A Comparison of Different Approaches to Nonparametric Inference for Subdistributions

Transcription

1 A Comparison of Different Approaches to Nonparametric Inference for Subdistributions Johannes Mertsching Master Thesis Supervision: dr. Ronald B. Geskus prof. Chris A.J. Klaassen Korteweg-de Vries Institute for Mathematics Faculty of Sciences University of Amsterdam June 23

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3 Abstract Geskus 2 conjectured that nonparametric inference for subdistributions in the presence of both left truncation and right censoring may be done as for proper distributions after reweighing the underlying data. As a consequence, standard software from survival analysis could be used for competing risks analysis. In this thesis, two estimation problems variance estimation for cause-specific cumulative incidence estimators; K-sample test for competing risks data are examined in different right censoring and left truncation frameworks. The asymptotic properties and the small sample performance of estimators obtained using Geskus approach are scrutinized and compared to other estimators from the literature. Information Title: A Comparison of Different Approaches to Nonparametric Inference for Subdistributions Author: Johannes Mertsching, Johannes.Mertsching@gmail.com, Supervisors: dr. R.B. Geskus, prof.dr. C.A.J. Klaassen Second reviewer: dr. A.J. van Es Date of submission: June 7, 23 Korteweg-de Vries Institute for Mathematics University of Amsterdam Science Park 94, 98 XH Amsterdam

4 Contents Introduction Foundation 2. Survival analysis without competing risks Competing Risks Censoring frameworks Essential concepts Weak convergence of random functions Preliminaries from martingale theory Supplementary results Variance Estimation for Cause-Specific Cumulative Incidence Estimators 7 2. A variance estimator under complete information Generalization to censoring complete information Variance estimation under random censoring information The Betensky variance estimator Gray s variance estimator Geskus variance estimator A failure term A simulation study Hypothesis Testing in Competing Risks Models A one-sample test under complete information K-sample tests under complete information Testing under censoring complete information Testing under random censoring information Gray s approach Geskus approach An approach by Bajorunaite and Klein A simulation study Miscellaneous ii

5 4 Generalization to Left Truncated Data 5 4. Truncation frameworks Martingale considerations under truncation complete information Estimation under random truncation information Estimation under both left truncation and right censoring Summary and Suggestions for Further Research 55 Appendices A 59 A. Equivalence of the PLE and the AJE under complete information 59 A.2 Equivalence of the Betensky and the Geskus variance estimator when there is complete information A.3 An example showing σrc 2 σ2 CC for the variance estimation of the product-limit estimator A.4 An example showing that σcc 2 t σ2 RC t for the K-sample test 67 B Relevant R Code 7 C Detailed Simulation Results 75 C. Replication of simulation in Braun and Yuan C.2 Simulation for counterexample C.3 LogRank test simulation for counterexample D Popular summary 87 iii

6 Introduction This thesis is settled in the area of survival analysis where one tries to estimate time-to-event distributions. Applications are widely spread and are for example found in medicine where default is often associated with death but for instance also in finance e.g. credit default probabilities and actuarial sciences e.g. loss payment probabilities. Often, default occurs due to different causes and one is interested in estimating cause-specific quantities. An exemplary quantity of interest is the probability of experiencing an event of a specific type by a given time. Frequently, we find ourselves in a framework where the occurrence of the event of interest cannot be observed if another event happened before. Such risks are called competing risks. In their presence, we encounter subdistributions whose estimation we shall consider here. In practice, inference is further complicated by right censoring and left truncation. Right censoring is the mechanism based on the idea that some individuals are lost to observation at one point in time before experiencing an event. Left truncation is the problem that some individuals are not observed at all since they experience an event before they could enter the study. We shall assume that the right censoring and left truncation mechanisms are independent of the times to event. We are going to compare several approaches to nonparametric inference for subdistributions for different kinds of censoring and truncation. In Geskus 2, it is conjectured that inference for subdistributions may be done similarly to inference for proper distribution after reweighing the underlying data. As a consequence, software from standard survival analysis could be applied to competing risks problems where available software is not as rich. We aim at investigating this conjecture in two applications. In chapter 2, we survey the point-wise inference for estimators of the cause-specific cumulative incidence function, i.e. the probability to experience an event of interest by a given time. Several approaches to estimate its variance have been introduced and asymptotic properties and small-sample performance shall be investigated here. Next, in chapter 3, we will do the same for approaches in simultaneous inference by considering K-sample tests. Estimation up to this point will consider different kinds of right censoring. In chapter 4, we will additionally allow for left truncation and show how the given arguments can be extended. A summary and suggestions for further research are given thereafter.

7 Chapter Foundation In this chapter, we want to lay the groundwork for later chapters. In a first section, relevant quantities from survival analysis without competing risks are introduced and their standard estimators are presented. Thereafter, the same is done for the competing risks framework. We shall consider two kinds of right censored data. Both types are introduced in this chapter as well. Concepts we will frequently use throughout this thesis are defined next. Our notation, which will to a large extent follow that in Gray 988, is introduced along the way.. Survival analysis without competing risks We will be concerned with the analysis of time-to-event data in a sample of size n. For individual i, we denote his event time by T i. Without explicitly mentioning it in the sequel, we will always assume that the event times are independent identically distributed so that there is one common event time distribution. We will refer to the distribution function as the cumulative incidence function and denote it by F t := P T i t. Its complement, the survival function, will be denoted by St := F t. We will also write F t := F t and employ a similar notation for any distribution function. Consequently, St = F t. Unless mentioned otherwise, we will always assume that F t is absolutely continuous with respect to Lebesgue measure and denote its density by ft. Based thereon, we define the hazard rate λt by P t T i t + h T i t λt := lim = h h 2 ft F t

