Consistency of the Semi-parametric MLE under the Cox Model with Linearly Time-dependent Covariates and Interval-censored Data

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1 Journal of Advanced Statistics, Vol. 4, No. 1, March Consistency of the Semi-parametric MLE under the Cox Model with Linearly Time-dependent Covariates and Interval-censored Data Qiqing Yu * and Qinggang Diao Department of Mathematical Sciences, SUNY, Binghamton, NY 13902, USA qyu@math.binghamton.edu Abstract Yu and Diao [1] studied the estimation problem under the Cox model with linearly time-dependent covariates and with interval-censored (IC) data under the distribution-free set-up. They proposed a modified semi-parametric MLE (MSMLE) and the simulation results suggest that the MSMLE is consistent. In this paper, we establish the consistency of the MSMLE. Keywords: Cox s model, time-dependent covariates, semi-parametric MLE, consistency. 1 Introduction In this paper, we establish the consistency of the semiparametric estimator proposed in Yu and Diao [1] under the proportional hazards (PH) model with a special continuous time-dependent covariates and with interval-censored data. Let Y be a continuous random variable, with the cumulative distribution function (cdf) F Y, where F Y (t) = P (Y t). Its survival function is denoted by S Y (t) = 1 F Y (t), its density function by f Y (t), f Y (t) S Y (t ) and its hazard function by h Y (t) =. Let z be a p 1 covariate vector. We say that (z, Y ) follows the PH (or Cox) model if the conditional hazard satisfies h(t z) = h Y z(t z) = h o (t)e βz, for t < τ, (1.1) where βz = β z, β is the transpose of the p 1 vector β, τ = supt : h o (t) > 0}, and h o is the hazard function of Y (z = 0). The PH model has been extended to the time-dependent covariates PH (TDCPH) model (see [2] p.113). For instance, replace z in (1.1) by z = z(t) = uβg(t), where u is a time-independent covariate, and g(t) is a function of the time t, e.g., g(t) = (t a)1(t a), where 1(A) is the indicator function of an event A (see [2] p.113). We shall call the latter case the PH model with linearly timedependent covariates (LDCPH model). Interval-censored (IC) data are (L i, R i ), i = 1,..., n, where the true survival time Y i (L i, R i ]. A realistic model for the IC data is the mixed case interval censorship model (see [3]). One of its special case is the case 2 interval censorship model (C2 model) (see [4]). The TDCPH model including the LDCPH model has been commonly used for right-censored (RC) data. The semi-parametric estimation with IC data under the LDCPH model was first studied by Yu and Diao [1], that is, h(t z(t)) satisfies Eq. (1.1) with Z(t) = U (t a)1(t a), where U is a time-independent covariate vector, (1.2) a is a real number and both β and S o are unknown. As explained in their paper, this covariate Z(t) is very typical and share the light on how to estimate (β, h o ) under the PH model with IC data and with other types of time-dependent covariates. Under the semi-parametric set-up, the typical estimation approach for right-censored data is the partial likelihood estimation. However, it is well known that this approach only works for right-censored data, but not for IC data (see, for example, Wong and Yu [5]). The main findings about the LDCPH model in Yu and Diao [1] are as follows. (1) Even if the parameter β is not a vector, β may not be identifiable if the support set (of the observable random vector) contains only finitely many elements. This is quite different from the case of the PH model with time-independent covariates, under which β is identifiable even if the support set contains only one point.

