Doubly Robust Inference for Hazard Ratio under Informative Censoring with Machine Learning

Abstract

Randomized clinical trials with time-to-event outcomes have traditionally used the log-rank test followed by the Cox proportional hazards (PH) model to estimate the hazard ratio between the treatment groups. These are valid under the assumption that the right-censoring mechanism is non-informative, i.e. independent of the time-to-event of interest within each treatment group. More generally, the censoring time might depend on additional covariates, and inverse probability of censoring weighting (IPCW) can be used to correct for the bias resulting from the informative censoring. IPCW requires a correctly specified censoring time model conditional on the treatment and the covariates. Doubly robust inference in this setting has not been plausible previously due to the non-collapsibility of the Cox model. However, with the recent development of data-adaptive machine learning methods we derive an augmented IPCW (AIPCW) estimator that has the following doubly robust (DR) properties: it is model doubly robust, in that it is consistent and asymptotic normal (CAN), as long as one of the two models, one for the failure time and one for the censoring time, is correctly specified; it is also rate doubly robust, in that it is CAN as long as the product of the estimation error rates under these two models is faster than root-$n$. We investigate the AIPCW estimator using extensive simulation in finite samples.

Full text

Doubly Robust Inference for Hazard Ratio under Informative Censoring
with Machine Learning
Jiyu Luo∗and Ronghui Xu †
Abstract
Randomized clinical trials with time-to-event outcomes have traditionally used the log-rank test followed by
the Cox proportional hazards (PH) model to estimate the hazard ratio between the treatment groups. These
are valid under the assumption that the right-censoring mechanism is non-informative, i.e. independent of the
time-to-event of interest within each treatment group. More generally, the censoring time might depend on
additional covariates, and inverse probability of censoring weighting (IPCW) can be used to correct for the bias
resulting from the informative censoring. IPCW requires a correctly specified censoring time model conditional
on the treatment and the covariates. Doubly robust inference in this setting has not been plausible previously
due to the non-collapsibility of the Cox model. However, with the recent development of data-adaptive machine
learning methods we derive an augmented IPCW (AIPCW) estimator that has the following doubly robust (DR)
properties: it is model doubly robust, in that it is consistent and asymptotic normal (CAN), as long as one of the
two models, one for the failure time and one for the censoring time, is correctly specified; it is also rate doubly
robust, in that it is CAN as long as the product of the estimation error rates under these two models is faster
than root-n. We investigate the AIPCW estimator using extensive simulation in finite samples.
Keywords: Cox proportional hazards model; Rate doubly robust; AIPCW.
1 Introduction
In the analysis of time-to-event data, the Cox proportional hazards (PH) model (Cox,1972) has been widely used
to estimate the hazard ratio (HR) between two treatment groups in a randomized clinical trial, for example. The
∗Herbert Wertheim School of Public Health and Human Longevity Science, University of California, San Diego, La Jolla, CA 92093-
0112, USA. E-mail: jil130@ucsd.edu.
†Herbert Wertheim School of Public Health and Human Longevity Science, Department of Mathematics and Halicioglu Data Science
Institute, University of California, San Diego, La Jolla, CA 92093-0112, USA. E-mail: rxu@health.ucsd.edu.
1
arXiv:2206.02296v1 [stat.ME] 6 Jun 2022

validity of the maximum partial likelihood estimator (MPLE) under the PH model relies on the non-informative
censoring assumption (Fleming and Harrington,1991); that is, the censoring time random variable is independent
of the failure time random variable within each treatment group. In practice, this assumption can be violated which
leads to informative censoring, and the censoring time may well depend on additional covariates. This issue was
recently highlighted in Van Lancker et al. (2021), who aimed to develop procedures to select baseline covariates in
order to be adjusted for in the Cox regression model. Such adjustment, however, changes the effect estimand, making
it difficult to compare across different adjustment sets. Alternatively, the crude or marginal hazard ratio, as it is
often referred to in the medical literature, between the two groups can still be consistently estimated using inverse
probability of censoring weighting (IPCW) under the relaxed censoring assumption that the censoring time and the
failure time are independent given the additional covariates.
IPCW was proposed in Robins and Finkelstein (2000) to correct for bias resulting from informative censoring of
the log-rank test and, prior to that, in Robins (1993). Up until then, the main body of literature in both applied
and theoretical survival analysis had assumed non-informative censoring, given the predictors in a regression model
(Fleming and Harrington,1991). A separate line of research where IPCW was called for, was under violation of the
PH assumption, where it was recognized that the MPLE gave rise to a population quantity that involved the nuisance
censoring distribution (Xu,1996;Xu and O’Quigley,2000). A series of work has since been done to correct for this
bias using IPCW approaches, including Boyd et al. (2012); Hattori and Henmi (2012); Nguyen and Gillen (2017);
Nu˜no and Gillen (2021). We note that the terminology ‘IPCW’ was not always mentioned in some of these works,
which used the (conditional) survival distribution increments as weights in each risk set; but these are algebraically
equivalent to the inverse probability of censoring weights.
The censoring distribution used in IPCW is often modeled parametrically or semiparametrically, and the resulting
IPCW estimator is consistent and asymptotically normal (CAN) if the model is correctly specified. Nguyen and Gillen
(2017) proposed a survival tree approach to estimate the conditional censoring distribution given the covariates, but
with no theoretical guarantee for inference. In fact, it is known that the resulting estimator is typically biased
(Belloni et. al.,2013).
Doubly robust (DR) approaches were developed when handling missing data (Robins,1993;Robins et al.,1995;
Scharfstein et al.,1999;Robins et al.,2000b;Robins and Rotnitzky,2001;van der Laan and Robins,2003;Bang
and Robins,2005;Tsiatis,2006). It is called doubly robust because two working models are involved, one for the
outcome of interest, and one for the missing data mechanism, and the estimator is consistent as long as one of the
two working models are correctly specified. When IPW is used to handle the missingness (referred to as coarsening),
this usually comes down to augmentation with the coarsened data and the resulting DR estimator is an augmented
IPW (AIPW) estimator (Tsiatis,2006).
2

