Visualization and dynamics of multidimensional health-related quality-of-life-adjusted overall survival: a new analytic approach

Abstract

Background

Vesnarinone Trial (VesT) was a three-armed, placebo-controlled, randomized clinical trial designed to study the effects of 30 mg or 60 mg/day vesnarinone. Certain contradictory results involving patient health-related quality-of-life (HRQOL) and overall survival (OS) have made a definitive and unified conclusion difficult.

Methods

To reconcile these findings, we have focused on the HRQOL-adjusted OS, commonly known as quality-adjusted life years (QALYs). Currently, analyses of QALYs incorporate a single HRQOL subscale. However, the VesT HRQOL instrument had two subscales: physical (PHYS) and emotional (EMOT). We have developed new ways to visualize and compare EMOT- and PHYS-adjusted OS.

Results

In each VesT arm, there was an increased probability of superior EMOT-adjusted OS, compared to PHYS-adjusted OS. The magnitude of these findings was comparable across trial arms. Despite inferior survival and superior EMOT and PHYS scores, the 60-mg/day arm presents similar EMOT- and PHYS-adjusted OS compared to the placebo arm.

Conclusions

We have provided a fresh perspective on the complex interactions between multiple HRQOL dimensions and OS. These novel methods address the burgeoning need for robust information on the interplay between OS and HRQOL from a patient, clinical care and public policy perspective.

Appendix

Commonly, individual $i=1,\ldots, n$ time to event T _i is subject to independent right-censoring by C _i , hence the follow-up time $\widetilde{T}_i={\text{minimum}}(T_ i, C_i)$ and the censoring indicator $\Updelta_i=I(T_i \leq C_i)$. Both T _i and C _i are assumed to be continuous. HRQOL is measured using a preference-based questionnaire with several subscales. For simplicity, assume it only has two subscales, p = 1, 2. HRQOL on the pth scale, is quantified by a continuous-time stochastic process $V^{(p)}(\cdot)$, with state space $\mathcal{S}_p=\{0,1,\ldots, S_p\}$. Assume that the health states in $\mathcal{S}_p$ are ordered increasingly, “0” being the worst and “S _p ” the best state. For each individual $i=1,\ldots, n$, assume that $V_{i}^{(p)}(\cdot)$ and C _i are independent and let a non-decreasing, known function $Q^{(p)}(\cdot)$ assign utilities between 0 (state “0”) and 1 (state “S _p ”) to each state in $\mathcal{S}_p$. For example, one could think of $V^{(1)}(\cdot)$ and $V^{(2)}(\cdot)$ as being the emotional and physical health subscales in the VesT example. When using a generic pair $\{V(\cdot),Q(\cdot)\}$, the HRQOL-adjusted lifetime is defined as

$$QT=\int\limits_{0}^{T}Q\{V(t)\}{\text {d}}t$$

and its observed version is

$$\widetilde{QT}=\int\limits_{0}^{\widetilde{T}}Q\{V(t)\}{\text {d}}t.$$

Since T is adjusted on two different HRQOL scales, the resulting Q ⁽¹⁾ T and Q ⁽²⁾ T are correlated. Their joint distribution is necessary to perform testing for significance.

Assume the existence of a constant L > 0 such that P(T < L) = 1 and P(C > L) > 0. A technical assumption, first, we estimate the joint distribution of Q ⁽¹⁾ T and Q ⁽²⁾ T, namely F _1,2(q ₁, q ₂) = P(Q ⁽¹⁾ T ≤ q ₁, Q ⁽²⁾ T ≤ q ₂). Define H _1,2(q ₁, q ₂) as P(Q ⁽¹⁾ T > q ₁, Q ⁽²⁾ T > q ₂) and note that

$$F_{1,2}(q_1,q_2)= H_{1,2}(0,0)-H_{1,2}(q_1,0)-H_{1,2}(0,q_2)+H_{1,2}(q_1,q_2).$$

