The NPMLE in a class of doubly censored current status data models

The NPMLE in a class of doubly censored current status data models with application to AIDS partner studies Mark J. van der Laan and Chris Andrews Division of Biostatistics University of California Berkeley, CA 94720 [email protected] [email protected]

Summary The California Partners' Study is an ongoing investigation of heterosexual HIVtransmission in partners of infected index cases (Padian, et al., 1987; Shiboski & Jewell, 1990). In addition to the HIV-status of the partner at the recruiting time one also observes the initial time of the partnership and a lower bound for the infection time of the index case. Following Jewell, Malani & Vittingho (1994) we assume that the infection time of the index case is uniformly distributed over the interval de ned by the lower bound and the recruiting time, but no further assumptions are made. We consider an NPMLE of the distribution of the time T the partner is exposed to an infected index partner till infection. We show that the model is a doubly censored current status data model as introduced in Jewell, Malani & Vittingho (1994) with a special known distribution of the origin. We provide a modi ed Iterative Weighted 1

Pool Adjacent Violator Algorithm for computation of the NPMLE. It is shown that the NPMLE converges at rate n?1=3. In addition, we propose con dence intervals for smooth functionals of the distribution of T . Simulations show good performance of the algorithm, con dence intervals and provide a practical comparison of this NPMLE with the NPMLE if all partnerships are already in existence at the infection time of the index case as used in Shiboski & Jewell (1990). We apply our methodology to the California Partners' Study. We discuss the implications of our results for doubly censored current status data models with other known distributions of the origin. Keywords: AIDS Partner Study, Current Status Data, Iterative Weighted Pool Adjacent Violator Algorithm.

1 Doubly censored current status data In biostatistical applications interest often focuses on the estimation of the distribution of the time T between two consecutive events. If the initial event time I is observed and the subsequent event time J is only known to be larger or smaller than a monitoring (recruitment) time B , then the data (
= 1(T  B ? I ); B ? I ) is described by the well understood singly censored current status model. Singly censored current status data is also known as interval censored data, case I (Groeneboom, 1991). In this model it is assumed that B ? I is independent of T . The quali er \singly censored" serves to dierentiate this structure from the data structure given below. Previous work and examples of current status data can be found in Diamond, McDonald & Shah (1986), Jewell & Shiboski (1990), Diamond & McDonald (1991), and Keiding (1991). In this case, one observes n independent and identically distributed observations Yi = (Ci;
i), where Ci  Bi ? Ii and
i  1(Ji  Bi) = 1(Ti  Ci). We have

pr(
= 1 j C = c) = G(c) and pr(
= 0 j C = c) = 1 ? G(c): 2

(1)

Hence, if C has marginal cumulative distribution function H Lebesgue, then the density of the distribution PG of (C;
) w.r.t. H is given by G pG (c; )  dP dH (c; ) = G(c) + (1 ? G(c))(1 ? ):

Here H is unspeci ed, but it should be noticed that the model for H does not aect maximum likelihood estimation for G and it does not aect the information bound for smooth functionals of G either. Uniform consistency, pointwise convergence to a limit distribution at rate n?1=3 and eciency of smooth functionals of the nonparametric maximum likelihood estimator (NPMLE) Gn of G is established in Groeneboom & Wellner (1992). Alternative shorter proofs of eciency of the NPMLE of smooth functionals of G (e.g., its moments) are given by van de Geer (1994) and Huang & Wellner (1995). In van de Geer (1993) it is proved that the NPMLE Gn converges at rate n?1=3 in L2(H )-norm. Jewell, Malani & Vittingho (1994) extended this current status model by allowing the initial time I to be unobserved but known to be between an observed lower bound A and the monitoring time B , with its conditional distribution over the observed given interval [A; B ] known for each subject. In addition, one needs to assume that I is independent of T , given A; B , and T is independent of (A; B ). Thus for subject i one observes (
i = 1(Ji  Bi); Ai; Bi) and the model is determined by the known distributions Fi of Ii, given Ai; Bi, i = 1; : : :; n. Note that if Fi puts probability one at Ai, then this model is the singly censored current status data model above. For a given choice of Fi we will refer to the model as the Fi-doubly censored current status data model. A general algorithm for computing the NPMLE and asymptotic properties of this NPMLE have not been established in the literature. The uniform-doubly censored current status data model, which corresponds with assuming I , given A; B , is uniformly distributed over [A; B ], has been stressed by the above authors as an interesting special case. They have used it to analyze AIDS-partner study data. Consider this latter special case for the moment. If we de ne C = B ? A, 3

then by using that T is independent of I , given A; B , and T is independent of A; B we obtain for the uniform-doubly censored current status data model ZC pr(
= 1 j A; B ) = pr(
= 1 j C ) = F (C )  1 G(t)dt: (2) G

C

0

Thus for inference on G we can reduce the data to Y = (C;
). Let H be the unknown sampling cumulative distribution of C = B ? A. The density of the data Y = (C;
) w.r.t. H is

pG (c; ) = FG(c) + (1 ? FG(c))(1 ? ): Because FG is a distribution function this is a submodel of the (singly censored) current status data model. Because of the latter fact, van der Laan, Bickel & Jewell (1997) propose using the explicit regularized NPMLE, which happens to be a fraction of kernel density estimators, for the nonparametric current status data model to estimate FG and then use the relation G = (d=dx)(xFG (x)) to obtain an estimate of G. It is shown that this procedure yields ecient estimates of smooth functionals of G, in spite of the fact that it does not exploit the special structure of FG. Jewell, Malani & Vittingho (1994) used the uniform-doubly censored current status data model to analyze data from California Partners' Study. The California Partners' Study is an ongoing investigation of heterosexual HIV-transmission in partners of infected index cases (Padian, et al., 1987, Shiboski & Jewell, 1992). Participants are recruited through referrals from physicians, research studies, and local departments of public health. Upon entry, serum samples are drawn from study subjects to determine status with regard to HIV-infection of the subject and his/her partner. In addition, detailed medical, contraceptive, and behavioral histories are obtained. From the latter information one can obtain a lower bound for the infection time of the index partner. Let Iinf be the infection time of the index partner and let J be the infection time of the other partner. Let B be the random recruiting or monitoring time at which both 4