8 Since the cumulative incidence function is a non-decreasing function, the hazard rate is non-negative. It is also referred to as the hazard function or the incidence function. Due to the above definition of the hazard rate, we see that St = exp The cumulative hazard is defined by Λt := λsds. λsds and it is immediately clear that St = exp Λt. This relation is the reason for the importance of the product integral in survival analysis. The product integral is defined as the limit of products over partitions with a decreasing mesh size just as the ordinary integral is defined as the limit of sums. We present the definition of the one-dimensional product integral as a special case of the multidimensional definition in Andersen et al. 993, section II.6. Definition. Product integral Let Xt be a cadlag function of locally bounded variation on [, τ]. We define for t [, τ] t Y t = + dxs := lim t i t i i= Here, = t <... < t n = t is a partition of [, t]. n + Xt i Xt i. Existence, important properties and further key results related to the product integral are presented in Andersen et al. 993, section II.6. Using the product integral, St = exp Λt can equivalently be expressed as t St = dλs.. In the sequel, we briefly want to motivate the form of the estimators of the above introduced quantities. Furthermore, we will state their asymptotic properties since they will be used throughout this thesis. We will employ the following notation from standard survival analysis without competing risks: r i t := [Ti t] = if individual i has not experienced an event before time t, N i t := [Ti t] = if individual i experienced an event by time t. For any process Xt, we will define Xt := lim s t Xs. Using this, r i t = N i t. 3

9 Based on the above definition, we introduce the following counting processes: rt := Nt := n r i t = number at risk at time t, i= n N i t = total number of observed events by time t. i= Estimation will usually be complicated by right censoring and left truncation. We denote by C i the censoring time for individual i and by L i his truncation time and assume that C i and L i are independent of T i. The corresponding distributions are denoted by We redefine r i t in the following way: Gt := P C i t, Ht := P L i t. r i t := [Ti C i t>l i ]. Hence, an individual is at risk only if it has neither experienced an event nor censoring and if it entered the study. Furthermore, an event can only be observed if the individual entered the study and has not experienced censoring. Thus, we define LN c i t = LN c t = [Li <u C i ]dn i u n LNi c t. i= We denote the ordered event times by t j and assume that there are dt j events at time t j. We note that dt j = L N c t j L N c t j. Recall that P t T i t + h T i t λt = lim. h h For small h, λth can therefore be interpreted as the conditional probability of experiencing an event in [t, t + h] given that there was no event before time t. Thus, the hazard rate is estimated by ˆλt = dt rt. Based thereon, the cumulative hazard is estimated by ˆΛt = t j t 4 dt j rt j.

10 Frequently, we shall write this using an integral notation as ˆΛt = rs d LN c s. This estimator is referred to as the Nelson-Aalen estimator. In section.4.2, we shall introduce a martingale central limit theorem. Using it, the asymptotics of the Nelson-Aalen estimator can be established. Let us denote by T the support of St, i.e. a time interval of the form [, τ] or [, τ for some given τ, ]. The following theorems are proved in Andersen et al. 993, p. 9, 9 and shall only be stated here. Theorem. Asymptotics Nelson-Aalen estimator I Let t T be given and define Js := [rs>]. Suppose, for n, Js rs λsds P and Then sup s [,t] Jsλsds P. ˆΛs Λs P as n. Theorem.2 Asymptotics Nelson-Aalen estimator II For t T, suppose there is a non-negative function r such that λ r is integrable on [, t]. Then, σ 2 t = λu r u du is well-defined. Furthermore assuming that, as n, n n s Ju λu ru du P σ 2 s for all s [, t],.2 ru >ɛ Ju λu ru [ Ju n we get the following convergence: ] du P for all ɛ >, t n Ju λudu P, n ˆΛ Λ D U on D[, t] as n. Here, U is a Gaussian martingale see formula.2 for a definition with U = and V arut = σ 2 t. 5

11 Here, D[, t] denotes the space of cadlag functions on [, t] endowed with the Skorokhod topology, which is explained in more detail in section.4.. Note that.2 implicitly defines r. The most popular estimator of the survival function, the Kaplan-Meier estimator, is based on the relation in. and given by Ŝt := t i t dt t i = dˆλs..3 rt i Its asymptotics can be established similarly to those of the Nelson-Aalen estimator by employing martingale properties. The proof of the following statement can be found in Andersen et al. 993, page 26 ff.. Theorem.3 Asymptotics Kaplan-Meier estimator I Suppose the conditions in Theorem. hold true. Then sup Ŝs Ss P as n. s [,t] Theorem.4 Asymptotics Kaplan-Meier estimator II Suppose the conditions in Theorem.2 hold true. Then, we get the following convergence: n Ŝ S D SU on D[, t] for n where U is again a Gaussian martingale with U = and V arut = σ 2 t. Again, we refer to section.4. for an explanation of weak convergence on D[, t]. In order to estimate the censoring distribution, we will use a product-limit estimator that we obtain after swapping the roles of T i and C i. For this purpose, let us denote the ordered observed censoring times by c j and the number of censorings at time c j by mc j. We assume that c j > t k in case they happen at the same time point, i.e. if there are censorings and events at the same time points, the censored individuals still remain in the risk set at this particular point and only pass out after. A product-limit estimator of G is then given by ˆḠt = c j t mc j rc j dc j A similar approach yields an estimator of the truncation distribution Geskus, 2. We denote by l j the ordered truncation times and assume that there are wl j individuals getting censored at l j. We assume that l j > t k c k in case they happen at the same time. A product-limit estimator is obtained by reversing the time axis since L i is right truncated by T i C i. Therefore, L i is left truncated by T i C i and the known product-limit estimator now reads as Ĥt = wl j = wl j. rl l j < t j rl l j >t j. 6

