Partitioned log-rank tests for the overall homogeneity of hazard rate functions

Abstract

In survival analysis, it is routine to test equality of two survival curves, which is often conducted by using the log-rank test. Although it is optimal under the proportional hazards assumption, the log-rank test is known to have little power when the survival or hazard functions cross. To test the overall homogeneity of hazard rate functions, we propose a group of partitioned log-rank tests. By partitioning the time axis and taking the supremum of the sum of two partitioned log-rank statistics over different partitioning points, the proposed test gains enormous power for cases with crossing hazards. On the other hand, when the hazards are indeed proportional, our test still maintains high power close to that of the optimal log-rank test. Extensive simulation studies are conducted to compare the proposed test with existing methods, and three real data examples are used to illustrate the commonality of crossing hazards and the advantages of the partitioned log-rank tests.

Appendix: Assumptions and Proofs

1.1 Assumptions and Lemma

We impose the following three assumptions, which correspond respectively to condition (1) of Corollary 7.2.1, and conditions (2) and (3) of Theorem 6.2.1 in Fleming and Harrington (1991). Let $\xi = \sup \{ t: \prod _{k=0}^p \pi _k(t)>0 \}$.

Assumption 2

Assume that W(s) converges in probability to $W_0(s)$ uniformly on [0, t] for any $t\in [0, \xi ]$, where $W_0(s)$ is a nonnegative, left-continuous function with right-hand limits such that $W_0(s)<\infty$ for any $s\le \xi$, $W_0(s)=0$ for $s>\xi$, and $W_0(s+)\equiv \lim _{u\downarrow s} W_0(u)$ has bounded variation on each closed subinterval of $[0, \xi ]$.

Assumption 3

When $\xi \notin \{ t: \prod _{k=0}^p \pi _k(t)>0 \}$, it holds for $k=0,1, \ldots , p$ that

• (a) $\int _{(0, t]} W^2_0(s) \pi _k(s)\{ 1-\Delta \Lambda _k(s) \} d\Lambda _k(s) <\infty$, and

• (b) $\lim \limits _{t\uparrow \xi } \limsup \limits _{n\rightarrow \infty } P\left\{ \int _{(t, \xi ]} n^{-1}W^2(s) Y_k(s) d\Lambda _k(s) >\epsilon \right\} =0$ for any $\epsilon >0$.

Assumption 4

When $\xi <\infty$, it holds for $k=0,1, \ldots , p$ that

$$\begin{aligned} \lim _{n\rightarrow \infty } P\left\{ \int _{\xi }^{\infty } n^{-1}W^2(s) Y_k(s) d\Lambda _k(s) >\epsilon \right\} =0 \end{aligned}$$

for any $\epsilon >0$.

Under the above assumptions, we present a lemma below which plays a fundamental role in the proofs of all theorems in this paper. For group k ($k=0, 1, \ldots , p$), recall that $S_k(t)$ and $L_k(t)$ denote the respective survival functions of $T_{k}$ and $C_{k}$, $\Lambda _k(t) = - \int _{(0, t]} {dS_k(s)}/\{1-S_k(s)\}$, and $\pi _{\cdot }(t)= \sum _{k=0}^p \rho _k \pi _k(t)$, where $\rho _k \equiv \lim _{n\rightarrow \infty } n_k/n \in (0, 1)$ and $\pi _k(t) = S_k(t) L_k(t)$. Let $M(t) = N(t) - \int _{(0, t]} Y_*(s) d\Lambda (s)$ with $\Lambda (t) = (\Lambda _0(t), \ldots , \Lambda _p(t))^{{\mathrm {T}}}$ and

$$\begin{aligned} \tilde{Z}(t)= & {} n^{-1/2} \int _{(0, t]} W(s) \left\{ I_{p+1} - Y(s) J^{{\mathrm {T}}} / Y_{\cdot } (s) \right\} d M(s). \end{aligned}$$

(1)

Define $\widetilde{\Sigma }(t)\equiv (\tilde{\sigma }_{kl}(t))_{0\le k, l\le p}$ as

$$\begin{aligned} \widetilde{\Sigma }(t) = \int _{(0, t]} \left[ \left\{ I - \frac{\pi (s)J^{{\mathrm {T}}}}{ \pi _{\cdot }(s)} \right\} W_0^2(s) \pi _*(s)\left\{ I- \Delta \Lambda _*(s) \right\} d \Lambda _*(s) \left\{ I - \frac{ J\pi ^{{\mathrm {T}}}(s)}{ \pi _{\cdot }(s)} \right\} \right] , \end{aligned}$$

where $\pi (s) = (\rho _0 \pi _0(s), \ldots , \rho _p \pi _p(s))^{{\mathrm {T}}}$.