partners are tested on being HIV-positive. At time B we observe
 1(J  B ). Thus
= 1 corresponds with the event that both partners are infected while
= 0 if the second partner is not yet infected. Suppose that it is known that Iinf 2 [A0; B ] and that Iinf , given A0; B , is uniformly distributed. (The notation A0 will become useful in a moment.) Jewell, Malani & Vittingho (1994) studied estimation of the distribution of the time between Iinf and J based on observing n independent and identically distributed copies of the data structure (A0; B;
). Here it is implicitly assumed that the partnership was already in existence at the moment that the index partner is infected. They analyzed the data set consisting of the so called long-term partnerships where this assumption is true. However, for many partnerships in the California Partners' Study (the so called short-term partnerships) it is known that the partnership started at time A with A0 < A  B . In this case the initial time I of interest is the rst point in time at which the partners have contact and the index partner is infected. Now, the time-variable of interest is the transmission time T = J ? I . Note that if Iinf 2 [A0; A], then I = A, and if Iinf 2 [A0; B ], then I = Iinf . Thus I , given A0; A; B , has the following conditional distribution:

pr(I = A j A0; A; B )  p = BA??AA and pr(I 2 dx j A0; A; B ) = Bdx ?A for x 2 [A; B ]: 0

0

(3)

0

For each partnership we observe

Y = (A; B;
 1(J  B ); A0):

(4)

Thus we can view this application as doubly censored current status data (A; B;
), but where the conditional distribution of I over the interval [A; B ] is a mixture of a pointmass at A and a uniform distribution over [A; B ], where the pointmass is subject dependent (i.e., depends on A0 as well). In other words, the above model is a Fidoubly censored current status data model introduced in Jewell, Malani & Vittingho (1994), with Fi de ned by (3). We will refer to this model as the MPU-doubly censored 5

current status data model, where MPU is an abbreviation of \Mixture of Pointmass and Uniform". This practical example shows that it is important to have a good estimator available for a general (subject dependent) choice of known distribution Fi of Ii, given Ai; Bi, in the doubly censored current status data model. In fact, even if we restrict attention to long-term partnerships in the above example (so that Iinf = I ), then one might still have dierent information on the infection time Iinf of each subject so that the uniform distribution is not appropriate. In addition, it is of interest to understand what the eect of the choice of this conditional distribution is on the estimation problem of G. Estimation of G in the uniform-doubly censored current status data model occurs at rate n?1=5 (van der Laan, Bickel & Jewell, 1997) and if we let Fi have probability one at Ai (i.e., it is now simply current status data), then G is estimated at rate n?1=3. In the above example, Fi is a mixture of a pointmass at Ai and a uniform distribution. Note that the MPU-doubly censored current status data model covers the range of models between the uniform-doubly censored current status data model (where the distribution of the origin is non-informative) and the current status data model (where the origin is observed) by letting p in (3) vary from 0 to 1. It is of obvious interest to understand practically and theoretically how the pointmass in Fi aects the estimation problem for G; in particular, does it improve the rate n?1=5 for estimation of G? In this paper we will focus in detail on the NPMLE in the MPU-doubly censored current status data model for the data structure (4). The model has important practical applications and it covers an interesting range of Fi-doubly censored current status data models. In the discussion we will indicate the generalizations of our results to the general Fi-doubly censored current status data model. For example, our proposed algorithm for computation of the NPMLE can be applied to any Fi-doubly censored current status data model. Before we give the organization of this paper we will need to de ne the loglikelihood 6

and the NPMLE in the MPU-doubly censored current status data model. Under the assumption that I is independent of T , given (A; B; A0), and that T is independent of (A; B; A0) we have

pr(
= 1 j A; B; A0) = pr(J  B j A; B; A0) = pr(T  B ? I j A; B; A0) ZB pr(T  B ? x; I 2 dx j A; B; A0) = A ZB pr(T  B ? x j A; B; A0)pr(I 2 dx j A; B; A0) = A = pr(I = A j A; B; A0)pr(T  B ? A j A; B; A0) ZB + pr(T  B ? x j A; B; A0)pr(I 2 dx j A; B; A0) A ZB 0 A ? A 1 = B ? A0 G(B ? A) +Z B ? A0 A G(B ? x)dx 0 C = C C?0 C G(C ) + C10 G(y)dy (5) 0 where C  B ? A and C 0  B ? A0. Thus for inference on G we can reduce the data to (C; C 0;
). We de ne ZC 0 FG(C; C 0)  C C?0 C G(C ) + C1 0 0 G(y)dy: (6) Let H be the bivariate distribution of (C; C 0). The density of the data (C; C 0;
) is given by:

pG (c; c0; ) = fFG(c; c0)dH (c; c0)g f(1 ? FG(c; c0))dH (c; c0)g1? : Thus the likelihood of n i.i.d. observations is given by n Y L(G j Y1; : : :; Yn ) = fFG(Ci; Ci0)dH (Ci; Ci0)g
i f(1 ? FG(Ci; Ci0))dH (Ci ; Ci0)g1?
i : i=1

For convenience, we will rank the observations such that the Ci's are ordered; we will denote these ordered monitoring times with t1 < : : : < tn. Without loss of generality (as in Groeneboom & Wellner, 1992) we can make the convention that
1 = 1 (corresponding with the rst monitoring time C1 = t1 and
n = 0: otherwise replace 7

this set of ordered monitoring times by the largest inner set of monitoring times for which it is true. Let Gn be the maximizer of this likelihood over all discrete cumulative distribution functions G with jumps at 0 < t1 < : : : < tn?1. Existence of this NPMLE of G follows trivially from the fact that a continuous function on a compact set has a maximum. By our convention, this NPMLE is attained at a Gn with Gn (0) > 0 and Gn (tn?1) < 1. We remark that the true NPMLE of G can be obtained by maximizing over all discrete G with support contained in the set of all Ci; Ci0, i = 1; : : :; n, and the true NPMLE might be dierent from our sieved NPMLE Gn . Our proposed algorithm in section 2 also applies to the true NPMLE. In this paper we will prove theoretical results for the true NPMLE. However, in our simulations and data analysis we prefer our sieved NPMLE because it is easier to compute, it will be very close to the true NPMLE and it has the same asymptotic properties, though we will not formally prove the latter. The organization of the paper is as follows. In section 2 we discuss a general Iterative Weighted Pool Adjacent Violator Algorithm (IWPAVA) for minimizing a convex function under monotonicity constraints and apply it to our (negative) loglikelihood. In section 3 we rst prove that the NPMLE FGn converges in L2(H ) at rate n?1=3 to FG, using general theory for maximum likelihood estimation in van de Geer (1993). Subsequently, we will show that 1) if G is continuous, then Gn is uniformly consistent on the support of HC and 2) if for a proportion of the partnerships p > 0, then this implies that Gn converges in L2(HC ) to G at the same rate n?1=3. Here HC is the marginal distribution of C . The proofs of the theorems are deferred to the appendix. In section 4 we propose a simple to implement likelihood method for obtaining con dence intervals for asymptotically normally distributed smooth functionals of Gn . In section 5 we carry out a simulation study to test the performance of the algorithm, the NPMLE, the con dence intervals and we compare the NPMLE of G in the MPUdoubly censored current status data model for a range of values of p 2 [0; 1] with the 8

NPMLE in a uniform doubly censored current status data model and the NPMLE in the current status data model. We conclude the paper with a discussion.