12 .2 Competing Risks In this thesis, we want to investigate competing risks models. In such models, there are multiple failure types and we are interested in estimating cause-specific quantities. We will denote the observed event type for individual i by D i and it is assumed that there are m competing events. For any quantity, we will indicate failure type by subscript. For example, keeping in mind that the overall number of events was denoted by Nt, we will denote the counting process that is counting the number of events of type k that happened by time t by N k t. As another example, we will use T ik = event time for the event of type k for individual i. If there was an event of type k, we will assign T il the value for l k. Consequently, we for example obtain the nice property F k t = P T i t, D i = k = P T ik t. We assume that F k has density f k. To abbreviate notation, we will frequently use T k instead of T ik which should not be confused with the overall event time of individual k. We are aware that the same ambiguity arises for other quantities, for example N k t could as well be the indicator denoting whether individual k experienced an event by time t. However, it will always be clear from the context how N k, T k and similar expressions should be interpreted. In the presence of competing risks, several types of hazards exist. The marginal hazard of an event is the hazard in the situation when other events do not occur. Estimation of the marginal hazard is only possible when the times to event are independent for different risk types and this independence cannot be tested for Tsiatis, 975. We shall not consider this hazard type further in this thesis. A second hazard type is the cause-specific hazard which is defined in the following way: P t T ik t + δ T i t λ k t := lim = f kt δ δ F t. Hence, we note that for small δ, δλ k t can be interpreted as the conditional probability of experiencing an event of type k in [t, t + δ] given there was no event of any kind before time t. We denote by d k t i the number of observed events of type k at time t i. The definition of λ k t suggests to estimate it by ˆλ k t = d kt rt. Based thereon, the cumulative cause-specific hazard Λ k t := λ ksds can be estimated by ˆΛ k t = rs dn ks. 7

13 The cause-specific hazard treats competing events as censorings. However, it is important to note that this type of hazard does not correspond to the causespecific cumulative incidence function F k t = P T k t. In fact, this observation gave rise to the definition of a third type of hazard, namely the subdistribution hazard. It is defined by P t T ik t + δ T ik t γ k t := lim = δ δ f k t F k t. Here, δγ k t can now for small δ be interpreted as the conditional probability to experience an event of type k in [t, t + δ] given there was no event of interest! before time t. Therefore, in order to estimate this quantity, a redefined risk set is used. Let us for now assume that there is neither right censoring nor left truncation. We will refer to this situation as complete information. We additionally consider right censoring in section.3 and left truncation in chapter 4. Then, we do also consider individuals to be at risk with respect to this changed meaning if they experience a competing event. They only leave the risk set if they experience the event of interest. Based on these considerations, we define and R ik t = [Tik t] R k t = n R ik t. i= Since this quantity is only needed for the event of interest, we shall write Rt = R k t. We note that individuals that experience a competing event will then always be at risk which implies Rt rt. This at risk definition might appear counter intuitive at first sight but it is useful in the analysis of the subdistribution hazard since we obtain the following estimator of the subdistribution hazard for failure type k: ˆγ k t = d kt Rt. Naturally, the cumulative subdistribution hazard Γ k t := γ ksds can be estimated by ˆΓ k t = Rs dn ks. It can further be shown that F k t = t dγ k t. A product-limit type estimator of the cause-specific cumulative incidence function is therefore given by ˆF P L k t t = dˆγ k t. 8

14 An alternative estimation approach is based on the relationship F k t = Su dλ k u. We get an estimator of F k t when replacing S and Λ k by their estimators. This estimator, commonly referred to as the Aalen-Johansen estimator, is given by ˆF k AJ Ŝu t = ru dn ku..4 Under complete information, we show in part A. that k t = ˆF k P L t. However, it will turn out to be crucial in the sequel that there are censoring types where this does not hold true anymore. Under complete information, it can be shown that the following process is a local square integrable martingale see section.4.2 with respect to F k = {F k t} t, F k t = σ{n ik u, u t, i =,..., n} Gray, 988: M ik t = N ik t ˆF AJ γ k sr ik sds..5 Note that F k is the filtration that does not contain information about the times to the competing events. Then, M k t = n M ik t = N k t i= γ k srsds is a local square integrable martingale with respect to F k as well..3 Censoring frameworks In this thesis, we are concerned with two different types of right censoring. For both types, an event for individual i can only be observed if it is under observation not censored. This motivates the definition of N c i t := [Ci >u]dn i u. Similarly, we define Nik c u. The two censoring frameworks differ with respect to available information about censoring times, i.e. there are different underlying filtrations. The first situation is commonly called censoring complete information or administrative censoring. In here, we assume that censoring times are known for all! individuals, even for those who experienced an earlier event. An individual that experienced a competing event will still be considered to be at risk, but only up to its censoring time. After, it leaves the risk set as any censored individual does. We note that all the individuals that did not experience the 9

