Estimation of a Concordance Probability for Doubly Censored Time-to-Event Data

Abstract

Evaluating the relationship between a response variable and explanatory variables is important to establish better statistical models. Concordance probability is one measure of this relationship and is often used in biomedical research. Concordance probability can be seen as an extension of the area under the receiver operating characteristic curve. In this study, we propose estimators of concordance probability for time-to-event data subject to double censoring. A doubly censored time-to-event response is observed when either left or right censoring may occur. In the presence of double censoring, existing estimators of concordance probability lack desirable properties such as consistency and asymptotic normality. The proposed estimators consist of estimators of the left-censoring and the right-censoring distributions as a weight for each pair of cases, and reduce to the existing estimators in special cases. We show the statistical properties of the proposed estimators and evaluate their performance via numerical experiments.

Appendix

We shall show the consistency and asymptotic normality of $\widehat{C}_{\mathrm{D}}^{(1)}(\hat{h})$ in this section. Since we can also show these properties for $\widehat{C}_{\mathrm{D}}^{(2)}(h)$ in the same way, the corresponding proofs are omitted.

To show the weak consistency, note that

$$\begin{aligned}&\mathrm{E}\left[ \varGamma _i\varDelta _i\varGamma _j\left\{ F(X_i)\bar{G}(X_i)^2F(X_j)\right\} ^{-1}\mathbb {I}_{{\varvec{\tau }}}\left\{ X_i<X_j\right\} \right] \\&\quad = \mathrm{E}\left[ \mathbb {I}_{{\varvec{\tau }}}\left\{ L_i<T_i\right\} \mathbb {I}_{{\varvec{\tau }}}\left\{ T_i<R_i\right\} \mathbb {I}_{{\varvec{\tau }}}\left\{ L_j<T_j\right\} \left\{ F(X_i)\bar{G}(X_i)^2F(X_j)\right\} ^{-1}\mathbb {I}_{{\varvec{\tau }}}\left\{ T_i<\widetilde{X}_j\right\} \right] \\&\quad = \mathrm{E}\left[ \mathbb {I}_{{\varvec{\tau }}}\left\{ L_i<T_i\right\} \mathbb {I}_{{\varvec{\tau }}}\left\{ T_i<R_i\right\} \mathbb {I}_{{\varvec{\tau }}}\left\{ L_j<\widetilde{X}_j\right\} \left\{ F(T_i)\bar{G}(T_i)^2F(\widetilde{X}_j)\right\} ^{-1}\mathbb {I}_{{\varvec{\tau }}}\left\{ T_i<\widetilde{X}_j\right\} \right] \\&\quad = \mathrm{E}\Big [\left\{ F(T_i)\bar{G}(T_i)^2F(\widetilde{X}_j)\right\} ^{-1}\mathbb {I}_{{\varvec{\tau }}}\left\{ T_i<\widetilde{X}_j\right\} \\&\quad \mathrm{E}\left[ \mathbb {I}_{{\varvec{\tau }}}\left\{ L_i<T_i\right\} \mathbb {I}_{{\varvec{\tau }}}\left\{ T_i<R_i\right\} \mathbb {I}_{{\varvec{\tau }}}\left\{ L_j<\widetilde{X}_j\right\} \Big |T_i,\widetilde{X}_j\right] \Big ]\\&\quad = \mathrm{E}\left[ \left\{ F(T_i)\bar{G}(T_i)^2F(\widetilde{X}_j)\right\} ^{-1}\mathbb {I}_{{\varvec{\tau }}}\left\{ T_i<\widetilde{X}_j\right\} F(T_i)\bar{G}(T_i)F(\widetilde{X}_j)\right] \ \ (\mathrm{by independence})\\&\quad = \mathrm{E}\left[ \bar{G}(T_i)^{-1}\bar{G}(T_i)\mathbb {I}_{{\varvec{\tau }}}\left\{ T_i<T_j\right\} \right] = \mathrm{E}\left[ \mathbb {I}_{{\varvec{\tau }}}\left\{ T_i<T_j\right\} \right] , \end{aligned}$$

where $\widetilde{X}_j=\max (T_j,R_j)$. Then, the denominator of $\widehat{C}_{\mathrm{D}}^{(1)}(h)$,