2 Iterative Weighted Pool Adjacent Violator Algorithm 2.1 The modi ed IWPAVA to minimize a convex function under order restrictions From reading Barlow, et al. (1972), Groeneboom & Wellner (1992) and Jongbloed (1995) one learns that the maximizer of a convex function (e.g., negative loglikelihood) over the set of all cumulative distribution functions can be computed with a (possibly modi ed) IWPAVA or equivalently with a (possibly modi ed) iterative convex minorant algorithm. In this subsection we state this general result and in the following subsection we apply it to our negative loglikelihood in the MPU-doubly censored current status data model. An elegant proof of the following theorem is the proof of proposition 1.1 in Groeneboom & Wellner (1992) (see also Jongbloed, 1995).

Theorem 2.1 Let  : IRn ! IR be a convex function and let A = f~x : 0 < x1 


 xn < 1g. Let t1 <


< tn be a given set of time-points. For a given cumulative distribution function G we de ne (G)  (G(t1 ); : : :; G(tn)). Let j (G)  (d=dxj )(~x)j~x=G be the j -th partial derivative of  at G. Then Gn = (Gn (t1); : : :; Gn (tn)) minimizes  over A if and only if

X

j i n X j =1

j (Gn )  0 for i = 1; : : :; n and j (Gn )Gn(tj ) = 0:

(7)

If  is strictly convex on A, then  has a unique minimum Gn .

9

An immediate corollary of theorem 2.1 is the following.

Theorem 2.2 Let , j , A, and t1; : : :; tn be as in Theorem 2.1

We have that Gn = (Gn (t1); : : : ; Gn(tn )) minimizes  over A if and only if Gn minimizes G ! (G j Gn ) over A where is de ned by

G!

n X

j =1

fG(tj ) ? Yj;Gn g2wj;Gn ;

YGn ;j = wj (Gn ) + Gn (tj ) j;Gn

and wj;Gn , j = 1; : : : ; n, is an arbitrary sequence of positive numbers.

This theorem is proved by applying theorem 2.1 to and noting that it gives the same characterizing inequalities and equality constraints given in (7). For a given estimator G0n the isotonic regression problem, i.e., minimizing (
j G0n ) over A, is solved with the simple and fast WPAVA (Barlow, et al., 1972). Therefore theorem 2.2 suggests the following IWPAVA: Set G0n equal to an initial estimator and iterate Gkn+1 = min?A1 (
j Gkn ), k = 0; 1; : : :. This algorithm corresponds with the iterative convex minorant algorithm proposed in Groeneboom & Wellner (1992) for interval censored data who highlight the fast practical performance. They suggest setting wj;G to be equal to the diagonal elements of the matrix of second order partial derivatives of . In this way the algorithm can be viewed as an approximation of the Newton-Raphson algorithm because each step corresponds with minimizing the second order Taylor approximation of  at Gkn , but where all the o diagonal elements of the matrix of second order derivatives of  are set equal to zero (Jongbloed 1995, section 2.3). The IWPAVA also converged very fast in the current status data type of application considered in van der Laan, Jewell & Peterson (1997). Jongbloed (1995) considers the convergence properties of the iterative convex minorant algorithm and thus of the IWPAVA. He shows that if the above algorithm converges, then it converges to the maximum. However, he also points out that there 10

is no guarantee that the algorithm converges. In fact, it appears that in our model the algorithm did not converge in the rst example we ran. Jongbloed (1995) shows that the following modi cation of the algorithm guarantees the desired convergence. Step 1: Set G0n equal to an initial estimator and let  2 (0; 0:5) be given. Step 2: Gnk+1; = min?A1 (
j Gkn). Step 3: If (Gnk+1; ) < (Gkn ) + (1 ? )d(Gkn )(Gnk+1; ? Gkn ), then set Gkn+1 = Gnk+1;. Else nd the minimizer n of  ! (Gkn + (1 ? )Gnk+1; ) and set Gkn+1 = n Gnk+1; + (1 ? n )Gkn . Step 4 Repeat step 2 and 3 till convergence. Because of practical reasons Jongbloed (1995) proposes the following economical alternative to step 3 in the above algorithm and proves the convergence still holds. Step 3': If (Gnk+1; ) < (Gkn ) + (1 ? )d(Gkn )(Gnk+1; ? Gkn ), then set Gkn+1 = Gnk+1;. Else nd  such that G  Gnk+1; + (1 ? )Gkn satis es: (1 ? )d(Gkn )(G ? Gkn )  (G) ? (Gkn )  d(Gkn )(G ? Gkn ):

(8)

Set Gkn+1 = G. An appropriate G can be found with the following binary search: Initialize low = 0 and high = 1. For  = (low + high)=2, if the left inequality in expression (8) fails set low = ; if the right inequality fails set high = . Repeat until both inequalities in (8) hold.

