Cox s proportional hazards model and Cox s partial likelihood
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1 Cox s proportional hazards model and Cox s partial likelihood Rasmus Waagepetersen October 12, / 27
2 Non-parametric vs. parametric Suppose we want to estimate unknown function, e.g. survival function. Approaches: Non-parametric using Kaplan-Meier. Advantage: no assumption regarding type of distribution. Disadvantage: requires iid observations. Parametric model. Advantage: we only need to estimate a few parameters that completely characterize distribution (e.g. exponential or Weibull) - gives low variance of estimates. Can be extended to non-iid observations using regression on covariates. Disadvantage: assumed model class may be (or always is) incorrect leading to model error or in other words, bias. Possible to combine the best of two approaches? 2 / 27
3 Semi-parametric approach - Cox s proportional hazards model Sir David Cox in a ground-breaking paper ( Regression models and life tables, 1972) suggested the following model for the hazard function given covariates z R p : h(t; z) = h 0 (t) exp(z T β), β R p. Here h 0 ( ) completely unspecified function except that it must be non-negative. Thus model combines great flexibility via non-parametric h 0 ( ) with the possibility of introducing covariate effects via exponential term exp(z T β) This model has become standard in medical statistics. 3 / 27
4 Some properties Cumulative hazard: H(t; z) = exp(z T β) Survival function t 0 h 0 (u)du = exp(z T β)h 0 (t) S(t; z) = S 0 (t) exp(zt β) S 0 (t) = exp( H 0 (t)) Proportional hazards: h(t; z) h(t; z ) = exp((z z ) T β) i.e. constant hazard ratio for two different subjects - curves can not cross! - this should be checked in any application. 4 / 27
5 Estimation - partial likelihood Model useless if we can not estimate parameter β. Problem: we can not use likelihood when h 0 ( ) unspecified. Second break-through contribution of Cox: invention of partial likelikehood for estimating β. Suppose we have observations (t i, δ i ) as well as (fixed) covariates z 1,..., z n, i = 1,..., n. We assume no ties (all t i distinct) and define D {1,..., n} as D = {l δ l = 1} - i.e. the index set of death times. For any t 0 we further define the risk set R(t) = {l t l t} i.e. the index set of subjects at risk at time t. 5 / 27
6 The partial likelihood The partial likelihood is L(β) = exp(zl T β) l D k R(t l ) exp(zt k β) Cox suggested to estimate β by maximizing L(β). does not depend on h 0 does not depend on actual death times - only their order censored observations only appear in risk set (as for Kaplan-Meier) Cox s idea has proven to work very well - but why? Lots of people have tried to make sense of this partial likelihood. 6 / 27
7 Cox s intuition Consider for simplicity the case of no censoring and let t (1),..., t (n) denote the set of ordered death times. We can equivalently represent data as the set of inter-arrival times v i = t (i) t (i 1) (taking t 0 = 0) together with the information r 1, r 2,..., r n about which subject died at each time of death - i.e. r i = l if subject l was the ith subject to die. Cox then factored likelihood of (v 1,..., v n, r 1,..., r n ) as (using generic notation for densities and probabilities) f (v 1 )p(r 1 v 1 )f (v 2 v 1, r 1 )p(r 2 v 1, v 2, r 1 ) f (v n v 1,..., v n 1, r 1,..., r n 1 )p(r n v 1,..., v n, r 1,..., r n 1 ) 7 / 27
8 Cox argued that terms f (v i...) could not contribute with information regarding β since the interarrival times can be fitted arbitrary well when h 0 is unrestricted - we can essentially just choose h 0 to consist of spikes at each death time. Thus estimation of β should be based on remaining factors L(β) = n p(r i H i ) i=1 where H i = {v 1,..., v i, r 1,..., r i 1 } history/previous observations. Here p(r i H i ) is the probability that subject r i is the ith person to die given the previous observations. 8 / 27
9 More precisely, let R i denote the random index of the ith subject that dies (R i = l means that T l is the ith smallest death time, i.e. T Ri = T (i) ). Assume that p(l H i ) only depends on H i through the knowledge that the ith death happens at time t (i) and that R(t (i) ) are the ones at risk at time t (i). Thus p(l H i ) = P(R i = l T Ri [t (i), t (i) + dt[, R(t (i) ) = A) This is the probability that l is the ith person to die given that the ith death happens at time t (i) and that the persons in A are at risk at time t (i) (thus probability is zero if l A) 9 / 27
