Presmoothing the Aalen-Johansen estimator in the illness-death model

One major goal in clinical applications of multi-state models is the estimation of transition probabilities. The usual nonparametric estimator of the transition matrix for non-homogeneous Markov processes is the Aalen-Johansen estimator (Aalen and Johansen 1978 [1]). In this paper we propose a modification of the Aalen-Johansen estimator in the illness-death model based on presmoothing. The consistency of the proposed estimators is formally established. Simulations show that the presmoothed estimators may be much more efficient than the Aalen-Johansen estimator. A real data illustration is included.


Introduction
The analysis of survival data may be described by the Markov process with two states, 'alive' and 'dead' and a single transition between them.This is known as the multi-state mortality model.Multi-state models (Andersen et al., 1993 [3]; Meira-Machado et al., 2009 [25]) may be considered a generalization of survival analysis where survival is the ultimate outcome of interest but where intermediate (transient) states are identified.For example, in cancer studies more than one endpoint may be defined such as 'local recurrence', 'distant metastasis' and 'dead'.A simple multi-state model is obtained by splitting the 'alive' state of the mortality model into two transient states.For example, the illness-death model is fully characterized by three states and three transitions between them, see Figure 1.Graphically, multi-state models are represented by diagrams with rectangular boxes and arrows between them indicating respectively possible states and possible transitions for a given patient.
A multi-state model is a stochastic process (X(t), t ∈ T ) with a finite state space, where X(t) represents the state occupied by the process at time t ≥ 0. For two states i,j and s < t, introduce the so-called transition probabilities p ij (s, t) = P (X(t) = j|X(s) = i) .
Estimating these quantities is interesting, since they allow for long-term predictions of the process.The inference in multi-state models is traditionally performed under the Markov assumption, which states that past and future are independent given the present state.Aalen and Johansen (1978) [1] introduced a nonparametric estimator of p ij (s, t) for non-homogeneous Markov models.Their estimation method extends the time-honored Kaplan-Meier estimator (Kaplan and Meier, 1958 [22]) to Markov chains.As for the Kaplan-Meier, the standard error of the Aalen-Johansen estimator may be large when censoring is heavy, particularly with a small sample size.
Interestingly, the variance of the Kaplan-Meier estimator may be reduced by presmoothing.The idea of presmoothing (Dikta, 1998 [15]) involves replacing the censoring indicators by some smooth fit before the Kaplan-Meier formula is applied.This preliminary smoothing may be based on a certain parametric family such as the logistic (thus leading to a semiparametric estimator), or on a nonparametric estimator of the binary regression curve.Successful applications of presmoothed estimators include nonparametric curve estimation (Cao and Jácome, 2004 [6]), regression analysis (de Uña-Álvarez and Rodríguez-Campos, Presmoothing the Aalen-Johansen estimator 1493 2004 [12]; Yuan, 2005 [27]), and the estimation of the bivariate distribution of censored gap times (de Uña-Álvarez and Amorim, 2011 [11]).The main goal of the present work is to propose a presmoothed version of the Aalen-Johansen estimator for the transition matrix of a Markov illness-death model, and to investigate its statistical properties.The proposed estimator is different to that in Amorim et al. (2011) [2], who considered presmoothed transition probabilities for possibly non-Markov models.In general, Markov and non-Markov approaches lead to completely different estimators, so markovian estimators can not be obtained as particular cases of non-markovian estimators, and vice-versa.
The rest of the paper is organized as follows.In Section 2 we introduce the new estimator and we formally establish its consistency.In Section 3 we compare by simulations the proposed estimator to the original Aalen-Johansen curve.In Section 4 we illustrate the proposed method using data from the Stanford heart transplant study.Main conclusions and discussion are reported in Section 5.The Appendix contains the technical proofs.