Since right censoring in survival data may be framed as a type of coarsening (Tsiatis,2006), Rotnitzky and Robins
(2005) developed an augmented IPCW (AIPCW) approach for censored survival data. For the PH model, however,
this approach is not straightforward to apply to. As will be seen later, this is mainly due to the non-collapsibility of
the Cox model (Martinussen and Vansteelandt,2013;Tchetgen Tchetgen and Robins,2012;Rava,2021).
In this paper, we consider simultaneously the regression parameter and the nuisance baseline hazard function
under the PH model. This naturally gives rise to full data estimating equations that are sums of independent and
identically distributed (i.i.d.) martingales. The augmentation leads to working models for the failure time and the
censoring time given the group indicator and the covariates. To specify a conditional failure time model that is
compatible with the original (marginal) PH model given the group membership only, data adaptive machine learning
(ML) or nonparametric methods are needed. With cross-fitting (Chernozhukov et al.,2018), the resulting AIPCW
estimator has doubly robust properties not only in the classical sense, which is referred to as model doubly robust, but
also rate doubly robust (Smucler et al.,2019;Hou et al.,2021). Here, rate double robustness refers to an estimator
being CAN when the product of the estimation error rates under the two working models is faster than root-n, while
either one of them is allowed to be arbitrarily slow.
The rest of the paper is organized as follows. In Section 1.1, we state the model and assumption about censoring.
In Section 2, we take a missing data approach by constructing the AIPCW score from the full data score, and
provide a detailed algorithm for the cross-fitted AIPCW estimator. Asymptotic properties of the AIPCW estimator
are described in Section 3. In Section 4, we conduct simulations for the AIPCW estimator using different nuisance
estimators, and also compare them with the IPCW estimators. Finally, we conclude with discussion in Section 5.
Additional materials are provided in the Appendix.
1.1 Model and assumption
Let Tand Cbe the failure time and the censoring time, respectively. Denote X= min(T , C), and ∆ = I(T≤C).
Denote also Y(t) = I(X≥t) the at-risk process, and N(t) = I(X≤t, ∆ = 1) the failure event counting process.
We consider the two-group survival setting where Ais a binary group indicator. For a randomized trial, this can be
the treatment groups. Let Zbe a p-dimensional vector of baseline covariates. We assume that the data consist of n
independent and identically distributed (i.i.d.) copies of the random vectors O= (X, ∆, A, Z).
Assumption 1. (informative censoring) C⊥T|(A, Z).
We assume the PH model for the two-group survival:
λ(t|A) = λ0(t) exp(βA),(1)
where λ(t|A) denotes the group-specific hazard function of T,βis the log hazard ratio, and λ0(t) is the baseline
3

hazard function.
2 Doubly robust inference
In this section following Tsiatis (2006) we treat right censoring as a coarsened data problem. We start with a set
of full data score functions under the PH model, and show that when IPCW is applied to this set of full data
score functions we obtain the familiar IPCW estimator under the Cox model (Boyd et al.,2012). We then mimic
the approach of Rotnitzky and Robins (2005) to augment the IPCW score functions and arrive at a doubly robust
AIPCW estimator. Finally, for inference purposes we introduce cross-fitting and describe the implementation of the
cross-fitted AIPCW estimator.
2.1 Full data score functions
The full data vector is (T, A, Z ). Following the commonly used NPMLE approach for the semiparametric PH model,
the unknown parameters are βand Λ0(t) = Rt
0λ0(u)du, the cumulative baseline hazard, which is discretized to jumps
at the observed event times only (Nielson et al.,1992).
Following Fleming and Harrington (1991), define the full data counting process NT(t) = I(T≤t) and the full
data at-risk process YT(t) = I(T≥t). Let
MT(t;β, Λ0) = NT(t)−Zt
0
YT(u)eβAdΛ0(u).(2)
Then MT(t;β, Λ0) is the full data martingale with respect to the full data filtration
Ff
t={NT(u), YT(u+), A, Z; 0 ≤u≤t}under model (1).
We have the following full data score functions for a single copy of the data:
Df
1(β, Λ0, t) = dMT(t;β , Λ0),
Df
2(β, Λ0) = Zτ
0
AdMT(t;β, Λ0).
where τis the maximum follow-up time. Note that Df
1(β, Λ0, t) is a martingale difference that is often used in
survival analysis; see for example, Lu and Ying (2004). For each t, the true values of the parameters βand Λ0satisfy
E{Df
1(β, Λ0, t)}= 0 and E{Df
2(β, Λ0)}= 0.(3)
4

2.2 IPCW score functions
In survival analysis, it’s common to consider the quantity
M(t) = N(t)−Zt
0
Y(u)eβAdΛ0(u).(4)
Note that it is not a martingale under informative censoring. We define Sc(t|A, Z) = P(C≥t|a, Z) the conditional
survival function of C,e
∆(t) = I(min(T, t)< C), and denote
dMw(t;β, Λ0, Sc) =Sc(t|A, Z)−1e
∆(t)dMT(t;β, Λ0)
=Sc(t|A, Z)−1dN (t)−Y(t)eβAdΛ0(t).(5)
Note that expression (5) gives the IPCW score functions:
Dw
1(β, Λ0, t;Sc) = dM w(t;β, Λ0, Sc),(6)
Dw
2(β, Λ0;Sc) = Zτ
0
AdMw(t;β, Λ0, Sc).(7)
With ncopies of i.i.d. data, this gives the following IPCW weighted estimating equations:
1
n
n
X
i=1
Dw
1i(β, Λ, t;Sc)=0,
1
n
n
X
i=1
Dw
2i(β, Λ; Sc)=0.
After some algebra, the above estimating equations can be combined to give the IPCW partial likelihood score
equation (Boyd et al.,2012):
n
X
i=1 Zτ
0b
Sc(t|Ai, Zi)−1(Ai−e
S(1)(β , t;b
Sc)
e
S(0)(β , t;b
Sc))dNi(t) = 0,(8)
where e
S(l)(β, t;Sc) = Pn
j=1 Al
jSc(t|Aj, Zj)−1Yj(t)eβAjfor l= 0,1, and b
Sc(t|A, Z) is some consistent estimator of
Sc(t|A, Z).
2.3 AIPCW score functions
The consistency of the IPCW estimator relies critically on Sc(t|A, Z) being correctly specified. When it is misspecified,
the IPCW estimator is biased. Rotnitzky and Robins (2005) provides an augmentation approach for an IPCW
estimator in survival analysis, so that it has the doubly robust property to be detailed later. However, their approach
cannot be directly applied because we have not only different weights for different individuals in the data set, but also
different weights for each risk set. To this end, it is helpful to augment the martingale increment in (5) as follows.
Denote Nc(t) = I(X≤t, ∆ = 0) the counting process for the censoring event, and Λc(t|A, Z) = Rt
0Sc(u|A, Z)−1d{1−
Sc(u|A, Z)}the cumulative hazard function of Cgiven A, Z. Then Mc(t;Sc) = Nc(t)−Rt
0Y(u)dΛc(u|A, Z) is the
5

