wth E( Z 0 D d+σz If a beetle s gven a dose, the probablty that t wll de s P( D P( d+ σz P[ Z ( d / σ] F[( d / σ] F( β + β 0 d / where β σ and β / σ If Z ~ (0, we obtan the probt model If Z has df z z F( z e / ( + e 0 For the st dstrbuton we have we obtan the st model var( Z / 3 he t and probt lnks are symmetr around /, n the sense that lnk( lnk( so the response urve s symmetr around / An alternatve asymmetr lnk funton s obtaned from F( z ep[ ep( z] whh gves ep[ ep( β + β ] 0 he orrespondng lnk funton s the omplementary - lnk he st model s omparable wth the probt model f [ ( ] β0+ β we replae σ by σ / 3 See R ode on the web-page for results for the 4 hs orresponds to dvdng the estmates by / 3.8 beetle data

2 Classfaton and ROC urves From a ftted model wth bnary data we may obtan a predton rule (typally to be used for new data: fˆ > 0 yˆ 0 fˆ 0 0 For a partular ut-off we have true postve rate (tpr P( yˆ y false postve rate (fpr P( yˆ y 0 A ROC (reever operatng haraterst urve s a plot of tpr as a funton of fpr for all possble ut-offs A measure of the predtve power of the model s the area under the ROC urve, denoted AUC Predtve power: orrelaton For a normal lnear model t s ommon to use See R ommands on web-page for llustraton 5 6 R ( µ ˆ y ( y y as a measure of «eplaned varaton» One may show that (f the model has an nterept, see page 55 n tet book ( y ( ˆ ˆ y µ µ orr( yμ, ˆ ( y y ( µ ˆ µ ˆ hs motvates that we also for GLMs may use as a measure of predtve power of a model R orr( yμ, ˆ Cohort and ase-ontrol data Consder a ohort study where we reord the ourrene (Y or not ourrene (Y0 of a dsease for n eposed (X and n 0 non-eposed (X0 ndvduals X X0 Y Y0 a b d otal n n 0 Consder then a ase-ontrol study where we reord eposure status (X0, among a sample of m ases (Y and m 0 ontrols (Y0 X X0 otal Y Y0 a b d m m 0 We may then estmate the dsease odds rato: a b β P( Y X / P( Y 0 X e ɵ n n P( Y X 0 / P( Y 0 X 0 d n n 0 0 ad b 7 From suh a ase-ontrol study we may dretly estmate the eposure odds rato for ases versus ontrols: a P( X Y / P( X 0 Y ɵ m ad m P( X Y 0 / P( X 0 Y 0 b d b m m 0 0 8

3 Usng Bayes theorem we fnd that the dsease odds rato e for eposed versus non-eposed equals the eposure odds rato for ases versus ontrols: β P( Y X / P( Y 0 X e P( Y X 0 / P( Y 0 X 0 P ( X D / P ( X 0 D P( X D 0 / P( X 0 D 0 hus we may estmate the dsease odds rato from ase-ontrol data β 9 Consder now a ohort study, where we regster a vetor of eposures and other ovarates at entry, and reord ourrenes of dsease (Y durng a gven perod of follow-up We assume a st regresson model for the ohort P( Y ep( β + β ep( β 0 + β + 0 Consder then a ase-ontrol study, we selet samples of ases and ontrols and observe ther X vetors Let Z f an ndvdual s seleted to the ase-ontrol study and Z 0 f t s not seleted Assume that P( Z Y 0, X ρ 0 P( Z Y, X ρ do not depend on 0 hen by Bayes theorem: P( Z Y, P( Y P( Y Z, P( Z Y y, P( Y y where y 0 β β + ( ρ / ρ * ρ ep( β0 + β ρ + ρ ep( β + β 0 0 ep( β + β + ep( + β * 0 * β0 hus we may use st regresson for ase-ontrol data «as f they were ohort data» Multnomal response We onsder a stuaton wth response ategores We may dstngush between nomnal (unordered and ordnal (ordered responses We onsder ndependent subjets, and let denote the probablty that subjet has response n ategory j We have j j he response for subjet may be gven by the vetor ( Y,..., Y, where Y f the response of subjet j s n ategory j, and all other Y 0 he pmf of ( Y,..., Y takes the form p( y,..., y h y y j

