Doubly robust weighted log-rank tests and Renyi-type tests under non-random treatment assignment and dependent censoring

Abstract

The log-rank test is widely used to test difference in event time distribution between treatment groups. However, if subjects are not randomly assigned to treatment groups, which is often the case in observation studies, the log-rank test is not asymptotically correct for detecting group survival difference due to the imbalance of confounding variables between groups. We develop a class of modified weighted log-rank tests and Renyi-type tests for two-sample survival comparison under non-random treatment assignment. The new tests can also account for non-random censoring that depends on baseline covariates. The proposed methods involve building working models for treatment assignment, cause-specific hazard of dependent censoring, and the time to event. We prove that, when either the models for treatment assignment and dependent censoring or the model for the event time is true, the new tests are asymptotically correct, i.e. being doubly robust. Numerical experiments demonstrate the tests’ double-robustness property in finite samples of realistic sizes, and also show that the doubly robust log-rank test is at least as powerful as the regular log-rank test when the treatment assignment is random and there is no dependent censoring. An application to a kidney transplant data set illustrates the utility of the proposed methods.

Keywords

Double robustness weighted log-rank tests Renyi-type tests dependent censoring confounding

1 Introduction

The log-rank test is the most popular method for two-sample survival comparison, and it is the most powerful nonparametric test when the two hazard functions differ in a proportional fashion. Its variants, weighted log-rank tests (Fleming and Harrington,¹ Chapter 7) and the Renyi-type tests,² are powerful to detect various types of hazard difference other than proportionality, such as early departure and crossing. For all these tests to be valid, the independent censoring assumption (Condition (3.7) in Fleming and Harrington¹) must be satisfied in each of the two samples. When the tests are used to compare two treatments with respect to a survival endpoint, the treatment assignment must be random for having a causal interpretation of the tests’ results.

In observational studies, whether or not a subject receives certain treatment often depends on some variables of the subject, e.g. age, health condition and insurance coverage. These variables are also often influential to the event time of interest. Thus they are so-called confounders for evaluating the treatment effect. Xie and Liu³ developed an inverse-probability-of-treatment weighted (IPTW) log-rank test to deal with confounding in two-sample survival comparison. However, as shown in Li,⁴ the IPTW test is conservative due to not accounting for the variability of the estimated weights. The independent censoring assumption is also violated in many applications. For instance, in studies with long follow-ups, there is usually a considerable percentage of censoring due to loss to follow-up rather than administrative reasons, and the lost subjects often differ from the remaining group in some baseline characteristics such as age, education, marital status, physical fitness, and socioeconomic status that have effects on the even time of interest. Robins and Finkelstein⁵ proposed an inverse-probability-of-censoring weighted (IPCW) log-rank test to deal with dependent censoring in two-sample survival comparison. Unfortunately, their test also does not account for the variability of the estimated weights, and thus is expected to be conservative, similar to the IPTW log-rank test.

For the situation where both non-random treatment assignment and dependent censoring are present, Schaubel and Wei⁶ and Zhang and Schaubel⁷ proposed methods for estimating the cumulative treatment effect on time-to-event outcomes. Specifically, Schaubel and Wei⁶ developed double inverse-weighted (inverse-probability-of-treatment-and-censoring weighting) estimators for the ratio of cumulative hazards, relative risk and difference in restricted mean lifetime; Zhang and Schaubel⁷ developed a doubly robust estimator for the difference in restricted mean lifetime, which is consistent when either the even time model given adjustment covariates is correctly specified or the treatment assignment and dependent censoring models given adjustment covariates are correctly specified. For the same situation, Li⁴ proposed a class of adjusted weighted log-rank tests and their Renyi-type versions using the idea of double inverse weighting.⁶ Li’s⁴ methods can test for the difference between treatment-specific hazards across an arbitrary time window, while the methods of Schaubel and Wei⁶ and Zhang and Schaubel⁷ can only infer the cumulative treatment effect from the time origin to certain time point.

In this paper, we develop a class of doubly robust weighted log-rank tests and their Renyi-type versions for two-sample survival comparison in the presence of confounders and dependent censoring. Viewing treatment assignment and censoring as a monotone coarsening (Tsiatis,⁸ Chapter 8), the development of the tests is similar to constructing doubly robust estimators for parameters in semiparametric models with coarsened data (Tsiatis,⁸ Chapter 10). In particular, the tests involve building working models for treatment assignment, dependent censoring and the event time given a set of adjustment covariates. Double robustness here means that the proposed tests are asymptotically correct when either the even time model is true or the treatment assignment and dependent censoring models are true. The new tests are not only more robust than the adjusted weighted log-rank tests,⁴ which require both the assumed models for treatment assignment and dependent censoring to hold, but also expected to be more powerful when all the three working models are true, as implied by Theorem 10.1 of Tsiatis.⁸ Li⁴ showed via simulations that the adjusted weighted log-rank tests are as powerful as the regular log-rank test in the absence of confounders and dependent censoring. We expect that the doubly robust tests also enjoy this property and may be more powerful than the log-rank test if the event time working model utilizes some covariates that are informative of the survival experience.

The rest of the article is organized as follows. In Section 2, we develop a class of doubly robust weighted log-rank statistics and give their asymptotic distributions, based on which we then develop the doubly robust weighted log-rank and Renyi-type tests and establish their double robustness property. As by-products, a doubly robust cumulative hazard estimator and a doubly robust survival estimator are also developed. The former has a similar form to the doubly robust estimator in Zhang and Schaubel,⁷ but is derived under a weaker assumption. Section 3 presents numerical experiments to assess the Type-I errors and powers of the doubly robust tests in finite samples. The proposed methods are applied to a kidney transplant data set in Section 4. We conclude the article with some discussion on future research directions. The proof of the asymptotic distribution for the doubly robust weighted log-rank statistic is delegated to a Web Appendix.

2 Methods

2.1 Data and assumptions

We aim to compare the population-average survival between two treatments labelled 0 and 1. Specifically, letting $T^{(j)}$ denote the event time of a subject randomly selected from the target population if she/he underwent treatment j (j = 0, 1), we want to compare the increment of the cumulative hazard function of $T^{(j)}, d Λ_{(j)} (t) \equiv P (t^{-} < T^{(j)} \leq t^{-} + dt | T^{(j)} > t^{-})$ , between the two treatments across an arbitrary time window $T$ . Let A indicate the treatment group. The potentially observed event time is then $T \equiv (1 - A) T^{(0)} + {AT}^{(1)}$ , which is subject to right censoring, denoted by C. In addition, we observe a vector of baseline covariates Z. Define $U = min (T, C)$ and $Δ = I (T \leq C)$ . The observed data from a random sample of n subjects may be represented by ${O_{i}}_{i = 1}^{n} \equiv {U_{i}, Δ_{i}, A_{i}, Z_{i}}_{i = 1}^{n}$ , which are independent and identically distributed across the n subjects.

A standard weighted log-rank test builds on increments of the Nelson–Aalen estimator for the treatment-specific cumulative hazard. If there were no censoring and $T^{(j)}$ were observed for the entire sample, we would obtain a data set of ${Q_{i}^{(j)}}_{i = 1}^{n} \equiv {T_{i}^{(j)}, Z_{i}}_{i = 1}^{n}$ , based on which the Nelson–Aalen estimator consistently estimates the treatment-specific cumulative hazard $Λ_{(j)} (t)$ (j = 0, 1). We call ${Q_{i}^{(j)}}_{i = 1}^{n}$ full data for estimating $d Λ_{(j)} (t)$ . The actually observed data ${O_{i}}_{i = 1}^{n}$ consist of ${O_{i}^{(0)}}_{i = 1}^{n} \equiv {A_{i 0}, A_{i 0} U_{i}, A_{i 0} Δ_{i}, Z_{i}}_{i = 1}^{n}$ and ${O_{i}^{(1)}}_{i = 1}^{n} \equiv {A_{i 1}, A_{i 1} U_{i}, A_{i 1} Δ_{i}, Z_{i}}_{i = 1}^{n}$ , where $A_{ij} = I (A_{i} = j)$ (j = 0, 1). The Nelson–Aalen estimator applied to ${O_{i}^{(j)}}_{i = 1}^{n}$ is not consistent for $Λ_{(j)} (t)$ if $T_{i}^{(j)}$ is dependent on A_i, i.e. non-random treatment assignment, or

P (t^{-} < T \leq t^{-} + dt | A = j, T > t^{-}) \neq P (t^{-} < T \leq t^{-} + dt | A = j, T > t^{-}, C > t^{-}),

i.e. dependent censoring. Note that

{O_{i}^{(j)}}_{i = 1}^{n}

are a coarsened version of

{Q_{i}^{(j)}}_{i = 1}^{n}

(see the definition of coarsening in Section 7.1, Tsiatis⁸) with a coarsening variable

