b Example: Shrinked blood vessels Y : shrinked: yes (1) / no (0) erkl.: Breath Volume (Vol) and Frequency (Rate) Ziel: P Y = 1 Vol,Rate modellieren!

Transkript

1 Logstc Regresson. Introducton Only partally translated at ths tme b Example: Shrnked blood vessels Y : shrnked: yes () / no () erkl.: Breath Volume (Vol) and Frequency (Rate) Zel: P Y = Vol,Rate modelleren!

2 c P Y= = h x (), x (2),..., x (m)

3 Rate Vol 2

4 3 P Y= = h x (), x (2),..., x (m) d Why s an ordnary lnear regresson nadequate? Y = β + βx () + β2x (2) βmx (m) + E What s the error term E? E Y = β + βx () + β2x (2) βmx (m) We have P Y= = E Y. Same form o.k. But: Estmated values may become < and >! Transformaton of Y? 2 values reman 2 values! Transformaton of E Y = P Y=!

5 4. e Modell. Logt-Funkton g π = log π π = log π log π g P Y= = log P Y = log P Y= = η η = β + βx () + β2x (2) βmx (m). η: lnearer Prädktor. f Bespel: g P Y = = Vol Rate.

6 5 Y

7 6. g Dskrmnanzanalyse: Y Gruppen-Zugehörgket X (j) multvarate Beobachtungen. Logstsche Regresson:. Schätzen: ˆπ 2. Zuordnen: Ŷ =, wenn ˆη > (ˆπ >.5)

8 . h Further Applcatons: 7 Toxkology: Toxc matter deadly for mce? What concentraton? Medcne: Treatment successful? Falure of (techncal) devces, Bugs n (techncal) products, Occurence of characterstcs n anmals or plants, clent scorng, General: 2 Groups.

9 8.2 Consderatons about the Model a Same flexblty as lnear regresson. Frequently: factors (nomnal varables) as explanatory v. b Example: Assessment of work stuaton. Y happy (), unhappy () X (j) Regon, Age, Gender, Race Only factor 2 k-cross table

10 NE Md-Atl. S Mdwest NW SW Pacfc total 9 unzufreden zufreden total

11 Hgher dmenson log-lnear models c Grupperte Daten: ml Beob. Y zu glechen x = xl: Ỹl = : x = xl Y Ỹk B mk, πk E Ỹl/ml = πl Logstsche Regresson: g πl = ηl

12 d Bespel Überleben von Frühgeburten. 247 Säuglnge. Erklärende Varable: Geburtsgewcht. Klassen von je g n Surv.no Surv.yes Weght

13 Y log(gewcht)

14 .2 e Transformerte Beobachtungen. E Ỹl/ml = πl, g πl = lnearer Prädktor. g Ỹl/ml lnearer Prädktor. 3 Was tun mt Yl/ml = oder =? g =, g =. Abhlfe: Emprsche Logts Zl = log Ỹl +.5 ml Ỹl +.5. Gewöhnlche multple Regresson mt Zl? Näherung.

15 emp.logt(y) 2 2 Max.Lkelhood Klenste Quadrate log(gewcht) Y

16 .2 f Interpretaton of Coeffcents? Need followng concepts: odds odds = P Y = P Y= π = /4 : odds :3 ( falure s 3 more frequent ) 5 log(odds)= g Y =, g: Logt-Funkton. log(odds)= η Wahrsch. π = g η = exp η +exp η. G : logstsche Funkton. Logstsche Regresson: log(odds) = lnearer Prädktor j β jx (j). π = logstsche Funkton j β jx (j).

17 .2 g Odds rato (Doppelverhältns): Verglech zweer Beobachtungen odds x log = log odds x log odds x2 odds x2 = η η2 = (x x2)β Koeffzent βj: Vergrösserung von x (j) um erhöht odds rato um Faktor e β j. 6 h Bespel Ader-Verengung: Wert für Vol =.5, Rate =.75 log(odds) = = 2.85 odds =.578, g ( 2.85) =.546 Verglech Vol =.5, Rate =.75: odds rato: e 3.88 = 48.4 odds = = 2.8, 2.8/3.8 =.73

18 7.2 Model wth Latent Varable = Schwellenwert-Modell x latente V. c

19 8 Z = x T β + E π = P Y= = P Z c = P = F E c x T β c ( ) β + β jx (j) j F : kumulatve Vertelungsfunkton des Zufallsfehlers E β = [ β c, β,..., βm] P Y= = g x T β mt g η = F η E logstsche Vt.: logstsche Regresson E Normal-Vt.: Probtmodell E Extremwertvt.: Komplementäres log-log Modell

20 9.3 Estmaton and Tests a Method of Maxmal Lkelhood. There are programs! b Log-Lkelhood: (m ) ll ỹ; β = log P Ỹ l = yl = log l π y l l l l ( π l) m l yl yl (m ) = log l + (y l log πl + (ml yl)log πl ) l l mt logt πl = x T β yl Ungrupp. Daten: ml =. ll ỹ; β = y= log π + y= log π.