4 Multnomal models for nomnal response For llustraton we onsder a data set on food hoe of 9 allgators n four Florda lakes (seton 6.3. We wll onsder the multnomal t model (or baselne-ategory t model usng ategory as a referene he model s gven by (for j,..., j ep + ep h p ( β jk k k p ( β k hk k h ( β j ep ep + ( β where (,..., p s a vetor of ovarates for subjet and β j ( β j,..., β jp s a vetor of ovarates for the probabltes of the j-th ategory h Here the response ategores (food hoes are nomnal he formulas above are also vald for f we let (R ode for the eample s gven on the ourse web 3 4 ote that we have (for j,..., and ote also that j p β j β k jk k a b a b ( βa βb t[ P( Y Y or Y ] j j P( Yj Yj or Y P ( Yj Yj or Y j 5 Multvarate GLMs he multnomal t model s an eample of a multvarate generalzed lnear model For a multvarate GLM we have ndependent vetors Y,..., Y wth a dstrbuton from the followng multvarate generalzaton of the eponental dsperson famly { } f ( y ; θ, φ ep [ y θ b( θ ] / a( φ + ( y, φ where θ s the (vetor-valued natural parameter, and s the dsperson parameter For the response vetor Y wth mean vetor μ E( Y, we have a vetor g of lnk funtons suh that gμ ( Xβ φ 6

5 When we wrte the multnomal t model as a multvarate GLM, we let Y ( Y,..., Y, and note that Y ( Y + + Y, Here we have μ (,...,, and the vetor g ( g,..., of lnk funtons s gven by g µ j g j ( μ ( µ + + µ, he lnear predtors take the form 0 0 β 0 0 β Xβ 0 0 β 7 Fttng the multnomal t model he ontrbuton to the -lkelhood from the -th subjet s j yj j y + y j j j j j j yj + j so ( j / are the natural parameters and provde the anonal lnks for the multnomal t model Remember that j h ( β j ep ep + ( β ep( h and + β h h 8 hus the -lkelhood may be wrtten yj L( ; j β y j yj + j j y ( + ep β ( β j j j j j p β jk k yj + ep( j j k β j he sore funtons takes the form L( β; y k ep( β j β k yj jk + ep( l β l y k j k j 9 he lkelhood equatons beome ( j,..., ; k,..., p k j k j y where j l ( β j ep ep l + he elements of the Hessan matr are gven by L( β; y k ep( β j β jk β jk β jk + ep( l β l ( β ( β ep( β + ep( l k ep( β β k k j k j l ( k k j j + ep l l 0

6 and for j j L( β; y β β jk j k l ( β ep( β j ep k k j + ep( l β k k j j he epeted (observed nformaton matr onssts of ( bloks of sze p p, and blok ( j, j takes the form L( β; y j I( j j j β j β j he nformaton matr s postve sem-defnte, so the -lkelhood funton s onave Devane and Pearson statst Assume now that we have grouped data, and that we have observatons orrespondng ovarate vetor (,..., Y j p Let be the proporton of the observatons that are n ategory j n yj hen the lkelhood s gven by l( ; y j j he devane beomes n yj ˆ j j ma lkelhood for atual model G ma lkelhood for saturated model n yj y j j n yj n yj j n ˆj n n he Pearson statst s gven by X ( n y nˆ j j j nˆ j If the ftted model holds, and we replae yj by Yj n the formulas above, the devane and the Pearson statst are appromately h-squared dstrbuted wth df #(parameters n saturated model #(parameters n ftted model ( p( ( p( 3