The Probability of Being in Response Function and Its Applications

Abstract

Cancer clinical trials usually have two or more types of related clinical events (i.e. response, progression and relapse). Hence, to compare treatments, efficacy is often measured using composite endpoints. Temkin (Biometrics 34: 571–580, [18]) proposed the probability of being in response as a function of time (PBRF) to analyze composite endpoints. The PBRF is a measure which considers the response rate and the duration of response jointly. In this article, we develop, study and propose estimators of PBRF based on multi-state survival data.

Appendix

Proof of Theorem 1 (Consistency)

Let the joint probability functions (pdf) of (T ⁰, X ⁰) be f(t, x). Define the two sub-marginal functions f ₁(t) and f ₂(x), respectively, as

$$ f_{1} (t) = \int\limits_{0}^{t} {f(t,x)dx} $$

and

$$ f_{2} (x) = \int\limits_{x}^{\infty } {f(t,x)dt} . $$

Using the method which is similar to Tsai et al. [17], we may prove that $\tilde{S}_{T} (t)$ converges almost surly to

$$ S_{T}^{*} (t) = \exp \left\{ { - \int\limits_{0}^{t} {\frac{{f_{1} (s)S_{C} (s)}}{{R_{o} (s)}}ds} } \right\} = \exp \left\{ { - \int\limits_{0}^{t} {\frac{{f_{1} (s)}}{R(s)}ds} } \right\}. $$

The Kaplan–Meier estimator $\hat{S}_{Y} (x)$ will converges almost surely to $S_{Y} (x)$. Combining these and the properties of empirical cumulative distribution function, we can show that $\hat{R}_{3} (t)$ will converge almost surely to

$$ R_{3} (t) = S_{T}^{*} (t)\int\limits_{0}^{t} {\frac{{f_{2} (x)}}{{S_{T}^{*} (x)}}} dx. $$

However, one may easily verify that

$$ \left( {\frac{R(t)}{{S_{T}^{*} (t)}}} \right)^{{\prime }} = \frac{{f_{2} (t)}}{{S_{T}^{*} (t)}}, $$

which implies that

$$ R(t) = S_{T}^{*} (t)\int\limits_{0}^{t} {\frac{{f_{2} (x)}}{{S_{T}^{*} (x)}}} dx $$

and the consistency result follows.

Proof of Theorem 2 (Asymptotic Normality)

By Taylor expansion,

$$ \sqrt n \left( {\hat{R}_{3} (t) - R(t)} \right) = \sqrt n A_{1} (t) + \sqrt n A_{2} (t) + \sqrt n A_{3} (t) + o_{p} (1), $$

where $A_{1} (t) = n^{ - 1} \sum\nolimits_{i = 1}^{n} e_{31i} (t),A_{2} (t) = n^{ - 1} \sum\nolimits_{i = 1}^{n} e_{32i} (t),A_{3} (t) = n^{ - 1} \sum\nolimits_{i = 1}^{n} e_{33i} (t)$ and for $i = 1, \ldots ,n$

$$ \begin{aligned} e_{31i} (t) = & \frac{{\delta_{Xi} I\left( {X_{i} \le t} \right)S_{T}^{*} (t)}}{{S_{T}^{*} \left( {X_{i} } \right)S_{C} \left( {X_{i} } \right)}} - R(t), \\ e_{32i} (t) = & \frac{{\delta_{Xi} \delta_{Ti} I\left( {T_{i} \le t} \right)S_{T}^{*} (t)R_{3} \left( {T_{i} } \right)}}{{S_{T}^{*} \left( {T_{i} } \right)r_{o}^{e} \left( {T_{i} } \right)}} - \int\limits_{0}^{t} {\frac{{S_{T}^{*} (t)R(s)I\left( {X_{i} \le s \le T_{i} } \right)f_{11} (s)ds}}{{S_{T}^{*} (s)\left[ {r_{o}^{e} (s)} \right]^{2} }}} \\ e_{33i} (t) = & \frac{{S_{T}^{*} (t)}}{{n_{T} (T_{i} )}}\left\{ {\frac{R(t)}{{S_{T}^{*} (t)}} - \frac{{R_{3} \left( {T_{i} } \right)}}{{S_{T}^{*} \left( {T_{i} } \right)}}} \right\}I\left( {\delta_{Ti} = 0,T_{i} \le t} \right) \\ & \quad - \int\limits_{0}^{t} {\frac{{S_{T}^{*} (t)}}{{n_{T}^{2} (s)}}\left\{ {\frac{R(t)}{{S_{T}^{*} (t)}} - \frac{R(s)}{{S_{T}^{*} (s)}}} \right\}f_{2} (s)ds} \\ n_{T} (t) = & E\{ N_{T} (t)\} /n \\ f_{11} (s) = & {\text{pr}}(X \le T = s,\delta_{X} = 1,\delta_{T} = 1) \\ f_{2} (s) = & {\text{pr}}(T = s,\delta_{T} = 0) \\ \end{aligned} $$

Since $A_{j} (t),\;j = 1,2,3$ are all sum of i.i.d. random variables, $A_{1} (t) + A_{2} (t) + A_{3} (t)$ will converge to a mean zero Gaussian process $\Re_{3} \left( t \right)$ as $n \to \infty$ with the variance covariance matrix $Cov\left( {\Re_{3} (s),\Re_{3} (t)} \right)$ that can be consistently estimated by

$$ n^{ - 1} \sum\limits_{i = 1}^{n} {\left[ {\hat{e}_{31i} (s) - \hat{e}_{32i} (s) + \hat{e}_{33i} (s)} \right]\left[ {\hat{e}_{31i} (t) - \hat{e}_{32i} (t) + \hat{e}_{33i} (t)} \right],} $$

where $\hat{e}_{3ji} (t)$ is the estimate of the $e_{3ji} \left( t \right)$ when the unknown functions are substituted by their estimates, $j = 1,2,3$.