15 event of interest are treated similarly. This is an important observation since estimation of the subdistribution hazard does therefore not require knowledge of competing event times. These considerations are reflected in the following filtration Fine and Gray, 999, p. 499: F k,cc = {F k,cc t} t with F k,cc t = σ { [Ci >u], [Ci >u]n ik u, u t, i =,..., n }. We define R ik,cc u := [Ci >u] [ N ik u ]. Individual i is at risk with respect to the above meaning iff R ik,cc u =. Again, we will commonly leave out the k in the subscript and denote the total number at risk in this setting by R CC t = n R i,cc t. i= Under censoring complete information, M c ik t := N c ik t R i,ccuγ k udu is a local square integrable martingale Fine and Gray, 999. Andersen et al., 993, p. 39 ff. showed that M ik in.5 preserves its local martingale property with respect to F k = {F k t} t given by F k t = σ { [Ci >u], N ik u, u t, i =,..., n } when independence of censoring and event times is assumed. Note that M ik t = dn ik u Mik c t = [Ci >u]dn ik u R ik uγ k udu, [Ci >u]r ik uγ k udu. The predictability of [Ci >t] with respect to Fk and its boundedness therefore imply that Mik c is local square integrable martingale with respect to F k as well. Note that F k,cc t Fk t. Hence, the innovation theorem Andersen et al., 993, p. 79 ff. implies that Nik c has the following intensity process with respect to F k,cc : E [γ k tr i,cc t F k,cc t ] = γ k tr i,cc t. The latter equality follows since γ k u is a deterministic function and R i,cc is predictable with respect to F k,cc t. Therefore, we conclude that the intensity process and hence the compensator of Nik c with respect to F k,cct is the same as with respect to Fk t. Thus, M ik c is a local square integrable martingale with respect to F k,cc t. The second censoring situation we encounter is random censoring information. Censoring times are only known if they happened before the event times. For an individual that experiences an event before it gets censored, the censoring time is unknown. If all the individuals that experienced a competing event stayed in the redefined at risk set forever, this would not account for the censoring that individuals would have experienced if they had not passed out before. However, since censoring is assumed to be independent of the event times,

16 the unobserved censoring times can be estimated. Fine and Gray 999 introduced such an estimator that assigns time-dependent weights to all individuals. Individual i has weight R i,rc t given by R i,rc t = ˆḠt ˆḠt T ik..6 Here, T ik denotes the time at which individual i experiences an event other than that of type k. It is important to note that Mik c t := N ik c t R i,rcuγ k udu is not a local martingale with respect to a filtration not containing information about the competing event times. This particularly distinguishes the two censoring information frameworks. Under censoring complete information, Mik c essentially is a local martingale since only information about the independent censoring times has to be added to F k to make R i,cc predictable. This is different for R i,rc. Only individuals that experienced a competing event may contribute to R RC with a weight different from. Hence, information about competing event times has to be available to make R i,rc predictable. For example, one may try to consider F k = { F k t} t with { } F k t = F k t σ [Ci >u], Ĝ u T ik, u t, i =,..., n..7 Note that this filtration contains information about competing event times. However, augmenting the filtration by information about competing event times invalidates the martingale property of M ik in.5 which is the crucial starting point of the reasoning under censoring complete information. We conclude that the same martingale arguments cannot be applied anymore. One may wonder whether Mik c is a local martingale with respect to a different filtration. Such a filtration does not exist. This is a consequence of asymptotic results we establish in chapters 2 and 3 with the help of a counterexample in section A.3. However, it remains unclear what compensator Nik c has with respect to F k in.7. In order to obtain predictability of R i,rc, we associate the following filtration with the random censoring information situation: F RC = {F RC t} t with F RC t = σ { D i N c i u, [Ci >u] N i u, u t, i =,..., n }. Based thereon, R i,rc t can be determined. The total number at risk in the random censoring information framework is then given by R RC t = n R i,rc t. i= Later on, we will frequently make use of the relationship n R CCt R RC t a.s. as n.8

17 which we shall prove here. For this purpose, we first note that Beyersmann et al. 22, p. 29, 3 showed lim E[R i,cct] E[R i,rc t] =..9 n Furthermore, since max {E[ R i,cc t ], E[ R i,rc t ]} <, an application of the strong law of large numbers yields for n n R RCt E[R i,rc t] a.s.,. n R CCt E[R i,cc t] a.s... Combining.9,. and., we obtain.8. Hence, we see that R RC t is a good approximation in the above sense indeed and a transition from random censoring information to censoring complete information seems feasible. Asymptotics in the censoring complete information framework can be established using martingale arguments and it is tempting to apply similar techniques under random censoring information. If this were true indeed, any function of the subdistribution hazard could be estimated as in the standard survival setting after assigning time-dependent weights to all the individuals. This approach we will refer to as the reweighing approach. As a consequence, standard software could be used for competing risks analysis. The applicability of the reweighing approach shall be investigated in the sequel..4 Essential concepts In this part, we briefly want to state concepts and theorems that we will use frequently in later chapters. First and foremost, this is a martingale central limit theorem that will later allow us to derive asymptotic properties of estimators elegantly. In order to make the convergence therein explicit, weak convergence of random functions is briefly introduced first. Further results that we will use in later parts are stated thereafter..4. Weak convergence of random functions Let us denote by DT the space of cadlag functions on T where T now is the interval [, τ or [, τ] for some given τ, ]. We define a metric on DT in the following way Jakubowski, 27: Let Λ be the space of strictly increasing functions from T onto itself. For two elements x, y DT, we define d s x, y := inf λ Λ sup t T λt t + sup xλt yt. t T This metric was introduced by Skorokhod and is therefore referred to as the Skorokhod metric. We aim at endowing DT with a metric that makes it a 2