10 We now express the conditional probability in terms of the hazard function: P(R i = l, T Ri [t (i), t (i) + dt[ R(t (i) ) = A) =P(T l [t (i), t (i) + dt[, T k > T l, k A R(t (i) ) = A) = h 0 (t (i) ) exp(zl T β)dt (1 h 0 (t (i) ) exp(zk T β)dt) k A\{l} Note = is because we actually replace T k > T l by T k > t (i) + dt. This does not really matter since dt infinitesimal. 10 / 27
11 Finally, P(R i = l T Ri [t (i), t (i) + dt[, R(t (i) ) = A) = P(R i = l, T Ri [t (i), t (i) + dt[ R(t (i) ) = A) P(T Ri [t (i), t (i) + dt[ R(t (i) ) = A) P(R i = l, T Ri [t (i), t (i) + dt[ R(t (i) = A)) = j R(t (i) ) P(R i = j, T Ri [t (i), t (i) + dt[ R(t (i) ) = A) h 0 (t (i) ) exp(zl T β)dt k R(t (i) )\{l} = (1 h 0(t (i) ) exp(zk T β)dt) j R(t (i) ) h 0(t (i) ) exp(zj T β)dt k R(t (i) )\{j} (1 h 0(t (i) ) exp(zk Tβ)dt) exp(zl T β) = k R(t (i) ) exp(zt k β) Note: last = follows after cancelling h 0 (t (i) )dt and noting that (1 h 0 (t (i) ) exp(zk T β)dt) tends to one when dt tends to zero. NB: denominator is hazard for minimum of T k, k R(t (i) ) (exercise 18) 11 / 27
12 Conditional likelihood for matched case-control study Cox s idea very closely related to conditional likelihood for matched case-control studies. Let X denote a binary random variable (e.g. sick/healthy) for an individual in a population. We want to study the impact of a covariate z on X. Assume that the population can be divided into homogeneous groups (strata) so that probability of being ill is given by a logistic regression P(X = 1) = p i (z) = exp(α i + βz) 1 + exp(α i + βz) for an individual in the ith strata and with the covariate z. 12 / 27
13 Suppose X 1 = 1 with covariate z 1 is observed for a sick person in the ith stratum. In a matched case-control study this observation is paired with an observation X 2 = 0 with covariate z 2 for a randomly selected healthy person in the same stratum. The conditional likelihood is now based on the conditional probabilities P(X 1 = 1 X 1 = 1, X 2 = 0 or X 1 = 0, X 2 = 1) = This reduces to p i (z 1 )(1 p i (z 2 )) p i (z 1 )(1 p i (z 2 )) + (1 p i (z 1 ))p i (z 2 ) exp(βz 1 ) exp(βz 1 ) + exp(βz 2 ) which is free of the strata specific intercept α i. Note α i is a nuisance parameter when we are just interested in β. 13 / 27
14 Invariance argument Again consider the case of no censoring. Kalbfleisch and Prentice noticed that if one applies a strictly monotone differentiable function g to the survival times T 1,..., T n then T i = g(t i ) again follow a proportional hazards model with a completely unspecified hazard function h 0 (exercise 17). Hence estimation problem for β the same regardless of whether we consider T i s or T i s. They thus concluded that only the ordering (ranks) of the survival times and not the magnitudes of the survival times could matter for inference on β. One can verify that for the ranks R i, P(R 1 = r 1,..., R n = r n ) = P(T r1 < T r2 < < T rn ) is precisely Cox s partial likelihood. 14 / 27
15 Profile likelihood Cox s partial likelihood can also be derived as a profile likelihood. Consider likelihood (assuming no ties) n [h 0 (t i )dt exp(zl T β)] δ i exp[ exp(zi T β) i=1 ti 0 h 0 (u)du]. Let s try to maximize wrt h 0. First, we need h 0 (t l ) > 0 for l D. At the same time we should take h 0 (u) = 0 between death times. So we let h 0 (t)dt = α l in very small intervals around death times, [t l, t l + dt[, l D, and zero elsewhere. Note likelihood does not inform about h 0 (t) for t larger than max i t i. 15 / 27
16 Then likelihood becomes ( ) L(α, β) = α l exp[zl T β] exp( l D ( ) = α l exp[zl T β] exp( l D l D α l n exp(zi T β) α l ) l D:t l t i i=1 i R(t l ) Taking log and differentiating wrt α l we obtain log L(α, β) = 1 α l α l j R(t l ) Setting equal to zero and solving wrt α l gives exp(z T i β)) exp(z T j β) ˆα l (β) = 1 j R(t l ) exp(zt j β) 16 / 27
17 Plugging in ˆα l (β) for α l we finally obtain profile likelihood: ( ) exp(zl T β) L p (β) = L(ˆα, β) = exp( D ) j R(t l ) exp(zt j β) l D which is Cox s partial likelihood. As a byproduct we obtain the Breslow estimate of H 0 : Ĥ 0 (t) = 1 l D: j R(t l ) exp(zt j β) t l t where we replace β by partial likelihood estimate ˆβ. This reduces to Nelson-Aalen estimator if β = 0. Note Ĥ 0 (t) is discontinuous in contrast to H 0 (t) = t 0 h 0(u)du. Ĥ 0 (t) limiting case of H 0 with mass increasingly concentrated around death times. 17 / 27