The estimator: Main result
In this paper we consider the (progressive) illness-death model depicted in Figure 1.We assume that all subjects are in State 1 ('healthy') at time t = 0, and that they may either visit State 2 ('diseased') at some time point; or not, going directly to the absorbing ('dead') state.Given two time points s < t, there are in essence three different transition probabilities to estimate: p 11 (s, t), p 12 (s, t), and p 22 (s, t).The two other transition probabilities (p 13 (s, t) and p 23 (s, t)) can be obtained from p 13 (s, t) = 1 − p 11 (s, t) − p 12 (s, t) and p 23 (s, t) = 1 − p 22 (s, t).
The irreversible illness-death model is fully characterized by three transitions represented by the arrows in Figure 1.Let T ij denote the potential transition time from State i to State j.In this model we have two competing transitions 1 → 2 and 1 → 3. Therefore, we denote by ρ = I(T 12 ≤ T 13 ) the indicator of visiting state 2 at some time, and introduce Z = T 12 ∧ T 13 , the sojourn time in state 1.A subject visiting State 2 will arrive at the absorbing 'dead' state at time T 12 + T 23 , while this time will be T 13 for those not visiting State 2 (i.e.ρ = 0).Finally, let T = Z + ρT 23 denote the total survival time of the process.However, because of follow-up limitations, lost cases and so on, rather than (Z, T, ρ) one observes ( Z, T , Here C denotes the potential censoring time, which we assume to be independent of the process (that is, C and (Z, T ) are assumed to be independent).Under continuity, the information provided by ∆ 1 ρ is superfluous since we have ∆ 1 ρ = I( Z < T ).With this notation, the transition probabilities are written as Under the Markov assumption, all these quantities are estimated nonparametrically using Aalen-Johansen estimators.Explicit formulae of the Aalen-Johansen estimator for the illness-death model are available (Borgan, 1998 [5]).Here we give alternative expressions for this estimator suitable to motivate our method of presmoothing below.Assume that we have a sample of n individuals from the population under study.
Let ( Z i , T i , ∆ 1i , ∆ i , ∆ 1i ρ i ), i = 1, . . ., n be the corresponding sampling information.The Aalen-Johansen estimate of the transition probability p 11 (s, t) is the Kaplan-Meier estimator where Then, Aalen-Johansen estimate of the transition probability p 22 (s, t) is the Kaplan-Meier estimator where Finally, the estimator for p 12 (s, t) is given by  Dikta, 1998 [15]).Then, the corresponding presmoothed Aalen-Johansen (P-AJ) estimator is given by where m 0n (z) stands for an estimator of the binary regression function m 0 (z).Then, m 0 ( Z) is the conditional probability of the event ∆ 1 = 1 given Z.Since the pair Z, ∆ 1 is observable, the function m 0 (z) can be estimated by standard methods.For example, logistic regression may be performed.Consider now the presmoothed version of (2.2) given by where m 1n (z, t) stands for an estimator of the binary regression function ) is the conditional probability of the event ∆ = 1 given ( Z, T ) and given that transition 1 → 2 is observed (∆ 1 ρ = 1).The estimator m 0n (z) is based on the whole sample, while m 1n (z, t) is based on the subsample i : ∆ 1i ρ i = 1.We assume that these two empirical functions approximate well their targets in a uniform sense; more specifically, set Conditions under which U 1 and U 2 can be fulfilled were investigated in a number of papers, including Dikta (1998 [15], 2000 [16]), Devroye (1978a [13], b [14]), Mack and Silverman (1982) [23] and Härdle and Luckhaus (1984) [19].The uniform consistency of p P AJ 11 (s, t) will hold on 0 ≤ s < t ≤ τ , where τ is strictly smaller than the upper bound of the support of Z. Put M 1 (y) = P ( Z < y ≤ T ).For the uniform consistency of p P AJ 22 (s, t) and p P AJ 12 (s, t) we will refer to the following assumption: This condition allows to handle some denominators which appear in the proofs.It can be interpreted as a ǹon empty risk set' assumption for the transition from State 2 to State 3. By force, τ 0 > 0, while τ 1 is (similarly as for τ ) strictly smaller than the upper bound of the support of T .We have the following result.The proof is deferred to the Appendix.