2 2 Journal of Advanced Statistics, Vol. 4, No. 1, March 2019 (2) The generalized likelihood function needs to be modified, as it must be of the form of hazard functions under the semi-parametric set-up in (1.2). Otherwise, there is no consistent estimator of β. (3) Several naive modifications on the generalized likelihood function do not lead to consistent estimators. (4) A modified semi-parametric MLE (MSMLE) was proposed and their simulation studies suggested that the MSMLE of β is consistent. In this paper, we prove the consistency of the MSMLE. The main difficulty in the proof is that there may not exist a convergent subsequence of the estimates of h o in (1.1), but the MSMLE is in the form of the estimate of h o. The paper is orgainized as follows. The MSMLE is introduced in Section 2. The consistency of the MSMLE is established in Section 3. The proofs of some lemmas are put in Section 4. 2 The MSMLE The generalized likelihood function with IC data (L i, R i ) s is often given by L = n i=1 µ S( )(I i Z i ), where µ S( ) (I i Z i ) = S(L i Z i ) S(R i Z i ) and I i = (L i, R i ]. (2.1) L depends on the survival function S(t z). Yu and Diao [1] showed that if S(t z) satisfies the PH model and is absolutely continuous, and Z = Z(t) = (t a)u1(t a), then t S(t z(t)) = exp( e βu(x a)1(t a) h o (x)dx) 0 S o (t) if t a or u = 0 = S o (a) exp( t a eβu(x a) h o (x)dx) if t > a and u 0. (2.2) Hereafter, abusing notations, we write S(t u) = S(t z(t)) and h(t u) = h o (t) exp(βu1(t a)(t a)). Notice that the two hazard functions h o (t) = 1(t > 0) and h 1 = 1(t (0, 2) (2, )) both lead to S o (t) = exp( t) if t > 0. Since h o (or f o ) can differ on a set A satisfying A df o(t) = 0, where F F o = 1 S o, we define f o (t) = o(t) if F o(t) exists for identifiability of f o and h o. Let S F be the 0 otherwise support set of the cdf F or a measure df, in the sense that x S F iff F (x + ϵ) F (x ϵ) > 0, ϵ > 0. Proposition 1. (Yu and Diao [1]). The survival function S(t u) is identifiable if t S FL S FR. An identifiability condition for β under the mixed case IC model is that S FL S FR contains infinitely many points t j } j 1 with a limiting point, say t o j t j in (a, ), provided that F o(t o ) > 0. An interval A is called an innermost interval (II) if it is an intersection of the observed intervals I 1,..., I n, and if A I i = A or for each I i. It is well known (see [8]) that under the PH model with time-independent covariates in order to maximize L, it suffices to put the weights of S o to the II s. Moreover, the weight to each II is uniquely determined, but not the distribution of the weight within the II. Let A 1,..., A m be all the II s induced by I i s and let (v j, w j ) be the pair of endpoints of A j, where w 0 = < w 1 < w 2 < < w m. For each i, let ξ i = 1(R i < w m ) and define l i and r i by w ri R i < w ri +1 and w li L i < w li +1. Then the likelihood function in (2.1) becomes L 1 (β, S o ) = (S o (w li ) S o (w ri )) R i a or u i=0 L i<a<r i,u i 0 [exp( w j (a,l i ] S o (w li ) ξ i S o (a) exp( wj w j [a,r i ] wj v j e β(x a) h o (x)dx) ξ i exp( v j e β(x a) h o (x)dx)} w j (a,r i ] wj v j L i>a,u i 0 e β(x a) h o (x)dx)]. S o (a) Since h o is a function of x and needs to be properly defined on [v j, w j ] for all j < m (note that S o (w m ) = 0), Yu and Diao proposed to modify the likelihood as follows. First, let h o (x) = 0 if x / k (v k, w k ], where (v 1, w 1 ],..., (v m, w m ] are all the II s. Moreover, notice that each (v k, w k ] will be contained by several modified observed intervals I i s (see Remark 2 in Yu and (2.3)