Thus, to estimate $F_{1,2}(\cdot,\cdot)$ it suffices to estimate $H_{1,2}(\cdot,\cdot)$. In the absence of censoring and based on a sample of size n, H _1,2(q ₁, q ₂) can be estimated by the empirical mean of the corresponding indicators of interest,

$$n^{-1}\sum\limits_{i=1}^{n}I\{ Q^{(1)}T_i > q_1, Q^{(2)}T_i > q_2 \}.$$

However, when censoring is present, define $m^{(p)}_i(t)={\text {infimum}} \left[s \geq 0 ; \int_{0}^{s}Q^{(p)}\{V^{(p)}_{i}(u)\}{\text{d}}u \geq t \right]$ and D ^(p) _i (t) = m ^(p) _i (t) ∧ T _i , for p = 1,2 and $i=1,\ldots, n$. By convention, D ^(p) _i (0) = 0. This expression states that D ^(p) _i (t) marks the first time when the ith individual has accumulated at least t HRQOL-adjusted lifetime on the $Q^{(p)}(\cdot)$ scale. Should this not happen until time T _i , then D ^(p) _i (t) is assigned the value T _i . The key to producing a consistent estimator for the quantity of interest is to note that if $\widetilde{Q^{(p)}T_i} > q_p$, then Q ^(p) T _i  > q _p and censoring time C _i does not occur prior to D ^(p) _i (q _p ).

Consequently, if A ^(p) _i (q _p ) = I{Q ^(p) T _i  > q _p , C _i  > D ^(p) _i (q _p )} and $B^{(p)}_i(q_p)=I\{\widetilde{Q^{(p)}T}_i > q_p \}$, then A ^(p) _i (q _p ) = B ^(p) _i (q _p ), where p = 1,2. Here, $I\{ \cdot \}$ represents the indicator function. Define D ^(1,2) _i (q ₁, q ₂) to be the maximum of D ⁽¹⁾ _i (q ₁) and D ⁽²⁾ _i (q ₂). An inverse probability-of-censoring weighted estimator becomes available after assessing when the indicator functions involved are completely observed. Further, should G(u) = P(C > u) be known

$$\widetilde{H}_{1,2}(q_1,q_2)=n^{-1}\sum\limits_{i=1}^{n} \frac{A^{(1)}_i(q_1)A^{(2)}_i(q_2)}{G\{ D^{(1,2)}_i(q_1,q_2)\}}$$

would represent an unbiased estimator for H _1,2(q ₁, q ₂).

This assertion is true because

$$\begin{aligned} E\left[\frac{A^{(1)}(q_1)A^{(2)}(q_2)}{G\{ D^{(1,2)}(q_1,q_2) \}}\right]&= E\left(E\left[\left.\frac{A^{(1)}(q_1)A^{(2)}(q_2)}{G\{ D^{(1,2)}(q_1,q_2)\}}\right| T,V^{(1)}(\cdot),V^{(2)}(\cdot) \right]\right) \\ &\quad=E \left(I\{ Q^{(1)}T > q_1, Q^{(2)}T > q_2 \} E\left[\left.\frac{I\{C > D^{(1,2)}(q_1,q_2) \}}{G\{ D^{(1,2)}(q_1,q_2)\}}\right| T, V^{(1)}(\cdot), V^{(2)}(\cdot) \right] \right) \\ &\quad=E\{ I( Q^{(1)}T > q_1, Q^{(2)}T > q_2 ) \}=H_{1,2}(q_1,q_2). \end{aligned}$$

However, G(u) being unknown, it is estimated by the Kaplan–Meier estimator $\widehat{G}(u)$ of the censoring time survival function computed using $\{(\widetilde{T}_i,1-\Updelta_i);i=1,\ldots, n\}.$

Estimator consistency

A consistent estimator for H _1,2(q ₁, q ₂) is

$$\widehat{H}_{1,2}(q_1,q_2)=n^{-1}\sum\limits_{i=1}^{n} \frac{B^{(1)}_i(q_1)B^{(2)}_i(q_2)}{\widehat{G}\{ D^{(1,2)}_i(q_1,q_2)\}},$$

from which a consistent estimator $\widehat{F}_{1,2}(q_1,q_2)$ of F _1,2(q ₁, q ₂) becomes immediately available. Consistency of H _1,2(q ₁, q ₂) is shown using arguments along the lines of [21] and [22]. Note that-1

$$\begin{aligned} \widehat{H}_{1,2}(q_1,q_2)-H_{1,2}(q_1,q_2)&= \widetilde{H}_{1,2}(q_1,q_2)-H_{1,2}(q_1,q_2) +\widehat{H}_{1,2}(q_1,q_2)-\widetilde{H}_{1,2}(q_1,q_2) \\ &=n^{-1}\sum\limits_{i=1}^{n} \left[ \frac{B^{(1)}_i(q_1)B^{(2)}_i(q_2)} {\widehat{G}\{D^{(1,2)}_i(q_1,q_2)\}}-H_{1,2}(q_1,q_2)\right] \end{aligned}$$