{\tilde{C}}_{ij}

defined as

{\tilde{C}}_{i j} = {\begin{array}{l} - 1 & if A_{i j} = 0 \\ r & if A_{i j} = 1, C_{i} = r and T_{i}^{(j)} > r, where 0 \leq r < \infty \\ \infty & if A_{i j} = 1 and C_{i} \geq T_{i}^{(j)} \end{array}

and a coarsening function

G_{r} {Q_{i}^{(j)}} \equiv {T_{i}^{(j)} I (T_{i}^{(j)} \leq r), Z_{i}}

. That is,

{O_{i}^{(j)}}_{i = 1}^{n} = {{\tilde{C}}_{i j}, G_{{\tilde{C}}_{i j}} {Q_{i}^{(j)}}}_{i = 1}^{n}

. We call

{O_{i}^{(j)}}_{i = 1}^{n}

coarsened data for estimating

d Λ_{(j)} (t)

Throughout the article, we make the coarsening at random (CAR) assumption

P ({\tilde{C}}_{ij} = r | Q_{i}^{(j)}) = ϖ_{{\tilde{C}}_{j}} [r, G_{r} {Q_{i}^{(j)}}]

(1)

i.e. the probability of coarsening depends on

Q_{i}^{(j)}

only through a function of the observed data. Since the coarsening of

{Q_{i}^{(j)}}_{i = 1}^{n}

{O_{i}^{(j)}}_{i = 1}^{n}

is monotone (see the definition of monotone coarsening in Section 8.1, Tsiatis⁸), it is more convenient to work with the coarsening hazard function defined as

λ_{r}^{{\tilde{C}}_{j}} {T_{i}^{(j)}, Z_{i}} = P ({\tilde{C}}_{i j} = r | {\tilde{C}}_{i j} \geq r, T_{i}^{(j)}, Z_{i})

(2)

in describing the coarsening probabilities. The right-hand side of equation (2) is a slight abuse of notation, since if the distribution function of

{\tilde{C}}_{ij}

is continuous at r,

λ_{r}^{{\tilde{C}}_{j}} {T_{i}^{(j)}, Z_{i}}

should be defined in terms of a continuous-time hazard function, namely

λ_{r}^{{\tilde{C}}_{j}} {T_{i}^{(j)}, Z_{i}} = \lim_{h \to 0 +} h^{- 1} P (r \leq {\tilde{C}}_{i j} < r + h | {\tilde{C}}_{i j} \geq r, T_{i}^{(j)}, Z_{i})

However, it will be convenient to continue with this abuse of notation, and it will not invalidate the exposition of the methods in the sequel. Under monotone coarsening, the CAR assumption is equivalent to that $λ_{r}^{{\tilde{C}}_{j}} {T_{i}^{(j)}, Z_{i}}$ is a function of $G_{r} {Q_{i}^{(j)}}$ . Noting that

λ_{r}^{{\tilde{C}}_{j}} {T_{i}^{(j)}, Z_{i}} = P (C_{i} = r | A_{i} = j, C_{i} \geq r, T_{i}^{(j)}, Z_{i}, T_{i}^{(j)} > r) I (T_{i}^{(j)} > r)

for

0 \leq r < \infty

, we can equivalently express the CAR assumption as the following two

\begin{array}{l} T_{i}^{(j)} ⊥ A_{i} | Z_{i} \\ P (C_{i} = r | A_{i} = j, C_{i} \geq r, T_{i}^{(j)}, Z_{i}, T_{i}^{(j)} > r) = P (C_{i} = r | A_{i} = j, C_{i} \geq r, T_{i}^{(j)} > r, Z_{i}) \end{array}

The right-hand side of (A2) is the cause-specific hazard of censoring at time r given A_i = j, the baseline covariates

Z_{i}

, and that both

T_{i}^{(j)}

and C_i are at least r. In the sequel, we denote it by

λ^{C} (r | A_{i} = j, Z_{i})

. (A2) is similar to the fundamental assumption (1) in Robins and Finkelstein⁵ for correcting bias due to dependent censoring in a clinical trial with inverse probability of censoring weighted log-rank tests.

2.2 A Doubly robust estimator for the treatment-specific survival function

To compare treatment-specific survivals, one usually plots estimated survival curves at first. In the presence of non-random treatment allocation or dependent censoring, the popular Kaplan–Meier estimator no longer consistently estimates the treatment-specific survival function $S_{(j)} (t) \equiv P (T^{(j)} > t)$ . Since the treatment assignment and censoring can be treated as a monotone coarsening of the full data ${Q_{i}^{(j)}}_{i = 1}^{n}$ and there is a full data unbiased estimating equation for $d Λ_{(j)} (t)$

\sum_{i = 1}^{n} m (t; Q_{i}^{(j)}, d Λ_{(j)}) \equiv \sum_{i = 1}^{n} d N_{i}^{T^{(j)}} (t) - Y_{i}^{T^{(j)}} (t) d Λ_{(j)} (t) = 0

(3)

where

N_{i}^{T^{(j)}} (t) = I (T_{i}^{(j)} \leq t)

and

Y_{i}^{T^{(j)}} (t) = I (T_{i}^{(j)} \geq t)

, under the CAR assumption, we can follow the principle of adaptive estimation with monotone coarsening (Tsiatis,⁸ Section 10.3) to construct a doubly robust estimator

{\hat{Λ}}_{(j)} (t)

for

Λ_{(j)} (t)

and then estimate

S_{(j)} (t)

exp {- {\hat{Λ}}_{(j)} (t)}

. Specifically, we consider the estimator for

d Λ_{(j)} (t)

that solves the doubly robust estimating equation derived from Equation (10.60) in Tsiatis⁸

\begin{matrix} \sum_{i = 1}^{n} (\frac{I ({\tilde{C}}_{ij} \geq t) m (t; Q_{i}^{(j)}, d Λ_{(j)})}{π_{t}^{(j)} [G_{t} {Q_{i}^{(j)}}, {\hat{ψ}}_{j}]} \\ + \sum_{r < t} [\frac{I ({\tilde{C}}_{ij} = r) - λ_{r}^{(j)} [G_{r} {Q_{i}^{(j)}}, {\hat{ψ}}_{j}] I ({\tilde{C}}_{ij} \geq r)}{K_{r}^{(j)} [G_{r} {Q_{i}^{(j)}}, {\hat{ψ}}_{j}]}] E {m (t; Q_{i}^{(j)}, d Λ_{(j)}) | G_{r} {Q_{i}^{(j)}}, {\hat{ξ}}_{j}}) = 0 \end{matrix}

(4)

where

λ_{r}^{(j)} [G_{r} {Q_{i}^{(j)}}, ψ_{j}]

is a model posited for the coarsening hazard

λ_{r}^{{\tilde{C}}_{j}} {T_{i}^{(j)}, Z_{i}}

with unknown parameters

ψ_{j}, {\hat{ψ}}_{j}

is any efficient estimator for

ψ_{j}

based on the observed data,

π_{t}^{(j)} [G_{t} {Q_{i}^{(j)}}, {\hat{ψ}}_{j}] = \underset{r < t}{Π} [1 - λ_{r}^{(j)} [G_{r} {Q_{i}^{(j)}}, {\hat{ψ}}_{j}]]

K_{r}^{(j)} [G_{r} {Q_{i}^{(j)}}, {\hat{ψ}}_{j}] = \underset{r' \leq r}{Π} [1 - λ_{r'}^{(j)} [G_{r'} {Q_{i}^{(j)}}, {\hat{ψ}}_{j}]]

, and

E {m (t; Q_{i}^{(j)}, d Λ_{(j)}) | G_{r} {Q_{i}^{(j)}}, {\hat{ξ}}_{j}}

is obtained under a model

p^{Q^{(j)}} (\cdot, ξ_{j})

posited for the full data

Q_{i}^{(j)}

with unknown parameters

ξ_{j}

, which are substituted with any observed data

\sqrt{n}

-consistent estimator

{\hat{ξ}}_{j}

. According to Section 10.3 of Tsiatis,⁸ the solution

d {\hat{Λ}}_{(j)} (t)

to equation (4) is a consistent estimator for

d Λ_{(j)} (t)

(under suitable regularity conditions) if either the posited model

λ_{r}^{(j)} [G_{r} {Q_{i}^{(j)}}, ψ_{j}]

or the posited model

p^{Q^{(j)}} (\cdot, ξ_{j})

is true, i.e. being doubly robust.

Under monotone coarsening and the CAR assumption, $λ_{r}^{(j)} [G_{r} {Q_{i}^{(j)}}, ψ_{j}]$ can be specified by positing models for (i) treatment A given the baseline covariates Z and (ii) the cause-specific hazard of censoring $λ^{C} (t | A = j, Z)$ . $p^{Q^{(j)}} (\cdot, ξ_{j})$ can be specified by positing a model for (iii) the hazard of event time $T^{(j)}$ given the baseline covariates $Z, λ^{T^{(j)}} (t | Z) \equiv {lim}_{Δ t \to 0 +} Δ t^{- 1} P (t \leq T^{(j)} < t + Δ t | Z, T^{(j)} \geq t)$ . For (i), we assume a logistic regression model

logit {P (A = 1 | Z)} = θ' {\tilde{X}}^{A}

(5)

where

{\tilde{X}}^{A} = (1, X^{A'})'

and

X^{A}

is a vector made up of (possibly transformed) elements of Z.