2.2 Application to the MPU-doubly censored current status data model Recall that we ordered the observations (Ci; Ci0;
i) in the sense that C1 <


< Cn and that we made the convention
1 = 1 and
n = 0. Let C0  0. The negative loglikelihood is given by: n X (G) = ? log(FG(Ci; Ci0))
i + log(F G (Ci; Ci0))(1 ?
i) i=1

11

where F G  1 ? FG. We restrict  to a function on A = f(G(C0); : : :; G(Cn?1 )) : 0 < G(C0) 


 G(Cn?1 ) < 1)g where an element of A corresponds with a step function G with jumps at C0 < : : : < Cn?1 . Note this proposed NPMLE minimizes  over A so we can apply the general modi ed IWPAVA given in the previous section. We provide the expressions for the rst and second order partial derivatives of  with respect to xj = G(Cj ), j = 0; : : : ; n ? 1. First note, for a cumulative distribution function G in A we can write 0 Xi F (C ; C 0) = Ci ? Ci x + 1 x (C ? C ); i = 0; : : : ; n:

Ci0 k=1 k?1 k k?1 Thus for j = 0; : : :; n ? 1, and i = 1; : : :; n, ) ( d F (C ; C 0) = 1(j  i) 1(j = i) Ci0 ? Ci + 1(j < i) Cj+1 ? Cj : dxj G i i Ci0 Ci0 The derivative with respect to xj of  is given by ! n X
d F (C ; C 0) 1 ?
i i j (G) = ? ? G i i 0 0 i=1 FG (Ci ; Ci ) F G (Ci ; Ci ) ! dxj n X
i ? 1 ?
i (9) = ? 0 0 i=j FG (Ci ; Ci ) F G (Ci ; Ci ) ( ) 0 ? Ci C C j +1 ? Cj i  1(j = i) C 0 + 1(j < i) C 0 : i i We will set the weights wj of the IWPAVA equal to the diagonal elements of the matrix of second order derivatives of . Because (d2=dx2j )FG = 0 we have 2 d wj;G  dx2 (~x) j ~x=G ! n X
1 ?
i i (10) = 0 2 + F G (Ci ; C 0)2 i i=j FG (Ci ; Ci ) ( )2 0 ? Ci C C j +1 ? Cj i  1(j = i) C 0 + 1(j < i) C 0 i i The NPMLE Gn can now be computed with the modi ed IWPAVA given in the G

i

i

i

Ci0

preceding subsection with k Y k = j (Gn ) + Gk (C ) Gn ;j

wj;Gkn

n

j

12

and j (Gkn ), wj;Gkn de ned in (9) and (10), respectively. This algorithm with Step 3' has been implemented by us and as predicted by theory (Jongbloed, 1995) it has converged quickly in each simulation we have run.

3 L2-consistency and uniform consistency of the NPMLE

Theorem 3.1 Assume that pr(C 0 < M ) = 1 for an M < 1 and R R y?1dH (x; y) < 1. Let Gn be the NPMLE of G in the MPU-doubly censored current status data model. Then

kFGn ? FGkH 

sZ

(FGn ? FG )2(x; y)dH (x; y) = OP (n?1=3):

The following results translate this consistency result for FGn into consistency results for Gn . The proofs require a study of the relation between FG and G. We have the following simple direct relation between G and FG:

G(x) = dyd (yFG(x; y)):

Because the right-hand side does not depend on y we also have for any conditional distribution H1 of y, given x, with y  x: Z1 d G(x) = x dy (yFG(x; y))dH1(y j x):

Alternatively, we have by integration by parts for any x < y: Zx Zx y y G(x) = y ? s FG(ds; y) = y ? x FG (x; y) ? 0 F (s; y) (y ?y s)2 ds 0

and thus also for any conditional distribution H1 (
j x): Z 1( y G(x) = x y ? x FG(x; y)? ) Zx y Zx y  0 F (s; y) (y ? s)2 ds 0 y ? s F (ds; y) dH1 (y j x): The following theorem establishes uniform consistency of Gn . 13

(11)

Theorem 3.2 Assume pr(C 0 < M ) = 1 for an M < 1, R R y?1dH (x; y) < 1, dHC (x) = hC (x)dx and that G is continuous. If hC > 0 on [0;  ), then supx2[0; ) j Gn (x) ? G(x) j! 0 almost surely. We can also use theorem 3.1 to prove the following rate result for the NPMLE. Theorem 3.3 Assume (1) pr(C 0 < M ) = 1 for an M < 1, (2) R R y?1dH (x; y) < 1, (3) there exists  > 0 such that for some  > 0 H (x +  j x)  pr(C 0 > x +  j C = x) >  > 0 for FC -almost every x, (4) for this same  > 0 we have that the conditional distribution of C , given C 0 , is absolute continuous with respect to Lebesgue RR measure on [0; C 0 ? ) with density h(
j C 0), (5) y>x+ M 2 (x; y)dH (x; y) < 1, where M (x; y)  Rxy? h(s j y)ds=h(x j y). Let Gn be the NPMLE of G. Then sZ kGn ? GkHC  (Gn ? G)2(x)dHC (x) = OP (n?1=3): Condition (3) excludes the case that pr(C 0 = C ) = 1 which corresponds with the uniform doubly censored current status data model, where Gn converges at rate n?1=5. Condition (4) says that either C = C 0 or C is distributed continuously over [0; C 0). This is the kind of condition one expects to hold in the California Partners' Study; if the partnership is already in existence at infection of the index case, then C = C 0, but otherwise one expects C to be continuously distributed over [0; C 0).

4 Con dence intervals for smooth functionals In spite of the fact that in each Fi-doubly censored current status data model G is not root-n estimable, smooth functionals of the NPMLE Gn are typically root-n-consistent and asymptotically normal. In van der Laan, Bickel & Jewell (1997) it is shown that a large class of functionals of the form  = R r(t)(1 ? G(t))dt, for a given r, are root-n-estimable in the uniform-doubly censored current status data model and thus 14

certainly in the more informative MPU-doubly censored current status data model. In this section we provide a method for obtaining con dence intervals for such smooth functionals based on the NPMLE Gn . Let 0 = t0 <


< tk be the support points of the NPMLE Gn . Consider the parametric submodel of the nonparametric MPU-doubly censored current status data model, which only allows step functions G with support points given by 0 = t0 <