Simulation study
In this section, we compare by simulations the presmoothed Aalen-Johansen estimator for the transition probabilities to the original Aalen-Johansen estimator.More specifically, the AJ and P-AJ type estimators p 11 (s, t), p 12 (s, t) and p 22 (s, t) introduced in Section 2 are considered.As presmoothing function we always take a parametric (logistic) family, so we actually have a semiparametric Aalen-Johansen estimator.
To simulate the data in the illness-death model, we followed the work of Amorim et al. (2011) [2].We assume that all individuals are in State 1 ("healthy") at time t = 0. Therefore, the patient's history (or course) may be divided into two groups according to whether the disease occurred (that is, passing through State 2) (1 → 2 → 3) or not (1 → 3).We separately consider these two possible subgroups of individuals.For the first subgroup of individuals (ρ = 1), the successive gap times (Z, T − Z) are simulated according to the bivariate distribution with unit exponential margins.The parameter θ controls for the amount of dependency between the gap times (Z, T − Z) and was set to 0 and 1, corresponding to 0 and 0.25 correlation between Z and T − Z.For the second subgroup of individuals (ρ = 0), the value of Z is simulated according to an exponential with rate parameter 1.In summary the simulation procedure is as follows: Step 1 Draw ρ ∼ Ber(p) where p is the proportion of subjects passing through State 2.
Step 2 If ρ = 1 then: Step 3 If ρ = 0 then: In our simulation we consider that 70% of the individuals were in the first group.The follow-up time was subjected to right censoring, C, according to uniform models U [0, 4] and U [0, 3].The first model results in 24% of censoring on the first gap time Z, and in 47% of censoring on the second gap time T − Z, for those individuals with ρ = 1.The second model increases these censoring levels to 32% and about 57%, respectively.
After some algebra, it is seen that the function m 1 (z, t) and where λ G (.) and λ 1 T |Z=z (.|z) stand respectively for the hazard rate of the censoring variable and the hazard rate of T given Z = z under restriction ρ = 1.Note that λ G (t) = 1/(τ G − t) when C ∼ U [0, τ G ] and that λ 1 T |Z=z (t|z) is given by being 1 when θ = 0.The function m 1 (z, t) belongs to the logistic family with some preliminary transformation of the conditioning variables, namely we have (for β 0 = 0 and β 1 = 1) .
This is the parametric model we fit to m 1 (z, t) in the simulations.For m 0 (z) = P ∆ 1 = 1| Z = z , we have and where λ Z (z) stands for the hazard function of Z.
Similarly as above, we also perform logistic presmoothing for the function m 0 (z), with the variable Z transformed by −ln(τ G − Z).This function belongs to the logistic family with some preliminary transformation.To estimate the function m 0 (z) in the simulations, we fit the logistic model which contains the true presmoothing function m 0 as a special case (γ 0 = 0, γ 1 = 1).
The β parameter in model m 1 (.; β) is estimated via maximization of the conditional likelihood of the ∆ i 's given the ( Z i , T i )'s, for those subjects with ∆ 1 ρ = 1 (see Dikta (1998 [15], 2000 [16])).Similarly, the γ parameter in model m 0 (.; γ) is estimated via maximization of the conditional likelihood of the ∆ 1i 's given the Z i 's.Note that the β parameter is needed for estimating p 22 (s, t) and p 12 (s, t), while γ enters the estimation of p 11 (s, t) and (again) p 12 (s, t).The aim of this simulation study is to compare the Aalen-Johansen estimator (1978) [1] and the new estimator based on presmoothing (P-AJ).Again, for measuring the estimates' relative performance, we followed the work of Amorim et al. (2011) [2].As in Amorim et al. (2011) [2], we computed the integrated absolute bias, integrated variance and the integrated MSE of the estimates.For each simulated setting (θ = 0 and θ = 1) we derived the analytic expression of p ij (s, t) so that the bias and the MSE of the estimator could be examined.K = 1000 data sets were generated, with three different sample sizes n = 50, n = 100 and n = 200.
Let pk ij (s, t) denote the estimated transition probability based on the kth generated data set.For each fixed (s, t) we obtained the mean for all generated data sets, pij (s, t) We then computed the pointwise estimates of the bias, variance, MSE and L1 distance as: To summarize the results we also calculated the integrated absolute bias (BIAS), integrated variance (VAR), integrated MSE (IMSE) and the integrated L1 distance (L1), defined in Table 1.We fixed the values of s using the quantiles 0.25, 0.5 and 0.75 of the exponential distribution with rate 1.The results given in Tables 3 to 6 were obtained by numerical integration on the interval [s, t 1 ] with t 1 = 4, taking a grid of step δ = 0.05.
In Tables 3 to 6 we report the results for the summary statistics attained by the proposed estimator when based on several presmoothing functions (P-AJ), for all scenarios.In all tables, the row labeled with m corresponds to presmoothing with the true function which is unrealistic in practice, because this function will be typically unknown.However, this row represents a 'gold standard' the other methods can be compared to.The row labeled with m(.; β, γ) corresponds to a semiparametric estimator which is obtained using a presmoothing based on a parametric family which contains the true m.Specifically, we consider a logistic model with the preliminary transformation of the conditioning variables Z = z, T = t shown before.In order to investigate the robustness of the proposed estimator with respect to misspecifications of the binary regression family, we considered also presmoothing via standard logistic models, without any preliminary transformation of the gap times.This is labeled with m(., ξ).Note that the true m does not belong to this parametric family.Finally, we also report the results pertaining to the Aalen-Johansen estimator, which corresponds to the situation with no presmoothing at all.This is labeled in the Tables as AJ.
It is obvious from the analysis of Tables 3 to 6, that presmoothing leads to estimators with smaller variance and thus attaining better results with regard to the integrated MSE also true for the L1 distance.As expected, the (integrated) MSE, bias, L1 norm and variance of the estimated transition probabilities always decrease with an increasing sample size, while they increase with the censoring degree.The estimator which makes use of the true m is the one with the best performance.However, this estimator is unrealistic since in practice one has to estimate the function m.In general, the lowest errors among the realistic versions of the estimators correspond to the estimator based on the correctly specified parametric family, m(.; β, γ).However, the presmoothed estimator based on the wrong parametric model m(.; ξ) is still (much) better than AJ.This means that it is worthwhile doing some presmoothing even when we are not completely sure about the parametric family.
Results shown in the Tables 3 to 6 support the idea that presmoothing leads to variance improvement.When compared to the estimators based on presmoothing, the relative efficiency (defined as the quotient between the two integrated MSEs) of the Aalen-Johansen estimator is always below 1.For higher values of s, where the censoring effects are stronger, the relative efficiency can drop below 50%.These findings agree with the results obtained by Amorim et al. (2011) [2] and support the intuition that the use of presmoothing for the estimation of transition probabilities will be more clearly seen in the presence of large censoring degrees.
In general, presmoothing introduces some bias in estimation, while reducing the variance.This bias component is larger when there is some misspecification in the chosen parametric model.Our simulation results serve to illustrate this issue too.Indeed, it is seen that, despite of offering a smaller IMSE, the bias associated to the semiparametric Aalen-Johansen estimator is sometimes larger than that of the original Aalen-Johansen.
Tables 3 and 4 show a systematic bias for all estimators of the transition probabilities p 12 (s, t) and p 22 (s, t).This is because these tables report the re- sults attained when generating data from a dependency scenario and therefore reflects the failure of the Markov assumption.To illustrate these features we present in Figures 2 and 3 the graphical average results for the two methods (AJ and P-AJ corresponding to presmoothing via standard logistic models, m(., ξ)).These figures plot the data generating functions and pointwise 95% oscillation limits of the estimates p 22 (s, t), for sample sizes of n = 200 with percentages of censored data obtained using C ∼ U [0, 3].The good performance of the resulting estimates (for both methods) is evident for independent gap times (θ = 0), recovering the functional forms of the corresponding true curves very successfully.However, a systematic bias of p 12 (s, t) (not shown) and p 22 (s, t) in the dependent scenario (θ = 1) is also clear, see Figure 2.This bias is much more evident when s is large, in agreement with the amount of false information introduced by the Markov condition (which increases with s).In all scenarios, the use of the presmoothing yields estimators with less variability.We have also considered different scenarios with different proportions of individuals passing through state 2. A larger value of p = P (ρ = 1) is favorable for the estimation of p 22 (s, t) (lower values for IMSE, BIAS, L1 norm and variance), whereas a smaller value of p lead to better estimates for p 12 (s, t).When comparing the two methods (with and without presmoothing) similar conclusions were obtained and therefore they are not reported here.