martingale corresponding to the censoring event counting process with respect to its natural history filtration. Also
denote S(t|A, Z) = P(T≥t|A, Z ), and F(t|A, Z) = 1 −S(t|A, Z ). Define
dMaug (t;β, Λ0, S, Sc) = dMw(t;β, Λ0, Sc) + Zt
0
E{dMT(t;β, Λ0)|A, Z, T ≥u}dMc(u;Sc)
Sc(u|A, Z)(9)
=dN(t)−Y(t)dΛ0(t)eβA
Sc(t|A, Z)−J(t;S, Sc)dS (t|A, Z ) + S(t|A, Z)eβA dΛ0(t),(10)
where J(t;S, Sc) = Rt
0S(u|A, Z)−1Sc(u|A, Z )−1dMc(u;Sc). The last ‘=’ above used the fact that, for u≤t,
E{NT(t)|A, Z, T ≥u}=P(T≤t|A, Z, T ≥u) = F(t|A, Z )
S(u|A, Z),(11)
E{YT(t)|A, Z, T ≥u}=P(T≥t|A, Z, T ≥u) = S(t|A, Z )
S(u|A, Z).(12)
The above leads to the AIPCW score functions:
D1(β, Λ0, t;S, Sc) = dM aug(t;β , Λ0, S, Sc),(13)
D2(β, Λ0;S, Sc) = Zτ
0
A·dMaug (t;β, Λ0, S, Sc).(14)
In Theorem 1below, we will show that (13) and (14) are doubly robust score functions. We use superscript o
to denote the truth; for example, So(t|A, Z ), So
c(t|A, Z) and Λo
c(t|A, Z) denote the true S(t|A, Z ), Sc(t|A, Z) and
Λc(t|A, Z), respectively. Also let βoand Λo
0denote the true values of the parameters of interest. We assume the
following:
Assumption 2. So(τ|a, z)> c for a∈ {0,1}, z ∈ Z and some c > 0.
Assumption 3. So
c(τ|a, z)> c for a∈ {0,1}, z ∈ Z and some c > 0.
Theorem 1. Under Assumptions 1-3, if either S=Soor Sc=So
c,
E{D1(βo,Λo
0, t;S, Sc)}=E{D2(βo,Λo
0;S, Sc)}= 0.(15)
The above theorem states that the scores (D1, D2) identifies the true parameters (βo,Λo
0),as long as one of the
two survival functions, S(t|A, Z) and Sc(t|A, Z ), is true.
Given ni.i.d. data points, we estimate βo,Λoby solving
1
n
n
X
i=1
D1i(β, Λ0, t;S, Sc)=0,(16)
1
n
n
X
i=1
D2i(β, Λ0;S, Sc)=0.(17)
Solving for (16) gives
e
Λ0(β, t;S, Sc) = Zt
0
1
nPn
i=1 Sc(u|Ai, Zi)−1dNi(u)−Ji(u;S, Sc)dS(u|Ai, Zi)
S(0)(β , u;S, Sc),(18)
6

where
S(l)(β, t;S, Sc) = 1
n
n
X
i=1
Al
ieβAi{Sc(u|Ai, Zi)−1Yi(t) + Ji(t;S, Sc)S(t|Ai, Zi)}(19)
for l= 0,1. Further define ¯
A(β, t;S, Sc) = S(1) (β, t;S, Sc)/S(0)(β , t;S, Sc). After plugging (18) into (17), we have:
U(β;S, Sc) = 1
n
n
X
i=1 Zτ
0{Sc(t|Ai, Zi)−1dNi(t)−Ji(u;S, Sc)dS(t|Ai, Zi)}{Ai−¯
A(β, t;S, Sc)}= 0.(20)
It’s worth noting that like the partial likelihood score equation, (20) is not a sum of i.i.d terms due to ¯
A(β, t;S, Sc).
As seen from the derivation leading to (10), the augmentation to the weighted martingale increment, which is linear
in N(t) and Y(t), is the result of augmentation to the weighted N(t) and Y(t), respectively. It is apparent that
Sc(t|Ai, Zi)−1dNi(t)−Ji(t;S, Sc)dS(t|Ai, Zi) is the augmented weighted dNi(t), and the augmented weighted Yi(t)’s
give rise to the quantities S(l)(·) and ¯
A(·), which are the analogies of similar quantities under the usual Cox model.
For example, ¯
A(β, t;S, Sc) corresponds to the empirical mean of the treatment random variable Aamong subjects
who fail at time t, which we may denote by ρ(β, t).
The quantity ρ(β, t) was implied in Rotnitzky and Robins (2005), as a nuisance parameter, based on the partial
likelihood score function. It would, however, not be straightforward to construct compatible models for ρ(β, t), which
is defined on nested risk sets over time. The set of full data estimating functions we consider here, simultaneously
for βand Λ0, on the other hand, lead naturally to models for Sand Sc.
2.4 Cross-fitted AIPCW estimator
In practice, both survival functions S(t|A, Z) and Sc(t|A, Z ) are unknown and need to be estimated by some estimator
b
S(t|A, Z) and b
Sc(t|A, Z). Parametric and semiparametric models, like the Cox model and the accelerated failure time
(AFT) model, are often applied since their theoretical properties are well-studies and with little requirement on the
computing power. However, these models can be misspecified, especially for S(t|A, Z ) due to the non-collapsibility
of the Cox model. ML or nonparametric methods, like splines (Gary,1992;Kooperberg et al.,1995a) and random
survival forest (Ishwaran et al.,2008), offer a good alternative. ML or nonparametric estimators, however, do not
have root-nconvergence rate, which makes it difficult to conduct inference. We will show that the asymptotic
normality can be established if we also apply cross-fitting, where the entire sample is first split into kfolds, and for
each fold, we estimate the nuisance functions using only the out-of-fold sample. Details of the cross-fitted AIPCW
estimator b
βare described in Algorithm 1. Heuristically, cross-fitting works by inducing independence between the
nuisance parameter estimators and the rest of the quantities in the scores, thereby allowing asymptotic normality to
be established (Smucler et al.,2019;Hou et al.,2021).
Quantities involving cross-fitting would be slightly different from quantities without cross-fitting, and will involve
7

