LIKELIHOOD QUANTITIES FORBINARY DATA• Some review of Chapter 3 (and Chapter4).• H-conditional likelihood for binary datathat encompasses the conditional logistic(Poisson) likelihoods.• Expressions for score and information.1SETTING• Grouped time (shrink intervals to get con-tinuous time).• Focus on a single risk set (not necessarilyevent time).• General odds model (not necessarily POM(or PHM).2PROBABILITY STRUCTURE• λi= pi/(1 − pi) = p(Zi; Z)/[1 − p(Zi; Z)] -subject i Z-odds.• λd=Qi∈dλi.3CONDITIONAL EVENT ANDPROBABILITYLet H be a (random) event (that we will con-dition on). Define:• pd|h= pr(D = d|H = h, Z) = pr(D = d|H = h, Z)- H-conditional D-probability (or intensity).• π(h|d) = pr(H = h|Dd= 1, Z) = pr(H = h|D =d, Z) (e.g., H =eR or H =eR, |D|)4CONDITIONAL PROBABILITY GIVENAN EVENT H = hTheorem: Let D be generated according tothe independent binary data probability struc-ture. I.e.,pd|R= pr(D = d) = λdqR.Then, the H-conditional |D|-probability ispd|h= pr(D = d|H = h, Z) =λdπ(h|d)Ps⊂Rλsπ(h|s).Proof: Bayes Theorem5GENERAL ODDS MODELConsider a general odds models (not just POM):• λ(t, z; β) - model for odds as a functionof t and z.a• λi(β) = λ(t, Zi(t); β) - model odds for i atβ.• λs(β) =Qi∈sλi(β).• Zi(β) =λ0i(β)λi(β)- “working covariate” for iat β.baNote that for POM β is a vector that has both base-line odds λ(t) and odds ratio β.bλ0i(β) is the derivative of λ(β) wrt β.6H-CONDITIONAL LIKELIHOODLikelihood H-conditional odds model “plugsin” the odds model λ(t, Zi(t); β) for the “true”subject odds λi:l(β; D|H) = pD|H(β) =λD(β) π(H|D)Ps⊂Rλs(β) π(H|s).7DEFINITIONS• “Working covariate” for set d at β:Zd(β) =λ0d(β)λd(β).• “Conditional expectation” of the working covari-ate at β,Eh(β) =Xd⊂RZd(β) pd|h(β).• Probability that i is a case at β:pi|h(β) =Xd⊂R:d3ipd|h(β)• Probability that i and j are cases at β:pi,j|h(β) =Xd⊂R:d3i,jpd|h(β)8USEFUL IDENTITIES: WORKINGCOVARIATE FOR A SETLemma: Zd(β) =λ0d(β)λd(β)=Pi∈dZi(β) whereλ0d(β) is the derivative of λd(β) with respectto β.Proof: : Homework9USEFUL IDENTITIES: THECONDITIONAL EXPECTATION OFTHE WORKING COVARIATE Eh(β)Lemma: Eh(β) ==Pd⊂RZd(β) pd|h(β) =Pi∈RZi(β) pi|h(β).Proof:Suppressing the β,Eh=Xd⊂RZdpd|h=Xd⊂R(Xd3iZi) pd|h=Xi∈RZiXd⊂R:d3ipd|h=Xi∈RZipi|h.10LIKELIHOOD QUANTITIES11NEW NOTATIONCase set indicator:• Let Dd= I(D = d) (indicator for theevent D = d). The events Dd, d ⊂ Rindicate possible case sets.• Note D is a set of indices, Ddis an indi-cator variable.12H-CONDITIONAL LIKELIHOOD ANDLOG-LIKELIHOODl(β; D|H) =λD(β) π(H|D)Ps⊂Rλs(β) π(H|s)L(β; D|H) = log λD(β) − logXs⊂Rλs(β) π(H|s)13SCORE EXPRESSIONSU(β; D|H) = U(β) = ZD(β) − EH(β)=Xd⊂RZd(β)[Dd− pd|h(β)]=Xi∈RZi(β)[Di− pi|h(β)]Suppressing β,U(β) =λ0DλD−Ps⊂Rλ0sπ(H|s)Pu⊂Rλuπ(H|u)= ZD−Xs⊂R λ0sλs!