18 Polish space. Endowed with Skorokhod s metric d s, DT becomes a Banach space only. However, Billingsley 968 proved that there is an equivalent metric d s that makes the Banach space separable and thus a Polish space. Based on the above considerations, we now introduce weak convergence of random functions. Suppose X n, X DT. We say X n converges weakly to X and write X n D X, if for all bounded, real-valued and continuous functions f: EfX n EfX as n where the latter convergence is to be understood with respect to d s. A generalization of the Skorokhod topology to multidimensional spaces and hence a generalization of weak convergence to that of multidimensional random functions is sketched in Jakubowski Preliminaries from martingale theory Many arguments throughout this thesis are based on martingale theory. The key in most of the arguments providing large sample properties is a martingale central limit theorem that we want to introduce in this section. However, in order to do so, we shall briefly recap martingale-related definitions first. We find ourselves in a filtered probability space where we denote the filtration by F = {Ft, t T }. A cadlag process M is called a martingale on T with respect to F if it is integrable on T, adapted to F and if it additionally satisfies the martingale property, i.e. for s t T : E[Mt Fs] = Ms. It is called square integrable if it additionally satisfies sup E[Mt 2 ] <. t T A stochastic process is said to satisfy a property locally if that property is satisfied along a fundamental or localizing sequence of stopping times. A sequence T n of stopping times is called fundamental if it is non-decreasing and if t T : P T n t = as n. For example, we say that M is a local square integrable martingale if there is a fundamental sequence T n such that M T n is a square integrable martingale. Here, we employ the notation M T n t := MT n t and call M T n the stopped process. For a square integrable martingale M, it is known that M 2 is a non-negative submartingale. An application of the Doob-Meyer theorem guarantees the existence of a predictable, increasing process Spreij, 23, p. 9 that we will denote by < M > such that M 2 < M > 3

19 is a martingale. < M > is called the quadratic variation process. Based thereon, it will turn out to be useful to define a predictable covariation process for two square integrable martingales M and M 2 by < M, M 2 >:= 4 [< M + M 2 > < M M 2 >]. It can be shown Spreij, 23 that < M, M 2 > is the unique process that can be written as the difference of two natural processes such that M M 2 < M, M 2 > is a martingale. Here, a process X is called natural if it is increasing and if it further satisfies the following equality for any right-continuous, bounded martingale M: [ ] [ ] E MsdXs = E Ms dxs.,t],t] An extension of quadratic variation and predictable covariation to local martingales is feasible. For a local martingale M, we define its quadratic variation process as the unique finite variation cadlag predictable process such that M 2 < M > is a local martingale. A similar extension of the predictable covariation process from martingales to local martingales is straightforward Andersen et al., 993, p. 68. A process X is a semimartingale if there is an increasing, predictable process A such that M = X A is a local martingale. A is frequently called the compensator of X and the Doob-Meyer decomposition theorem Spreij, 23, p. gives conditions under which A exists. For a semimartingale X, we define its quadratic covariation by < X >:=< M >. Similarly, we define the predictable covariation process for two semimartingales. For a cadlag process X, let us define the jump process Xs = Xs Xs. Hence, if X is left-continuous, X. For a local square integrable martingale M and ɛ > given, define A ɛ t := s t Ms [ Ms >ɛ]. Denote its compensator by Ãɛ. We define M ɛ := A ɛ Ãɛ. It can be shown that M M ɛ has jumps smaller than 2ɛ only Rebolledo, 98. In order to state a multi-dimensional martingale central limit theorem, we consider a k-dimensional sequence of local square integrable martingales, M n = M n,..., M n k 4.

20 Denote by < M n > the k k-matrix containing the quadratic covariation processes < M n i, M n j >. Suppose that V is a k k-dimensional continuous, deterministic, positive semidefinite matrix-valued function on T. Let M be a continuous Gaussian vector martingale with < M >= V. Such a martingale is characterized by two properties: M t M s N k, Vt Vs,.2 M t M s is independent of M u, u s..3 V is chosen such that M always exists Andersen et al., 993. We are now able to introduce the desired convergence theorem. The martingale central limit theorem can be viewed as a generalization of the standard central limit theorem. Whereas the standard central limit theorem makes a statement about the convergence of the sum of independent identically distributed random variables, the martingale central limit theorem generalizes this from random variables to stochastic processes, namely martingales. Different versions exist that differ with respect to restrictions on the martingale. The following version from Andersen et al. 993, p. 83 is the one that is used frequently in the context of survival analysis and that will turn out to be sufficient for our purposes. Theorem.5 Martingale central limit theorem Suppose T T. Moreover, let M n be defined as above. Furthermore, denote by M n ɛ the K-dimensional vector where the i th component equals M n iɛ. If we assume that, as n, < M n > t P Vt for all t T, < M n ɛi > t P for all t T, i {,..., K}, ɛ >, the following convergence holds true as n : M n t,..., M n D t l M t,..., M t l for t,..., t l T. If T is dense in T and if it contains τ if T = [o, τ], we get M n D M in DT k as n and < M n > converges uniformly on compact subsets of T in probability to V. DT k is the space of R k -valued cadlag functions on T endowed with the Skorokhod topology. The importance of the above martingale central limit theorem in survival analysis is based on the observation that Mt := Nt λsrsds is a local square integrable martingale with respect to an adequate filtration Andersen et al., 993, p. 73. Furthermore, when integrating with respect to Mt, it is known that the stochastic integral is a local martingale if the integrand is a locally bounded predictable process. If M is a local square integrable martingale, 5