θ

can be estimated by the maximum likelihood estimator

\hat{θ}

, which solves the estimating equation

\sum_{i = 1}^{n} {\tilde{X}}_{i}^{A} {A_{i} - expit (θ' {\tilde{X}}_{i}^{A})} = 0

(6)

where

expit (x) = exp (x) / {1 + exp (x)}

. If model (5) is correct,

\hat{θ}

consistently estimates the true parameter

θ

; otherwise, under regularity conditions (b) and (c) (given in the Web Appendix),

\hat{θ}

converges in probability to a limit

θ^{*}

defined in condition (c).⁹

For the cause-specific hazard of censoring $λ^{C} (t | A = j, Z)$ , we assume a proportional hazards model

λ^{C} (t | A = j, Z) = λ_{0 j}^{C} (t) exp {α_{j}' X^{C}}, j = 0, 1

(7)

where

λ_{0 j}^{C} (t)

is an unspecified baseline cause-specific hazard function of censoring and

X^{C}

is a vector of derived covariates that are functions of Z. Estimators for

α_{j}

and

Λ_{0 j}^{C} (t) \equiv \int_{0}^{t} λ_{0 j}^{C} (s) d s

can be obtained by the maximum partial likelihood estimator

{\hat{α}}_{j}

and the Breslow estimator

{\hat{Λ}}_{0 j}^{C} (t)

, respectively. Specifically,

{\hat{α}}_{j}

is the solution to the estimating equation

\sum_{i = 1}^{n} \int_{0}^{\infty} {X_{i}^{C} - \frac{\sum_{i = 1}^{n} X_{i}^{C} exp {α_{j}' X_{i}^{C}} Y_{ij} (t)}{\sum_{i = 1}^{n} exp {α_{j}' X_{i}^{C}} Y_{ij} (t)}} d N_{ij}^{C} (t) = 0, j = 0, 1,

where

N_{ij}^{C} (t) = A_{ij} I (U_{i} \leq t, Δ_{i} = 0)

and

Y_{ij} (t) = A_{ij} I (U_{i} \geq t)

, and the Breslow estimator for

Λ_{0 j}^{C} (t)

is defined as

{\hat{Λ}}_{0 j}^{C} (t) = \int_{0}^{t} \frac{\sum_{i = 1}^{n} d N_{ij}^{C} (s)}{\sum_{i = 1}^{n} exp {{\hat{α}}_{j}' X_{i}^{C}} Y_{ij} (s)}, j = 0, 1 .

If model (7) is correct, ${\hat{α}}_{j}$ and ${\hat{Λ}}_{0 j}^{C} (t)$ consistently estimate $α_{j}$ and $Λ_{0 j}^{C} (t)$ , respectively. Otherwise, under regularity conditions (a), (d), (e) and (f) in the Web Appendix, ${\hat{α}}_{j}$ and ${\hat{Λ}}_{0 j}^{C} (t)$ converge in probability to limits $α_{j}^{*}$ and $Λ_{0 j}^{C *} (t)$ defined in the Web Appendix.¹⁰

For the hazard of event time $T^{(j)}$ given the baseline covariates Z, we also assume a proportional hazards model

λ^{T^{(j)}} (t | Z) = λ_{0 j} (t) exp {β_{j}' X}, j = 0, 1

(8)

where

λ_{0 j} (t)

is an unspecified baseline hazard function and X is a vector of derived covariates that are functions of Z. Similar to estimating

α_{j}

and

Λ_{0 j}^{C} (t), β_{j}

and

Λ_{0 j} (t) \equiv \int_{0}^{t} λ_{0 j} (s) d s

can be estimated by the maximum partial likelihood estimator

\hat{β_{j}}

and the Breslow estimator

{\hat{Λ}}_{0 j} (t)

respectively. Specifically,

{\hat{β}}_{j}

is the solution to the estimating equation

\sum_{i = 1}^{n} \int_{0}^{\infty} {X_{i} - \frac{\sum_{i = 1}^{n} X_{i} exp {β_{j}' X_{i}} Y_{ij} (t)}{\sum_{i = 1}^{n} exp {β_{j}' X_{i}} Y_{ij} (t)}} d N_{ij} (t) = 0, j = 0, 1,

where

N_{ij} (t) = A_{ij} I (U_{i} \leq t, Δ_{i} = 1)

, and the Breslow estimator for

Λ_{0 j} (t)

is defined as

{\hat{Λ}}_{0 j} (t) = \int_{0}^{t} \frac{\sum_{i = 1}^{n} d N_{ij} (s)}{\sum_{i = 1}^{n} exp {{\hat{β}}_{j}' X_{i}} Y_{ij} (s)}, j = 0, 1 .

If model (8) is correct, ${\hat{β}}_{j}$ and ${\hat{Λ}}_{0 j} (t)$ consistently estimate $β_{j}$ and $Λ_{0 j} (t)$ , respectively. Otherwise, under regularity conditions (a), (d), (g) and (h) in the Web Appendix, ${\hat{β}}_{j}$ and ${\hat{Λ}}_{0 j} (t)$ converge in probability to limits $β_{j}^{*}$ and $Λ_{0 j}^{*} (t)$ defined in the Web Appendix.¹⁰

Define $p_{ij} (θ) = exp {(- 1)^{j + 1} θ' {\tilde{X}}_{i}^{A}} / {1 + exp {(- 1)^{j + 1} θ' {\tilde{X}}_{i}^{A}}}, w_{ij} (θ) = A_{ij} / p_{ij} (θ), κ_{i} (t) = I (C_{i} \geq T_{i} or C_{i} \geq t), {\hat{Λ}}_{ij}^{C} (t) = exp {{\hat{α}}_{j}' X_{i}^{C}} {\hat{Λ}}_{0 j}^{C} (t)$ and ${\hat{Λ}}_{ij} (t) = exp {{\hat{β}}_{j}' X_{i}} {\hat{Λ}}_{0 j} (t)$ . Under the CAR assumption and models (5), (7) and (8), equation (4) can be re-expressed with these notations. Specifically, $I ({\tilde{C}}_{ij} \geq t) = A_{ij} κ_{i} (t)$ and

π_{t}^{(j)} [G_{t} {Q_{i}^{(j)}}, {\hat{ψ}}_{j}] = p_{i j} (\hat{θ}) \exp {- {\hat{Λ}}_{i j}^{C} (t \land T_{i}^{(j)})}

Thus

\begin{matrix} \frac{I ({\tilde{C}}_{ij} \geq t) m (t; Q_{i}^{(j)}, d Λ_{(j)})}{π_{t}^{(j)} [G_{t} {Q_{i}^{(j)}}, {\hat{ψ}}_{j}]} = w_{ij} (\hat{θ}) exp {{\hat{Λ}}_{ij}^{C} (t)} κ_{i} (t) d M_{i}^{T} (t; d Λ_{(j)}) \\ = w_{ij} (\hat{θ}) exp {{\hat{Λ}}_{ij}^{C} (t)} {{dN}_{i} (t) - Y_{i} (t) d Λ_{(j)} (t)} \end{matrix}

(9)

where

{dM}_{i}^{T} (t; d Λ_{(j)}) = {dN}_{i}^{T} (t) - Y_{i}^{T} (t) d Λ_{(j)} (t), N_{i}^{T} (t) = I (T_{i} \leq t), Y_{i}^{T} (t) = I (T_{i} \geq t), N_{i} (t) = I (U_{i} \leq t, Δ_{i} = 1)

and

Y_{i} (t) = I (U_{i} \geq t)

. In addition

I ({\tilde{C}}_{ij} = r) = \begin{matrix} {\begin{matrix} 1 - A_{ij} & if r = - 1 \\ A_{ij} {dN}_{i}^{C} (r) & if 0 \leq r < \infty \end{matrix} \end{matrix}

(10)

where

N_{i}^{C} (t) = I (U_{i} \leq t, Δ_{i} = 0)

λ_{r}^{(j)} [G_{r} {Q_{i}^{(j)}}, {\hat{ψ}}_{j}] I ({\tilde{C}}_{i j} \geq r) = {\begin{array}{l} 1 - p_{i j} (\hat{θ}) & if r = - 1 \\ A_{i j} I (C_{i} \geq r) I (T_{i}^{(j)} > r) d {\hat{Λ}}_{i j}^{C} (r) & if 0 \leq r < \infty \end{array}

(11)

K_{r}^{(j)} [G_{r} {Q_{i}^{(j)}}, {\hat{ψ}}_{j}] = p_{i j} (\hat{θ}) \exp {- {\hat{Λ}}_{i j}^{C} (r \land T_{i}^{(j)})}

(12)

\begin{matrix} E {m (t; Q_{i}^{(j)}, d Λ_{(j)}) | G_{- 1} {Q_{i}^{(j)}}, {\hat{ξ}}_{j}} = E {m (t; Q_{i}^{(j)}, d Λ_{(j)}) | Z_{i}, {\hat{ξ}}_{j}} \\ = E {{dN}_{i}^{T^{(j)}} (t) | Z_{i}, {\hat{ξ}}_{j}} - E {Y_{i}^{T^{(j)}} (t) | Z_{i}, {\hat{ξ}}_{j}} d Λ_{(j)} (t) \\ = exp {- {\hat{Λ}}_{ij} (t)} d {\hat{Λ}}_{ij} (t) - exp {- {\hat{Λ}}_{ij} (t)} d Λ_{(j)} (t) \end{matrix}