An example from Heart Transplant data
For illustration purposes, we apply the proposed methods of Section 2 to data from the Stanford Heart Transplant Study.The data are available as part of the R survival package, and they are also reported in Crowley and Hu (1977) [8].This study covers the period from October 1967 to April 1974.It includes 103 patients enrolled in the Stanford Heart transplant program, from which 69 received a heart transplant and among these 45 died.The total number of deaths was 75 (30 without transplantation); the remaining 28 patients contributed with censored survival times.The transplant can be considered as an associated state of risk, and we may use the so-called illness-death model with states "own heart", "new heart" (or transplant) and "dead".In most applications, a Markov model is often assumed for the multi-state model.A Cox model (Cox, 1972 [7]) can be used to test this assumption (Hougaard, 1999 [21]; Andersen et al., 2000 [4]).This is usually performed by including covariates depending on the history, such as the time of transition to the current state or the time since entry into the current state.This assumption was verified for the Stanford Transplant Study, e.g. by Hougaard (1999) [21], which conclude that there is no effect of time since transplant on mortality, and thus that the Markov model is satisfactory.This is important, because otherwise, the consistency of the Aalen-Johansen estimator and the new estimator based on presmoothing cannot be ensured.On the other  hand, if markovianity is fulfilled, the use of these methods is a wise choice.To deal with ties, a re-definition of the empiricals M 0n (y) and M 1n (y) is needed.Put Z i:n for the i-th ordered Z-statistics.Similarly, put is the i-th concomitant (i.e. the Z-value attached to T i:n ).When there are no ties, these empiricals reduce to those introduced in section 2.
Our aim with this application is to illustrate the differences between the estimated transition probabilities from Aalen-Johansen estimator (AJ) and the semiparametric estimator based on presmoothing (P-AJ).The semiparametric estimator was obtained using standard logistic regression for m 0n (z) = P (∆ 1 = 1| Z = z) and m 1n (z, t) = P (∆ = 1| Z = z, T = t, ∆ 1 ρ = 1).Figure 4 displays these functions for the Stanford heart data.The noise around displayed line comes from the fact that the variable z is omitted in the plot while it is present in the model.In Table 2 we present the summary (coefficients, standard errors between brackets and p-value) of the two presmoothing functions.In this case the influence of Z is not statistically significant on m 1 (z, t).The goodness-offit test that we used for testing the parametric presmoothing functions is an application of the Kolmogorov-Smirnov type version of the model-based bootstrap approach described in Dikta et al (2006) [18].The Kolmogorov-Sminorv test was used for testing the parametric logistic presmoothing functions m 0n (z), m 1n (z, t).In both cases the test was not able to reject the logistic model (respectively p-values of 0.638 and 0.237).We also show the goodness-of-fit test proposed by Hosmer and Lemeshow (1989) [20] was used for testing the parametric logistic presmoothing functions m 0n (z), m 1n (z, t).In both cases the test was not able to reject the logistic model (without reaching statistical significance, p-value=0.218and p-value=0.566).The bootstrap estimates were used to obtain the 95% limits for the confidence interval of p 11 (s, t), p 12 (s, t) and p 22 (s, t).The value s was chosen to be the median of the total time (s = 90 days).As expected, the P-AJ estimator has less variability than the AJ estimator, which has fewer jump points as t increases.For example, the extra jump points of the presmoothed AJ estimator of p 22 (s, t) correspond to transplanted patients with censored values of the total β 0 = 2.1148(0.5052)2.83e-05 β 1 = −0.0089(0.0058)0.1281 β 2 = −0.0025(0.0007)0.0006 time.However, both methods provide similar point estimates for all values of time.In sum, the new approach provides more reliable curves with less variability and accordingly narrower pointwise confidence bands.