21 integrating with respect to Mt even yields a local square integrable martingale. Under suitable conditions the martingale central limit theorem becomes applicable and statements about the asymptotic behavior of a sequence of such integrals can be made..4.3 Supplementary results An equation that we will encounter in connection with product integrals is the Duhamel equation Andersen et al., 993, p. 9. It relates two product integrals and later-on allows us to apply martingale arguments. Without proving it, we shall state its one-dimensional version here. Lemma. Duhamel equation Suppose Y t = t + dxs, t Y t = + dx s and further assume that Y is non-singular. Then Y t t Y t = Y s Y s dxs X s. In particular, if Xt X t is a local square integrable martingale, Duhamel s equation implies that Y t Y s Y t is a local square integrable martingale if Y s is locally bounded and predictable. 6

22 Chapter 2 Variance Estimation for Cause-Specific Cumulative Incidence Estimators In this chapter, we aim at estimating the variance of estimators of the causespecific cumulative incidence function. Based thereon, we are able to do inference at fixed time points. For example, we can then construct confidence intervals for this very estimator. First, we are going to explain variance estimation under complete information followed by an examination of the censoring complete information case. Both settings allow for a straightforward derivation using martingale arguments. The random censoring information situation is considered thereafter. We will briefly show how estimators from the literature can be derived and thereafter explain why an application of the reweighing approach suggested in Geskus 2 is not feasible. 2. A variance estimator under complete information Let us first consider the complete information setting, i.e. we assume that there is neither right censoring nor left truncation. Using a martingale central limit theorem, we derive an estimator having the desired asymptotic properties. The given arguments are entirely taken from the proof by Andersen et al. 993, p. 256 ff. for the standard survival setting without competing risks. However, we shall use a different underlying martingale. The most crucial change to the derivation in Andersen et al. 993 for this purpose is that we are in the filtered probability space where the filtration is now given by F k = {F k t} t with F k t = σ{n ik u, u t, i =,..., n}. This is the filtration that does not contain information about the time to the competing events. Let us for now assume that Γ k t is absolutely continuous so 7

23 that γ k t exists. Recall that in this setting, M k t = N k t γ k srsds 2. is a local square integrable martingale with respect to F k Gray, 988. Such a process is the starting point in Andersen et al. 993 and from here we proceed exactly as done there. We define S k as the survival of event type k -function, i.e. S k t := F k t = exp Γ k t. We denote the support of S k by T. When at least 2 event types happen with a positive probability, F k = P D i = k < and T = [, ]. Under the assumption given above formula 2., S k t is a continuous function. Using the definition of the product integral, we write this as S k t = t dγ k s. Hence, a product limit estimator of S k t is given by t Ŝk P L t = dˆγ k s. We keep in mind that the cumulative subdistribution hazard can be estimated by a Nelson-Aalen type estimator, i.e. ˆΓ k t = with Jt := [Rt>]. Therefore, Js Rs dn ks t Ŝk P L t = Js dn ks. Rs The following theorem gives conditions for uniform consistency of this estimator. It can be established as done in Andersen et al., 993, theorem IV.3. using M k instead of M. The conditions are therefore similar to those in Theorem.. Theorem 2. Asymptotics of S k I Suppose t T. Furthermore, assume that, as n, Then ŜP L k Js Rs γ ksds P, Jsγ k sds P. is uniformly consistent on [, t], i.e. sup ŜP k L s S k s P as n. s [,t] 8

24 To proceed, we define in analogy with in Andersen et al. 993 and based thereon Γ k t := Juγ k udu S k t := exp Γ k t = t dγ k s. Let us note that Γ k t Γ kt and consequently Sk t S kt. Later on, we shall use the relationship ˆΓ k t Γ k t = = Js Rs dn ks Js Rs dm ks. Jsγ k sds We observe that Sk t > by the way we defined it and therefore the Duhamel equation from section.4.3 becomes applicable. It yields which we rewrite as Ŝk P L t Sk t = Ŝk P L t Sk t = s Sk s dˆγ k Γ k s Ŝ P L k Ŝk P L s Js Sk srs dm k s. 2.2 Ŝkt The integrand is bounded by S k t and we therefore conclude that Sk t is a local square integrable martingale with expectation zero. Hence, ] [ŜP L k t E Sk t =. Since we observed earlier that Sk t S kt, we therefore conclude that EŜP k L t S k t. The same holds true in the standard survival setting Andersen et al., 993, p To get an estimator for the variance of S k t, we use formula 2.2 again and note that < M k > is the compensator of Mk 2. Hence, } 2 ŜP L k t {ŜP L V ar Sk t k t = E Sk t = E < ŜP k L > t. Using the predictability of Ŝk P Ls Js Sk srs S k and further employing the martingale property in 2., we compute the quadratic variation of ŜP k Lt Sk t : < ŜP k L Sk > t = {ŜP L k Sk s s } 2 Js Rs γ ksds. 9