$\displaystyle \frac{1}{n(n-1)}\sum _{i,j}\varGamma _i\varDelta _i\varGamma _j\mathbb {I}_{{\varvec{\tau }}}\left\{ X_i<X_j\right\} \left( \widehat{F}(X_i)\widehat{\bar{G}}(X_i)^2\widehat{F}(X_j)\right) ^{-1}$, converges to $\mathrm{E}\, [ \mathbb {I}_{{\varvec{\tau }}}\{T_1<T_2\}]$ in probability by the uniform consistency of $\widehat{F}(\cdot )$ and $\widehat{\bar{G}}(\cdot )$ [2] and a law of large numbers for U-processes [15]. Similarly, it can also be shown that

$$\begin{aligned}&\frac{1}{n(n-1)}\sum _{i,j}\varGamma _i\varDelta _i\varGamma _j\mathbb {I}\left\{ \hat{h}({\varvec{Z}}_i)>\hat{h}({\varvec{Z}}_j)\right\} \mathbb {I}_{{\varvec{\tau }}}\left\{ X_i<X_j\right\} \left( \widehat{F}(X_i)\widehat{\bar{G}}(X_i)^2\widehat{F}(X_j)\right) ^{-1} \\&\quad \overset{p}{\rightarrow }\mathrm{E}\left[ \mathbb {I}\left\{ h^*({\varvec{Z}}_1)>h^*({\varvec{Z}}_2)\right\} \mathbb {I}_{{\varvec{\tau }}}\left\{ T_1<T_2\right\} \right] . \end{aligned}$$

Then, the consistency of $\widehat{C}_{\mathrm{D}}^{(1)}(\hat{h})$ follows from Slutsky’s lemma.

To show the asymptotic normality of $\widehat{C}_{\mathrm{D}}^{(1)}(\hat{h})$, we first consider asymptotic behavior of the estimator with fixed $h=h(\cdot ;{\varvec{\beta }})$. For notational brevity, hereafter the following symbols are used.

$$\begin{aligned}&\varLambda _{ij}=\varGamma _i\varDelta _i\varGamma _j,\ J_{ij}(h)=\mathbb {I}\left\{ h({\varvec{Z}}_i)>h({\varvec{Z}}_j)\right\} ,\ I_{ij}^{{\varvec{\tau }}}=\mathbb {I}_{{\varvec{\tau }}}\left\{ X_i<X_j\right\} ,\\&W_{ij}=\left\{ F(X_i)\bar{G}(X_i)^2F(X_j)\right\} ^{-1}, \widehat{W}_{ij}=\left\{ \widehat{F}(X_i)\widehat{\bar{G}}(X_i)^2\widehat{F}(X_j)\right\} ^{-1}. \end{aligned}$$

For $C^*(h)=\mathrm{P}\left[ h({\varvec{Z}}_1)>h({\varvec{Z}}_2)|T_1<T_2, T_1<\tau _{\mathrm{U}}, \tau _{\mathrm{L}}<T_2\right]$, note that

$$\begin{aligned} \mathcal {W}(h):= & {} \sqrt{n}\left( \widehat{C}_{\mathrm{D}}^{(1)}(h)-C^*(h)\right) \\= & {} \sqrt{n}\left( \frac{\sum _{i,j} \varLambda _{ij}I_{ij}^{{\varvec{\tau }}}\{J_{ij}(h)-C^*(h)\} W_{ij}}{\sum _{i,j}\varLambda _{ij}I_{ij}^{{\varvec{\tau }}}\widehat{W}_{ij}}\right) \\&+ \,\sqrt{n}\left( \frac{\sum _{i,j} \varLambda _{ij}I_{ij}^{{\varvec{\tau }}}\{J_{ij}(h)-C^*(h)\} (\widehat{W}_{ij}-W_{ij})}{\sum _{i,j}\varLambda _{ij}I_{ij}^{{\varvec{\tau }}}\widehat{W}_{ij}}\right) \\=: & {} \mathcal {W}_1(h) + \mathcal {W}_2(h). \end{aligned}$$

Remark that $\mathcal {W}_1(h)$ and $\mathcal {W}_2(h)$ correspond to the variance component of the U-statistic and weights $\widehat{W}_{ij}$, respectively. It follows from the uniform consistency of $\widehat{F}$ and $\widehat{\bar{G}}$ and a functional limit theorem for U-processes [16] that

$$\begin{aligned} \mathcal {W}_1(h) = n^{-3/2}\frac{\sum _{i,j}\varLambda _{ij}I_{ij}^{{\varvec{\tau }}}(J_{ij}(h)-C^*(h))}{p({\varvec{\tau }})} + o_p(1),\quad n\rightarrow \infty , \end{aligned}$$