(13)

and for

0 \leq r < t

\begin{matrix} E {m (t; Q_{i}^{(j)}, d Λ_{(j)}) | G_{r} {Q_{i}^{(j)}}, {\hat{ξ}}_{j}} \\ = E {m (t; Q_{i}^{(j)}, d Λ_{(j)}) | T_{i}^{(j)} \leq r, Z_{i}, {\hat{ξ}}_{j}} I (T_{i}^{(j)} \leq r) + E {m (t; Q_{i}^{(j)}, d Λ_{(j)}) | T_{i}^{(j)} > r, Z_{i}, {\hat{ξ}}_{j}} I (T_{i}^{(j)} > r) \\ = E {m (t; Q_{i}^{(j)}, d Λ_{(j)}) | T_{i}^{(j)} \leq r, Z_{i}, {\hat{ξ}}_{j}} I (T_{i}^{(j)} \leq r) \\ + [\frac{\exp {- {\hat{Λ}}_{i j} (t)} d {\hat{Λ}}_{i j} (t)}{\exp {- {\hat{Λ}}_{i j} (r)}} - \frac{\exp {- {\hat{Λ}}_{i j} (t)} d Λ_{(j)} (t)}{\exp {- {\hat{Λ}}_{i j} (r)}}] I (T_{i}^{(j)} > r) \end{matrix}

(14)

Plugging equations (9) to (14) into equation (4), we obtain

\begin{matrix} \sum_{i = 1}^{n} w_{ij} (\hat{θ}) exp {{\hat{Λ}}_{ij}^{C} (t)} {{dN}_{i} (t) - Y_{i} (t) d Λ_{(j)} (t)} \\ + {1 - w_{ij} (\hat{θ})} [exp {- {\hat{Λ}}_{ij} (t)} d {\hat{Λ}}_{ij} (t) - exp {- {\hat{Λ}}_{ij} (t)} d Λ_{(j)} (t)] \\ + \int_{0}^{t} w_{ij} (\hat{θ}) exp {{\hat{Λ}}_{ij}^{C} (r)} d {\hat{M}}_{ij}^{C} (r) [\frac{exp {- {\hat{Λ}}_{ij} (t)} d {\hat{Λ}}_{ij} (t)}{exp {- {\hat{Λ}}_{ij} (r)}} - \frac{exp {- {\hat{Λ}}_{ij} (t)} d Λ_{(j)} (t)}{exp {- {\hat{Λ}}_{ij} (r)}}] \\ = 0 \end{matrix}

(15)

where

d {\hat{M}}_{ij}^{C} (r) = {dN}_{ij}^{C} (r) - Y_{ij} (r) d {\hat{Λ}}_{ij}^{C} (t)

. Solving equation (15) for

d Λ_{(j)} (t)

, we obtain the following doubly robust estimator for

Λ_{(j)} (t)

{\hat{Λ}}_{(j)} (t) = \int_{0}^{t} \frac{\sum_{i = 1}^{n} [w_{ij} (\hat{θ}) exp {{\hat{Λ}}_{ij}^{C} (s)} {dN}_{i} (s) + exp {- {\hat{Λ}}_{ij} (s)} d {\hat{Λ}}_{ij} (s) {1 - w_{ij} (\hat{θ}) G_{ij} (s; {\hat{Λ}}^{C}, \hat{Λ})}]}{\sum_{i = 1}^{n} [w_{ij} (\hat{θ}) exp {{\hat{Λ}}_{ij}^{C} (s)} Y_{i} (s) + exp {- {\hat{Λ}}_{ij} (s)} {1 - w_{ij} (\hat{θ}) G_{ij} (s; {\hat{Λ}}^{C}, \hat{Λ})}]}

(16)

where

\begin{matrix} G_{ij} (s; {\hat{Λ}}^{C}, \hat{Λ}) = 1 - \int_{0}^{s} exp {{\hat{Λ}}_{ij}^{C} (r) + {\hat{Λ}}_{ij} (r)} d {\hat{M}}_{ij}^{C} (r), {\hat{Λ}}^{C} = {{\hat{Λ}}_{ij}^{C} : i = 1, \dots, n, j = 0, 1}, {\hat{Λ}}_{ij}^{C} (\cdot) \\ = {\hat{Λ}}_{0 j}^{C} (\cdot) exp ({\hat{α}}_{j}' X_{i}^{C}), \hat{Λ} = {{\hat{Λ}}_{ij} : i = 1, \dots, n, j = 0, 1}, \end{matrix}

and

{\hat{Λ}}_{ij} (\cdot) = {\hat{Λ}}_{0 j} (\cdot) exp ({\hat{β}}_{j}' X_{i})

. Zhang and Schaubel⁷ proposed a similar doubly robust estimator for

Λ_{(j)} (t)

under non-random treatment assignment and dependent censoring. In their paper, they made Assumption (A1) and another assumption that

T ⊥ C | (A, Z)

, which is stronger than (A2).

Theorem 1 shows that, if either models (5) and (7) are correct or model (8) is correct, ${\hat{Λ}}_{(j)} (t)$ is a uniformly consistent and asymptotically normal estimator for $Λ_{(j)} (t)$ under suitable regularity conditions. The proof can follow that in Zhang and Schaubel⁷ for the consistency and asymptotic normality of their doubly robust estimator for the treatment-specific cumulative hazard and thus is omitted.

Theorem 1

Set $π_{j} (t) = P (Y_{ij} (t) > 0), I_{j} = {t : π_{j} (t) > 0}$ and $u_{j} = sup I_{j}$ (j = 0, 1). Under the CAR assumption, the regularity conditions (a)–(j) in the Web Appendix and the assumption that either models (5) and (7) are correct or model (8) is correct, ${\hat{Λ}}_{(j)} (t)$ converges almost surely and uniformly to $Λ_{(j)} (t)$ for $t \in [0, u_{j}]$ , and $\sqrt{n} ({\hat{Λ}}_{(j)} (t) - Λ_{(j)} (t))$ converges weakly to a zero-mean Gaussian process in $D [0, u_{j}]$ , the space of cadlag functions on $[0, u_{j}]$ , with covariance function $σ_{Λ_{(j)}} (s, t) = E {Φ_{ij} (s) Φ_{ij} (t)}$ , where

\begin{matrix} Φ_{ij} (t) = Φ_{ij 1} (t) + Φ_{ij 2} (t) + Φ_{ij 3} (t) + Φ_{ij 4} (t), \\ Φ_{ij 1} (t) = \int_{0}^{t} \frac{{dQ}_{ij} (s; θ^{*}, Λ^{C *}, Λ^{*}) - D_{j} (s; θ^{*}, Λ^{C *}, Λ^{*}) d Λ_{(j)} (s)}{D_{j} (s; θ^{*}, Λ^{C *}, Λ^{*})}, \\ Φ_{ij 2} (t) = \int_{0}^{t} {\frac{d K_{j}' (s; θ^{*}, Λ^{C *}, Λ^{*})}{D_{j} (s; θ^{*}, Λ^{C *}, Λ^{*})} - \frac{{dQ}_{j} (s; θ^{*}, Λ^{C *}, Λ^{*}) B_{j}' (s; θ^{*}, Λ^{C *}, Λ^{*})}{D_{j}^{2} (s; θ^{*}, Λ^{C *}, Λ^{*})}} \\ V^{- 1} (θ^{*}) {\tilde{X}}_{i}^{A} {A_{ij} - p_{ij} (θ^{*})}, \\ Φ_{ij 3} (t) = \int_{0}^{t} {\frac{d_{j}' (s; θ^{*}, Λ^{C *}, Λ^{*})}{D_{j} (s; θ^{*}, Λ^{C *}, Λ^{*})} - \frac{{dQ}_{j} (s; θ^{*}, Λ^{C *}, Λ^{*})}{D_{j}^{2} (s; θ^{*}, Λ^{C *}, Λ^{*})} ξ_{j}' (s; θ^{*}, Λ^{C *}, Λ^{*})} Ω_{Cj}^{- 1} (α_{j}^{*}) U_{ij}^{C} (α_{j}^{*}) \\ + \int_{0}^{t} \int_{u}^{t} {\frac{d ψ_{j} (s, u; θ^{*}, Λ^{C *}, Λ^{*})}{D_{j} (s; θ^{*}, Λ^{C *}, Λ^{*})} - \frac{γ_{j} (s, u; θ^{*}, Λ^{C *}, Λ^{*}) {dQ}_{j} (s; θ^{*}, Λ^{C *}, Λ^{*})}{D_{j}^{2} (s; θ^{*}, Λ^{C *}, Λ^{*})}} \frac{d M_{ij}^{C *} (u)}{r_{Cj}^{(0)} (u; α_{j}^{*})}, \\ Φ_{ij 4} (t) = \int_{0}^{t} {\frac{d ρ_{j}' (s; θ^{*}, Λ^{C *}, Λ^{*})}{D_{j} (s; θ^{*}, Λ^{C *}, Λ^{*})} - \frac{{dQ}_{j} (s; θ^{*}, Λ^{C *}, Λ^{*})}{D_{j}^{2} (s; θ^{*}, Λ^{C *}, Λ^{*})} ζ_{j}' (s; θ^{*}, Λ^{C *}, Λ^{*})} Ω_{j}^{- 1} (β_{j}^{*}) U_{ij} (β_{j}^{*}) \\ + \int_{0}^{t} [\int_{u}^{t} {\frac{d ω_{j} (s, u; θ^{*}, Λ^{C *}, Λ^{*})}{D_{j} (s; θ^{*}, Λ^{C *}, Λ^{*})} - \frac{δ_{j} (s, u; θ^{*}, Λ^{C *}, Λ^{*}) {dQ}_{j} (s; θ^{*}, Λ^{C *}, Λ^{*})}{D_{j}^{2} (s; θ^{*}, Λ^{C *}, Λ^{*})}} \\ + \frac{μ_{j} (u; θ^{*}, Λ^{C *}, Λ^{*})}{D_{j} (u; θ^{*}, Λ^{C *}, Λ^{*})}] \frac{d M_{ij}^{*} (u)}{r_{j}^{(0)} (u; β_{j}^{*})}, \end{matrix}