Conclusions and final remarks
There has been several recent contributions for the estimation of the transition probabilities in the context of multi-state models.However, the Aalen-Johansen estimator is still the standard method for estimating these quantities in Markov models.In this paper we propose a modification of Aalen-Johansen estimator in the illness-death model, based on a preliminary estimation (presmoothing) of the censoring probability for the total time (respectively, of the sojourn time in state 1), given the available information.An interesting open question is if this idea can be generalized (and how) to more complex multi-state models.
We have derived the consistency of the proposed estimators.The consistency result is not restricted to parametric presmoothing, but it also includes the possibility of using some nonparametric estimators to this end.We verified through simulations that the method based on the presmoothing may be much more efficient than the original Aalen-Johansen estimators, even when there is some misspecification in the chosen parametric family.To this regard, it is worth mentioning that possible misspecifications in the presmoothing model will introduce some bias, while still allowing for a variance reduction.The size of the bias will depend on the misspecification level of the chosen presmoothing model, and on the amount of censored information.Dikta et al. (2005) [17] studied this problem under a misspecified parametric model, showing that the bias component increases with the model's misspecification degree and the proportion of censored observations.
In a different context, the relative importance of introducing parametric information with censored data was investigated by Miller (1983) [26].Similarly, in our scenario, relative advantages of presmoothing are more clearly seen with an increasing censoring degree and at the distribution's right tail.In such a case, standard corrections for censoring typically exhibit a large variance; however, presmoothing functions, when accurately estimated, offer a joint control of both the bias and the variance in estimation.Importantly, the validity of a given model for presmoothing can be checked graphically or formally, by applying a goodness-of-fit tests (e.g.Dikta et al., 2006 [18] and Hosmer and Lemeshow, 1989 [20] for the logistic model).This implies that the risk of introducing a large bias through a misspecified model can be controlled in practice.We illustrated the proposed methodology and all this preliminary investigation of the presmoothing model using data from the Stanford Heart transplant study.
We have not investigated the semiparametric efficiency of the proposed presmoothed Aalen-Johansen estimator.Indeed, there is some lack of research in this line even for the basic estimators introduced in the seminal papers on semiparametric censorship models (Dikta, 1998 [15]; Dikta et al., 2005 [17]).As an exception, we point out that efficiency results are available for some particular family of semiparametric censorship models (see e.g.Zhang, 2004 [28]).We wonder if these type of results can be derived also for the semiparametric Aalen-Johansen estimator.This is an interesting topic for our future research.
In this paper we have not dealt with the possible effect of covariates on the transition probabilities.However, it is possible to include covariates in the presmoothed estimator following the usual approach for Markov models.For this, one just considers each transition probability as a certain transformation of the transition intensity functions.Then, transition intensities may be allowed to depend on covariates following Cox-type regression models.See e.g.Andersen et al. (2000) [4].In order to estimate the regression parameters and the baseline transition intensities, one needs however to adapt the likelihood function to the new setting of presmoothing in which some parametric information on the conditional probability of uncensoring is available.Details are not obvious and will be considered in our future research.
The original and the presmoothed AJ estimators are consistent in Markov models.If the Markov property is violated, then the consistency of the timehonored Aalen-Johansen estimator and of its presmoothed version can not be ensured in general.Exceptions to this are the estimator for p 11 (s, t) (for which the Markov assumption is empty) or for p ij (0, t) (the so-called stage occupation probabilities, see Datta and Satten, 2001 [9]).Alternative estimators of the transition probabilities not relying on the Markov condition were recently proposed (Meira-Machado et al., 2006 [24]; Amorim et al., 2011 [2]).As a drawback, these alternative methods will suffer from a larger variance in estimation, particularly when the sample size is small and there is a large censoring degree.Consequently, AJ-type estimators will be preferred when there is no strong evidence against the Markov condition.
by Portuguese Funds through FCT -Fundação para a Ciência e a Tecnologia, within the Project Est-C/MAT/UI0013/2011.The authors are grateful to the reviewers for their helpful comments.