25 From 2., it is known that sup s [,t] ŜP k L s Sk s P. Furthermore, since we assume that S k t is continuous, we know that sup s [,t] ŜP k L s Ŝk P L s P as n. Therefore, we conclude that ŜP L V ar k Js Sk Rs γ P ksds as n. Thus, replacing γ k s by its natural estimate ˆγ k s, we get a variance estimator ŜP L k t t Js V ar = S k t R 2 s dn ks. If we denote the variance of ŜP k L t, the product-limit estimator, by σp 2 L t, we therefore conclude that 2 t ˆσ P 2 L Js Lt = ŜP k t R 2 s dn ks. This estimator, which is commonly called the Aalen-type estimator, turned out not to be the preferred one in the standard survival setting without competing risks Therneau, 999. An estimator that was found to perform preferably in small samples is obtained when extending the model. It is no longer assumed that Γ k t = γ ksds and S k t is therefore not necessarily continuous. Using this more general approach, we may work with M k t = N k t RsdΓ k s. Using the predictability of R and equation in Andersen et al. 993, we conclude that this local martingale has quadratic variation process Hence, < M k > t = Rs Γ k sdγ k s. < ŜP k L Sk > t = {ŜP L k Sk s s } 2 Js Rs Γ ksdγ k s. By replacing S k V ar by ŜP L k and Γ k by ˆΓ k we then obtain t t = S k t ŜP L k Js RsRs N k s dn ks. This yields the Greenwood-type estimator for the variance of the estimator of the cause specific cumulative incidence function: ˆσ Gwt 2 = ŜP k L t 2 RsRs N k s dn ks. 2

26 The large sample properties now follow similarly to those of the Kaplan-Meier estimator in Andersen et al. 993, theorem IV.3.2. when replacing M by M k. Note that the required conditions are therefore similar in nature to those in Theorem.2. Without a proof, we shall first state the necessary conditions and then give the convergence results we get for an increasing sample size n. Even though this is not explicitly indicated in the notation, we note that all the quantities will depend on n. Theorem 2.2 Asymptotics of S k II For some t T, suppose there is a non-negative function R such that γ ks is integrable on [, t]. Then, Rs σ 2 P Lt := S 2 k t γ k s Rs ds is well-defined. If we further assume that, as n, s Ju s [, t] : n Ru γ kudu P σ2 P L s, 2.3 s we get ɛ > : n and n S 2 k Js Rs γ ks [ Js n Rs >ɛ Juγ k sds P, ] ds P, n Ŝ P L k t S k t D N, σ 2 P Lt for n. Finally, the above derived variance estimates are uniformly consistent in the following sense: sup nˆσ Gws 2 σp 2 Ls P as n, s [,t] sup nˆσ Aas 2 σp 2 Ls P as n. s [,t] Note that Rt is implicitly defined by 2.3. To summarize, we derived a variance estimator for ŜP k L t. Since S k t = F k t, this is then obviously a variance estimator for ˆF k P L t := ŜP k L t as well. In section A., we show that under complete information, ˆF AJ k t = ˆF k P L t and their variances evidently equal. Therefore, the above derived variance estimator also estimates the variance of the Aalen-Johansen estimator. 2.2 Generalization to censoring complete information In a next step, we want to generalize the above setting slightly to the censoring complete information situation. In this setting, an individual that experienced a competing event is considered to be at risk to experience the event of interest up 2

27 to his censoring time. After, it leaves the risk set as any censored individual does. Recall from section.3 that in this setting, the underlying filtration is given by F k,cc = {F k,cc t} t and that Mik c t := N ik c t R i,ccuγ k udu is a local square integrable martingale with respect to F k,cc. Define Nk ct = n i= N ik c t. We conclude that Mk c t := N k c t R CC uγ k udu is a local square integrable martingale with respect to F k,cc as well. Hence, we can apply martingale properties as done above in the complete information situation. Again, we will obtain a variance estimator of ŜP k L t since the derivation makes use of the product limit form t dγ k s = exp γ k udu. In the censoring complete information situation, Γ k u is estimated by ˆΓ k t = Js R CC s dn c k s and the product-limit estimator is therefore given by t dn k cs. R CC s However, in the censoring complete information situation, Ŝ P L k t = t dn k cs Ŝs R CC s rs dn k c s = ŜAJ k t. The martingale derivation remains valid but we will eventually get a variance estimator for t Ŝk P L t := dn k cs. R CC s The obtained variance estimator will not estimate the variance of the Aalen- Johansen estimator. However, for later purposes, let us summarize the asymptotic properties that we get under conditions that are similar to those in section 2. with Rt replaced by R CC t. The product-limit estimator ŜP k L t will again be uniformly consistent and in the censoring complete information setting and it will further satisfy n Ŝk P L t S k t D N, σcct 2 as n with σ 2 CCt := S k t 2 γ k s R CC s ds. 22

28 An estimator of the variance will then be given by ˆσ CCt 2 = ŜP k L t 2 c R CC sr CC s Nk csdn k s and again, the variance estimate is uniformly consistent in the following sense: sup nˆσ CCs 2 σccs 2 P as n. s [,t] 2.3 Variance estimation under random censoring information In section.3, we have argued that Mk c t = N k ct R RCsγ k sds is not a local martingale under random censoring information. Hence, arguments different from the censoring complete information framework have to be applied to variance estimation. Various ways of estimating the variance of the Aalen-Johansen estimator under random censoring information have been introduced. A brief overview is given in Braun and Yuan 27. We will call the estimator that was found to perform best in simulation studies under random right censoring information Braun and Yuan, 27 and in the presence of additional left truncation Allignol et al., 2 the Betensky estimator. We will introduce truncation in section 4.. In the present section, the Betensky estimator is derived first from the multistate model. Thereafter, an alternative approach by Gray 988 is presented. Next, we will show that the variance estimator proposed by the approach in Geskus 2 does not have the desired asymptotics. A possible correction term is discussed afterwards The Betensky variance estimator The Betensky variance estimator was derived by Betensky and Schoenfeld 2 and Gaynor et al. 993 using different approaches. Allignol et al. 2 showed that their estimator is equivalent to the Greenwood-type estimator from the multistate model. We want to sketch how the Greenwood-type estimator can be derived from the multistate model by following the arguments in Andersen et al For this purpose, we briefly explain the multistate model first: Recall that there are m competing risks. We now consider an m + - dimensional state space, say,,..., m and model the movement of an individual as a Markov process with initial state in t =. The probability to experience a transition from state a to state b in [s, t] is denoted by P ab s, t. We denote by Ps, t the m+ m+-matrix collecting the various transition probabilities, Ps, t = {P ab s, t}, a, b =,..., m. 23