V^{- 1} (θ^{*}) {\tilde{X}}_{i}^{A} {A_{ij} - p_{ij} (θ^{*})} (- 1)^{j + 1}, Ω_{Cj}^{- 1} (α_{j}^{*}) U_{ij}^{C} (α_{j}^{*})

and

Ω_{j}^{- 1} (β_{j}^{*}) U_{ij} (β_{j}^{*})

are the influence functions of

\hat{θ}, \hat{α_{j}}

and

\hat{β_{j}}

, respectively (of which the explicit expressions are given in the Web Appendix),

\begin{matrix} Λ^{C *} = {Λ_{ij}^{C *} : i = 1, \dots, n, j = 0, 1}, Λ_{ij}^{C *} (\cdot) = Λ_{0 j}^{C *} (\cdot) exp (α_{j}^{*'} X_{i}^{C}), \\ Λ^{*} = {Λ_{ij}^{*} : i = 1, \dots, n, j = 0, 1}, Λ_{ij}^{*} (\cdot) = Λ_{0 j}^{*} (\cdot) exp (β_{j}^{*'} X_{i}), \\ d M_{ij}^{C *} (u) = {dN}_{ij}^{C} (u) - Y_{ij} (u) d Λ_{ij}^{C *} (u), d M_{ij}^{*} (u) = {dN}_{ij} (u) - Y_{ij} (u) d Λ_{ij}^{*} (u), \end{matrix}

and

\begin{matrix} {dQ}_{ij} (s; θ^{*}, Λ^{C *}, Λ^{*}), D_{j} (s; θ^{*}, Λ^{C *}, Λ^{*}), d K_{j} (s; θ^{*}, Λ^{C *}, Λ^{*}), \\ {dQ}_{j} (s; θ^{*}, Λ^{C *}, Λ^{*}), B_{j} (s; θ^{*}, Λ^{C *}, Λ^{*}), d_{j} (s; θ^{*}, Λ^{C *}, Λ^{*}), \\ ξ_{j} (s; θ^{*}, Λ^{C *}, Λ^{*}), d ψ_{j} (s, u; θ^{*}, Λ^{C *}, Λ^{*}), γ_{j} (s, u; θ^{*}, Λ^{C *}, Λ^{*}), \\ r_{Cj}^{(0)} (u; α_{j}^{*}), d ρ_{j} (s; θ^{*}, Λ^{C *}, Λ^{*}), ζ_{j} (s; θ^{*}, Λ^{C *}, Λ^{*}), d ω_{j} (s, u; θ^{*}, Λ^{C *}, Λ^{*}), \\ δ_{j} (s, u; θ^{*}, Λ^{C *}, Λ^{*}), μ_{j} (u; θ^{*}, Λ^{C *}, Λ^{*}) \end{matrix}

and

r_{j}^{(0)} (u; β_{j}^{*})

are defined in the Web Appendix.

The uniform consistency of the treatment-specific survival estimator ${\hat{S}}_{(j)} (t) \equiv exp {- {\hat{Λ}}_{(j)} (t)}$ and the weak convergence of $\sqrt{n} ({\hat{S}}_{(j)} (t) - S_{(j)} (t))$ follow from Theorem 1, the continuous mapping theorem and the functional delta method.

Corollary 1

Under the CAR assumption, the regularity conditions (a)–(k) in the Web Appendix and the assumption that either models (5) and (7) are correct or model (8) is correct, ${\hat{S}}_{(j)} (t)$ converges almost surely and uniformly to $S_{(j)} (t)$ for $t \in [0, u_{j}]$ , and $\sqrt{n} ({\hat{S}}_{(j)} (t) - S_{(j)} (t))$ converges weakly to a zero-mean Gaussian process in $D [0, u_{j}]$ with covariance function $σ_{S_{(j)}} (s, t) = S_{(j)} (s) S_{(j)} (t) E {Φ_{ij} (s) Φ_{ij} (t)}$ .

One can consistently estimate the covariance function $σ_{S_{(j)}} (s, t)$ by ${\hat{S}}_{(j)} (s) {\hat{S}}_{(j)} (t) n^{- 1} \sum_{i = 1}^{n} {\hat{Φ}}_{ij} (s) {\hat{Φ}}_{ij} (t)$ , where ${\hat{Φ}}_{ij}$ is the empirical version of $Φ_{ij}$ obtained by replacing the unknown parameters in $Φ_{ij}$ by their estimators and the limiting terms therein with the corresponding sample averages.

2.3 Doubly robust weighted log-rank tests

We propose a class of doubly robust weighted log-rank tests based on the stochastic process of cumulative weighted differences between the doubly robust estimators $d {\hat{Λ}}_{(j)} (t)$ (j = 0, 1)

W_{DR} (t; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ}) = \int_{0}^{t} K (s; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ}) (d {\hat{Λ}}_{(1)} (s) - d {\hat{Λ}}_{(0)} (s))

(17)

where

\begin{matrix} K (s; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ}) = \sqrt{n} W (s) \frac{{\hat{D}}_{0} (s; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ}) {\hat{D}}_{1} (s; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ})}{{\hat{D}}_{0} (s; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ}) + {\hat{D}}_{1} (s; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ})} \\ {\hat{D}}_{j} (s; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ}) = n^{- 1} \sum_{i = 1}^{n} [w_{ij} (\hat{θ}) exp {{\hat{Λ}}_{ij}^{C} (s)} Y_{i} (s) + exp {- {\hat{Λ}}_{ij} (s)} {1 - w_{ij} (\hat{θ}) G_{ij} (s; {\hat{Λ}}^{C}, \hat{Λ})}] \end{matrix}

(18)

for j = 0 and 1, and

W (\cdot)

is a nonnegative, bounded and predictable process. In our simulations and application to real data, we adopt Fleming and Harrington’s¹¹ class of weight functions

W (s) = {\hat{S} (s -)}^{ρ} {1 - \hat{S} (s -)}^{γ}, ρ \geq 0, γ \geq 0,

where

\hat{S} (s) = exp [- \int_{0}^{s} \frac{\sum_{i = 1}^{n} \sum_{j = 0}^{1} [e^{{\hat{Λ}}_{ij}^{C} (u)} {dN}_{ij} (u) + e^{- {\hat{Λ}}_{ij} (u)} d {\hat{Λ}}_{ij} (u) \int_{0}^{u} e^{{\hat{Λ}}_{ij}^{C} (v) + {\hat{Λ}}_{ij} (v)} d {\hat{M}}_{ij}^{C} (v)]}{\sum_{i = 1}^{n} \sum_{j = 0}^{1} [e^{{\hat{Λ}}_{ij}^{C} (u)} Y_{ij} (u) + e^{- {\hat{Λ}}_{ij} (u)} \int_{0}^{u} e^{{\hat{Λ}}_{ij}^{C} (v) + {\hat{Λ}}_{ij} (v)} d {\hat{M}}_{ij}^{C} (v)]}]

is a doubly robust estimator for the overall survival function.

W_{DR} (t; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ})

with

(ρ, γ) = (0, 0)

and

(ρ, γ) = (1, 0)

are referred to as the doubly robust log-rank and Prentice–Wilcoxon¹² statistics, respectively.

Theorem 2 in the following shows the asymptotic representation of the doubly robust weighted log-rank statistic (17), based on which we will develop the doubly robust weighted log-rank tests and Renyi-type tests.