Algorithm 1 k-fold Cross-fitted AIPCW estimation of β
Input: A sample of nobservations that are split into kfolds of equal size with index sets
I1,I2,...,Ik.
for each fold indexed by mdo
obtain estimated nuisance functions (b
S(−m),b
S(−m)
c) using the out-of-fold sample indexed by I−m:={1, . . . , n}\
Im.
end for
Output: b
β, the solution to
1
n
n
X
i=1
D1i(β, Λ0, t;b
S(−m(i)),b
S(−m(i))
c)=0,(21)
1
n
n
X
i=1
D2i(β, Λ0;b
S(−m(i)),b
S(−m(i))
c)=0,(22)
where m(i) maps observation ito index of the fold it belongs to.
the estimated nuisance parameters (b
S(−1),b
S(−1)
c),...,(b
S(−k),b
S(−k)
c). Specifically, solving for (21), we will have
e
Λcf
0(β, t;b
S, b
Sc) = Zt
0
1
nPn
i=1 b
S(−m(i))
c(u|Ai, Zi)−1dNi(u)−Ji(u;b
S(−m(i)),b
S(−m(i))
c)db
S(−m(i))(u|Ai, Zi)
S(0)(β , u;b
S, b
Sc),(23)
with
S(l)
cf (β, t;b
S, b
Sc) = 1
n
n
X
i=1
Al
ieβAi{b
S(−m(i))
c(t|Ai, Zi)−1Yi(t) + Ji(t;b
S(−m(i)),b
S(−m(i))
c)b
S−m(i))(t|Ai, Zi)}(24)
for l= 0,1.
Also, ¯
Acf (β, t;b
S, b
Sc) = S(1)
cf (β, t;b
S, b
Sc)/S(0)
cf (β, t;b
S, b
Sc), and after plugging (23) into (22), we have the final cross-
fitted AIPCW estimating equation:
Ucf (β;b
S, b
Sc)
=1
n
n
X
i=1 Zτ
0{b
S(−m(i))
c(t|Ai, Zi)−1dNi(t)−Ji(t;b
S(−m(i)),b
S(−m(i))
c)db
S(−m(i))(t|Ai, Zi)}{Ai−¯
A(β, t;b
S, b
Sc)}.(25)
We solve the cross-fitted AIPCW estimating equation (25) using the Newton-Ralphson algorithm.
3 Asymptotic Properties
We will now describe the asymptotic properties of the proposed cross-fitted AIPCW estimator, when estimated using
a random sample of size n. We first list a few additional assumptions.
8

Assumption 4. There exist S∗(t|a, z)and S∗
c(t|a, z)with S∗(τ|a, z)> c and S∗
c(τ|a, z)> c for some c > 0, such
that
sup
t∈[0,τ],a∈{0,1},z∈Z |b
S(t|a, z)−S∗(t|a, z)|=Op(an),
sup
t∈[0,τ],a∈{0,1},z∈Z |b
Sc(t|a, z)−S∗
c(t|a, z)|=Op(bn),
for some an=o(1) and bn=o(1).
Assumption 5. For the limits S∗and S∗
c, there exists a neighbourhood Bof βoand functions s(l)(β, t;S∗, S∗
c)for
l= 0,1defined on B × [0, τ ]such that supt∈[0,τ],β∈B |S(l)(β , t;S∗, S∗
c)−s(l)(β, t;S∗, S ∗
c)|=op(1).
Assumption 6. For l= 0,1,s(l)(β, t;S∗, S∗
c)are continuous functions of β∈ B, uniformly in t∈[0, τ ]and are
bounded on B × [0, τ ].s(0)(β, t;S∗, S ∗
c)is bounded away from zero in B × [0, τ]. For all β∈ B,t∈[0, τ ]:
s(1)(β , t;S∗, S∗
c) = ∂
∂β s(0) (β, t;S∗, S∗
c) = ∂2
∂β2s(0)(β, t;S∗, S ∗
c).(26)
In addition, let ¯a=s(1)/s(0) and v= ¯a−¯a2. We have ν(βo;S∗, S ∗
c) = Rτ
0v(βo, t;S∗, S∗
c)s(0)(βo, u;S∗, S ∗
c)dΛo
0(t)>0.
Assumption 4assumes that both b
Sand b
Scconverge to some limiting function S∗and S∗
cthat are not necessarily
the truth. Here, we do not make the root-nconvergence assumption for each of b
Sand b
Scthat often limits us to
parametric or semiparametric models. This assumption also implies that b
S(−m)and b
S(−m)
cconverge to S∗and S∗
c
at the same rate. Assumptions 5and 6are similar to regularity assumptions that are typically made under the PH
models (Anderson and Gill,1982).
The asymptotic properties of the cross-fitted AIPCW estimator b
βdefined in Algorithm 1are summarized in
Theorems 2and 3below.
Theorem 2. Under Assumptions 4-6, if either S∗=Soor S∗
c=So
c, then b
βp
→βo.
Theorem 3. Under Assumptions 4-6, if any of the following conditions hold:
(a) (Rate Double Robustness) S∗=So, S∗
c=So
cand anbn=o(n−1/2);
(b) (Model Double Robustness) S∗=Soand an=O(n−1/2). In particular, there exists an influence function
ξ(t, a, z)such that b
S(t|a, z)−S∗(t|a, z) = Pn
j=1 ξj(t, a, z)/n +op(n−1/2);
(c) (Model Double Robustness) S∗
c=So
cand bn=O(n−1/2). In particular, there exists an influence function
η(t, a, z)such that b
Sc(t|a, z)−S∗
c(t|a, z) = Pn
j=1 ηj(t, a, z)/n +op(n−1/2),
then we have
√n(b
β−βo) = 1
√n
n
X
i=1
ν(βo, S∗, S ∗
c)−1ψi(βo,Λo
0, S∗, S ∗
c) + op(1),(27)
9