3 Journal of Advanced Statistics, Vol. 4, No. 1, March Diao [1]) with J ( 1) distinct values of u i s, where J depends on k. There are two types of (v k, w k ): (1) w k v k 0, or w k a, or w k a 0 and a (v k, w k ]; (2) otherwise. For k < m, define constant on (v k, w k ] if (v k, w k ) belongs to type (1) h o (x) = J-piece-wise constant on (v k, w k ] if (v k, w k ) belongs to type (2) (in particular, if (v k, w k ) belongs to type (2), then h o (x) = J h kj 1(x (v kj, w kj ]) for x (v k, w k ], (2.4) j=1 wk v k where v k = v k1, w k1 = v k2, w k2 = v k3,..., w kj = w k, w kj v kj = J if a (v k, w k ], j 1,..., J} w k v k J 1 if a / (v k, w k ], j 2,..., J} and h kj s are constant. If k = m, simply define S o (w m ) = 0 (h o can be arbitrary on (v m, w m ], provided that h o 0 and wm v m h o (x)dx = ). Abusing notations, let (a j, b j ] be the interval in which h o is constant, as specified in (2.4). Then h o (x) = j h j1(x (a j, b j ]), where h j is a constant and (a j, b j ] may not be an II. Then L 1 in (2.3) becomes L(β, S o ) = n (S(L i U i ) S(R i U i )) i=1 = exp( } h j [b j a j ])[1 exp( h j [b j a j ])] ξ i R i a, or u i=0 b j L i b j (L i,r i] S o (a) exp( e auiβ h j [e uiβbj e uiβaj ]) u i β L i a,u i 0 b j (a,l i ] [1 exp( e au iβ } h j [e u iβb j e u iβa j ])] u i β b j (L i,r i] exp( h j [b j a j ] ξi ) L i <a<r i,u i 0 b j L i [1 exp( h j (b j a j ) e au iβ } h j [e u iβb j e u iβa j ])] ξ i. u i β b j (L i,a] b j (L i,r i ] (2.5) The MSMLE maximizes L over all h j s and β. It is well known that the Newton Raphson method does not work for deriving the SMLE or the MSMLE under the PH model with IC data (see Wong and Yu [5]). Yu and Diao [1] suggested to use the steepest decent method. 3 Consistency In order to simplify the presentation of the proof of consistency, we shall only make use of the C2 model and assume that β (, ). The C2 model assumes that C 1 and C 2 are two random followup times, (L, R) = (, C 1 )1(Y C 1 ) + (C 1, C 2 )1(Y (C 1, C 2 ]) + (C 2, )1(Y > C 2 ), (C 1, C 2 ) and (Y, U) are independent, and P (C 1 < C 2 ) = 1. Define a measure µ on the Borel σ-field B on R 1 by µ(b) = P (C 1 B)+P (C 2 B), B B. Assume that (L i, R i, U i ) s are i.i.d. from F L,R,U ( def = Q). Denote ˆQ = ˆQ(x, y, z) = n i=1 1(L i x, R i y, U i z)/n. Theorem 1. Assume that the censoring satisfies the C2 model, S(t u) satisfies (2.2), the identifiable condition in Proposition 1 holds, and U takes on at least two distinct values. Then Ŝn S o dµ a.s. 0 and a.s. ˆβ n β, where (Ŝn, ˆβ n ) is obtained by maximizing L in (2.5). Let Ω be the sample space. In order to prove Theorem 1, we shall make use of the next 3 lemmas.