Theorem 2

Set $I = {t : π_{0} (t) π_{1} (t) > 0}$ and $t_{sup} = sup I$ . Under the CAR assumption, conditions (a)–(j) and (l) in the Web Appendix and the assumption that either models (5) and (7) are correct or model (8) is correct

W_{DR} (t; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ}) = \int_{0}^{t} K (s; θ^{*}, Λ^{C *}, Λ^{*}) {λ_{(1)} (s) - λ_{(0)} (s)} d s + n^{- 1 / 2} \sum_{i = 1}^{n} φ_{i} (t) + o_{p} (1)

(19)

where

λ_{(j)} (s) = d Λ_{(j)} (s) / ds

(j = 0, 1),

o_{p} (1)

represents a term that converges in probability to zero in

D [0, t_{sup}]

equipped with the uniform norm, and

φ_{i} (t) = φ_{i 1} (t) + φ_{i 2} (t) + φ_{i 3} (t) + φ_{i 4} (t)

where

\begin{matrix} φ_{i 1} (t) = A_{i 1} (t; θ^{*}, Λ^{C *}, Λ^{*}) - A_{i 0} (t; θ^{*}, Λ^{C *}, Λ^{*}), \\ φ_{i 2} (t) = \sum_{j = 0}^{1} L_{j}' (t; θ^{*}, Λ^{C *}, Λ^{*}) V^{- 1} (θ^{*}) {\tilde{X}}_{i}^{A} {A_{ij} - p_{ij} (θ^{*})}, \\ φ_{i 3} (t) = \sum_{j = 0}^{1} {P_{j}' (t; θ^{*}, Λ^{C *}, Λ^{*}) Ω_{Cj}^{- 1} (α_{j}^{*}) U_{ij}^{C} (α_{j}^{*}) + \int_{0}^{t} H_{j} (u, t; θ^{*}, Λ^{C *}, Λ^{*}) \frac{d M_{ij}^{C *} (u)}{r_{Cj}^{(0)} (u; α_{j}^{*})}}, \\ φ_{i 4} (t) = \sum_{j = 0}^{1} {F_{j}' (t; θ^{*}, Λ^{C *}, Λ^{*}) Ω_{j}^{- 1} (β_{j}^{*}) U_{ij} (β_{j}^{*}) + \int_{0}^{t} J_{j} (u, t; θ^{*}, Λ^{C *}, Λ^{*}) \frac{d M_{ij}^{*} (u)}{r_{j}^{(0)} (u; β_{j}^{*})}}, \end{matrix}

$A_{ij} (t; θ^{*}, Λ^{C *}, Λ^{*}) = \int_{0}^{t} w (s) D_{1 - j} (s; θ^{*}, Λ^{C *}, Λ^{*}) D^{- 1} (s; θ^{*}, Λ^{C *}, Λ^{*}) d M_{i (j)} (s)$ (j = 0, 1) with mean zero, and w(s), $D (s; θ^{*}, Λ^{C *}, Λ^{*}), d M_{i (j)} (s), L_{j} (t; θ^{*}, Λ^{C *}, Λ^{*}), P_{j} (t; θ^{*}, Λ^{C *}, Λ^{*})$ , $H_{j} (u, t; θ^{*}, Λ^{C *}, Λ^{*}), F_{j} (t; θ^{*}, Λ^{C *}, Λ^{*})$ and $J_{j} (u, t; θ^{*}, Λ^{C *}, Λ^{*})$ are defined in the Web Appendix.

The proof of Theorem 2 is delegated to the Web Appendix. Its basic idea is to write $W_{DR} (t; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ})$ as

\begin{matrix} W_{DR} (t; θ^{*}, Λ^{C *}, Λ^{*}) + {W_{DR} (t; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ}) - W_{DR} (t; θ^{*}, {\hat{Λ}}^{C}, \hat{Λ})} \\ + {W_{DR} (t; θ^{*}, {\hat{Λ}}^{C}, \hat{Λ}) - W_{DR} (t; θ^{*}, Λ^{C *}, \hat{Λ})} + {W_{DR} (t; θ^{*}, Λ^{C *}, \hat{Λ}) - W_{DR} (t; θ^{*}, Λ^{C *}, Λ^{*})} \end{matrix}

(20)

obtain the asymptotically linear representations of

\sqrt{n} (\hat{θ} - θ^{*})

using the maximum likelihood theory under possibly misspecified models⁹ and of

\sqrt{n} {{\hat{Λ}}_{ij}^{C} (\cdot) - Λ_{ij}^{C *} (\cdot)}

and

\sqrt{n} {{\hat{Λ}}_{ij} (\cdot) - Λ_{ij}^{*} (\cdot)}

using the maximum partial likelihood theory under a possibly misspecified Cox model¹⁰ or by following the proofs in Sections 2 and 4 of the Web Appendix of Zhang and Schaubel,⁷ and finally apply the (functional) delta methods to the second, third and fourth summands in equation (20).

Remark 1

Theorem 1, Corollary 1 and Theorem 2 are proved under the condition that the censoring time C is merely a dependent censoring time. If there is also an independent censoring time that is independent of the event time and the dependent censoring time given the treatment, one should just model the cause-specific hazard of the dependent censoring time given the treatment and the baseline covariates, and Theorem 1, Corollary 1 and Theorem 2 still hold. The proofs with independent censoring are very similar to those without independent censoring and thus omitted.

The doubly robust weighted log-rank test for

H_{0} : λ_{(0)} (t) = λ_{(1)} (t) forall t \in I

(21)

adopts the test statistic:

Z_{DR} (U^{†}) \equiv W_{DR} (U^{†}; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ}) / {\hat{σ}}_{W} (U^{†})

, where

U^{†} = sup {t : {\bar{Y}}_{\cdot 0} (t) {\bar{Y}}_{\cdot 1} (t) > 0}, {\bar{Y}}_{\cdot j} (t) = \sum_{i = 1}^{n} Y_{ij} (t)

(j = 0, 1), and

{\hat{σ}}_{W}^{2} (t) = n^{- 1} \sum_{i = 1}^{n} {\hat{φ}}_{i}^{2} (t)

where

{\hat{φ}}_{i}

is the empirical version of

φ_{i}

obtained by replacing the unknown parameters in

φ_{i}

by their estimators and the limiting terms therein with the corresponding sample averages. In this article, we only study the properties (Type-I error and power) of the two-sided test based on

Z_{DR} (U^{†})

, i.e. rejecting H₀ if

| Z_{DR} (U^{†}) | \geq z_{α / 2}

, where

z_{α / 2}

is the standard normal’s

α / 2

-th upper quantile. One-sided tests can be investigated in a similar way. Theorem 2 implies that, when H₀ is true,

W_{DR} (U^{†}; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ})

can be represented asymptotically by a sum of independent and identically distributed zero-mean variates

φ_{i} (t_{sup})

(i = 1, \dots, n)

. Thus, by the central limit theorem, the doubly robust weighted log-rank test based on

Z_{DR} (U^{†})

has asymptotic significance level α. Equations (19) and (18) imply that the power of the two-sided test approaches one as

n \to \infty

under an alternative

H_{A} : \int_{0}^{t_{sup}} w (s) \frac{D_{0} (s; θ^{*}, Λ^{C *}, Λ^{*}) D_{1} (s; θ^{*}, Λ^{C *}, Λ^{*})}{D_{0} (s; θ^{*}, Λ^{C *}, Λ^{*}) + D_{1} (s; θ^{*}, Λ^{C *}, Λ^{*})} {λ_{(1)} (s) - λ_{(0)} (s)} d s \neq 0

(22)

Hence, the doubly robust weighted log-rank test is asymptotically correct.

To compare the treatment-specific hazards over an arbitrary subset $T \subset I$ , we develop a doubly robust weighted log-rank test for

H_{0}^{T} : λ_{(0)} (t) = λ_{(1)} (t) forall t \in T

(23)

by choosing a weight function W(t) such that W(t) = 0 for any

t \notin T

. Similar arguments to the above can show that this test is also asymptotically correct.

2.4 Doubly robust Renyi-type tests

In view of the original Renyi-type test,² we define the doubly robust Renyi-type test statistic as

Z_{DR}^{R} (U^{†}) = \frac{{sup}_{0 \leq t \leq U^{†}} | W_{DR} (t; \hat{θ}, {\hat{Λ}}^{C}, \hat{Λ}) |}{{\hat{σ}}_{W} (U^{†})}

(24)

It is hard to derive the asymptotic distribution of $Z_{DR}^{R} (U^{†})$ under H₀. Nonetheless, the multiplier central limit theorem (van der Vaart and Wellner,¹³ Section 2.9) provides a bypass around this difficulty. Specifically, when n is large, we can approximate the distribution of $n^{- 1 / 2} \sum_{i = 1}^{n} φ_{i} (t)$ by the conditional distribution of $n^{- 1 / 2} \sum_{i = 1}^{n} ω_{i} {\hat{φ}}_{i} (t)$ given $(U_{i}, Δ_{i}, A_{i}, Z_{i})$ $(i = 1, \dots, n)$ , where ω_i’s are independent standard normal variables. Therefore, the asymptotic null distribution of $Z_{DR}^{R} (U^{†})$ can be simulated. The resulting doubly robust Renyi-type test is as follows.

Calculate $Z_{DR}^{R} (U^{†})$ based on the data.

Generate m independent sets of independent standard normal variables ${ω_{i}^{(l)}}_{i = 1}^{n}$ $(l = 1, \dots, m)$ and calculate the corresponding ${\tilde{Z}}_{DR}^{R (l)} (U^{†}) \equiv {sup}_{0 \leq t \leq U^{†}} | n^{- 1 / 2} \sum_{i = 1}^{n} ω_{i}^{(l)} {\hat{φ}}_{i} (t) | / {\hat{σ}}_{W} (U^{†})$ .

Compute the simulation-based p-value: ${\tilde{p}}_{R} \equiv m^{- 1} \sum_{l = 1}^{m} I {{\tilde{Z}}_{DR}^{R (l)} (U^{†}) \geq Z_{DR}^{R} (U^{†})}$ .