where the expression for ψi(βo,Λo
0, S∗, S ∗
c)is provided in Appendix A.
Theorem 3establishes both the model double robustness and the rate double robustness properties. Traditionally,
doubly robust inference is established assuming both working models are parametric or semiparametric. Model
double robustness here allows estimation under the possibly wrong model to converge at any rate. The theorem also
establishes rate double robustness, which states that if the estimators under both working models converge to the
truth and that their product rate is faster than root-n, the proposed AIPCW estimator is CAN even if one of the
nuisance estimators converges arbitrarily slowly. This result permits more flexible ML or nonparametric methods
with valid inference.
The asymptotic variance of the proposed estimator is simplified under condition (a). In this case, we provide an
estimator of the asymptotic variance, which is given in the Theorem 4below.
Theorem 4. Under Assumptions 4-6, if condition (a) of Theorem 3holds, i.e. if S∗=So,S∗
c=So
cand anbn=
op(n−1/2), then bν−2K/n is a consistent estimator for the asymptotic variance of b
β, where bνand Kare provided in
Appendix A.
When one of the working models is misspecified, the asymptotic variance is rather complicated. In this case,
resampling methods such as bootstrap (Efron,1979) may be used to estimate the variance since the AIPCW estimator
is asymptotically linear.
4 Simulation
In this section, we compare the performance of the cross-fitted AIPCW estimators b
βusing different working models,
against different IPCW estimators and the MPLE. We consider sample sizes n= 500 and n= 1000, and 1000 data
sets are simulated for each setting, which corresponds to margin of error of about +/−1.35% for the coverage
probability of nominal 95% confidence intervals. Five-fold cross-fitting is used.
For data generation, we first follow the diagram in Figure 1(a) and generate U1∼Unif (-1, 1), A∼Bernoulli
(0.5), Z1∼N(0.5U1,1), Z2∼N(U2
1,0.09), and T=−log(0.5U1+ 0.5)e−A. Here, Tfollows the PH model (1) with
βo=−1 and λo
0(t) = 1.
We consider two scenarios of data generating process for the censoring time C, as described in Figure 1(b). Both
scenarios have around 25% samples administratively censored at τ= 1, and 40% of the remaining samples censored
during follow-up. Note that administrative censoring works in the same way for Tand C, i.e. those events are
consider as ‘censored’ for both the estimation of Sand the estimation of Sc. It is obvious that Scenario 1 can be
correctly modeled. Scenario 2 is designed such that most commonly used semiparametric models fail. As it turns
10

out, under Scenario 2 Sc(τ|A, Z) can be very close to zero for some values of Aand Z, leading to possible violation of
Assumptions 2and 3. This echoes the argument made in D’Amour et al. (2021) that the overlap assumption needed
for DR estimates often fails in practice.
T C
A
Z
U1
U2Scenario Data generating process for C
1: Cox PH λc(t) = exp(−1+2Z2)
2: Mixture
log(U2)∼N(0,1)
Z1>0 : log(C) = −0.2A−2p|Z2|+ 0.3U2
Z1≤0 : log(C)=2.4−0.3A+ 0.5p|Z1|
+0.5p|Z2| − U2
(a) (b)
Figure 1: (a) Variable diagram. (b) Data generating process for C.
We consider three types of working models: PH model using the R package ‘survival’; splines (Kooperberg et al.,
1995a) using the R package ‘polspline’; and random survival forest (RSF) (Ishwaran et al.,2008) using the R package
‘randomForestSRC’. We set splitrule = ’bs.gradient’ for RSF, while keeping all the others settings as default. We
study 7 different combinations of working models for the proposed AIPCW estimator: Cox-Cox, Cox-spline, Cox-
RSF, spline-Cox, RSF-Cox, spline-spline, and RSF-RSF, where the first part in the names denotes the model for S
and the second part denotes the model for Sc. It is worth noting that due to the non-collapsibility of the Cox model,
a semiparametric conditional model for Sis almost always misspecified. Therefore the consistency of AIPCW-Cox-
Cox, AIPCW-Cox-spline and AIPCW-Cox-RSF relies on the correct specification of the censoring model. We also
note that the convergence rate of the spline and RSF is largely unknown, which depends on the choice of tuning
parameters. See Discussion for more on this.
We also investigate the performance of MPLE and various IPCW estimators: IPCW-Cox, IPCW-spline, IPCW-
RSF, IPCW-A and IPCW-1. More specifically, IPCW-A estimates Scusing the product-limit estimator for each
group indicated by A, while IPCW-1 estimates Scusing the product-limit estimator on the entire sample. Robust
variance estimator from Boyd et al. (2012) is used to estimate the model standard errors of the IPCW estimators.
Standard errors for the cross-fitted AIPCW estimators are estimated using Theorem 4, which assumes both Sand
Scmodels are correctly specified.
To avoid numerical problems, we impose a minimum on b
S(−m)(t|A, Z) and b
S(−m)
c(t|A, Z) in the above, so that
values below 0.01 are trimmed to be 0.01. Finally, as a benchmark, we also fit model (1) to the full data without
11