4 4 Journal of Advanced Statistics, Vol. 4, No. 1, March 2019 Lemma 1. Suppose that µ n } n 1 is a sequence of measures on the measurable space (A, Σ) such that µ n (B) µ(b), B Σ. Let f n } n 1 be a sequence of non-negative integrable functions. Then A lim f n dµ lim A f n dµ n. n n Lemma 2. P (Ω ˆQ) = 1, where Ω ˆQ = ω Ω : sup x,y,z ˆQ(x, y, z)(ω) Q(x, y, z)(ω) 0}. Lemma 3. Given ω Ω, for each sub-sequence of Ŝn( )} n 1, there exists a further subsequnce, say Ŝn j ( )} j 1 and a function S ( ) such that Ŝn j (t u) S (t u) for each (t, u). Lemma 1 is Fatou s Lemma with varying measures. It is almost the same as Proposition 17 in [7] (page 231), and so is its proof. Thus we skip the proof of Lemma 1. Lemma 2 is the multvariate version of the Glivenko - Cantelli theorem. Eddy and Hartigan [8] presented a similar version of Lemma 2 with certain addtional regularity conditions such as P ((L, R, U) B) = 0, where B is the boundary of S FL,R,U. However, P (B) > 0 under the assumptions in this paper. Thus their result is not applicable here. Lemma 3 is a variation of Helly s selection theorem. The proofs of Lemmas 2 and 3 are relegated to Section 4 for a better presentation. Proof of Theorem 1. We shall give the proof in 3 steps. Step 1 (preliminary). Let S (ub) (t) = S (t u) = exp( t 0 eu(s a)b1(s>a) h (s)ds), where h is a hazard function and let S = S (0) and F = 1 S. By Eq. (2.5), the normalized generalized log-likelihood is L n (S, b) = 1 n n j=1 log(s (ujb) (L j ) S (ujb) (R j )). By the strong law of large numbers (SLLN), L n (S, b) a.s. L o (S, b)( def = E(L n (S, b))). Let w S (c 1, c 2, U) = F (c 1 U) log F (c 1 U) + S(c 2 U) log S (c 2 U) + (S(c 1 U) S(c 2 U)) log(s (c 1 U) S (c 2 U)). Then L o (S, b) =E(L n (S, b)) = E(log(S (L U) S (R U))) =E(E(log(S (L U) S (R U))) U) =E(E(w S (C 1, C 2, U) U)). Step 2 : L o (S, b) is maximized iff S (t u) = S(t u) for each (t, u) S µ S FU and b = β. It is easy to check that the expression w S (c 1, c 2, u) is maximized by a conditional survival function S ( ), if and only if S (c i u) = S(c i u), i 1, 2}. Since sup p log p : 0 p 1} 1, w S (c 1, c 2, u) is bounded by 3, we see that L o (S, b) is finite. Thus S( ) maximizes L o (S, b). Moreover, by Eq. (2.2), S(t u) = exp( t 0 eβu(x a)1(t a) h o (x)dx) is continuous in (t, u), thus any other S ( ) that maximizes L o (S, b) satisfies S ( u) = S( u) on S µ S FU. Furthermore, notice that one can write S(t u) = exp t 0 e(u u o)β(s c)1(s>c) h 1 (s)ds}, where u o S FU and h 1 (s) = e u oβ(s c)1(s>c) h o (s). Without loss of generality (WLOG), we can assume that 0 S FU. Thus S (t 0) = S(t 0) for t S µ. Now, S (t u) = exp t 0 eub(s c)1(s>c) h o (s)ds}, We shall show that b = β. Since the identifiable conditions in Proposition 1 hold, for t o and t k S FC1 S FC2 satisfying t k t o and F o(t o ) > 0, log(s h (t o ) (t k 0)) ( log(s (t o 0)) log(s o (t k )) ( log(s o (t o )) tk 0 h o(s)ds t o 0 h o(s)ds = h o (t o ).

5 Journal of Advanced Statistics, Vol. 4, No. 1, March By assumption, there exists some u 0, then e ub(t o a)1(t o a) log(s(t k 0)) ( log(s(t o 0)) h (t o ) log(s o (ub) (t k )) ( log(s o (ub) (t o )) tk 0 eu(s a)β1(s>a) h o (s)ds t o 0 eu(s a)β1(s>a) h o (s)ds = e u(t o a)β1(t o a) h o (t o ). Thus, we must have e u(to a)b1(to a) h(t o ) = e u(to a)β1(to a) h o (t o ) and h(t o ) = h o (t o ), as F o(t o ) > 0. Then h o (t o ) > 0, which means b = β. That is, L o (S, b) is maximized by (S o, β). This proves the claim. Step 3 (final