(4)

where $p({\varvec{\tau }})=\mathrm{P}\left[ T_1<T_2,T_1<\tau _{\mathrm{U}}, \tau _{\mathrm{L}}<T_2\right]$. Moreover note that

$$\begin{aligned} \mathcal {W}_2(h) = \int _{0}^{\tau _{\mathrm{U}}}\int _{\tau _{\mathrm{L}}}^{M}\sqrt{n}\left( \frac{\widehat{W}(s,t)}{W(s,t)}-1\right) \mathrm{d}{\hat{\gamma }}(s,t,h), \end{aligned}$$

(5)

where $W(s,t)=\left( F(s)\bar{G}(s)^2F(t)\right) ^{-1}$, $\widehat{W}(s,t)=\left( \widehat{F}(s)\widehat{\bar{G}}(s)^2\widehat{F}(t)\right) ^{-1}$, and

$$\begin{aligned} {\hat{\gamma }}(s,t,h)=\sum _{i,j}\varLambda _{ij}I_{ij}^{{\varvec{\tau }}}(J_{ij}(h)-C^*(h))W_{ij}\mathbb {I}\left\{ X_i\le s,X_j\le t\right\} \Big /\sum _{i,j}\varLambda _{ij}I_{ij}^{{\varvec{\tau }}}\widehat{W}_{ij}. \end{aligned}$$

By a uniform law of large numbers for U-processes [15] and the uniform consistency of $\widehat{F}$ and $\widehat{\bar{G}}$, we obtain that

$$\begin{aligned} \sup \{ |{\hat{\gamma }}(s,t,h)-\gamma (s,t,h)|\,:\, s\in [0,\tau _{\mathrm{U}}], t\in [\tau _{\mathrm{L}},M], {\varvec{\beta }}\}\overset{p}{\rightarrow }0, \end{aligned}$$

(6)

where $\gamma (s,t,h)=p((s,t)')(\widehat{C}_{\mathrm{D}}^{(1)}(h)-C^*(h))/p({\varvec{\tau }})$.

Therefore, the asymptotic distribution for $\widehat{C}_{\mathrm{D}}^{(1)}(h)$ is obtained if we show that $\sqrt{n}\left( \frac{\widehat{W}(s,t)}{W(s,t)}-1\right)$ converges in distribution to a zero-mean Gaussian process (indexed by s and t).

Note that

$$\begin{aligned} \sqrt{n}\left( \frac{\widehat{W}(s,t)}{W(s,t)}-1\right)= & {} \sqrt{n}\left( \frac{\widehat{F}(s)-F(s)}{F(s)}\right) \frac{F(s)\bar{G}(s)^2F(t)}{\widehat{F}(s)\widehat{\bar{G}}(s)^2\widehat{F}(t)}\\&+\,\sqrt{n}\left( \frac{\widehat{\bar{G}}(s)^2-\bar{G}(s)^2}{\bar{G}(s)^2}\right) \frac{\bar{G}(s)^2F(t)}{\widehat{\bar{G}}(s)^2\widehat{F}(t)}\nonumber \\&+\,\sqrt{n}\left( \frac{\widehat{F}(t)-F(t)}{F(t)}\right) \frac{F(t)}{\widehat{F}(t)}\nonumber \\= & {} \sqrt{n}\left( \frac{\widehat{F}(s)-F(s)}{F(s)} +2\frac{\widehat{\bar{G}}(s)-\bar{G}(s)}{\bar{G}(s)} +\frac{\widehat{F}(t)-F(t)}{F(t)}\right) \\&+\,\, o_p(1). \end{aligned}$$

As in Remark, $\sqrt{n}(\widehat{F}-F,\widehat{\bar{G}}-\bar{G})$ jointly converges in distribution to a two-dimensional zero-mean Gaussian process. Hence any finite-dimensional distributions of $\sqrt{n}(\widehat{F}-F,\widehat{\bar{G}}-\bar{G})$ converges in distribution to the corresponding multidimensional normal distribution. To complete the proof, we consider the asymptotic expansion of $\mathcal {W}(\hat{h})$ with respect to ${\hat{\beta }}$:

$$\begin{aligned} \mathcal {W}(\hat{h})=\mathcal {W}(h^*)+\nabla C^*(h^*)^\top \sqrt{n}(\hat{{\varvec{\beta }}}-{\varvec{\beta }}^*)+o_p(1), \end{aligned}$$

where $\nabla C^*(h^*)$ is the partial derivative of $C^*(h)$ with respect to ${\varvec{\beta }}$ evaluated at ${\varvec{\beta }}={\varvec{\beta }}^*$. By the same argument as in Appendix of [26]; see especially (A1) and (A2) in [26], we can conclude the asymptotic normality of ${{\mathcal {W}}}(\hat{h})$.