18 Estimating function point of view All previous derivations more or less heuristic. However, not crucial to understand Cox s partial likelihood as a likelihood or as derived from a likelihood. Just consider properties of associated estimating function. Score of partial likelihood is an estimating function which (see next slide) is unbiased (each term mean zero) sum of uncorrelated terms (gives CLT) - general theory for estimating functions suggests that partial likelihood estimates asymptotically consistent and normal. 18 / 27
19 Score function is sum of terms u(β) = d log L(β) dβ u i (β) = z Ri E[z Ri T Ri [t (i), t (i) + dt[, R(t (i) )]. Each term has mean zero: E[u i (β)] = E[E[u i (β) H i ]] = 0 Moreover, terms are uncorrelated. For i < j: E[u i (β)u j (β)] = E[u i (β)e[u j (β) H j ]] = 0 Thus good reason to believe that CLT works for score function. 19 / 27
20 Asymptotic properties of estimates and tests The observed information for the partial likelihood is j(β) = d dβ T u(β) = Var[z Ri T Ri [t (i), t (i) + dt[, R(t (i) )] = i D Var[u i (β) H i ] Information i(β) = Ej(β) = Var(u(β)) i D In analogy with usual asymptotic results we obtain for large n, ( ˆβ β) N(0, i(β) 1 ) In practice we estimate i(β) by j( ˆβ). This can be used for constructing confidence intervals in the usual way. Moreover, we can construct Wald tests, score-tests and likelihood-ratio tests in the usual way. 20 / 27
21 Asymptotic distribution - sketch Let β denote true value of regression parameter. First order (multivariate) Taylor: u(β ) = u( ˆβ) + where β β ˆβ β. Thus d dβ T u(β) β= β (β ˆβ) u(β ) = j( β)( ˆβ β ) i(β ) 1/2 u(β ) = i(β ) 1/2 j( β)( ˆβ β ) 21 / 27
22 Assume now as n tends to infinity, i(β ) 1/2 u(β ) N(0, I ) (CLT) (convergence in distribution) and i(β ) 1/2 j( β)i(β ) 1/2 I (convergence in probability). Since i(β ) 1/2 ( ˆβ β ) = (i(β ) 1/2 j( β)i(β ) 1/2 ) 1 i(β ) 1/2 u(β ) it follows by the assumptions above that i(β ) 1/2 ( ˆβ β ) N(0, I ) in distribution. 22 / 27
23 Consider H 0 : β = β 0. Several possibilities under H 0 : (Wald) j(β 0 ) 1/2 ( ˆβ β 0 ) N(0, I ) (Score test) j(β 0 ) 1/2 u(β 0 ) N(0, I ) ( likelihood ratio) 2 log(l(β 0 )/L( ˆβ)) χ 2 (p 1) See KM 8.3 and 8.5 for further details. NB: in the case of z i {0, 1} (two-group scenario), score-test for H 0 : β = 0 is equivalent with log-rank test (exercise 19). 23 / 27
24 Data with ties Suppose we have tied death times t 11 = t 12 = = t 1d 1 < t 21 = = t 2d 2 < < t r1 = = t rd r I.e. r distinct death times with d l deaths at the l distinct time. Let z lj be the covariate for the individual with death time t lj and let z l = d l j=1 z lj. Suppose we knew t l1 < t l2 < < t ld l, l = 1,..., r and let B l(j 1) consist of individuals who die at times t l1,..., t l(j 1). Then Cox s partial likelihood is r d l exp(β T zlj ) l=1 j=1 k R(tl1 )\B exp(zt l(j 1) k β) r exp(β T z = l ) dl l=1 j=1 [ k R(t l1 ) exp(zt k β) k B l(j 1) exp(zk Tβ)] 24 / 27
25 When we do not know the ordering of tl1,..., t ld l we can not compute term k B l(j 1) exp(zk Tβ). Breslow: simply ignore this sum. Resulting partial likelihood becomes r exp(β T zl ) l=1 ( k R(t l1 ) exp(zt k β))d l Efron: replace sum by j 1 times average, that is k B l(j 1) exp(z T k β) (j 1) 1 d l d l exp(β T zlk ) k=1 25 / 27
26 Cox s discrete time proportional odds model Reuse notation from actuarial estimate but introduce covariates: p k (z) = P(indiv. with covariates z dies in [u k 1, u k [ alive at time u k 1 ). Cox proposed proportional odds model: O k (z) = p k(z) 1 p k (z) = p k(0) 1 p k (0) exp(zt β) = O k (0) exp(z T β) Let D k be index set of d k individuals who die in [u k 1, u k [. Probability that precisely individuals in D k die given risk set R(u k 1 ) is p k (z l ) (1 p k (z l )) = O k (z l ) (1 p k (z l )) l D k l R(u k 1 )\D k l D k Probability that d k individuals die: O k (z l ) A R(u k 1 ): #A=d k l A l R(u k 1 ) l R(u k 1 ) (1 p k (z l )) 26 / 27
27 Discrete time partial likelihood Partial likelihood based on probabilities that individuals in D k die given d k individuals die and given R(u k 1 ). Only consider intervals with d k > 0 L(β) = k:d k >0 exp( l D k zl T β) A R(u k 1 ): exp( l A zt l β) #A=d k Note: O k (0) plays the same role as exp(α i ) in matched case control model. Different approaches to handling ties vary regarding computational complexity. On modern computers all options usually feasible. 27 / 27
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