conclusion). Let Ω o = ω : lim n L n (S o, β) = L o (S o, β)} Ω ˆQ. It follows from the SLLN and Lemma 2 that P (Ω o ) = 1. Hereafter, fix an ω Ω o. By Lemma 3, given any subsequence n i } i 1 of n} n 1, there exist a futher subsequence, say itself such that Ŝn i (t u) S (t u) for each (t, u). Moreover, since ˆβ ni (, ), there is a subsequence which converges to b [, ]. WLOG, we can assume that Ŝn( ) S ( ) and ˆβ n b. Then lim L n(ŝn, ˆβ n ) log(ŝn(l u) Ŝn(r u))d ˆQ(l, r, u) n n = lim log(ŝn(l u) Ŝn(r u))d ˆQ(l, r, u)) (3.1) n lim log(ŝn(l u) Ŝn(r u))d ˆQ(l, r, u) (by Lemma 1, n as (1) ˆQ(η) Q(η) for every Borel subset η of = (l, r, u) : l < r, u (, )} and (2) log(ŝn(l u) Ŝn(r u)) 0). Consequently, L o (S o, β) lim L n(ŝn, ˆβ n ) (as L n (S o, β) L n (Ŝn, ˆβ n )) n lim log(ŝn(l u) Ŝn(r u))d ˆQ(l, r, u) (by (3.1)) n = log(s (l u) S (r u))dq(l, r, u) (as Ŝn(t u) S (t u)) = L o (S, b) L o (S o, b) (by the claim in Step 2), ω Ω o. Notice that P (Ω o ) = 1. Thus S = S o a.s. µ and b = β by the claim in Step Proofs Proof of Lemma 2. Given ϵ = 1/k, there exist x 1,..., x m, y 1,..., y m, z 1,..., z m (, ] such that maxp (L (x i 1, x i )), P (R (y i 1, y i ))P (U (z i 1, z i ))} < ϵ for i 1,..., m}, where x 0 = y 0 = z 0 = and x m = y m = z m =. Let Ω k be the event that max i,j,h ˆQ(x i, y j, z h ) Q(x i, y j, z h ) + ˆF L (x i ) F L (x i ) + ˆF R (y j ) F R (y j ) + ˆF U (z h ) F U (z h ) } 0. Since m is finite, P (Ω k ) = 1. Let ω Ω k. There exists n o such that ˆQ(x i, y j, z h ) Q(x i, y j, z h ) < ϵ, ˆQ(x i,, ) Q(x i,, ) < ϵ, ˆQ(, y j, ) Q(, y j, ) < ϵ, ˆQ(,, z h ) Q(,, z h ) < ϵ, i, j, h 0, 1, 2,..., m}.

6 6 Journal of Advanced Statistics, Vol. 4, No. 1, March 2019 Now (x, y, z), some (i, j, h) such that (x, y, z) (x i 1, x i ] (y j 1, y j ] (z h 1, z h ]. If (x, y, z) = (x i, y j, z h ), then ˆQ(x, y, z) Q(x, y, z) < ϵ. Otherwise, there are 12 disjoint cases such as: (1) x (x i 1, x i ), y (y j 1, y j ) and z (z h 1, z h ); (2) x (x i 1, x i ), y = y j and z (z h 1, z h ); (3) x (x i 1, x i ), y (y j 1, y j ) and z = z h ; (4) x (x i 1, x i ), y = y j and z = z h ; etc. In each of the 12 cases, we can also show that ˆQ(x, y, z) Q(x, y, z) < ϵ. For instance, in Case (1), ˆQ(x, y, z) Q(x, y, z) ˆQ(x i, y j, z h ) Q(x i 1, y j 1, z h 1 ) + Q(x i, y j, z h ) ˆQ(x i 1, y j 1, z h 1 ) ˆQ(x i, y j, z h ) Q(x i, y j, z h ) + Q(x i, y j, z h ) Q(x i 1, y j 1, z h 1 ) + ˆQ(x i 1, y j 1, z h 1 ) Q(x i 1,zh 1, y j 1, z h 1 ) + Q(x i 1, y j 1, z h 1 ) Q(x i, y j, z h ) ˆQ(x i, y j, z h ) Q(x i, y j, z h ) + }} Q(x i, y j, z h ) Q(x i 1, y j 1, z h 1 ) }} ϵ (F L (x i ) F L (x i 1))+(F R (y j ) F R (y j 1))+(F U (z h ) F U (z h 1 )) + ˆQ(x i 1, y j 1, z h 1 ) Q(x i 1, y j 1, z h 1 ) + Q(x i 1, y j 1, z h 1 ) Q(x i, y j, z j ) 8ϵ, (as ω Ω k ). In Case (4), ˆQ(x, y, z) Q(x, y, z) ˆQ(x i, y j, z h ) Q(x i 1, y j, z h ) + Q(x i, y j, z h ) ˆQ(x i 1, y j, z h ) ˆQ(x i, y j, z h ) Q(x i, y j, z h ) + Q(x i, y j, z h ) Q(x i 1, y j, z h ) + ˆQ(x i 1, y j, z h ) Q(x i 1,zh, y j, z h ) + Q(x i 1, y j, z h ) Q(x i, y j, z h ) ˆQ(x i, y j, z h ) Q(x i, y j, z h ) + Q(x i, y j, z h ) Q(x i 1, y j, z h ) }}}} ϵ (F L (x i ) F L (x i 1 )) 4ϵ. + ˆQ(x i 1, y j, z h ) Q(x i 1, y j, z h ) + Q(x i 1, y j, z h ) Q(x i, y j, z j ) The