< tk . De ne j  G(tj ), j = 0; : : :; k. A density p in this model is indexed by the multivariate parameter ~ with components j  G(tj ), j = 0; : : : ; k, and the parameter space is given by   f~ : 0 


 k g. De ne ~n  (Gn (t0); : : :; Gn (tk )). Then ~n is the MLE in this parametric model and ~n is in the interior of the parameter space . Though the true G is not an element of this model, ~n can be viewed as the MLE of ~ where ~ maximizes R log(p~)(y)dPG (y). For a xed (in tj ) parametric model, under regularity conditions, the standardized MLE n1=2(~n ? ~) is asymptotically normal with its limit covariance matrix equal to the information matrix I (~). This fact inspires us to obtain con dence intervals for root-n-estimable smooth functionals of the true G by proceeding as if this parametric model is the true model. Before we proceed we remark that the well known Greenwood's formula for the estimate of the variance of the Kaplan-Meier estimator can be derived as the parametric information bound corresponding with the model only allowing discrete cumulative distribution function with the same support as the Kaplan-Meier estimator. Thus we are just stating that this method is applicable in great generality as long as one applies it to root-n-estimable parameters. Though formally proving this is still quite a challenging task which we do not pursue here. Let  be a smooth real valued functional of G and let n be the NPMLE of  obtained by substitution of Gn . Suppose that n1=2(n ? ) is asymptotically normally distributed. If we act as if the true G is an element of the parametric model, and can thus be identi ed by a vector ~, then we can write  = (~). Because Gn is the 15

MLE in the parametric model we have that n = (~n). In the xed model, under weak regularity conditions, parametric theory says that n1=2(n ? ) is asymptotically normal with mean zero and variance 2, where 2 can be consistently estimated with _ n)I^(~n)?1 ( _  n )> ; ^ 2 = ( _ ) = ((d=d0 ) (~); : : :; (d=dk ) (~)) is the gradient of at  and I^(~n) is the where ( observed information matrix. Thus in the xed parametric model one would use the 95% asymptotic con dence interval for  given by n  1:96^ n?1=2. In spite of the fact that the true G is certainly not an element of the parametric model and that the parametric model is not xed, we conjecture that this method yields asymptotically correct con dence intervals for root-n-estimable parameters of G. We oer the following heuristic argument. First, Gn can be viewed as an MLE of the Kullback-Leibner projection Gd , corresponding with ~ as de ned above, of the true G. For a xed (in tj ) model, parametric theory provides a normal limit distribution with covariance matrix I (~) of n1=2(n ? d ), where d corresponds with this projection Gd of G. Under appropriate conditions, one expects that this normal approximation holds uniformly in the sequence of models implied by the support points of the NPMLE Gn . In addition, because n converges to  at rate n?1=2 one should also have that the standardized approximation error n1=2(d ? ) of this parametric model converges to zero. To make our method concrete it remains to specify ^ . First, we will parameterize our likelihood by ~. Let j (i) 2 f0; : : : ; kg be de ned by the requirement Ci 2 [tj(i); tj(i)+1), i = 1; : : :; n. For a step-function G on the support points 0 = t0 <


< tk we have for i = 1; : : : ; n.

FG(Ci; Ci0) = F~(Ci; Ci0) 8j(i) 9 0 ? Ci < = X C 1  i C 0 j(i) + C 0 : l?1(tl ? tl?1) + j(i)(Ci ? tj(i)); : i i l=1 16

The parametric loglikelihood indexed by the parameter ~ = (0; : : : ; k ) is given by n     X (~) = n1 log F~(Ci; Ci0)
i + log F ~(Ci; Ci0) (1 ?
i): i=1

In the parametric model the information matrix I (~) can be consistently estimated with the observed information matrix I^p;p (~n)  ? dpdd2 p (~). Straightforward algebra shows 0 1 n X 1
1 ?
i i A I^p;p (~n) = n @ F 2 (C ; C 0) + 2 0 i F ( C ; C ) i ~n i=1 ~n i i d d  d F~(Ci; Ci0) d F~(Ci; Ci0) ; p ~=~n ~=~n p 0

0

0

0

where

d F (C ; C 0) = 1(C 2 [t ; t )) Ci0 ? tp + 1(C  t ) tp+1 ? tp : i p p+1 i p+1 dp ~ i i Ci0 Ci0 Note one obtains I^(~n) immediately at the last step of the modi ed IWPAVA for computing Gn (and thus ~n) described in section 2. As a consequence one obtains these con dence intervals for smooth functionals of G for free after having computed Gn .

This method has been implemented and we show the results in the simulation section.

5 Simulation study 5.1 General remarks The asymptotic results presented in this paper are applicable to moderate and even small size datasets. For several sample sizes we compare the performance of the NPMLEs of the MPU-model, the Uniform-model, and the singly censored current status data (CSD) model. Performance is measured by convergence of G^ to G in L1-norm, 17

convergence of (G^ ) to (G) for a smooth functional , and the length and coverage probabilities of con dence intervals of . The MPU model is intermediate to the CSD model and the Uniform model. For p (see equation (3)) near 1, the MPU model is similar to the CSD model; for p near 0, the MPU model is essentially the Uniform model. In each simulation we compare four MPU models corresponding to p = 1=10, 1=3, 2=3, and 9=10 with each other and to the Uniform and CSD models. In each study the true distribution, G, of T is exponential with mean 1. For each of three sample sizes (400, 2000, 10000) we generated C  Uniform(0; 4). For each of the four MPU models we let C 0 be such that (C 0 ? C )=C 0 = p. Finally for each of the six models we generated
jC; C 0 according to equation (1), (2), or (5) as appropriate. The results of the simulations are comparable across models because the marginal density of C is constant across the models. This generation procedure was repeated 1000 times. A typical result (Figure 1) shows the estimator of G based on singly censored CSD is closer to the truth and has more support points than the MPU-based estimator, which in turn outperforms the Uniform-based estimator.

5.2 Comparison of Eciencies of NPMLEs The asymptotic relative eciency of the NPMLE in the MPU model compared to the NPMLE in the Uniform model is in nity because the latter estimator converges at a slower rate (n1=5 rather than n1=3). Thus in nite samples the relative eciency will increase as the sample size increases. Asymptotically the rate at which the relative eciency tends to in nity is n(1=3)?(1=5) = n2=15. We measure the overall eciency of the six estimators by the average (L1) distance from the true distribution function. The Integrated Absolute Deviation (IAD) is Z X jG^ n (x) ? G(x)jdH (x)  jG^ n(ci) ? G(ci )j; 0

i

18

where H is the distribution of C which is the same in each model. The eciency in this L1-norm is a measure of overall performance of the estimator weighted by the density of the sampling distribution (Table 1). The rst set of three rows shows increasing eciency of MPU-based estimator relative to Uniform-based estimator as sample size increases. The eect is larger in columns on right where the MPU is similar to CSD. The second set of three rows shows the eciency of the MPU-based estimator relative to the CSD-based estimator is nearly constant as the sample size increases re ecting that the estimators converge at the same rate (n?1=3). The relative eciency is always less than unity but converges to 1 as p ! 1.