(1)

$$+n^{-1}\sum\limits_{i=1}^{n}B^{(1)}_i(q_1)B^{(2)}_i(q_2) \frac{G\{D^{(1,2)}_i(q_1,q_2)\}-\widehat{G} \{D^{(1,2)}_i(q_1,q_2)\}}{G\{D^{(1,2)}_i(q_1,q_2) \} \widehat{G}\{D^{(1,2)}_i(q_1,q_2)\}}.$$

(2)

The expression in (1) is a sum of zero-mean i.i.d. terms, so it converges to zero in probability. In addition, D ^(1,2) _i (q ₁, q ₂) < L < τ _C : = sup{t;G(t) > 0} and $\widehat{G}$ converges uniformly in probability to G on [0, τ _C ). The term in (2) is bounded from above, in absolute value, by

$$\frac{{\text {sup}}\{|\widehat{G}(u)-G(u)|;u \leq \tau_C \}} {\widehat{G}(\tau_C)G(\tau_C)},$$

so it converges to zero, in probability. This shows that $\widehat{H}_{1,2}(\cdot,\cdot)$ is a consistent estimator of $H_{1,2}(\cdot,\cdot)$; hence, an estimator for $F_{1,2}(\cdot,\cdot)$ can be derived easily.

Limiting distribution results

Let $M^{c}(u)=I\{C \leq u, C \leq T)\}- \int_0^{u}I(\widetilde{T} \geq s){\text{d}}\Uplambda^{c}(s)$, where $\Uplambda^{c}(s)=exp\{-G(s)\}$ represents the cumulative hazard function of the censoring distribution. Using Lemma 2.4 in [23], it follows that for all $t\leq {\text{max}}\{\widetilde{T}_{j};j=1,\ldots, n\}$, we have that

$$\frac{G(t)-\widehat{G}(t)}{G(t)}= \int\limits_{0}^{t} \frac{\widehat{G}(u-)}{G(u)} \frac{\sum\nolimits_{j=1}^{n}{\text{d}}{M_{j}}^{c}(u)} {\sum\nolimits_{j=1}^{n}I(\widetilde{T}_{j} \geq u)}.$$

When t = D ^(1,2) _i (q ₁, q ₂) in the previous formula, one obtains that

$$\frac{G\{D^{(1,2)}_i(q_1,q_2)\}-\widehat{G}\{D^{(1,2)}_i(q_1,q_2)\}} {G\{D^{(1,2)}_i(q_1,q_2)\}}= \int\limits_{0}^{L} I\{D^{(1,2)}_i(q_1,q_2) \geq u \}\frac{\widehat{G}(u-)}{G(u)} \frac{\sum\nolimits_{j=1}^{n}{\text{d}}{M_{j}}^{c}(u)} {\sum\nolimits_{j=1}^{n}I(\widetilde{T}_{j} \geq u)}.$$

Note that $W_{1,2}(q_1,q_2):=n^{1/2} \{\widehat{H}_{1,2}(q_1,q_2)-H_{1,2}(q_1,q_2)\} =n^{1/2}\{\widetilde{H}_{1,2}(q_1,q_2)-H_{1,2}(q_1,q_2)\} +n^{1/2}\{\widehat{H}_{1,2}(q_1,q_2)-\widetilde{H}_{1,2}(q_1,q_2)\}$

$$\begin{aligned} &=n^{-1/2} \sum_{i=1}^{n} \left[\frac{A^{(1)}_i(q_1)A^{(2)}_i(q_2)} {G\{D^{(1,2)}_i(q_1,q_2)\}}-H_{1,2}(q_1,q_2)\right] \\ &\quad+ n^{-1/2} \int\limits_{0}^{L}n^{-1}\sum_{i=1}^{n} \left[I\{D^{(1,2)}_i(q_1,q_2) \geq u\} \frac{A^{(1)}_i(q_1)A^{(2)}_i(q_2)} {\widehat{G}\{D^{(1,2)}_i(q_1,q_2)\}}\right] \frac{\widehat{G}(u-)}{G(u)}\frac{\sum_{j=1}^{n}{\text{d}}{M_{j}}^{c}(u)} {n^{-1}\sum_{j=1}^{n}I\{\widetilde{T}_{j} \geq u\}}. \end{aligned}$$