Lemma 1

Let $Q = (Q_0, Q_1, \ldots , Q_p)^{{\mathrm {T}}}$ be a $(p+1)$-variate Gaussian process. Suppose that all the components Q have independent increments, $Q_k(0)=0$ almost surely, for any $0\le s\le t$, ${ \mathbb {E}}\{Q_k(t)\} = 0$ and ${ \mathbb {E}}\{Q_k(t)Q_l(s)\} = \tilde{\sigma }_{kl}(s\wedge t)$, where $\tilde{\sigma }_{kl}(t)$’s are continuous functions. Under Assumptions 2–4, as n goes to infinity, $\tilde{Z}$ defined in (1) converges weakly to Q in $(\mathcal {D}[0, \infty ])^{p+1}$, where $\mathcal {D}[0, \infty ]$ is the space of functions on $[0, \infty ]$ that are right-continuous with finite left-hand limits.

Proof of Lemma 1

Along the lines of the proof of Theorem 6.2.1 in Fleming and Harrington (1991), the lemma can be proved by showing that under Assumptions 2, 3 and 4, the components of $\tilde{Z}(t)$ satisfy both (3.17) and (3.18) of Theorem 5.3.5 in Fleming and Harrington (1991). $\square$

1.2 Proofs of Theorems 1, 2, 3 and 4

We first prove Theorems 3 and 4. Theorems 1 and 2 follows immediately as they are the special cases of Theorems 3 and 4 with $p=1$.

Proof of Theorem 3

When the $p+1$ survivor functions are all equal, let $\Lambda _c$ be the common cumulative hazard function. Then $\widetilde{\Sigma }(t)$ reduces to

$$\begin{aligned}&\Sigma (t)\equiv (\sigma _{kl}(t))_{0\le k, l\le p} \nonumber \\&\quad = \int _{(0, t]} W_0^2(s) \left\{ \pi _*(s) - \frac{\pi (s)\pi ^{{\mathrm {T}}}(s)}{ \pi _{\cdot }(s)} \right\} \left\{ 1- \Delta \Lambda _c (s) \right\} d \Lambda _c (s). \end{aligned}$$

(2)

Thus, part (a) follows from Lemma 7.2.1 in Fleming and Harrington (1991).

It follows from Lemma 1 that under $H_0$, as $n\rightarrow \infty$, the multivariate stochastic process $Z_{{{\tiny [-1]}}}$ converges weakly to $Q_{{{\tiny [-1]}}}$ which has independent increments and variance–covariance matrix $\Sigma _{{{\tiny [-1,-1]}}}(t)$. This implies that under $H_0$, as $n\rightarrow \infty$,

$$\begin{aligned} n^{-1/2} D_l(t)= & {} Z_{{{\tiny [-1]}}}(t) \; \text{ converges } \text{ in } \text{ distribution } \text{ to }\; N(0, \Sigma _{{{\tiny [-1,-1]}}}(t)), \\ n^{-1/2} D_u(t)= & {} Z_{{{\tiny [-1]}}}(\infty )- Z_{{{\tiny [-1]}}}(t) \; \text{ converges } \text{ in } \text{ distribution } \text{ to }\; N(0, \Sigma _{{{\tiny [-1,-1]}}}(\infty )\\&-\Sigma _{{{\tiny [-1,-1]}}}(t)), \end{aligned}$$

and that $n^{-1/2} D_l(t)$ and $n^{-1/2} D_u(t)$ are asymptotically independent. Therefore, part (b) holds.

Accordingly, the stochastic process $\tilde{T}$ converges weakly to $A_1(t)$ with

$$\begin{aligned} A_1(t)= & {} Q_{{{\tiny [-1]}}}^{{\mathrm {T}}}(t) \Sigma _{{{\tiny [-1,-1]}}}^{-1}(t) Q_{{{\tiny [-1]}}} (t) \nonumber \\&+ \Big \{ Q_{{{\tiny [-1]}}} (\infty ) - Q_{{{\tiny [-1]}}}(t)\Big \}^{{\mathrm {T}}} \Big \{ \Sigma _{{{\tiny [-1,-1]}}}(\infty ) - \Sigma _{{{\tiny [-1,-1]}}}(t)\Big \}^{-1} \{ Q_{{{\tiny [-1]}}} (\infty )\nonumber \\&- Q_{{{\tiny [-1]}}}(t)\}, \end{aligned}$$

(3)

which implies part (c). $\square$

Proof of Theorem 4

Under Assumptions 1 and 2, W(t), $Y_k(t)/n$, $Y_{\cdot }(t)/n$ converge in probability to $W_0(s)$, $\rho _k\pi _{\cdot }(s)$ and $\pi _{\cdot }(s)$ uniformly on $[0, \xi )$, respectively. Under (iv) of Assumption 1, each element of $\Sigma (t)$ is finite, which means $\Sigma (t)$ is well defined. Consequently, $\widehat{\Sigma }(t)$ converges in probability to $\Sigma (t)$, which implies part (a).