5.3 Comparison of Eciencies of Smooth Functionals

All six estimators of smooth functionals,  = R R(t)dG(t), of G converge at rate n?1=2. Thus we should not expect to observe a diverging eciency of the MPU-based estimators relative to the Uniform-based estimator exhibited in the previous section. Although we might wish to estimate the mean of the distribution G, which corresponds to R(t) = t, this functional does not have nite variance in the Uniform-model when the support of G is unbounded (van der Laan, Bickel & Jewell, 1997). Thus we use the following smoothed truncated mean: 8 > > t t  1 > <   R(t) = 12+t + 22?1 sin  t2??11 1  t  2 (12) > > 1 +2 : 2  t 2 An obvious choice for 2 is 4 because we observe data up to but not beyond 4. The choice for 1 is somewhat arbitrary as it determines only the sharpness of the truncation at 2; we chose 1 = 2. Of course this somewhat elaborate functional is not necessary if the support of G is nite and we observe data from the entire support of G. 19

In addition to the IAD computed above in the rst simulation, we computed ^ = (G^ ) = R R(t)dG^ (t). The empirical standard deviations of ^ of the six estimators can be used to estimate the eciencies of the estimators relative to each other. The eciencies of the estimators based on MPU-doubly censored current status data relative to the CSD-based estimator are reported in Table 2 for sample sizes 400, 2000 and 10000. We note that the relative eciencies for n = 10000 will be very close approximations to the asymptotic relative eciencies. It if of interest to note that for p = 1=3 the information of a MPU-observation (where the origin is only known with a probability 1/3) already represents 77% in the limit and 90% for sample sizes under 2000 of the information in a current status observation (where the origin is known). This is a remarkably positive result for AIDS-partner studies.

5.4 Con dence intervals for smooth functionals We now report the practical performance of the con dence intervals for smooth funcR tionals proposed in section 4. Based on each ^ = R(t)dG^ (t) computed above, we constructed 95% and 50% con dence intervals for Z ?2 ?1  = R(t)dG(t) = 1 ? 2(1e + (+2e?1 )2)  0:945:  The estimated coverage probabilities (Table 3) show the nominal rate is achieved for nite samples in each of the six models. As expected, the length of the con dence intervals (Table 4) is shortest for the CSD-based estimator, longest for the Uniformbased estimator, and intermediate for the MPU-based estimators.

6 Data Analysis As described previously the California Partners' Study has the basic MPU data structure. We computed the NPMLE of G and from it a con dence interval for a smoothed 20

truncated mean (1 = 100, 2 = 165) for the entire dataset and several interesting subgroups (Table 5). We note the con dence intervals for the STD and No STD groups do not overlap, indicating a history of sexually transmitted disease in the female partner is a signi cant factor in the transmission of HIV.

7 Discussion We de ned the class of MPU-doubly censored current status data models which accounts for an important feature in AIDS-partner studies. Furthermore, as the pointmass p ranges from 0 to 1 the MPU model represents an interesting subclass of Fi-doubly censored current status data models ranging from the completely noninformative uniform-doubly censored current status data model to the singly-censored current status data model as other extreme. We proposed to use a fast IWPAVA for computing the NPMLE which yields con dence intervals for smooth functionals of the NPMLE for free at its last step. It is clear that this algorithm and the con dence intervals immediately generalize to any Fi-doubly censored current status data model. We proved that the NPMLE in the MPU-doubly censored current status data model with p > 0 converges at rate n?1=3 while for p = 0 it is known to converge at rate n?1=5. It will be no surprise that we believe that this rate n?1=5 will hold in any Fi-doubly censored current status data model with Fi continuous. We also conjecture that if Fi has a pointmass somewhere in [Ai; Bi], then the NPMLE will converge at rate n?1=3 as in our MPU-doubly censored current status data model. In practice this means that if Fi is quite informative in the sense that most of the distribution is concentrated in a small interval, then there will be a signi cant bene t for the purpose of estimation of G. Our simulations have shown that the n1=2-standardized smooth functionals of the NPMLE are asymptotically normally distributed and that the proposed asymptotic con dence intervals perform well. We believe that these ndings will hold for any Fi-doubly censored current status data model. 21

We also used simulations to compare for various values of p the information in the MPU-doubly censored current status data model for estimation of G and smooth functionals of G. The results give practical insight in the gain from having a pointmass in the Fi-doubly censored current status data model; in particular, it shows that reasonable values of p imply a signi cant gain relative to the uniform-doubly censored current status data model and it brings one already close to the singly-censored current status data model where the origin is known (p = 1). An important practical implication of this nding in AIDS-partner studies is that it is bene cial to collect information from the subjects on the distribution of the infection time. We conclude that our results and simulations for the MPU-model also provides methodology for and sheds light on the important Fi-doubly censored current status data model for general Fi.

Acknowledgments This research was supported by a FIRST award from the National Institute of General Medical Sciences, National Institute of Health and a National Science Foundation Mathematical Sciences Division Post-Doctoral Research Grant.

Appendix A: Proof of theorem 3.1 Our proof of theorem 3.1 follows closely the proof of van de Geer (1993) that the Hellinger-consistency rate is n?1=3 for the nonparametric current status data NPMLE (see example 4.8(a), page 35, van de Geer, 1993). For that purpose we will rst change our notation to her setting. For a cumulative distribution function G we de ne KG (x; y)  yFG(x; y) = (y ? x)G(x)+ R0x G(y)dy. We will parameterize the distributions in our model with  = KG so that the parameter space is given by  = fKG : G cumulative distribution functiong. Thus the model is given by fP :  2 g, where the bivariate cumulative distribution function H of 22

(C; C 0) is considered xed. Let  = 1   with d1(x; y) = y?1dH (x; y) and  the counting measure on f0; 1g. We note that  is a nite (by assumption) dominating measure for this model. For  2  we de ne p  dP =d as the density w.r.t. :

p (c; c0; ) = (c; c0) (c0 ? (c; c0))1? : The NPMLE of the true 0 is de ned by: Z ? 1 n = max2 log(p (c; c0; ))dPn(c; c0; ); where Pn is the empirical distribution function. Because the loglikelihood only depends on FG(Ci; Ci0), i = 1; : : : ; n, it will typically be the case that the maximum over all G is attained by a step function (piecewise constant) G with jumps at a Ci or Ci0, i = 1; : : : ; n. We have that n = KGn , where Gn is the NPMLE of G. Let Z s dPn s dP !2 1 2 H (Pn ; P )  2 d ? d d be the Hellinger-norm between Pn and P squared. Suppose that