Appendix: Technical proofs
In this Section we give the proof to Theorem 1.Throughout this Section p ij (s, t) stands for the presmoothed Aalen-Johansen estimator p P AJ ij (s, t).Theorem 1(a) is a consequence of Dikta (1998) [15].Now we prove Theorem 1(b), that is, the uniform strong consistency of where (recall) m 1n (z, t) is an estimator of m 1 (z, t) = P (∆ = 1| Z = z, T = t, Z < T ) and where (recall) M 1n (y) = n −1 n j=1 I( Z j < y ≤ T j ) is the empirical counterpart of M 1 (y) = P ( Z < y ≤ T ).Since continuity is assumed throughout, note that ∆ 1 ρ = I( Z < T ).The following notation will be used: With this notation, we have Note that p 22 (s, t) = 0 may happen; indeed, this is the case whenever n M 1n ( T i ) = 1 and m 1n ( Z i , T i ) = 1 for some i ∈ I(s, t).In order to avoid problems when taking logarithms, introduce the following approximation to p 22 (s, t): We will refer to the following Lemma, which follows from e.g.Corollary 5.2.3 in de la Peña and Giné (1999) [10].
Under condition M , from Lemma 1 we have eventually for y ∈ [τ 0 , τ 1 ] and some constant c > 0 and Note that p 22 (s, t) = exp(−Ψ(s, t)) because of the Markov condition, and that It will be shown that p 22 (s, t) = exp(−Ψ(s, t)) is indeed the limit of exp(−Ψ n (s, t)).This will follow from the mean-value theorem after proving the uniform strong consistency of Ψ n (s, t), which is the goal of the following Lemma.
By the mean-value theorem, exp(log p 22 (s, t)) − exp(−Ψ n (s, t)) for some ξ * n between Ψ n and − log p 22 .Now: where Note that x i is well defined for i ∈ I * (s, t) and that we have that the absolute value of II is bounded by (take In order to prove Theorem 1(c) write, with J(s, t) where M 0 (y) = P ( Z ≥ y).Since, because of the Markov condition, the SLLN gives IV (s, t) → p 12 (s, t) w.p. 1.Furthermore, by using Proposition 5.1.12in de la Peña and Giné (1999) [10] as in Lemma 2 above we get w.p. 1 It remains to show that I(s, t), II(s, t), and III(s, t) go to zero w.p. 1 uniformly on [0, τ ].But this is easily seen by using Theorem 1(a),(b), Glivenko-Cantelli, and the fact that M 0 is bounded away from zero on [0, τ ].