Reject H₀ at level α if ${\tilde{p}}_{R} < α$ .

Theorem 2, the multiplier central limit theorem, and the law of large numbers in conjunction imply that this doubly robust Renyi-type test has asymptotic significance level α.

Following the development of the doubly robust weighted log-rank test for $H_{0}^{T}$ , we construct a doubly robust Renyi-type test for $H_{0}^{T}$ by choosing a weight function W(t) such that W(t) = 0 for any $t \notin T$ and then applying the test procedure as above.

3 Numerical experiments

We conducted numerical experiments to check the Type-I errors and powers of the proposed tests under non-random treatment assignment and dependent censoring. In all the simulations, the nominal significance level for any test is 0.05 and the number of Monte Carlo repetitions is 1000. We used similar simulation senarios to Zhang and Schaubel⁷ and Li.⁴ For every Monte Carlo sample, we generated three baseline covariates X₁, X₂ and X₃ each from a standard normal distribution truncated at –0.5 and 0.5. The correlation between untruncated X₁ and X₃ equals 0.2, and all other pairwise correlations are 0. The treatment variable A was simulated from Bernoulli with probability $expit (- 0.5 X_{1} - 0.5 X_{2})$ of being one, leading to about 50% of the subjects getting treatment 1. We generated the event time T, the dependent censoring time C, and an independent censoring time $C_{ind}$ under the following scenarios:

Null:

\begin{matrix} λ (t | A = j, X) = exp (- 3 - X_{1} - 0.9 X_{2} - X_{3}) (j = 0, 1), \\ λ^{C} (t | A = 0, X) = exp (- 4.7 + X_{1} + 1.2 X_{2}), \\ λ^{C} (t | A = 1, X) = exp (- 4.2 - 0.2 X_{1} - 0.7 X_{2}), \\ C_{ind} \sim Uniform (39.5, 40.5); \end{matrix}

Nearly Proportional

\begin{matrix} λ (t | A = 0, X) = exp (- 2.8 - 0.95 X_{1} - 0.85 X_{2} - 0.95 X_{3}), \\ λ (t | A = 1, X) = exp (- 3 - X_{1} - 0.9 X_{2} - X_{3}), \\ λ^{C} (t | A = 0, X) = exp (- 4.7 + X_{1} + 1.2 X_{2}), \\ λ^{C} (t | A = 1, X) = exp (- 4.2 - 0.2 X_{1} - 0.7 X_{2}), \\ C_{ind} \sim Uniform (39.5, 40.5); \end{matrix}

Early Departure

\begin{matrix} λ (t | A = 0, X) = exp (- 1 - 1.5 X_{1} - X_{2} - 0.7 X_{3}) / (1 + t) + 0.01, \\ λ (t | A = 1, X) = exp (- 1.5 - X_{1} - 0.9 X_{2} - X_{3}) / (1 + t) + 0.01, \\ λ^{C} (t | A = 0, X) = exp (- 3.9 + X_{1} + 1.2 X_{2}), \\ λ^{C} (t | A = 1, X) = exp (- 3.9 - 0.2 X_{1} - 0.7 X_{2}), \\ C_{ind} \sim Uniform (39.5, 40.5); \end{matrix}

Crossing

\begin{matrix} λ (t | A = 0, X) = 0.21 exp (- 0.1 X_{1} - 0.1 X_{2} - 0.1 X_{3}) + 0.001 t, \\ λ (t | A = 1, X) = 0.1 exp (- 0.2 X_{1} - 0.05 X_{2} - 0.1 X_{3}) + 0.04 t, \\ λ^{C} (t | A = 0, X) = exp (- 3.1 + X_{1} + 1.2 X_{2}), \\ λ^{C} (t | A = 1, X) = exp (- 3.1 - 0.2 X_{1} - 0.7 X_{2}), \\ C_{ind} \sim Uniform (19.5, 20.5), \end{matrix}

where

λ (t | A = j, X)

stands for the conditional hazard of T given A = j and the three baseline covariates X. In the Null scenario, we have

λ_{(0)} (\cdot) = λ_{(1)} (\cdot)

. In the Nearly Proportional, we have

λ_{(1)} (t) / λ_{(0)} (t) \approx 0.85

for most t’s. In the Early Departure, the two hazard functions differ considerably when t is small, but the difference approaches zero as

t \to \infty

. In the Crossing, the two hazard functions cross each other. Figure 1 displays the plots of

λ_{(1)} (t)

and

λ_{(0)} (t)

versus t in the Nearly Proportional, Early Departure and Crossing scenarios. The dependent censoring rate is 26.1% in the Early Departure and around 20% in the other three scenarios. The independent censoring rate is 0.2% in the Crossing and around 10% in the other three scenarios.

To show double robustness, we considered the doubly robust tests for which only the coarsening model or the event time model is correctly specified. We also investigated the empirical Type-I errors of the log-rank test and the adjusted log-rank tests for which only the treatment assignment model or the dependent censoring model is true. For the treatment assignment model used in the doubly robust tests and the adjusted log-rank tests, the correct model was fitted using covariates X₁ and X₂, and the incorrect model was fitted using X₁ only. For the dependent censoring model used in the doubly robust tests and the adjusted log-rank tests, the correct model was fitted using covariates X₁ and X₂, and the incorrect model was fitted using X₂ only. For the event time model used in the doubly robust tests, the correct model was fitted using covariates X₁, X₂ and X₃, and the incorrect model was fitted using X₁ and X₃. We use ‘TTF’ and ‘FFT’ to, respectively, mean that only the treatment assignment and dependent censoring models are true and that only the event time model is true, for the doubly robust tests. Similarly, ‘TF’ and ‘FT’, respectively, mean that only the treatment assignment model is true and that only the dependent censoring model is true, for the adjusted weighted log-rank tests.

Table 1 shows the empirical Type-I errors and powers of the doubly robust log-rank, Prentice–Wilcoxon, and Renyi-type log-rank tests. One can see that the Type-I error rates of all the doubly robust tests are around the nominal level except a bit inflation for the doubly robust Prentice–Wilcoxon and Renyi-type log-rank tests with the smaller sample size 300. The empirical powers of all the tests increase with the sample size. As expected, the doubly robust log-rank, Prentice–Wilcoxon, and Renyi-type log-rank tests are, respectively, the most powerful in the Nearly Proportional, Early Departure and Crossing scenarios.

Table 1.

Type-I error rates and powers of the doubly robust log-rank (DRLR), Prentice–Wilcoxon (DRPW) and Renyi-type log-rank (DRRLR) tests.

			Scenario
n	Test	Working models	Null	Nearly proportional	Early departure	Crossing
300	DRLR	TTF	0.043	0.292	0.662	0.086
		FFT	0.059	0.280	0.636	0.078
	DRPW	TTF	0.065	0.276	0.801	0.169
		FFT	0.066	0.272	0.792	0.165
	DRRLR	TTF	0.061	0.288	0.768	0.180
		FFT	0.066	0.278	0.757	0.171
500	DRLR	TTF	0.047	0.428	1.000	0.135
		FFT	0.048	0.426	1.000	0.133
	DRPW	TTF	0.048	0.425	1.000	0.238
		FFT	0.050	0.420	1.000	0.242
	DRRLR	TTF	0.050	0.426	1.000	0.305
		FFT	0.053	0.425	1.000	0.297

Table 2 shows the Type-I errors of the log-rank test and the adjusted log-rank tests for which only one of the two working models is true in the Null scenario. One can see that the log-rank test and the adjusted log-rank tests all have inflated Type-I error, because they did not or failed to adjust for the non-random treatment assignment and dependent censoring.

Table 2.

Type-I error rates of the log-rank test (LR) and the adjusted log-rank tests (ALR) under non-random treatment assignment and dependent censoring (in the NULL scenario).

		ALR
n	LR	TF	FT
300	0.121	0.075	0.096
500	0.132	0.067	0.091

According to Theorem 10.1 of Tsiatis,⁸ when both the coarsening model and the event time model are correctly specified, the doubly robust estimator

d {\hat{Λ}}_{(j)} (t)

is more efficient than the double inverse-weighted hazard estimator

d {\tilde{Λ}}_{(j)} (t) \equiv \frac{\sum_{i = 1}^{n} w_{ij} (\hat{θ}) exp {{\hat{Λ}}_{ij}^{C} (t)} {dN}_{i} (t)}{\sum_{i = 1}^{n} w_{ij} (\hat{θ}) exp {{\hat{Λ}}_{ij}^{C} (t)} Y_{i} (t)},

on which the adjusted weighted log-rank tests⁴ are built. Therefore, the doubly robust tests with all the working models being true are expected to be more powerful than the adjusted weighted log-rank tests. To verify this, we conduct simulations to compare their powers in the Nearly Proportional scenario. Table 3 shows that the doubly robust tests with all the working models being true indeed have larger powers than the adjusted weighted log-rank tests.

Table 3.

Powers of the doubly robust (DR) log-rank (LR), Prentice–Wilcoxon (PW) and Renyi-type log-rank tests with all the working models being true and the adjusted log-rank (LR), Prentice–Wilcoxon (PW) and Renyi-type log-rank tests in the Nearly Proportional scenario.

