Nonparametric inference for the joint distribution of recurrent marked variables and recurrent survival time

Abstract

Time between recurrent medical events may be correlated with the cost incurred at each event. As a result, it may be of interest to describe the relationship between recurrent events and recurrent medical costs by estimating a joint distribution. In this paper, we propose a nonparametric estimator for the joint distribution of recurrent events and recurrent medical costs in right-censored data. We also derive the asymptotic variance of our estimator, a test for equality of recurrent marker distributions, and present simulation studies to demonstrate the performance of our point and variance estimators. Our estimator is shown to perform well for a wide range of levels of correlation, demonstrating that our estimators can be employed in a variety of situations when the correlation structure may be unknown in advance. We apply our methods to hospitalization events and their corresponding costs in the second Multicenter Automatic Defibrillator Implantation Trial (MADIT-II), which was a randomized clinical trial studying the effect of implantable cardioverter-defibrillators in preventing ventricular arrhythmia.

Appendix

We wish to find the influence function of $\hat{G}(t,m)$. To do so, we will first discuss the uniform consistency of $\hat{G}(t,m)$, then find the influence function of $\hat{\varLambda }(t,m)$, and finally use those results to get the final influence function of interest. We use the same notation and setup of Sect. 2 of the paper.

1.1 Uniform consistency of $\hat{\varLambda }(t,m)$ and $\hat{G}(t,m)$

Note that $\hat{F}(t,m)$ and $\hat{H}(t)$ are empirical processes, $E(\hat{F}(t,m))=F(t,m)$ and $E(\hat{H}(t))=H(t)$. Therefore, $\hat{F}(t,m)$ and $\hat{H}(t)$ are uniformly consistent estimators for F(t, m) and H(t) by the Glivenko–Cantelli theorem (Vaart 1998). Note that $\hat{\varLambda }(t,m)$ is a functional of $\hat{F}(t,m)$ and $\hat{H}$, and therefore, $\hat{\varLambda }(t,m)$ is uniformly consistent to $\varLambda (t,m)$ by Lemma A.1 in Lin et al. (2000). Also, $\hat{S}(t)$ is uniformly consistent according to Wang and Chang (1999). Since $\hat{G}(t,m)$ is a functional of $\hat{S}(t)$ and $\varLambda (t,m)$, it follows that $\hat{G}(t,m)$ is uniformly consistent to G(t, m) by Lemma A.1 in Lin et al. (2000).

1.2 Influence function: $\hat{\varLambda }(t,m)$

First, note that $\hat{\varLambda }(t,m)= \int _{[0,t]} \hat{F}(ds,m)/\hat{H}(s)$ depends on the pair ($\hat{F}(t,m)$, $\hat{H}(t)$) through two maps:

$$\begin{aligned} (A,B) \rightarrow \left( A,\frac{1}{B}\right) \rightarrow \int _0^t \frac{1}{B}dA. \end{aligned}$$

Then, the functional derivative of the maps at (F, H) is:

$$\begin{aligned} (\alpha ,\beta ) \rightarrow \left( \alpha ,-\frac{\beta }{H^2}\right) \rightarrow \int _0^t \frac{d\alpha }{H}- \int _0^t \frac{\beta dF}{H^2}, \end{aligned}$$

where $\alpha =\hat{F}-F$ and $\beta =\hat{H}-H$. Now, by the functional delta method (Vaart 1998) and simplification we have:

$$\begin{aligned} \hat{\varLambda }(t,m) -\varLambda (t,m)= & {} \int _0^t \frac{1}{H(s)}[\hat{F}(ds,m) - F(ds,m)]\\&-\,\int _0^t \frac{\hat{H}(s) - H(s)}{( H(s) )^2}F(ds,m) + o_p\left( \frac{1}{ \sqrt{n} } \right) \\= & {} \int _0^t \frac{\hat{F}(ds,m)}{H(s)} - \varLambda (t,m) - \int _0^t \frac{\hat{H}(s)F(ds,m)}{(H(s))^2}\nonumber \\&+ \varLambda (t,m) + o_p\left( \frac{1}{\sqrt{n}}\right) \\= & {} \frac{1}{n} \sum _{i=1}^n \left[ \frac{I(k_i \ge 2)}{k_i^*} \sum _{j=1}^{k_i^*}\left( I(y_{ij} \le t,m_{ij}\le m) \frac{1}{H(y_{ij})} \right) \right. \\&\left. -\, \int _0^t \frac{F(ds,m)}{(H(s))^2} \frac{1}{k_i^*} \sum _{j=1}^{k_i^*} I(y_{ij}\ge s) \right] \\&+\, o_p\left( \frac{1}{\sqrt{n}}\right) \\= & {} \frac{1}{n} \sum _{i=1}^n \phi _i(t,m)+ o_p\left( \frac{1}{\sqrt{n}}\right) \\ \end{aligned}$$

where

$$\begin{aligned} \phi _i(t,m)= & {} \frac{I(k_i \ge 2)}{k_i^*} \sum _{j=1}^{k_i^*}\left( I(y_{ij} \le t,m_{ij}\le m) \frac{1}{H(y_{ij})} \right) \\&-\, \int _0^t \frac{F(ds,m)}{(H(s))^2} \frac{1}{k_i^*} \sum _{j=1}^{k_i^*} I(y_{ij}\ge s) \end{aligned}$$

is the influence function of $\hat{\varLambda }(t,m)$ $-$ $\varLambda (t,m)$. Then a consistent estimator for $\phi _i(t,m)$, $\hat{\phi }_i(t,m)$, can be calculated by plugging in $\hat{H}(t)$ and $\hat{F}(t,m)$ for H(t) and F(t, m).

1.3 Influence function: $\hat{G}(t,m)$

Now, we wish to find the influence function for $\hat{G}(t,m)$, the estimated joint distribution of recurrent survival time and marked variable, where $\hat{G}(t,m) = \int _0^t \hat{S}(s-) \hat{\varLambda }(ds,m)$. The influence function for the recurrent survival function, $\hat{S}(t)$, as presented in Wang and Chang, is written as follows:

$$\begin{aligned} \phi _i^{WC}(t)= & {} S(t)\left[ \int _0^t\frac{1}{H^2(u)}\left( \frac{1}{k_i^*}\sum _{j=1}^{k_i^*}I(y_{ij}\ge u)\right) F(du)\right. \\&\left. -\,\frac{I(k_i \ge 2)}{k_i^*}\sum _{j=1}^{k_i^*} \frac{I(y_{ij}< t)}{H(y_{ij})}\right] \ . \end{aligned}$$

Now, completing steps similar to those in the previous section of the Appendix, we use both $\phi _i(t,m)$ and $\phi _i^{WC}(t)$ to obtain the influence function for G(t, m):

$$\begin{aligned} \hat{G}(t,m)-G(t,m)&= \int _0^t S(s-) \left[ \hat{\varLambda }(ds,m) - \varLambda (ds,m) \right] \\&\quad +\,\int _0^t\left[ \hat{S}(s-)-S(s-)\right] \varLambda (ds,m) + o_p\left( \frac{1}{\sqrt{n}}\right) \\&=\frac{1}{n} \sum _{i=1}^n \left[ \int _0^t S(s-) \phi _i(ds,m) + \int _0^t \phi _i^{WC}(s)\varLambda (ds,m) \right] \\&\quad +\,o_p\left( \frac{1}{\sqrt{n}}\right) \\&= \frac{1}{n} \sum _{i=1}^n \eta _i(t,m) + o_p\left( \frac{1}{\sqrt{n}}\right) \ . \end{aligned}$$

Then an estimator for $\eta _i(t,m)$, $\hat{\eta }_i(t,m)$, can be estimated by plugging in $\hat{S}(t)$, $\hat{\phi }_i(t,m)$, $\hat{\phi }_i^{WC}(t)$, and $\hat{\varLambda }(t,m)$ for S(t), $\phi _i(t,m)$, $\phi _i^{WC}(t)$, and $\varLambda (t,m)$.