Fig 5 .
Fig 5.Estimated transition probabilities for p ij (s, t) with s = 16 based on the Aalen-Johansen estimator (on the left) and based on the presmoothed Aalen-Johansen estimator (on the right) with the corresponding 95% pointwise confidence bands.Stanford Heart Transplant data.

Figure 5
Figure5plot, for the two methods, the estimated transition probabilities p ij (s, t), 1 ≤ i ≤ j ≤ 3 together with pointwise confidence bands based on the bootstrap.The bootstrap estimates were obtained for B = 1000 replicates, by randomly sampling the n items from the original data set with replacement.The bootstrap estimates were used to obtain the 95% limits for the confidence interval of p 11 (s, t), p 12 (s, t) and p 22 (s, t).The value s was chosen to be the median of the total time (s = 90 days).As expected, the P-AJ estimator has less variability than the AJ estimator, which has fewer jump points as t increases.For example, the extra jump points of the presmoothed AJ estimator of p 22 (s, t) correspond to transplanted patients with censored values of the total Now, we discuss how to introduce modified estimators based on presmoothing.Presmoothing the Aalen-Johansen (AJ) involves replacing the censoring indicators (in the transition probabilities p 11 (s, t) and p 22 (s, t)) by a smooth fit.The presmoothed version of p 11 (s, t) is obtained by replacing the ∆ 1i 's in (2.1) by some smooth fit to the binary regression function m 0 3) where p AJ 11 (s, t − ) = lim u↑t p AJ 11 (s, u)

Table 1
Summary statistics measuring bias, variance, mean square error and L1 distance

Table 2
Summary of the two presmoothing functions m 0n and m 1n based on logistic models

Table 3 .
Integrated absolute bias, integrated variance and the integrated MSE of pij (s, .)

Table 5 .
Integrated absolute bias, integrated variance and the integrated MSE of pij (s, .)