censoring.
The simulation results for Scenarios 1 and 2 are reported in Tables 1and 2, respectively. It is immediate that
under informative censoring, MPLE, IPCW-1 and IPCW-A have substantial bias leading to poor coverage of the
confidence intervals (CI). Under Scenario 1 where the censoring model is correctly specified as Cox, the other three
IPCW estimators (-Cox, -spline, -RSF) all appear to perform reasonably well. All seven AIPCW estimators also
perform well under Scenario 1, with AIPCW-Cox-RSF having larger bias compared to the rest.
Under Scenario 2, IPCW-Cox appears more biased than IPCW-spline and IPCW-RSF, as expected. But even for
the latter two estimators, their SE’s severely under-estimate the SD’s, leading to poor coverage of the CI’s. This also
points to the known fact that inference is not guaranteed when ML or nonparametric methods are used in IPCW,
as discussed earlier. AIPCW-Cox-Cox also has large bias under Scenario 2, as expected. The rest six AIPCW’s are
less biased. For the larger sample size n= 1000, AIPCW using two ML or nonparametric methods appears to have
the least bias, with close to nominal coverage probabilities. Finally we note that, under Scenario 2, spline-based
AIPCWs tend to have larger variance. This might be explained by the fact that splines are less stable near the
boundary τ, which under Scenario 2 has small b
Sc(τ|A, Z) for some values of Aand Zas mentioned earlier.
12

Table 1: Simulation results under Scenario 1. Data are generated following Figures 1(a) and (b) with
βo=−1. Red indicates that the model or approach is invalid.
Sample Size Estimators Bias SD SE CP
n= 500
AIPCW-Cox-Cox 0.002 0.196 0.191 0.94
AIPCW-Cox-spline -0.001 0.198 0.190 0.94
AIPCW-Cox-RSF 0.023 0.197 0.207 0.96
AIPCW-spline-Cox 0.005 0.185 0.177 0.94
AIPCW-RSF-Cox 0.005 0.189 0.178 0.94
AIPCW-spline-spline 0.002 0.185 0.177 0.94
AIPCW-RSF-RSF 0.002 0.192 0.190 0.95
IPCW-Cox -0.006 0.186 0.179 0.94
IPCW-spline -0.005 0.188 0.179 0.94
IPCW-RSF 0.008 0.190 0.177 0.93
IPCW-A-0.221 0.180 0.162 0.70
IPCW-1-0.221 0.179 0.162 0.70
MPLE -0.205 0.175 0.167 0.76
Full data 0.002 0.103 0.099 0.93
n= 1000
AIPCW-Cox-Cox -0.008 0.137 0.134 0.94
AIPCW-Cox-spline -0.010 0.138 0.133 0.94
AIPCW-Cox-RSF 0.019 0.141 0.153 0.97
AIPCW-spline-Cox 0.001 0.127 0.123 0.94
AIPCW-RSF-Cox 0.002 0.130 0.125 0.94
AIPCW-spline-spline 0.001 0.127 0.123 0.94
AIPCW-RSF-RSF -0.005 0.134 0.134 0.95
IPCW-Cox -0.009 0.130 0.128 0.94
IPCW-spline -0.007 0.135 0.128 0.94
IPCW-RSF 0.011 0.134 0.128 0.95
IPCW-A-0.225 0.126 0.114 0.51
IPCW-1-0.224 0.126 0.114 0.51
MPLE -0.207 0.122 0.118 0.58
Full data -0.003 0.069 0.07 0.94
SD: standard deviation; SE: standard error; CP: coverage probability of nominal 95% CI
13

Table 2: Simulation results under Scenario 2. Data are generated following Figures 1(a) and (b) with
βo=−1. Red indicates that the model or approach is invalid.
Sample Size Estimators Bias SD SE Coverage
n= 500
AIPCW-Cox-Cox -0.129 0.285 0.276 0.93
AIPCW-Cox-spline -0.029 0.604 0.623 0.97
AIPCW-Cox-RSF -0.064 0.249 0.243 0.93
AIPCW-spline-Cox -0.068 0.282 0.256 0.93
AIPCW-RSF-Cox -0.034 0.275 0.250 0.93
AIPCW-spline-spline 0.038 0.578 0.585 0.96
AIPCW-RSF-RSF -0.039 0.264 0.238 0.93
IPCW-Cox -0.114 0.266 0.174 0.77
IPCW-spline -0.046 0.452 0.192 0.68
IPCW-RSF -0.088 0.257 0.179 0.80
IPCW-A-0.227 0.184 0.170 0.74
IPCW-1-0.226 0.183 0.166 0.72
MPLE -0.216 0.179 0.174 0.77
Full data 0.002 0.103 0.099 0.93
n= 1000
AIPCW-Cox-Cox -0.127 0.195 0.192 0.90
AIPCW-Cox-spline -0.056 0.396 0.367 0.95
AIPCW-Cox-RSF -0.035 0.187 0.189 0.95
AIPCW-spline-Cox -0.056 0.191 0.180 0.93
AIPCW-RSF-Cox -0.021 0.185 0.178 0.92
AIPCW-spline-spline 0.008 0.344 0.332 0.95
AIPCW-RSF-RSF -0.020 0.198 0.179 0.93
IPCW-Cox -0.103 0.204 0.126 0.71
IPCW-spline -0.045 0.377 0.146 0.63
IPCW-RSF -0.047 0.202 0.134 0.78
IPCW-A-0.220 0.127 0.120 0.56
IPCW-1-0.219 0.127 0.117 0.53
MPLE -0.211 0.123 0.123 0.61
Full data -0.003 0.069 0.07 0.94
SD: standard deviation; SE: standard error; CP: coverage probability of nominal 95% CI
14