29 Suppose there are n independent individuals having the above behavior. For a b, the number of direct, observed transitions from a to b at [, t] is denoted by N ab t and it can be shown that N ab t has intensity process λ ab tr a t where λ ab t is the transition intensity for one individual and r a t is the total number of individuals in state a at time t. Based on these considerations, Andersen et al. 993 derive a product-limit estimator of Ps, t, the multidimensional Aalen-Johansen estimator, which is given by ˆPs, t = t I + d ˆΛu. s Here, ˆΛu is an estimator of Λu, i.e. of the matrix having as its entries the cumulative intensities Λ ab u = u λ absds. These intensities can be estimated by a Nelson-Aalen-type estimator, ˆΛ ab u = u r a s dn abs. The derivation of the Betensky variance estimator now reads fairly similar to that of the Greenwood estimator in the setting without competing risks. Andersen et al. 993 define Λ t as the matrix with components Λ abt = [ras>]λ ab sds and show that ˆΛt Λ t is a local square integrable martingale. Using a multidimensional form of the Duhamel equation from section.4.3, they then prove that ˆPs, tp s, t I = ˆPs, u d ˆΛt Λ t P s, u. Based on this martingale representation of the Aalen-Johansen estimator, one can derive a covariance estimator in a similar way as it was done in section 2. based on 2.2. However, the necessary computations are now multidimensional and hence somewhat more complex, see Andersen et al. 993, p. 29 ff. for the explicit formulas. We obtain the competing risks models by setting for a b, a all the transition intensities λ ab t equal to zero. Therefore, in the competing risks setting, the m + m + -dimensional transition matrix is of the form P s, t P s, t P 2 s, t... P m s, t... Ps, t = Consequently, their covariance estimator can be applied in order to estimate the variance of the Aalen-Johansen estimator in the competing risks setting as well. 24

30 2.3.2 Gray s variance estimator Gray 988 suggested a different variance estimator. Even though his variance estimate tends to overestimate the actual variance in small samples Braun and Yuan, 27, the following considerations give rise to an estimator that has a similar structure to the one we get using the reweighing approach. It shall further be of use in later chapters when considering K-sample tests. Gray makes use of the stochastic processes M i t = N c i t ruλ i udu. Here, i denotes failure type. He establishes that the M i are orthogonal square integrable local martingales with predictable variation processes < M i > t = ruλ i udu with respect to F RC Gray, 988. We shall employ the notation M k u := i k M iu and use a similar notation for any other quantity such as λ k t := i k λ it and F k t := i k F it. The following decomposition, that was later also used by Lin 997, is the starting point of his derivation: ˆF k t F k t = = ŜudˆΛ k u Ŝu ru dm ku + SudΛ k u Ŝu Su dλ k u. Recall that F k t = Su dλ ku. S is assumed to be continuous, thus F k t = SudΛ ku. Assume that St >. Then, ˆF k t F k t = Ŝu ru dm ku + Ŝu Su df k u Su An application of Duhamel s equation from section.4.3 yields Ŝt St St = Ŝu rusu dmu Ŝt St where Mu = m i= M iu. We emphasize that as a consequence, St is a continuous process of finite variation. This is important to note since accordingly the integration by parts formula from ordinary integration holds true. Hence, t Ŝu Su df k u Su Ŝt St t Ŝu Su = F k t F k u d St Su Ŝt St t F k = F k t + u Ŝu dmu. St rusu 25

31 Using the above mentioned integral representation for Ŝt St St, we rewrite this as t Ŝu Su df k u Su Ŝu t = F k t rusu dmu + F k u Ŝu dmu rusu [F k u F k = t]ŝu dmu. rusu Therefore, we conclude that ˆF k t F k t = Ŝu ru dm ku + [F k u F k t]ŝu dmu. rusu Let us assume that the conditions in theorem.3 hold true. Then, Ŝt is known to be a uniformly consistent estimator of St. Thus, asymptotically, ˆF k t F k t = = Su ru dm ku + m l= [F k u F k t] dmu ru { Su ru [l=k] + [F ku F k t] ru Recall that M k and M k are independent and further < M k > t = rsλ k sds. } dm l u. An application of the martingale central limit theorem yields asymptotic normality of n ˆFk t F k t. We further determine its variance. Recall that St = F k t F k t and that we assume F k to be continuous. Furthermore, using the predictability of the integrand, we conclude that σrc 2 t, the variance { } of lim n n ˆFk t F k t, is given by σrct 2 = P lim n m n = lim n n l= [ { Su ru [l=k] + [F } ku F k t] 2 d < M l > u ru F k u 2 F k u 2 λ k udu + λ k udu 2.4 ru ru + F k t 2 { 2F k t λ k u + λ k u du ru F k u λ k udu + ru }] F k u ru λ kudu. Replacing unknown quantities by their estimates, this gives rise to a variance estimate based on Gray