H (Pn ; P ) = OP (n?1=3); (13) R R Because (n1=2 ? 1=2)2dH (x; y)=y  H 2(Pn ; P ) this implies that (n1=2 ? 1=2)2dH (x; y)=y = OP (n?2=3). Now, note that R (FG1=n2 ? FG1=2)2dH (x; y) = R (n1=2 ? 1=2)2dH (x; y)=y = OP (n?1=3). Finally, Z Z q q q q 2 (FGn ? FG) dH = ( FGn ? FG)2( FGn + FG)2dH Z q q  4 ( FGn ? FG)2dH = n?2=3: Thus for proving the theorem it suces to prove (13). We are in the situation that we can apply the proof of van de Geer (1993, example 4.8a) to our model. First, the map  ! p is linear and  is convex. Note that p (x; y) = KG (x; y) is a known function y ? x times a cumulative distribution function 23

G plus a convex function R0x G(s)ds bounded on the support of H . The entropy for the set p1=2 :  2 g is thus dominated by the entropy of the class f(y ? x)G(x) : G cumulative distribution functiong. Because y ? x is a known xed bounded function on the support of H the entropy of this class is of the same order as the entropy of the class fG : G cumulative distribution functiong. Because this is contained in the class of monotone bounded functions we have the following entropy bound:

p HB (; fpp :  2 g; k
kP )  const 

uniformly in P , and also uniformly in any measure bounded by a xed constant (Birman & Solomak, 1967). Because  is nite, we thus have this bound for the entropy of fp :  2 g endowed with Hellinger metric to our disposal (see van de Geer, 1993):

p : HB (; fpp :  2 g; k
k )  const 

Furthermore, again because  is nite (i.e., p?01=2 is P0-square integrable) the same bound is valid for Gu = f(p =pu; )1=2 :  2 g, fu; = up + (1 ? u)p0 , equipped with the empirical metric k
kPn (see van de Geer, 1993, for the de nition). Now, application of theorem 4.5 of van de Geer (1993) yields H (Pn ; P ) = OP (n?1=3).

Appendix B: Proof of theorem 3.2 By Helly's selection theorem we have that Gn has a subsequence Gnk for which Gnk (x) ! G1 (x) for some non-decreasing function between 0 and 1 at every continuity point x of G1 . Consider now the term R j FGnk ? FG j (x; y)dH (x; y). From the de nition of FG (equation (6)) and the triangle inequality, this term is bounded by Z j Gnk ? G1 j (x) y ?y x dH (y j x)dHX (x) Z Z x + y1 (Gnk ? G1 )(s)ds dH (y j x)dHX (x): 0 1

24

The rst term can be written Z Z 1 y ?x y dH (y j x) j Gnk ? G1 j (x)dHX (x): x

It converges to zero by the dominated convergence theorem and the fact that Rx1(y ? x)y?1dH (y j x)  1. The integrand of the second term converges to zero for each (x; y) and is uniformly bounded by 1. Again, the dominated convergence theorem implies the R second term converges to zero as well. Thus j FGnk ? FG j dH ! 0. By theorem 3.1 we also know that R j FGnk ? FG j dH ! 0. Therefore R j FG ? FG j dH = 0, which implies FG = FG on the support of H . By the relation (11) this implies that G1 = G on Lebesgue-a.e. on [0;  ] if h > 0 on [0;  ]. Continuity of G implies continuity of G1 . As a consequence, we have G1 (x) = G(x) for each x 2 [0;  ]. So Gnk (x) ! G(x) for each x 2 [0;  ]. Because G is continuous and Gnk is monotone this proves that Gnk converges uniformly to G on [0;  ]. Finally, because we could have applied this proof to any subsequence of Gn (instead of the original sequence Gn ) we have proved that each subsequence of Gn has a uniformly convergent subsequence. Thus, Gn converges uniformly to G. 1

1

1

Appendix C: Proof of theorem 3.3 Let X = C and Y = C 0. Let dH1 (y j x)  dH (y j x)=H (x +  j x) on [x + ; 1) and zero on [x; x + ). Note that by condition (3) dH1(
j y) is not zero everywhere and is thus an actual conditional distribution function. Consider relation (11) with this choice of H1. Because y > x +  and y < M (condition (1)) it follows that Z j Gn ? G j (x)dHX (x) Z Z y = j y ? x (Fn ? F )(x; y)dH1(y j x) j dHX (x) Z Zx ? j 0 (Fn ? F )(s; y) (y ?y s)2 ds j dH1(y j x)dHX (x) ZZ y  y ? x j Fn ? F j (x; y)dH1(y j x)dHX (x) 25

+

Z Z Zx 0

j Fn ? F j (s; y) (y ?y s)2 dsdH1(y j x)dHX (x)

ZZ M   yx+ j Fn ? F j (x; y)dH (y j x)dHX (x) Zx MZ 2 +  j Fn ? F j (s; y)dsdH (y j x)dHX (x): yx+ 2 0

The rst term is OP (n?1=3) by theorem 3.1. By Fubini's theorem, condition (4) and dH (y j x)dHX (x) = dH (x j y)dHY (y) the second integral can be rewritten as Z Z y? j Fn ? F j (s; y)M (s; y)h(s j y)dsdHY (y): (14) s=0

For any pair of functions g; h we have Z g(x)h(x)d(x)  kgk;2khk;2  kgk1kgk;1khk;2;

(15)

where k
k;j denote the Lj ()-norm, j = 1; 2. Application of (15) bounds R0y? j Fn ? F j (s; y)M (s; y)h(s j y)ds by 1=2 Z y? Z y? 2 j Fn ? F j (s; y)h(s j y)ds: M (s; y)h(s j y)ds 2 0 0 R Represent the right-hand side as M1(y) 0y? j Fn ? F j (s; y)h(s j y)ds. Another application of (15), with g = R0y? j Fn ? F j (s; y)h(s j y)ds, h = M1 and d = dHY , R bounds M1(y) 0y? j Fn ? F j (s; y)h(s j y)ds by Z Z y? j Fn ? F j (s; y)h(s j y)dsdHY (y); 2kM1kHY ;2 0

where kM1kHY ;2 = kM kH;2 < 1 by condition (4). This proves that the second term (14) is bounded by R R kFn ? F j dH = OP (n?1=3) by theorem 3.1.

