What are those βs anyway? Understanding Design Matrix & Odds ratios

Transcription

1 Ral paramtr stimat WILD Wildlif Population Analysis of 6 What ar thos βs anyway? Undrsting Dsign Matrix & Odds ratios Rfrncs Hosmr D.W.. Lmshow Applid logistic rgrssion. John Wily & ons Inc. Nw York. Rviw of th dsign matrix logit link Link function usd to link data to th ral paramtrs of intrst. Rmmbr w ar in this class focusd on rlating our obsrvations back to th dmography of th populations w study. Thus th ral paramtrs of intrst ar survival succss rcruitmnt movmnt population growth rat distribution (occupancy) as wll as th nuisanc paramtrs (.g. rporting rcaptur r-sighting rats) includd in th maximum liklihood stimators w mploy. Th data includ not only th rspons variabl (.g. succss or failur) but th additional information w oftn collct about th individuals or groups that w sampl which may b usd to quantify important rlationships about thos dmographic paramtrs or to xplain th htrognity (i.. variation) among individuals or groups to obtain bttr stimats of th paramtrs of intrst. For xampl if I am trying to stimat th survival rat of a havily harvst population of a gam spcis whr thr ar sx- ag-rlatd diffrncs in natural mortality harvst thn sx ag ar important data in my obsrvations thy can b tratd as groups or covariats in my analysis but I dfinitly want to xplor th diffrncs in thos rats by linking thm to th data on sx ag. Th xampl from th known fat lctur lab usd group tim mass as covariats (data). Group tim took on valus of 0 or so w rfr to thm as indicator or dummy variabls. Covariats such as mass masurmnts ar rfrrd to as continuous variabls masurmnts or indicators that chang ovr tim ar rfrrd to as tim varying.. Logit link is th most commonly usd link function: Xβ Xβ xp Xβ xp Xβ Xbta (logit) a. Constrains ral paramtrs to 0- intrval 8Jan6

2 WILD Wildlif Population Analysis of 6 b. Robust whn usd in MLE ovr a rlativly wid rang of logit valus c. In th known fat xampl it is ssntially quivalnt to using logistic rgrssion.. Logit a. Equation that dscribs th rlationship of th data th stimatd paramtrs (βs) in th link function b. In MARK this is a linar combination (dosn t hav to b) This might includ paramtr or svral x β xβ + x β x β + x β + x β + x β c. Oftn writtn in gnralizd matrix notation as Xβ 3. Dsign matrix ) X is th matrix of data (covariats) usd to spcify th modl a) May rprsnt indicator variabls (0s s) b) May b valus (as in th tim variabl or group covariat xampls) c) May b variabl (covariat) nams ) β ar th stimatd paramtrs (i.. th cofficints in th linar combination) 3) via matrix multiplication: x x x 3β x x x β 3 Xβ β+ xβ + xβ 3+ x3β 4 x x x 3 β 3 x x x β 3 4 a. Th most powrful modl building tool in MARK b. Usd to input th logits ) Columns ar usd to spcify th βs to b stimatd ) Each row is usd to spcify th linar combination of βs that form th logit for on ral paramtr c. Exampl using indicator (dummy) variabls known fat data on group two tim priods. 8Jan6

3 WILD Wildlif Population Analysis 3 of 6 Null modl ( i ar qual i.. stimats a singl survival rat): Intrcpt : : o ubstituting into th link function: X= β= Xβ= Xβ= Xβ= xp. xp xp xp d. Modl whr th i diffr btwn th tims (stimats survival rats): Intrcpt B t : : 0 o X= 0 β= Xβ= 0 0 8Jan6

4 WILD Wildlif Population Analysis 4 of 6 ubstituting into th link function: X β= Xβ= xp xp xp xp. Exampl using continuous variabls (stimats an intrcpt trnd across intrvals). Exping th xampl to includ thr tim priods stimating th chang in survival as a trnd ovr tim. Considr th on group xampl again. Intrcpt B Tim : : 3: 3 o X= 3 β= Xβ= X β= X β = X β = 3 3 ubstituting into th link function: 8Jan6

5 WILD Wildlif Population Analysis 5 of 6 xp xp xp xp 3 3 xp 3 3 xp 3 f. Exampl using variabl nams (stimats an intrcpt trnd ovr covariat valus) Lt s introduc a continuous covariat spcifid in th data somthing lik mass; using th MARK dfault variabl nam Var. This tim it s th nam that is constant among individuals but th valus can diffr in th input fil. Considr th on group xampl again: Intrcpt B Var : Var : Var 3: Var o Var X= Var Var β= Var Var Xβ= Var Var 3 Var Var X β= Var X β = Var X β = Var 3 ubstituting into th link function: 8Jan6

