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.
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Appendix
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
and
Using the method which is similar to Tsai et al. [17], we may prove that \( \tilde{S}_{T} (t) \) converges almost surly to
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
However, one may easily verify that
which implies that
and the consistency result follows.
Proof of Theorem 2 (Asymptotic Normality)
By Taylor expansion,
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 \( i = 1, \ldots ,n \)
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
where \( \hat{e}_{3ji} (t) \) is the estimate of the \( e_{3ji} \left( t \right) \) when the unknown functions are substituted by their estimates, \( j = 1,2,3 \).
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Tsai, W.Y., Luo, X., Crowley, J. (2017). The Probability of Being in Response Function and Its Applications. In: Matsui, S., Crowley, J. (eds) Frontiers of Biostatistical Methods and Applications in Clinical Oncology. Springer, Singapore. https://doi.org/10.1007/978-981-10-0126-0_10
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DOI: https://doi.org/10.1007/978-981-10-0126-0_10
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