References Barlow, R. E., Bartholomew, D.J., Bremner, J.M. & Brunk, H.D. (1972). Statistical Inference Under Order Restrictions , New York, John Wiley. 26

Birman, M.S. & Solomak, M.Z. (1967). Piecewise-polynomial approximations of functions of the classes Wp. Mat. Sb., 73, 295{317. Diamond, I.D., McDonald, J.W. & Shah, I.H. (1986). Proportional hazards models for current status data: Application to the study of dierentials in age at weaning in Pakistan. Demography 23, 607{620. Diamond, I.D. & McDonald, J.W. (1991), The analysis of current status data. Demographic Applications of Event History Analysis, J. Trussell, R. Hankinson, & J. Tilton (eds.), Oxford: Oxford University Press. van de Geer, S. (1993). Hellinger-consistency of certain nonparametric maximum likelihood estimators. Ann. Statist. 21, 14{44. van de Geer, S. (1994). Asymptotic normality in mixture models, preprint University of Leiden, the Netherlands. Groeneboom, P. (1991). Nonparametric maximum likelihood estimators for interval censoring and deconvolution, Technical report nr. 378, Department of Statistics, Stanford University, California. Groeneboom, P. & Wellner, J.A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation, Boston, Birkhauser. Huang, J. & Wellner, J.A. (1995). Asymptotic normality of the NPMLE of linear functionals for interval censored data, case I. Statistica Neerlandica 49, 153{163. Homann-Jrgensen, J. (1984). Stochastic processes on Polish Spaces, Unpublished manuscript. Jewell, N.P. & Shiboski, S.C. (1990). Statistical analysis of HIV infectivity based on partner studies. Biometrics 46 1133{1150. Jewell, N.P. & Malani, H.M. & Vittingho, E. (1994). Nonparametric estimation for a form of doubly censored data with application to two problems in aids. J. Amer. Statist. Assoc. 89 7{18.

27

Jongbloed, G. (1995), Three statistical inverse problems, Ph.D. dissertation, Delft University of Technology, Delft. Keiding, K. (1991). Age-speci c incidence and prevalence (with discussion). J. R. Statist. Soc. A 154, 371{412. van der Laan, M.J., Bickel, P.J. & Jewell, N.P. (1997), Singly and doubly censored current status data; estimation, asymptotics and regression. Scand. J. Statist. 24, 289{308. van der Laan, M.J., Jewell, N.P. & Peterson, D.R. (1997). Ecient estimation of the lifetime and disease onset distribution. Biometrika, 84, 539{554. Padian, N., Marquis, L., Francis, D.P., Anderson, R.E., Rutherford, G.W., O'Malley, P.M. & Winkelstein, W. (1987). Male-to-Female transmission of Human Immunode ciency Virus. J. American Medical Assoc., 258, 788{790. Shiboski, S.C. & Jewell, N.P. (1990). Statistical Analysis of the time dependence of HIV infectivity based on partner study data. J. Amer. Statist. Assoc., 87, 360{372.

28

Relative Eciency of

Sample Size 400 MPU to Uniform 2000 10000 400 MPU to CSD 2000 10000

1=10 1.0 1.0 1.2 0.59 0.51 0.48

1=3 1.1 1.1 1.3 0.61 0.56 0.53

p

2=3 1.4 1.5 1.8 0.81 0.77 0.72

9=10 1.7 1.8 2.2 0.93 0.91 0.88

Table 1: Comparison of Integrated Absolute Deviations. First set of three rows shows eciency of MPU-based estimator relative to Uniform-based estimator. Second set of three rows shows the eciency of the MPU-based estimator relative to the CSD-based estimator.

29

Sample p Size Uniform 1=10 1=3 2=3 400 0.72 0.75 0.85 0.92 2000 0.72 0.70 0.81 0.98 10000 0.60 0.73 0.77 0.90

9=10 0.95 1.02 0.96

CSD 1.00 1.00 1.00

Table 2: Comparison of Standard Errors of a Smooth Functional. Table displays the eciency of the MPU-based estimator relative to the CSD-based estimator for estimation of the smooth functional de ned by equation 12). It is nearly constant as the sample size increases re ecting that the estimators converge at the same rate (n?1=2). The relative eciency is approximately 1 on the right but less on the left.

30

Nominal Sample Coverage Size Uniform 400 0.975 95% 2000 0.969 10000 0.958 400 0.537 50% 2000 0.534 10000 0.466

1=10 0.959 0.957 0.978 0.555 0.537 0.560

1=3 0.955 0.957 0.944 0.530 0.498 0.498

p

2=3 0.946 0.972 0.944 0.501 0.500 0.500

9=10 0.958 0.952 0.942 0.521 0.531 0.498

CSD 0.949 0.938 0.947 0.485 0.487 0.490

Table 3: Estimated coverage of 95% and 50% con dence intervals. The nominal rate is accurate for each estimator even for small samples.

31

Sample p Size Uniform 1=10 1=3 2=3 400 2.09 1.97 1.63 1.44 2000 2.14 2.07 1.68 1.44 10000 2.16 2.11 1.72 1.45

9=10 1.44 1.38 1.36

CSD 1.31 1.32 1.32

Table 4: Estimated Asymptotic Standard Error ^ . Con dence intervals (which have length proportional to the standard error) based on CSD estimator are shortest and those based on the Uniform estimator are longest.

32

Sample Group Size All 295 Condom 180 No Condom 115 STD 132 No STD 163

^

103.1 107.6 94.7 89.1 110.8

s.e. 3.9 4.6 5.9 7.1 4.9

90%-con dence interval 96.6 109.5 100.1 115.2 84.9 104.4 77.5 100.7 102.6 118.9

Table 5: Con dence intervals for truncated mean of distribution of time until transmission. Condom/No Condom groups divide all partnerships by whether a condom is ever used or never used. STD/No STD groups divide all partnerships by whether the female partner has a history of sexually transmitted disease.

33

34

Figure 1: Estimation of G using Uniform data, MPU data, and current status data. For the MPU model, p = 0:5. In each case the sample size is 400. The IAD (section 5.2) of the CSD-based estimator is 20% less than the MPU-based estimator and the IAD of the Uniform-based estimator is 250% more.

35