6 WILD Wildlif Population Analysis 6 of 6 Var xp Var xp Var Var Var xp Var Var xp Var 3 Var xp Var Var xp Var In this cas th logits quations ar all th sam MARK substituts th appropriat valu for th namd variabl for ach obsrvation (captur history) in th input fil. g. Exampl using additiv modl Now building on th xampls abov w crat an additiv modl by combining two modls for tim trnd in survival Var (tim Var). This is calld th additiv modl bcaus ths ffcts ar addd togthr in th logit which is anothr way of saying that thir affct is constant or indpndnt of th othr dpndnt variabls (covariats) in th modl. o in this xampl this modl stimats th trnd in survival ovr tim (β ) that is consistnt ovr all valus of Var vic-vrsa. Intrcpt B Var Tim : Var : Var 3: Var 3 o Var X Var Var 3 β= 3 Var Var 3 Xβ= Var Var 3 Var 3 3 Var 3 3 X β= Var 3 X β = Var 3 8Jan6

7 WILD Wildlif Population Analysis 7 of 6 ubstituting into th link function: Var 3 3 Var 3 xp Var 3 Var 3 xp Var 3 Var 3 xp Var 3 Var 3 xp Var 3 3 Var 3 xp Var 3 Var 3 xp Var 3 h. Exampl using intraction Furthr building on th xampls abov w add an intraction trm that stimats th chang in th trnd in survival ovr tim to chang across th valus of Var. W do this by adding anothr stimatd paramtr. Within that column in th dsign matrix w plac th product of Var th valu of tim. W can do this using th product function in th dsign matrix ( th Dsign Matrix Functions in th MARK hlp fil for mor on th product function information on additional functions for us in th dsign matrix) th product can b spcifid in svral diffrnt ways: Intrcpt B Var Tim Var*Tim : Var product(colcol3) : Var product(colcol3) 3: Var 3 product(colcol3) Intrcpt B Var Or Tim Var*Tim : Var product(varcol3) : Var product(varcol3) 3: Var 3 product(varcol3) Intrcpt Th rsult is th sam: Or B Var Tim Var*Tim : Var product(var) : Var product(var) 3: Var 3 product(var3) 8Jan6

8 WILD Wildlif Population Analysis 8 of 6 o Var Var X Var Var Var 3 3Var β= 3 4 Var Var Xβ= Var Var 3 Var 3 3Var 4 Var 3 Var4 Var 3 Var 4 Var 33 3Var 4 X β= Var Var 3 4 X β = Var Var 3 4 X β Var 3 3Var ubstituting into th link function: Var 3 Var 4 xp Var 3Var 4 Var 3 Var 4 xp Var 3 Var 4 Var 3 Var4 xp Var 3 Var 4 Var 3 Var4 xp Var 3 Var4 3 Var 33 3Var4 xp Var 33 3Var 4 Var 33 3Var4 xp Var 33 3Var 4 With any of th abov Dsign Matrics th dvianc liklihood numbr of stimatd paramtrs stimats of th β i AICc ar th sam. i. Intractions among indicator variabls Intractions btwn indicator polychotomous paramtrs (mor than two possibilitis) rquir ach column usd in spcifying th polychotomous paramtr to b multiplid by th columns spcifying th othr paramtr(s). 8Jan6

9 WILD Wildlif Population Analysis 9 of 6 For xampl considr xampls similar to th KM xampl with two groups only 4 tim priods Th dfault dsign matrix for two groups ((g)) is Int B g : : 3: 4: 5: 0 6: 0 7: 0 8: 0 Th dfault dsign matrix for th modl for (t) survival diffrnt among tims but not btwn groups Int B t t t3 : : 0 0 3: 0 0 4: 0 0 5: : 0 0 7: 0 0 8: 0 0 Th additiv modl with group tim ffcts ((t+g) (i.. thr is a diffrnc in survival btwn groups but it is constant ovr tim). Int B g t t B5 t3 : : 0 0 3: 0 0 4: : : : : 0 0 Finally th modl with ffcts of group tim that vary among tim priods ((t*g) is: 8Jan6

10 WILD Wildlif Population Analysis 0 of 6 Odds ratios Int B g B t t t3 B5 g*t B6 g*t B7 g*t3 0 : : : : : : : : Th cofficints in th logit (β i) ar intrprtabl as natural log of th odds ratios. Th odds of succss (survival) ar th ratio of th probability of succss to th probability of failur or Th odds ratio btwn groups is thn 4. Indicator variabls i. ( ) /( ). /( ) i Lt s us a simpl xampl of known fat data whr 50 individuals wr undr obsrvation for tim priods using tlmtry without staggrd ntry without cnsoring. In this xampl lt 40 individuals surviv (fat ) th first tim priod 30 surviv th scond tim priod. Tim priod Fat (survivd) (did) 0 0 Total urvival for priod is 40/50 = 0.80 survival in priod is 30/40 = Th odds of surviving priod ar 40/0 = 0.80/0.0 = 4/ (ln(4.0) =.3863) th odds of surviving priod ar 30/0= 0.75/0.5=3/ (ln(3.0) =.0986). Th chang in odds ratio of survival for th two tim priod is: ( ) ( ) 40 = 0 = = Jan6