λsπ(H|s)Pu⊂Rλuπ(H|u)= ZD−Xs⊂RZsps,H= ZD− EH,the first expression.14SCORE EXPRESSIONSContinuing...U(β) = ZD−Xs⊂RZsps|H=Xd⊂RZdDd−Xs⊂RZsps|H=Xd⊂RZd[Dd− pd|H],the second expression. Further, using theUseful Identities,ZD− EH=Xi∈DZi−Xi∈RZipi|H=Xi∈RZiDi−Xi∈RZipi|H=Xi∈RZi[Di− pi|H]giving the third expression.15OBSERVED AND EXPECTEDINFORMATION EXPRESSIONSvobs(β; D|H) = vobs(β) =Xs⊂Rps|H[Zs− EH]⊗2−Xs⊂RZ0s[Ds− ps|H]=Xi∈RZ⊗2ipi|Hqi|H+Xj6=k∈RZjZTk(pj,k|H− pj|Hpk|H)−Xi∈RZ0i[Di− pi|H]vexp(β; D|H) = vexp(β) =Xs⊂Rps,H[Zs− EH]⊗2=Xi∈RZ⊗2ipi|Hqi|H+Xj6=k∈RZjZTk(pj,k|H− pj|Hpk|H)β is suppressed in the information expressions.16OBSERVED INFORMATION: FIRSTEXPRESSIONvobs(β) =Xd⊂Rpd|H(β)[Zd(β) − EH(β)]⊗2−Xd⊂RZ0(β)[Dd− pd|H(β)].Proof: From the second score expression,vobs(β) = −ddβU(β)= −Xd⊂R{Z0(β)[Dd− pd|H(β)] − Zd(β)p0d|H(β)T}Now, recall that pd|H(β) =λd(β) π(H|d)Ps⊂Rλs(β) π(H|s). So, sup-pressing β,p0d|H=λ0dπ(H|d)Ps⊂Rλsπ(H|s)− λdπ(H|d)Ps⊂Rλ0sπ(H|s)[Ps⊂Rλsπ(H|s)]2= Zdpd|H− pd|HXs⊂RZsps|H.17OBSERVED INFORMATIONThus,Xd⊂RZd(β)p0d|H(β)T=Xd⊂RZ⊗2dpd|H− [Xd⊂RZsps|H]⊗2=Xd⊂RZ⊗2dpd|H− E⊗2H=Xd⊂Rpd|H(β)[Zd(β) − EH(β)]⊗2Putting the pieces together we get the result.18EXPECTED INFORMATIONEXPRESSIONSNote that these differ from the observed in-formation vobsin that they do not have:−Xs⊂RZ0s(β)[Ds− ps|H(β)] or−Xi∈RZ0i(β)[Di− pi|H(β)]parts.• Think of case-set ‘probability’ pd|H(β) orcase ’probability’ pi|H(β) under the model.• ‘Expectation’ over case-set Ddor case Diindicators using the model probabilities ofthe terms above is zero. I.e., very heuris-tically EIβ[Ds] − ps|H(β) = 0.This is heuristic. Theory shows that thesewill be asymptotically zero if model ‘fits’the data.19BASIC LIKELIHOOD PROPERTIESSuppose that• There exists a point in the model space β = β0that corresponds to the Z -intensity for failure D,i.e. p(t, y, z; β0) = yp(t, z; β0) = p(t; Y, Z).• Diare Z -conditional independent.Then• The (H-conditional) expectation of the score iszero.• The (H-conditional) variance of the score is equalto the (H-conditional) expectation of the observedor expected information.Asymptotic properties will require more specificassumptions.20H- conditional likelihood quantities in terms ofsets, set indicators, and subjectsSet notationl(β)λD(β) π(H|D)Ps⊂Rλs(β) π(H|s)L(β) log λD(β) − logPs⊂Rλs(β) π(H|s)U(β) ZD(β) − EH(β)vobs(β)Ps⊂Rps|H[Zs− EH]⊗2− [Z0D−Ps⊂RZ0sps|H]vexp(β)Ps⊂Rps|H[Zs− EH]⊗2Set indicator notationl(β)Qd⊂Rλd(β) π(H|d)Ps⊂Rλs(β) π(H|s)DdL(β)Pd⊂R[log λd(β)] − logPs⊂Rλs(β) π(H|s)U(β)Pd⊂RZd(β)[Dd− pd|H(β)]vobs(β)Ps⊂Rps|H[Zs− EH]⊗2−Ps⊂RZ0s[Ds− ps|H]vexp(β)Ps⊂Rps|H[Zs− EH]⊗2=Ps⊂Rps|HZ⊗2s− E⊗2HSubject indicatorsU(β)Pi∈RZi(β)[Di− pi|H(β)]vobs(β)Pi∈RZ⊗2ipi|Hqi|H+Pj6=k∈RZjZTk(pj,k|H− pj|Hpk|H)−Pi∈RZ0i[Di− pi|H]vexp(β)Pi∈RZ⊗2ipi|Hqi|H+Pj6=k∈RZjZTk(pj,k|H− pj|Hpk|H)21NOTES• Although there are a lot of different quan-tities and situations covered, the expres-sions are pretty consistent!• Z notation is very nice! In special case oflog linear, Z = Z (HW).• Setting H to e.g.,fR,fR, |D| orfR, D yieldslikelihood expressions for our case-controlCLL.• Sums will be over the appropriate sets ⊂fR because π(fR|s) will be zero forfR thatdo not