5 Discussion
For the analysis of two-group survival, including for randomized clinical trials, non-informative censoring is assumed.
When the simple PH model (1) is used with no covariates adjusted for, this requires the censoring distribution to be
independent of any covariates. When this assumption is violated, the commonly used MPLE is biased and typically
IPCW is used to correct that bias if the interest remains to estimate the marginal hazard ratio between the two
groups. IPCW, on the other hand, requires modeling the censoring distribution, which can be wrong unless ML or
nonparametric estimates are used. In this paper we have developed an AIPCW estimator that is both model DR
and rate DR. Rate double robustness allows us to get around the non-collapsibility of the Cox regression model using
more flexible ML or nonparametric methods for the conditional failure time model demanded by the DR construct,
because almost any parametric or semiparametric would otherwise be invalid.
The theoretical results require certain rate condition of the estimates of the nuisance parameters. These are not
always established for a given ML or nonparametric estimator. Cui et al. (2022) and Kooperberg et al. (1995b)
demonstrated that under certain conditions, rate better than n1/4can be achieved for random survival forest and
splines, which would lead to a faster than root-nproduct rate. Faster than n1/4rate are also shown to be attainable
for other ML methods, for example, regression trees (Wager and Walther,2015) and neural networks (Chen and
White,1999). The rates, of course, depend on the hyper-parameter values. In the simulations we used the default
settings for the spline and the random survival forest. Investigation of other ML or nonparametric methods, as well
as their tuning, in relationship with the performance of DR estimators, remains a topic of future work.
This work focused on two-group survival and a binary A. Generalization to continuous and/or multivariate Ais
conceptually straightforward although different algebra might be involved. In particular for continuous A, we would
no longer have A2=Aand additional quantities like S(2) need to be introduced.
Finally the models for Sand Scmay include additional and different sets of covariates for these two models, so
long as the failure time and the censoring time are independent given the common covariates Z.
The R codes for the cross-fitted AIPCW estimator as well as the simulation procedures investigated in this work
are available online in http://github.com/charlesluo1002/DR-Cox.
15

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20

Appendix
A Notation and Expressions
First, we list or repeat notations that will be used in the proofs. For iin 1, . . . , n, we define
Mci(t;Sc) = Nc(t)−Zt
0
Y(u)dΛc(u|Ai, Zi),
Ji(t;S, Sc) = Zt
0
S(u|Ai, Zi)−1Sc(u|Ai, Zi)−1dMc(u;Sc),
dNi(t;S, Sc) = Sc(t|Ai, Zi)−1dNi(t)−Ji(t;S, Sc)dS(t|Ai, Zi),
Γ(l)
i(β, t;S, Sc) = Al
ieβAi{Sc(t|Ai, Zi)−1Yi(t) + Ji(t;S, Sc)S(t|Ai, Zi)},
S(l)(β, t;S, Sc) = 1
n
n
X
i=1
Γ(l)
i(β, t;S, Sc),
dMaug
i(t;β, Λ0, S, Sc) = dNi(t, S, Sc)−Γ(0)
i(β, t;S, Sc)dΛ0(t),
¯
A(β, t;S, Sc) = S(1) (β, t;S, Sc)/S(0)(β , t;S, Sc),
U(β;S, Sc) = 1
n
n
X
i=1 Zτ
0
dNi(t;S, Sc){Ai−¯
A(β, t;S, Sc)},
V(β, t;S, Sc) = d¯
A(β, t;S, Sc)/dβ =¯
A(β, t;S, Sc)−¯
A(β, t;S, Sc)2,
¯a(β, t;S, Sc) = s(1) (β, t;S, Sc)/s(0) (β , t;S, Sc),
v(β, t;S, Sc) = ¯a(β , t;S, Sc)−¯a(β , t;S, Sc)2,
µ(β;S, Sc) = Zτ
0{¯a(βo, t;S, Sc)−¯a(β, t;S, Sc)}s(0)(βo, t;S, Sc)dΛo
0(t),
ν(β;S, Sc) = Zτ
0
v(β, t;S, Sc)s(0) (βo, t;S, Sc)dΛo
0(t).
21

Next are expressions used in the asymptotic results. We stated in Theorem 3that
√n(b
β−βo) = 1
√n
n
X
i=1
ν(βo, S∗, S ∗
c)−1ψi(βo,Λo
0, S∗, S ∗
c) + op(1).(28)
Here ψi(βo,Λo
0, S∗, S ∗
c) = ψ1i+ψ2i+ψ3iwhere
ψ1i=Zτ
0{Ai−¯a(βo, t;S∗, S∗
c)}dMaug
i(t;βo,Λo
0, S∗, S ∗
c),(29)
ψ2i=k
n(k−1) X
j∈I−m(i)Zτ
0{¯a(βo, t;S∗, S∗
c)−Ai}Ji(t;S∗, S∗
c){dξj(t, Ai, Zi) + eβoAiξj(t, Ai, Zi)dΛo
0(t)},
ψ3i=k
n(k−1) X
j∈I−m(i)Zτ
0{Ai−¯a(βo, t;S∗, S∗
c)} ηj(t, Ai, Zi)
S∗
c(t|Ai, Zi)2{dNi(t)−Yi(t)eβoAidΛo
0(t)}
−Zt
0S∗(u|Ai, Zi)
S∗
c(u|Ai, Zi)2ηj(u, Ai, Zi){dMci(u;S∗
c) + Yi(u)S∗
c(u|Ai, Zi)−1dS∗
c(u|Ai, Zi)}
+S∗(u|Ai, Zi)−1S∗
c(u|Ai, Zi)−2Yi(u)dηj(u, Ai, Zi){dS∗(t|Ai, Zi) + S∗(t|Ai, Zi)eβoAidΛo
0(t)}!.
The long expression above simplifies depending on which models are correctly specified. Under case (a) of Theorem 3,
when both Sand Scare correctly specified, ψ2i=ψ3i= 0. Under case (b), when b
Sis √n-consistent , ψ3i= 0. Under
case (c), when b
Scis √n-consistent, ψ2i= 0.
In Theorem 4, we stated that bν−2K/n is a consistent estimator for the asymptotic variance of b
βwhen both
S∗=Soand S∗
c=So
care correctly specified. The expression for Kand bνare as follows:
bν=1
n
n
X
i=1 Zτ
0
V(b
β, t;b
S, b
Sc)dNi(t;b
S(−m(i)),b
S(−m(i))
c),
K=1
n
n
X
i=1 e
ψ1i(b
β, e
Λ0(b
β, ·;b
S, b
Sc),b
S(−m(i)),b
S(−m(i))
c)2,
where
e
ψ1i(β, Λ0, S, Sc) = Zτ
0{Ai−¯
A(β, t;S, Sc)}dM aug
i(t;β, Λ0, S, Sc).
22