11 WILD Wildlif Population Analysis of Confirming what w illustratd abov β = =.33. Thus w could hav drawn th sam conclusion from xamination of th paramtrs stimatd in program MARK. Individuals wr.33 tims as likly to surviv priod as thy wr to hav survivd priod. 5. Intrcpt coding Th rason MARK uss intrcpt coding in th dfault dsign matrix is to allow for th intrprtation of th odds ratios among th logits usd in stimating th ral paramtrs of intrst. Considr two dsign matrics for th known fat Black Duck xampl (t): Int B t t t3 B5 t4 B6 t5 B7 t6 B8 t7 0 0 : : : : : : : : t B t t3 t4 B5 t5 B6 t6 B7 t7 B8 t8 0 0 : : : : : : : : Th stimats of th ral paramtrs ar idntical as ar th dvianc liklihood AICc. Howvr th stimatd paramtrs in th uppr (intrcpt codd) matrix ar intrprtabl as log odds of ffct sizs from th idntity matrix blow ar log odds of succss (survival). This concpt is asily xtndd to groups within th data. 6. Continuous dpndnt covariats Givn that w ar daling with linar logit quations in most cass th β i associatd with continuous dpndnt variabls ar intrprtd as changs in th log odds ratio pr unit chang in th valu of th variabl whil holding all othr variabls constant. Using th black duck xampl again considr th modl (min<0). In this modl as dscribd in th input fil th Conroy t al. manuscript min<0 is th numbr of days 8Jan6

12 WILD Wildlif Population Analysis of 6 blow frzing during ach of th intrvals btwn occasions. Ths wr ntrd as group covariats in th dsign matrix. Thus th dsign matrix th associatd logits ar: Intrcpt min<0 : 4 : 6 3: 7 4: 7 5: 7 6: 6 7: 5 8: 5 Logit Th stimats of th β i wr: Estimatd Paramtr Estimat β β Thus w can calculat th logits th ral paramtrs of intrst as: min<0 Logit urvival Odds ratio Jan6

13 Logit WILD Wildlif Population Analysis 3 of Logit urvival min<0 Our intrprtation should b that th odds of survival ach wk ar 0.54 ( lowr for ach additional (on) day blow frzing during that wk. 7. Multivariabl modls a. Additiv modls ) tims Whn mor than on variabl is includd in th modl th intrprtation of th odds must includ th diffrncs in th distributions of th valus for ach variabl. Considr th Black Duck xampl again th modl (min0+ag) whr ag = for juvnils ag = 0 for adults. Th dsign matrix th logits ar: Intrcpt B min<0 ag : 4 ag : 6 ag 3: 7 ag 4: 7 ag 5: 7 ag 6: 6 ag 7: 5 ag 8: 5 ag Logit 4 ag 3 6 ag 3 7 ag 3 7 ag 3 7 ag 3 6 ag 3 5 ag 3 5 ag 3 From th rsults th stimatd paramtrs: Indx Labl βi i Intrcpt min< ag W can calculat th xpctd survival rats using th link function th odd ratios btwn th groups as abov 8Jan6

14 Logit WILD Wildlif Population Analysis 4 of 6 min<0 Logit urvival Ag = 0 Ag = Odds of survival Logit urvival Odds of survival Odds ratio btwn ags Graphing ths rsults w s that th logits ar paralll but th graphs of survival ar not dspit th fact that th odds ratios rmain constant. Thus whn comparing survival rats for modls with mor than on variabl th valus of th paramtrs at which th rlationships ar valuatd must b spcifid xplicitly Logit Ag = 0 Logit Ag = urvival Ag = 0 urvival Ag = min< b. Modls including intractions Whn intractions ar stimatd in th modl vn mor car must b takn with th intrprtation hr s why. Considr th Black Duck xampl again th modl (min0*ag) whr ag = for juvnils ag = 0 for adults. Th dsign matrix th logits ar: Intrcpt B min<0 ag (min<0)*ag : 4 ag product(colag) : 6 ag product(colag) Logit 4 ag 4ag ag 6ag 3 4 8Jan6

15 WILD Wildlif Population Analysis 5 of 6 3: 7 ag product(colag) 4: 7 ag product(colag) 5: 7 ag product(colag) 6: 6 ag product(colag) 7: 5 ag product(colag) 8: 5 ag product(colag) 7 ag 7ag ag 7ag ag 7ag ag 6ag ag 5ag ag 5ag 3 4 From th rsults th stimatd paramtrs: Indx Labl Estimat E LCI UCI Intrcpt min< ag min<0*ag W can calculat th xpctd survival rats using th link function th odd ratios btwn th groups as abov Ag = 0 Ag = min<0 Logit urvival Odds of survival Logit urvival Odds of survival Odds btwn ags Graphing ths rsults w s that th logits ar NOT paralll (i.. thy hav diffrnt slops). Thus th odds ratio btwn ag groups is not constant. Thus it is vn mor important whn comparing survival rats for modls with mor than on variabl intractions that th valus of th paramtrs at which th rlationships ar valuatd ar spcifid xplicitly. 8Jan6

16 Logit urvival (wkly) WILD Wildlif Population Analysis 6 of Logit Ag = 0 Logit Ag = urvival Ag = 0 urvival Ag = min< min<0 Ag = 0 Ag = Odds Odds min<0 Logit urvival urvival min<0 Logit urvival urvival Ag min< Jan6