(3)

Because

$$\frac{\widehat{G}(u-)}{G(u)} \frac{1}{n^{-1}\sum\nolimits_{j=1}^{n}I(\widetilde{T}_{j} \geq u)} \buildrel{{\text {a.s.}}}\over{\longrightarrow} \left\{P(\widetilde{T} \geq u)\right\}^{-1},$$

it follows that the second term in (3) can be written as

$$\begin{aligned} &n^{-1/2} \int\limits_{0}^{L}n^{-1}\sum_{i=1}^{n} \left[\left\{P(\widetilde{T} \geq u)\right\}^{-1} I\{D^{(1,2)}_i(q_1,q_2) \geq u\}\frac{A^{(1)}_i(q_1)A^{(2)}_i(q_2)} {\widehat{G}\{D^{(1,2)}_i(q_1,q_2)\}}\right] \sum_{j=1}^{n}{\text{d}}{M_{j}}^{c}(u)+o_{P}(1) \\ &=n^{-1/2} \int\limits_{0}^{L} [P(\widetilde{T} \geq u)]^{-1}E\left[I\{D^{(1,2)}(q_1,q_2) \geq u\} \frac{A^{(1)}(q_1)A^{(2)}(q_2)}{\widehat{G}\{D^{(1,2)}(q_1,q_2)\}}\right] \sum_{j=1}^{n}{\text{d}}{M_{j}}^{c}(u)+o_{P}(1) \\ &=n^{-1/2}\sum_{i=1}^{n} \int\limits_{0}^{L} [P(\widetilde{T} \geq u)]^{-1}E\left[I\{D^{(1,2)}(q_1,q_2) \geq u \} \frac{A^{(1)}(q_1)A^{(2)}(q_2)} {G\{ D^{(1,2)}(q_1,q_2) \}}\right] {\text{d}}{M_{i}}^{c}(u)+o_{P}(1) \\ &=n^{-1/2}\sum_{i=1}^{n} \int\limits_{0}^{L} \frac{J_{1,2}(q_1,q_2,u)} {P(\widetilde{T} \geq u)}{\text{d}}{M_{i}}^{c}(u)+o_{P}(1), \end{aligned}$$

where

$$J_{1,2}(q_1,q_2,u):= E\left[I\{D^{(1,2)}(q_1,q_2) \geq u\}\frac{A^{(1)}(q_1)A^{(2)}(q_2)}{G\{ D^{(1,2)}(q_1,q_2) \}}\right].$$

Then, one can further write

$$\begin{aligned} W_{1,2}(q_1,q_2)&=n^{-1/2} \sum\limits_{i=1}^{n} \left[ \frac{A^{(1)}(q_1)A^{(2)}(q_2)} {G\{ D^{(1,2)}(q_1,q_2) \}}-H_{1,2}(q_1,q_2) \right] \\ & \quad +n^{-1/2}\sum\limits_{i=1}^{n} \int\limits_{0}^{L} \frac{J_{1,2}(q_1,q_2,u)} {P(\widetilde{T} \geq u)}{\text{d}}{M_{i}}^{c}(u)+o_{P}(1). \end{aligned}$$

Consequently, the asymptotic covariance of W _1,2(q ₁, q ₂) and W _1,2(r ₁, r ₂) is equal to

$$\begin{aligned} &E\left( \left[ \frac{A^{(1)}(q_1)A^{(2)}(q_2)}{G\{ D^{(1,2)}(q_1,q_2) \}}-H_{1,2}(q_1,q_2) +\int\limits_{0}^{L} \frac{J_{1,2}(q_1,q_2,u)}{P(\widetilde{T} \geq u)}{\text{d}}{M^{c}(u)} \right]\right. \\ &\times \left. \left[ \frac{A^{(1)}(r_1)A^{(2)}(r_2)}{G\{ D^{(1,2)}(r_1,r_2) \}}-H_{1,2}(r_1,r_2) +\int\limits_{0}^{L} \frac{J_{1,2}(r_1,r_2,u)}{P(\widetilde{T} \geq u)}{\text{d}}{M^{c}(u) }\right] \right) \\ &=E \left( \left[ \frac{A^{(1)}(q_1)A^{(2)}(q_2)}{G\{ D^{(1,2)}(q_1,q_2) \}}-H_{1,2}(q_1,q_2) \right] \left[ \frac{A^{(1)}(r_1)A^{(2)}(r_2)}{G\{ D^{(1,2)}(r_1,r_2) \}}-H_{1,2}(r_1,r_2) \right] \right) \\ & +E \left( \left[ \frac{A^{(1)}(q_1)A^{(2)}(q_2)}{G\{ D^{(1,2)}(q_1,q_2) \}}-H_{1,2}(q_1,q_2) \right] \int\limits_{0}^{L} \frac{J_{1,2}(r_1,r_2,u)}{P(\widetilde{T} \geq u)}{\text{d}}{M}^{c}(u) \right) \end{aligned}$$