		Test type
n	Test subtype	Adjusted	DR
300	LR	0.263	0.287
	PW	0.263	0.273
	Renyi	0.284	0.287
500	LR	0.390	0.429
	PW	0.386	0.420
	Renyi	0.396	0.430

In the Web Appendix B.1, we compared the Type-I errors and powers of the doubly robust tests versus the log-rank test in the absence of non-random treatment assignment and dependent censoring. The simulation results show that the doubly robust tests maintain the nominal Type-I error even if they make unnecessary adjustment for confounding and dependent censoring, and that they are at least as powerful as the log-rank test. In the Web Appendix B.2, we evaluated the finite sample performance of the doubly robust survival estimator ${\hat{S}}_{(j)} (t)$ . The simulation results show that it performs very well in terms of bias, variance and coverage of asymptotic confidence interval when the sample size is 300.

4 Analysis of the SRTR data

Kidney-alone (KA) transplant and simultaneous kidney-pancreas (SPK) transplant are two treatment options for end-stage renal disease (ERSD) in Type I diabetes patients. Compared to KA transplant, SPK has the advantage of treating both ERSD and the diabetes, but is a more complicated surgery, likely leading to more post-operative complications. It is of clinical interest to compare the post-transplant survival experience between these two treatments. To do that, we apply the proposed methods to a data set of Type I diabetes patients with ERSD who had a SPK or KA transplant at age $\geq 18$ during 1 January 2000–31 May 2016 in the USA. The survival endpoint is time to graft failure, defined as death or observed graft failure. The data set we analyzed was obtained from the Scientific Registry of Transplant Recipients (SRTR). The SRTR data system includes data on all donor, wait-listed candidates, and transplant recipients in the USA, submitted by the members of the Organ Procurement and Transplantation Network (OPTN). The Health Resources and Services Administration (HRSA), and the U.S. Department of Health and Human Services provide oversight to the activities of the OPTN and SRTR contractors.

We include only the subjects who received the transplant for the first time in the analysis. Subjects lost to follow-up or surviving through 31 May 2016 without observed graft failure were censored. We assume that the choice of transplant type and the censoring are non-random, and consider a common set of baseline covariates in the three working models for the transplant type, the censoring hazard and the survival outcome. These covariates are age at transplant, gender, race, blood type, pretransplant time on dialysis, and donor age, same as in Zhang and Schaubel⁷ and Li.⁴ The final sample contains 1930 KA and 1636 SPK transplant recipients without missing values in the outcome or covariates. The longest common follow-up time between the two groups of transplant recipients is 13.58 years. Seventy-eight percent of the subjects in the sample were censored. The fitted logistic model for transplant type shows that age at transplant, race and donor age are significant at 0.05 level. This model fits the data decently as reflected by a c-statistic of 0.841. In the fitted cause-specific hazard model for censoring, age at transplant is significant for KA recipients, and age at transplant, race and donor age are significant for SPK recipients. The Cox–Snell residual plots (Figure 1 in the Web Appendix) show that the model fits the censoring time data well. In the fitted Cox model for the event time, age at transplant and race are significant for KA recipients, and race and donor age are significant for SPK recipients. The Cox–Snell residual plots (Figure 2 in the Web Appendix) show that the model fits the event time data well except at the right tail of the event time distribution. The overlapping of some significant covariates among the three models suggests the existence of confounding and dependent censoring.

In the analysis, we first estimated the transplant-type-specific survival functions over the initial 10 years after transplant. The left panel of Figure 2 shows the doubly robust survival function estimates. As we can see, SPK recipients have higher cumulative probabilities of graft failure over the first 10 years than KA recipients. On the other hand, using a double inverse-weighted estimation approach, Li⁴ showed that the conditional graft-failure-free probabilities given no failure for the initial five years are higher in SPK recipients than in KA recipients. Based on these two results, we conjecture that the hazard of graft failure is higher in SPK recipients at the beginning of the 10-year post-transplant period but higher in KA recipients at the end. We are also interested in knowing whether the two hazard functions cross each other at a time point between 0 and 10 years or equal each other over a time interval around the middle of the 10-year period. To perform a finer comparison of the two transplant types than Li,⁴ we divide the 10-year time window into three sub-intervals,

[0, 4], [4, 6]

and

[6, 10]

, compute the doubly robust estimates of transplant-type-specific conditional survival functions given no failure for four years and six years, respectively, and carry out our proposed tests over each of the three sub-intervals. The middle panel of Figure 2 shows that the doubly robust estimates of the two transplant-type-specific conditional survival functions given no failure for four years are close to each other over the interval

[4, 6]

, suggesting that the two hazards may be equal over this interval. The right panel of Figure 2 shows that SPK recipients have a higher doubly robust estimate of conditional survival function given no failure for six years over the interval

[6, 10]

, compared to KA recipients. Table 4 tabulates the results of four doubly robust tests, the adjusted log-rank test and the log-rank test for comparing KA versus SPK recipients with respect to their graft failure hazard over the three time windows. In conjunction with Figure 2, all the tests confirmed our conjecture that the graft failure hazard for KA recipients is significantly higher at the beginning of the post-transplant period, specifically from Year 0 to Year 4, and is significantly lower from Year 6 to Year 10, compared to that for SPK recipients. All the tests showed no significant difference from Year 4 to Year 6. Although the log-rank test concurred with the doubly robust tests in every comparison, it is invalid for analyzing the SRTR data since we found confounding and dependent censoring based on the fitting results of the three working models for the doubly robust tests. It is expected that the adjusted log-rank test concurred with the doubly robust tests in every comparison, because the aforementioned model diagnostics suggested that the models for treatment assignment and dependent censoring were correctly specified.

Table 4.

The two-sided p-values for testing the KA versus SPK difference in the graft failure hazard over 0 to 4, 4 to 6, and 6 to 10 years after transplant.

	Test
Time window	DRLR	DRPW	DRRLR	DRRPW	ALR	LR
$[0, 4]$	<0.0001	<0.0001	0	0	<0.0001	<0.0001
$[4, 6]$	0.380	0.390	0.610	0.611	0.405	0.945
$[6, 10]$	0.019	0.020	0.024	0.029	0.015	0.009

DRLR: doubly robust log-rank; DRPW: doubly robust Prentice–Wilcoxon; DRRLR: doubly robust Renyi-type log-rank; DRRPW: doubly robust Renyi-type Prentice–Wilcoxon; ALR: adjusted log-rank; LR: log-rank.

5 Discussion

We developed a set of doubly robust weighted log-rank tests and Renyi-type tests for two-sample survival comparison under confounding and dependent censoring. Through asymptotic arguments and simulations, we showed that the new tests have the nominal Type-I error rate and are consistent when either the working models for treatment assignment and dependent censoring are true or the working model for the event time is true. Our simulations also showed that the doubly robust tests are more powerful than the adjusted weighted log-rank tests⁴ if all the assumed models for treatment assignment, dependent censoring and the event time are true, and that they maintain the nominal Type-I error and are at least as powerful as the log-rank test in the absence of confounding and dependent censoring.

The proposed tests apply to competing risks data for comparing population-average cause-specific hazards between two treatment groups. They also apply to left-truncated survival data. Furthermore, they apply to survival data with both dependent and independent censoring, where one should just model the cause-specific hazard of dependent censoring given the covariates, as demonstrated in the simulations.

We conclude by pointing out some future research directions. First, it is meaningful to extend our two-sample tests to multi-sample tests. This can be done by using a multinomial logistic regression model for treatment assignment. Second, either or both of the Cox models for dependent censoring and event time could be replaced by other hazard models, e.g. the additive hazards model.¹⁴ Third, we presume that the covariates needed to satisfy the CAR assumption are time-independent. It would be ideal to relax this assumption to allow time-varying covariates with (almost) known trajectories (e.g. under frequent monitoring). Theoretically, this can be achieved using time-dependent Cox models for dependent censoring and event time, but it is not easy to program the resulting tests, as the plug-in estimator for the asymptotic variance of the new doubly robust log-rank statistic is rather involved. Bootstrap might be helpful to reduce the programming effort here. Finally, how to construct a doubly robust test for comparing treatment-specific survivals in sub-groups would be an interesting topic in this precision medicine era. This subgroup analysis is not just to apply our methods exclusively to the sub-group data, but rather to borrow information from the other subjects in the sample to enhance the efficiency of the doubly robust tests under some reasonable model assumptions.

Supplemental Material

Supplemental material for Doubly robust weighted log-rank tests and Renyi-type tests under non-random treatment assignment and dependent censoring

Supplemental Material for Doubly robust weighted log-rank tests and Renyi-type tests under non-random treatment assignment and dependent censoring by Chenxi Li in Statistical Methods in Medical Research

Footnotes

Acknowledgements

The kidney transplant data used here have been supplied by the Minneapolis Medical Research Foundation (MMRF) as the contractor for the Scientific Registry of Transplant Recipients (SRTR). The interpretation and reporting of these data are the responsibility of the author and in no way should be seen as an official policy of or interpretation by the SRTR or the U.S. Government.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Supplemental material

Supplemental material for this article is available online.

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