B Proof of Double Robustness
Lemma 1. For any Sc(t|A, Z)with its corresponding censoring specific martingale Mc(t;Sc),
Zt
0
dMc(u;Sc)
Sc(u|A, Z)= 1 −Y(t)
Sc(t|A, Z)−N(t−)
Sc(X|A, Z),(30)
where N(t−) = I(X < t, T ≤C).
Note, this can be seen as a continuous version of Lemma 10.4 in Tsiatis (2006).
Proof
First note that
Zt
0
dNc(u)
Sc(u|A, Z)=Nc(t−)
Sc(X|A, Z),(31)
where NC(t−) = I(X < t, T > C). Next, since Sc(u|A, Z ) = exp{−Λc(u|A, Z)},
Zt
0
−Y(u)dΛc(u|A, Z)
Sc(u|A, Z)
=I(X≥t)Zt
0
dSc(u|A, Z)
Sc(u|A, Z)2+I(X < t)ZX
0
dSc(u|A, Z)
Sc(u|A, Z)2
=I(X≥t){−Sc(u|A, Z)−1}|u=t
u=0 +I(X < t){−Sc(u|A, Z )−1}|u=X
u=0
= 1 −Y(t)
Sc(t|A, Z)−I(X < t)
Sc(X|A, Z).(32)
Since I(X < t) = N(t−) + Nc(t−), (31) + (32) then gives the lemma.
Proof of Theorem 1
Recall that
dMaug (t;β, Λ0, S, Sc) = dMw(t;β, Λ0, Sc)−J(t;S, Sc)dS(t|A, Z ) + S(t|A, Z)eβAdΛ0(t),
where J(t;S, Sc) is also included in Appendix A.
a) Assume Sc=So
c.
23

We first consider dMw(t;βo,Λo
0, So
c). For h(A) = 1 or A,
E{h(A)dMw(t;βo,Λo
0, So
c)}
=Eh(A)So
c(t|A, Z)−1[dE {I(T≤t)I(C≥t)|T=t, A, Z }
−E{I(T≥t)I(C≥t)|T=t, A, Z} · eβoAdΛo
0(t)io
=E[h(A)So
c(t|A, Z)−1{dI (T≤t)P(C≥t|A, Z )
−I(T≥t)P(C≥t|A, Z)·eβoAdΛo
0(t)}]
=E{h(A)dMT(t;βo,Λo
0)}
= 0,
where the second ‘=’ above uses the informative censoring Assumption 1.
Next we consider J(t;S, Sc){dS(t|A, Z ) + S(t|A, Z)eβoAdΛo
0(t)}. Its expectation being zero follows immediately
from the fact that Mc(t;So
c) is a martingale.
b) Assume S=So.
Noting that YT(t)N(t−) = N(t−)dNT(t) = 0 and Y(t)dNT(t) = dN(t), we multiply (30) by
dMT(t) = dNT(t)−YT(t)eβoAdΛ0(t) giving:
dMT(t;βo,Λo
0)Zt
0
dMc(u;Sc)
Sc(u|A, Z)
=dNT(t)Zt
0
dMc(u;Sc)
Sc(u|A, Z)−YT(t)eβoAdΛo
0(t)Zt
0
dMc(u;Sc)
Sc(u|A, Z)
=dNT(t)−dNT(t)Y(t)
Sc(t|A, Z)−dNT(t)N(t−)
Sc(X|A, Z)−YT(t)eβoAdΛo
0(t) + Y(t)eβoAdΛo
0(t)
Sc(t|A, Z)+YT(t)N(t−)eβoAdΛo
0(t)
Sc(X|A, Z).
=dMT(t)−dMw(t).
Therefore
dMw(t;βo,Λo
0) = dMT(t;βo,Λo
0)−dMT(t;βo,Λo
0)Zt
0
dMc(u;Sc)
Sc(u|A, Z).
24

We note that (11) and (12) hold when S=So. From (9) then we have
E{dMaug (t;βo,Λo
0, So, Sc)}
=EdMw(t;βo,Λo
0, Sc) + Zt
0
E{dMT(t;βo,Λo
0)|A, Z, T ≥u}dMc(u;Sc)
Sc(u|A, Z)
=EdMT(t;βo,Λo
0)−dMT(t;βo,Λo
0)Zt
0
dMc(u;Sc)
Sc(u|A, Z)+Zt
0
E{dMT(t;βo,Λo
0)|A, Z, T ≥u}dMc(u;Sc)
Sc(u|A, Z)
=EZt
0hE{dMT(t;βo,Λo
0)|A, Z, T ≥u} − dMT(t;βo,Λo
0)idMc(u;Sc)
Sc(u|A, Z)
=EEZt
0hE{dMT(t;βo,Λo
0)|A, Z, T ≥u} − dMT(t;βo,Λo
0)idNc(u)
Sc(u|A, Z)A, Z, T ≥u, C =u
−EEZt
0hE{dMT(t;βo,Λo
0)|A, Z, T ≥u} − dMT(t;βo,Λo
0)iY(u)dΛc(u)
Sc(u|A, Z)A, Z, T ≥u, C ≥u
=EZt
0
dNc(u)
Sc(u|A, Z)hE{dMT(t;βo,Λo
0)|A, Z, T ≥u, C =u} − E{dMT(t;βo,Λo
0)|A, Z, T ≥u, C =u}i
−EZt
0
Y(u)dΛc(u)
Sc(u|A, Z)hE{dMT(t;βo,Λo
0)|A, Z, T ≥u, C ≥u} − E{dMT(t;βo,Λo
0)|A, Z, T ≥u, C ≥u}i
=0,
where in the 3rd line above E{dMT(t;βo,Λo
0)}= 0 because MT(t;βo,Λo
0) is a martingale.
The above also gives
EZt
0
AdMaug (t;βo,Λo
0, So, Sc)= 0.
25