(4)

$$+E \left( \left[ \frac{A^{(1)}(r_1)A^{(2)}(r_2)}{G\{ D^{(1,2)}(r_1,r_2) \}}-H_{1,2}(r_1,r_2) \right] \int\limits_{0}^{L} \frac{J_{1,2}(q_1,q_2,u)}{P(\widetilde{T} \geq u)}{\text{d}}{M}^{c}(u) \right)$$

(5)

$$+E\left\{ \int\limits_{0}^{L} \frac{J_{1,2}(q_1,q_2,u)}{P(\widetilde{T} \geq u)}{\text{d}}{M}^{c}(u) \int_{0}^{L} \frac{J_{1,2}(r_1,r_2,u)}{P(\widetilde{T} \geq u)}{\text{d}}{M}^{c}(u) \right\}.$$

(6)

Note that the expression in (4) is equal to

$$\begin{aligned} &E\left[ \frac{A^{(1)}(q_1)A^{(2)}(q_2)}{G\{ D^{(1,2)}(q_1,q_2) \}} \int\limits_{0}^{L} \frac{J_{1,2}(r_1,r_2,u)}{P(\widetilde{T} \geq u)}{\text{d}}{M}^{c}(u) \right] \\ &-E\left\{ \int\limits_{0}^{L} H_{1,2}(q_1,q_2) \frac{J_{1,2}(r_1,r_2,u)}{P(\widetilde{T} \geq u)}{\text{d}}{M}^{c}(u) \right\} \end{aligned}$$

(7)

Terms (4) and (5) are both equal to

$$-\int\limits_{0}^{L} \frac{J_{1,2}(q_1,q_2,u)J_{1,2}(r_1,r_2,u)}{P(\widetilde{T} \geq u)}{\text{d}}{\Uplambda}^{c}(u),$$

using arguments as in [24]. Standard results for stochastic integrals with respect to martingales lead to the fact that the expression in (6) is equal to

$$\begin{aligned} &E \left\{\int\limits_{0}^{L} \frac{J_{1,2}(q_1,q_2,u)J_{1,2}(r_1,r_2,u)}{P^{2}(\widetilde{T} \geq u)}d{\Uplambda}^{c}(u)d\langle M^{c}(u), M^{c}(u)\rangle \right\} \\ &=E\left\{\int\limits_{0}^{L} \frac{J_{1,2}(q_1,q_2,u)J_{1,2}(r_1,r_2,u)}{P^{2}(\widetilde{T} \geq u)}I(\widetilde{T} \geq u){\text{d}}\Uplambda^{c}(u) \right\} \\ &=\int\limits_{0}^{L} \frac{J_{1,2}(q_1,q_2,u)J_{1,2}(r_1,r_2,u)}{P(\widetilde{T} \geq u)}{\text{d}}\Uplambda^{c}(u). \end{aligned}$$

By applying the multivariate central limit theorem, it follows that $W_{1,2}(\cdot,\cdot)$ converges in finite-dimensional distribution to a 4-variate zero-mean Gaussian process whose covariance function is equal to

$$\begin{aligned} \sigma_{H_{1,2}}(q_1,q_2;r_1,r_2) &=E \left( \left[ \frac{A^{(1)}(q_1)A^{(2)}(q_2)}{G\{ D^{(1,2)}(q_1,q_2) \}}-H_{1,2}(q_1,q_2) \right] \left[ \frac{A^{(1)}(r_1)A^{(2)}(r_2)}{G\{ D^{(1,2)}(r_1,r_2) \}}-H_{1,2}(r_1,r_2) \right] \right) \\ & \quad -\int\limits_{0}^{L} \frac{J_{1,2}(q_1,q_2,u)J_{1,2}(r_1,r_2,u)}{P(\widetilde{T} \geq u)}{\text{d}}\Uplambda^{c}(u). \end{aligned}$$