To prove part (b), we recall that $Z(t) = \tilde{Z} (t) + \tilde{R}(t)$, where $\tilde{Z}$ is defined in (1) and

$$\begin{aligned} \tilde{R}(t) =n^{-1/2} \int _{(0, t]} W(s) \left\{ Y_*(s) - Y(s) Y^{{\mathrm {T}}} (s) / Y_{\cdot } (s) \right\} d \Lambda _c(s). \end{aligned}$$

Clearly, $\tilde{Z}$ satisfies the conditions of Lemma 1 and converges weakly to Q, which is defined in Lemma 1. Meanwhile, by the same arguments as those in proving (a),

$$\begin{aligned} \tilde{R}(t) = \int _{(0, t]} \frac{W(s)}{n} \left\{ Y_*(s) - \frac{ Y(s) Y^{{\mathrm {T}}} (s)}{ Y_{\cdot } (s)} \right\} \cdot n^{1/2} \left\{ \frac{d\Lambda (s)}{d\Lambda _c(s)} - J \right\} d\Lambda _c(s) \end{aligned}$$

converges in probability to

$$\begin{aligned} R(t) = \int _{(0, t]} W_0(s) \left\{ \pi _*(s) - \frac{ \pi (s) \pi ^{{\mathrm {T}}} (s)}{ \pi _{\cdot } (s)} \right\} \gamma (s) d\Lambda _c(s), \end{aligned}$$

(4)

where $\gamma (s) = (\gamma _0(s), \ldots , \gamma _p(s))^{{\mathrm {T}}}$. This in conjunction with the weak convergence of $\tilde{Z}$ implies that Z converges weakly to $Q + R$. Combining the weak convergence of $Z_{{{\tiny [-1]}}}$ and $\hat{\Sigma }_{{{\tiny [-1,-1]}}}$, we conclude that $\tilde{T}$ converges weakly to $A_2(t)$ over $t\in \Omega$, where

$$\begin{aligned} A_2(t)= & {} (Q + R)_{{{\tiny [-1]}}}^{{\mathrm {T}}}(t) \Sigma _{{{\tiny [-1,-1]}}}^{-1}(t) (Q + R)_{{{\tiny [-1]}}} (t) \nonumber \\&+ \{ (Q + R)_{{{\tiny [-1]}}} (\infty ) - (Q + R)_{{{\tiny [-1]}}}(t)\}^{{\mathrm {T}}} \{\Sigma _{{{\tiny [-1,-1]}}}(\infty ) - \Sigma _{{{\tiny [-1,-1]}}}(t) \}^{-1} \nonumber \\&\times \{ (Q + R)_{{{\tiny [-1]}}} (\infty ) - (Q + R)_{{{\tiny [-1]}}}(t)\}. \end{aligned}$$

(5)

Consequently, T converges in distribution to $\sup _{t\in \Omega } A_2(t)$. $\square$

Proof of Theorem 1

This theorem is a special case of Theorem 3 with $p=1$. It can be verified that $\Sigma _{{{\tiny [-1,-1]}}}(t) = \sigma (t)$ when $p=1$, therefore results (a) and (b) follow immediately.

We need only prove result (c). When $p=1$, part (c) of Theorem 3 implies that T converges in distribution to the supremum of

$$\begin{aligned} \frac{\{ Q_1(t) \}^2}{ \sigma (t)} + \frac{ \{ Q_1 (\infty ) - Q_1(t)\}^2}{ \sigma (\infty ) - \sigma (t)}, \end{aligned}$$

(6)

where $Q_1$ is a Gaussian process with independent increments and variance $\sigma$.

Let $\{B(t): t \in [0, \infty )\}$ denote a Brownian motion, then the supremum of (6) has the same distribution as

$$\begin{aligned}&\sup _{t\in \Omega } \left\{ \frac{[ B\{\sigma (t) \}]^2}{ \sigma (t)} + \frac{ [ B\{ \sigma (\infty ) \} - B\{ \sigma (t) \}]^2}{ \sigma (\infty ) - \sigma (t)} \right\} \\&\quad = \sup _{0<s<\sigma (\infty )} \left\{ \frac{\{ B(s) \}^2}{ s} + \frac{ [ B\{ \sigma (\infty ) \} - B(s)]^2}{ \sigma (\infty ) -s} \right\} \\&\quad = \sup _{0<t<1} \left[ \frac{\{ B'(t) \}^2}{t} + \frac{ \{ B'(1) - B'(t)\}^2}{ 1 -t} \right] \end{aligned}$$

with $B'(t) = B\{ \sigma (\infty ) s \}/\{\sigma (\infty )\}^{1/2}$. Since $B'(t)$ is still a Brownian motion, this leads to result (c). $\square$

Proof of Theorem 2

This theorem is a special case of Theorem 4 with $p=1$. It can be verified that $\Sigma _{{{\tiny [-1,-1]}}}(t)$, $R_{{{\tiny [-1]}}}(t)$ and $A_2(s)$ reduce respectively to the $\sigma (t)$, R(t) and A (s) defined in Theorem 2. This completes the proof. $\square$