21 .3 c* Schätzung: l ỹ; β / βj = y log πl l l βj = l = l ( yl πl + (ml yl) log π l βj ) πl (ml yl) πl βj yl( πl) (ml yl)πl πl( πl) = (y l mlπl) x (j) l l da dg η /dη = exp η /( + exp η ) 2 = π( π). 2 dg ηl x (j) l dηl Schätzglechung: l (y l ml πl) xl =

22 .3 f Bespel Ader-Verengung. Call: glm(formula = Y ~ Vol + Rate, famly = bnomal, data = d.adern) Devance Resduals:... Coeffcents: Value Std. Error z_appr. Pr(> z ) Sgnf (Intercept) ** Vol ** Rate ** 2 (Dsperson Parameter for Bnomal famly taken to be ) Null Devance: 54.4 on 38 degrees of freedom Resdual Devance: on 36 degrees of freedom Number of Fsher Scorng Iteratons: 5 Correlaton of Coeffcents: (Intercept) Vol Vol Rate

23 22.3 g Resduen-Devanz ( ) D y ; π = 2 ll (M) ll ỹ ; β. Maxmale errechbare Log-Lkelhood ( πl = yl/ml): ll (M) = l ( log (m ) l + yl log yl yl +(ml yl)log ml yl mllog ml ). h Modelle verglechen: Lkelhood-Rato-Tests. Test-Statstk: D y; π (K), π (G) = D y; π (K) D y; π (G) = 2(ll (G) ll (K) ) asymptotsch chquadrat-vertelt, wenn das klene Modell stmmt.

24 .3 Resduen-Devanz verglecht geschätztes Modell mt max. Mod. Anpassungstest Achtung: Geht nur be ncht zu klenen ml grupp. Daten. 23 j Klenstes Modell: π für alle Beobachtungen glech. ll () = (m l log ) l + log yl π l y l + nlog π mt π = l y l/n. Null-Devanz: D y; π = 2 ( ll (M) ll ()) Gesamt-Test für das Modell. (H: alle βs =!)

25 24.4 Resduen-Analyse a Rohe Resduen (response resduals) Rl = Ỹl/ml πl, πl = g x T l β Pearson resduals: R (P) l = Rl/ π l( πl)/ml Devance resduals: Betrag der -ten Beobachtung zur Devanz Workng resduals: Berechnung der logst. Regr. va teratv gewchtete Kl.Qu. (vgl. nchtln. Regr.) lneare Näherung Resduen : workng resduals. b Grafsche Darstellungen: Q-Q- (normal) plot mest unnütz!

26 Response Resduen geschätztes π lnearer Prädktor.5..5 Arbets Resduen 5 5 () () c Tukey-Anscombe-Dagramm: Rohe Res. / geschätzte π oder Arbetsres. / ln. Prädktor braucht Glättung..4 25

27 Response Resduen geschätztes π

28 .4 e 27 Partal resdual plots : Effekte von x(j) (= βjx (j) Konst.) plus geegnete Resduen gegen x (j).

29 28 Resduals 5 5 Resduals Apgar ph Resduals Gewcht Alter Resduals 5 5 Y ~ log(gewcht) + Alter + Apgar Jul 2,/2:34

30 29 regr regr(formula = Survval ~ Weght + Age + Apgar, data = t.d, famly = bnomal) Terms: coef stcoef t.rato df Ch2 p.value (Intercept) NA NA NA NA Weght Age Apgar devance df p.value Model Resdual NA Null NA Dsperson parameter taken to be. Famly s bnomal. AIC: Number of Fsher Scorng teratons: 5

31 3 res( Y ) res( Y ) sequence Gewcht Resduals Lnear Predctor hat dagonal st.res( Y ) Y~Gewcht + Alter + Apgar Jul 5,/5:4

32 3 Resduals Resduals Apgar ph Resduals Alter Apgar Resduals Y ~ Gewcht + Alter + Apgar Jul 5,/5:5

33 32 Call: regr(formula = cbnd(survval., Survval.) ~ Weght, data = t.d, famly = bnomal) Terms: coef stcoef t.rato df F p.value (Intercept) NA NA NA NA Weght devance df p.value Model Resdual Null NA Dsperson parameter estmated to be.555. Famly s bnomal. AIC: Number of Fsher Scorng teratons: 4

34 33 res( Y ) res( Y ) sequence Weght Resduals ftted hat dagonal 8 st.res( Y ) cbnd(survval., Survval.)~Weght Jun 4,/:43