Recall that F _1,2(q ₁, q ₂) = H _1,2(0, 0) −H _1,2(q ₁, 0) − H _1,2(0, q ₂) + H _1,2(q ₁, q ₂). Using the previous findings, it follows that $\sqrt{n}\{\widehat{F}_{1,2}(q_1,q_2)-F_{1,2}(q_1,q_2)\}$ converges in finite-dimensional distribution to a 4-variate zero-mean Gaussian process with covariance function

$$\begin{aligned} &\sigma_{F_{1,2}}(q_1,q_2;r_1,r_2)= {\text{Cov}} \left\{ H_{1,2}(0,0)-H_{1,2}(q_1,0)-H_{1,2}(0,q_2)+H_{1,2}(q_1,q_2), \right.\\ &\left.\quad H_{1,2}(0,0)-H_{1,2}(r_1,0)-H_{1,2}(0,r_2)+H_{1,2}(r_1,r_2) \right\} \end{aligned}$$

Define R _1,2(q ₁, q ₂, u) = J _1,2(0, 0, u) − J _1,2(q ₁, 0, u) − J _1,2(0, q ₂, u) + J _1,2(q ₁, q ₂, u) and

$$\begin{aligned} \varvec{\Upgamma}_{1,2}(q_1,q_2)&= \frac{A^{(1)}(0)A^{(2)}(0)}{G\{ D^{(1,2)}(0,0) \}}- \frac{A^{(1)}(q_1)A^{(2)}(0)}{G\{ D^{(1,2)}(q_1,0) \}}- \frac{A^{(1)}(0)A^{(2)}(q_2)}{G\{ D^{(1,2)}(0,q_2) \}}\\ & \quad + \frac{A^{(1)}(q_1)A^{(2)}(q_2)}{G\{ D^{(1,2)}(q_1,q_2) \}}- F_{1,2}(q_1,q_2). \end{aligned}$$

After involved, but direct, computations, it follows that σ _{F_1,2}(q ₁, q ₂;r ₁, r ₂) is equal to

$$E \left[\varvec{\Upgamma}_{1,2}(q_1,q_2) \varvec{\Upgamma}_{1,2}(r_1,r_2) \right] -\int\limits_{0}^{L}\frac{{\bf R}_{1,2}(q_1,q_2,u){\bf R}_{1,2}(r_1,r_2,u)}{P(\widetilde{T} \geq u)}{\text{d}}\Uplambda^{c}(u).$$

Variance estimation

Following standard arguments, the cumulative hazard $\Uplambda^{c}(t)$ of the censoring distribution is estimated by the Nelson–Aalen estimator

$$\widehat{\Uplambda}^{c}(t)=\int\limits_{0}^{t} \frac{\sum\nolimits_{i=1}^{n} (1-\Updelta_{i})dI(\widetilde{T}_{i} \leq u)} {\sum\nolimits_{i=1}^{n}I(\widetilde{T}_{i} \geq u)}.$$

The value of $P(\widetilde{T} \geq u)$ can be consistently estimated by $n^{-1} \sum\nolimits_{i=1}^{n} I(\widetilde{T}_{i}\geq u)$, while an estimator for J _1,2(q ₁, q ₂, u) is given by

$$\hat{J}_{1,2}(q_1,q_2,u)=n^{-1}\sum\limits_{i=1}^{n} I\{D^{(1,2)}_i(q_1,q_2) \geq u\} \frac{B^{(1)}_i(q_1)B^{(2)}_i(q_2)}{\widehat{G}\{D^{(1,2)}_i(q_1,q_2)\}}.$$