proofs for the other 10 cases are similar and are skipped. Notice ϵ = 1/k 0, P (Ω k ) = 1 and thus P ( k=1 Ω k) = 1. Proof of Lemma 3. Let u j } j 1 be the collection of all rational numbers. Since Ŝn(t u 1 ) is a sequence of bounded decreasing functions in t, by Helly s selection theorem, given any subsequence of n} n 1, there exists a further subsequence, say n 1i } i 1 such that Ŝn 1i (t u 1 ) converges for each t. Moreover, for j 2, there exists a subsequence, say n ji } i 1 of n (j 1)i } i 1 such that Ŝn ji (t u j ) converges for each t. It is easy to verify that Ŝn ii (t u j ) converges for each (t, u j ), say Ŝn ii (t u j ) S 1 (t u j ). In view of Eq. (2.2), write Ŝn(t u) = ˆβ n) Ŝ(u n (t) = exp( t 0 eu ˆβ n(s a)1(s>a) ĥ n (s)ds). Ŝ n (t u) is a bounded decreasing function of u ˆβ n. Thus S 1 (t u j ) is a bounded decreasing function of bu j. Since the set of rational numbers u j } j 1 is dense in (, ), S 1 (t u j ) can be extended to S 1 (t u) for each u (, ) by letting S 1 (t u) uj u S 1 (t u j ) if u is not a rational number. WLOG, we can assume b 0, then S 1 (t u) is a bounded decreasing function in u and is thus continuous except on a countable subset, say D. We shall show that S nii (t u) S 1 (t u) t and u / D. (A.1) It suffices to consider the case that u / D and u is not a rational number. Let w and v be two rationals such that w < u < v. Since the set of rationals are dense, w and v can be selected as close to u as possible. as i as v u Then S nii (t v) S nii (t u) S nii (t w). Verify that S nii (t v) S 1 (t v) S 1 (t u), as x = u is a continuous point of S(t x). Moreover, S nii (t w) S nii (t u) S 1 (t u). That is, (A.1) holds. as i S 1 (t w) as w u S 1 (t u). By the sandwich theorem,

7 Journal of Advanced Statistics, Vol. 4, No. 1, March Since D is countable, using the arguement similar to showing that Ŝn ii (t u j ) converges for each (t, u j ), we can show that there is a subsequence of n ii } i 1, say m k } k 1 n ii } i 1 such that Ŝm k (t u) converges for each u D. Consequently, Ŝm k (t u) converges for each (t, u). Acknowledgments. The authors thank the editor and a referee for their invaluable comments. References 1. Q.Q. Yu, and Q.G. Diao, The Proportional Hazards Model with linearly time-dependent covariates and interval-censored data", Journal of Advanced Statistics vol. 3, no. 4, 58-70, D.R. Cox, and D. Oakes, Analysis of Survival Data, Chapman & Hall, A. Schick, and Q.Q. Yu, Consistency of the GMLE with mixed case interval-censored data", Scandinavian Journal of Statistics, vol. 27, pp , P. Groeneboom, and J.A. Wellner, Information bounds and nonparametric maximum likelihood estimation, Birkhäuser Verlag, Basel, G.Y.C. Wong, and Q.Q. Yu, Estimation under the Lehmann regression model with interval-censored data", Communications in Statistics Simulation and Computation, vol. 41, , G.Y.C. Wong, Q.G. Diao, and Q.Q. Yu, Piecewise proportional hazards models with interval-censored data", Journal of Statistical Computation and Simulation, vol. 88, pp , H.L. Royden, Real Analysis, Second edition, New York: Macmillan, W. F. Eddy, and J. A. Hartigan, Uniform Convergence of the Empirical Distribution Function Over Convex Sets", The Annals of Statistics vol. 5, no. 2, pp , 1977.