Furthermore, $\varvec{\Upgamma}_{1,2}(q_1,q_2)$ and R _1,2(q ₁, q ₂, u) are estimated by

$$\begin{aligned} \widehat{\varvec{\Upgamma}}_{1,2}(q_1,q_2)&= n^{-1}\sum\limits_{i=1}^{n} \left[\frac{B^{(1)}_i(0)B^{(2)}_i(0)}{\widehat{G}\{ D^{(1,2)}_i(0,0) \}}- \frac{B^{(1)}_i(q_1)B^{(2)}_i(0)}{\widehat{G}\{ D^{(1,2)}_i(q_1,0) \}}- \frac{B^{(1)}_i(0)B^{(2)}_i(q_2)}{\widehat{G}\{ D^{(1,2)}_i(0,q_2) \}} \right. \\ & \quad +\left. \frac{B^{(1)}_i(q_1)B^{(2)}_i(q_2)}{\widehat{G}\{ D^{(1,2)}_i(q_1,q_2) \}} \right] -\widehat{F}_{1,2}(q_1,q_2) \end{aligned}$$

and ${\bf \widehat{R}}_{1,2}(q_1,q_2,u)= \hat{J}_{1,2}(0,0,u)- \hat{J}_{1,2}(q_1,0,u)- \hat{J}_{1,2}(0,q_2,u)+ \hat{J}_{1,2}(q_1,q_2,u)$, respectively.

In conclusion,

$$\begin{aligned} \widehat{\sigma}_{\widehat{H}_{1,2}}(q_1,q_2;r_1,r_2) &= n^{-1}\sum\limits_{i=1}^{n} \left( \left[ \frac{B^{(1)}_i(q_1)B^{(2)}_i(q_2)}{\widehat{G}\{ D^{(1,2)}_i(q_1,q_2) \}}- \widehat{H}_{1,2}(q_1,q_2) \right] \right. \\ & \quad \times \left. \left[ \frac{B^{(1)}_i(r_1)B^{(2)}_i(r_2)}{\widehat{G}\{ D^{(1,2)}_i(r_1,r_2) \}}-\widehat{H}_{1,2}(r_1,r_2) \right] \right) -\int\limits_{0}^{L} \frac{\hat{J}_{1,2}(q_1,q_2,u)\hat{J}_{1,2}(r_1,r_2,u)} {n^{-1}\sum\limits_{j=1}^{n}I(\widetilde{T}_{j} \geq u)}d\widehat{\Uplambda}^{c}(u) \end{aligned}$$

and

$$\widehat{\sigma}_{\widehat{F}_{1,2}}(q_1,q_2;r_1,r_2)= \widehat{\varvec{\Upgamma}}_{1,2}(q_1,q_2) \widehat{\varvec{\Upgamma}}_{1,2}(r_1,r_2) -\int\limits_{0}^{L} \frac{{\bf \widehat{R}}_{1,2}(q_1,q_2,u){\bf \widehat{R}}_{1,2}(r_1,r_2,u)} {n^{-1}\sum\nolimits_{j=1}^{n}I(\widetilde{T}_{j} \geq u)}d\widehat{\Uplambda}^{c}(u)$$

are consistent estimators of σ _{H_1,2}(q ₁, q ₂;r ₁, r ₂) and σ _{F_1,2}(q ₁, q ₂;r ₁, r ₂), respectively.

Confidence intervals

Based on these derivations, a 100 × (1 − α)-level confidence interval for TOF(t) is

$[\widehat{TOF}(t)-z_{1-\alpha} \sqrt{\widehat{\sigma}_{H_{1,2}}(t,0;0,t)/n}, \widehat{TOF}(t)+z_{1-\alpha} \sqrt{\widehat{\sigma}_{H_{1,2}}(t,0;0,t)/n}]$, where n is the group size and z _1-α is the upper α ^th quantile of the N(0, 1) distribution.

We now focus on the two-group setting. Let n ₁ and n ₂ be the sample sizes of the two groups, respectively. A 100 × (1 − α)-level confidence interval for TOF_{group 1}(t) − TOF_{group 2}(t) is

$$\begin{aligned} &{\left[\widehat{TOF}_{\rm group\,1}(t)-\widehat{TOF}_{\rm group\,2}(t)- z_{1-\alpha}\sqrt{\widehat{\sigma}_{H^{\rm group\,1}_{1,2}}(t,0;0,t)/n_1+\widehat{\sigma}_{H^{\rm group\,2}_{1,2}}(t,0;0,t)/n_2}, \right.}\\ &{\left.\widehat{TOF}_{\rm group\,1}(t)-\widehat{TOF}_{\rm group\,2}(t)+ z_{1-\alpha}\sqrt{\widehat{\sigma}_{H^{\rm group\,1}_{1,2}}(t,0;0,t)/n_1+\widehat{\sigma}_{H^{\rm group\,2}_{1,2}}(t,0;0,t)/n_2} \right].} \end{aligned}$$