Comparing center-specific cumulative incidence functions

Abstract

The competing risks data structure arises frequently in clinical and epidemiologic studies. In such settings, the cumulative incidence function is often used to describe the ultimate occurrence of a particular cause of interest. If the objective of the analysis is to compare subgroups of patients with respect to cumulative incidence, imbalance with respect to group-specific covariate distributions must generally be factored out, particularly in observational studies. This report proposes a measure to contrast center- (or, more generally group-) specific cumulative incidence functions (CIF). One such application involves evaluating organ procurement organizations with respect to the cumulative incidence of kidney transplantation. In this case, the competing risks include (i) death on the wait-list and (ii) removal from the wait-list. The proposed method assumes proportional cause-specific hazards, which are estimated through Cox models stratified by center. The proposed center effect measure compares the average CIF for a given center to the average CIF that would have resulted if that particular center had covariate pattern-specific cumulative incidence equal to that of the national average. We apply the proposed methods to data obtained from a national organ transplant registry.

Appendix

Proof of Theorem 2

The proof revolves around asymptotic expansions of the following quantities.

• 1. $n^{\frac{1}{2}}(\widehat{{\varvec{\beta }}}_k - {\varvec{\beta }}_k)$

• 2. $n^{\frac{1}{2}}\left\{ \widehat{{\varLambda }}^\#_{0jk}(t)-{\varLambda }^\#_{0jk}(t) \right\}$

• 3. $n^{\frac{1}{2}}\left\{ \widehat{{\varLambda }}^\#_{ijk}(t)-{\varLambda }^\#_{ijk}(t)\right\}$

• 4. $n^{\frac{1}{2}}\left\{ \widehat{S}_{ij}(t)-S_{ij}(t) \right\}$

• 5. $n^{\frac{1}{2}}\left\{ \widehat{F}_{ijk}(t) - F_{ijk}(t) \right\}$

• 6. $n^{\frac{1}{2}} \left\{ \widehat{\delta }_{jk}(t) - \delta _{jk}(t) \right\}$

[1]    $n^{\frac{1}{2}}(\widehat{{\varvec{\beta }}}_k - {\varvec{\beta }}_k)$

By a Taylor expansion of $U_k({\varvec{\beta }})$ around ${\varvec{\beta }}_k$,

$$\begin{aligned} n^{\frac{1}{2}}(\widehat{{\varvec{\beta }}}_k - {\varvec{\beta }}_k)= & {} {\varvec{I}}_{k}^{-1}({\varvec{\beta }}_k) n^{-\frac{1}{2}} \sum _{i=1}^n {\varvec{U}}_{ik}({\varvec{\beta }}_k) + o_p(1), \end{aligned}$$

where ${\varvec{U}}_{ik}({\varvec{\beta }})$ and ${\varvec{I}}_{k}({\varvec{\beta }})$ are as defined in the theorem. The result then follows from standard Martingale theory; e.g., Andersen and Gill (1982) and Fleming and Harrington (1991). Note that the processes $M_{ijk}(t; {\varvec{\beta }}_1)$, for $k=1,\ldots ,K$ are martingales with respect to the filtration

$$\begin{aligned} \mathcal{{F}}_{ij}(t) = \sigma \left\{ Y_{ij}(s), {\varvec{Z}}_i; s \in (0,t] \right\} . \end{aligned}$$

[2]    $n^{\frac{1}{2}} \left\{ \widehat{{\varLambda }}^\#_{0jk}(t)-{\varLambda }^\#_{0jk}(t) \right\}$

We decompose the quantity as follows,

$$\begin{aligned} n^{\frac{1}{2}} \left\{ \widehat{{\varLambda }}^\#_{0jk}(t)-{\varLambda }^\#_{0jk}(t) \right\}= & {} n^{\frac{1}{2}} \left\{ \widehat{{\varLambda }}^\#_{0jk}(t;\widehat{{\varvec{\beta }}}_k) - \widehat{{\varLambda }}^\#_{0jk}(t;{\varvec{\beta }}_k) \right\} \end{aligned}$$

(6)

$$\begin{aligned}&\quad +\,n^{\frac{1}{2}} \left\{ \widehat{{\varLambda }}^\#_{0jk}(t;{\varvec{\beta }}_k) - {\varLambda }^\#_{0jk}(t;{\varvec{\beta }}_k) \right\} . \end{aligned}$$

(7)

Since $\widehat{{\varLambda }}_{0jk}^\#(t)$ is the Breslow–Aalen analog of ${\varLambda }_{0jk}^\#(t)$, we adapt results derived for the Breslow–Aalen estimator (Fleming and Harrington 1991). From this perspective, we can write

$$\begin{aligned} (6)= & {} {\varvec{h}}_{jk}^T(t;{\varvec{\beta }}_k) {\varvec{I}}_k({\varvec{\beta }}_k)^{-1}n^{-\frac{1}{2}} \sum _{i=1}^n {\varvec{U}}_{ik}({\varvec{\beta }}_k)+o_p(1), \end{aligned}$$

where the last equality holds by Slutsky’s Theorem. With respect to (7), we can write

$$\begin{aligned} (7)= & {} n^{-\frac{1}{2}} \sum _{i=1} ^n \int _0^t r_{jk}^{(0)} (s;{\varvec{\beta }}_k) ^{-1} dM_{ijk}(s; {\varvec{\beta }}_k) + o_p(1), \end{aligned}$$

since $R_{jk}^{(0)}(t) {\rightarrow } r_{jk}^{(0)}(t)$ in probability. Putting (6) and (7) together, we get:

$$\begin{aligned}{}[2] = n^{-\frac{1}{2}} \sum _{i=1} ^n {\varPhi }_{ijk}(t;{\varvec{\beta }}_k) + o_p(1) , \end{aligned}$$

where $\displaystyle {\varPhi }_{ijk}(t;{\varvec{\beta }}) = - {\varvec{h}}_{jk}^T (t;{\varvec{\beta }}) {\varvec{I}}_k({\varvec{\beta }})^{-1} {\varvec{U}}_{ik}({\varvec{\beta }}) + \int _0^t r_{jk}^{(0)} (s;{\varvec{\beta }}) ^{-1} dM_{ijk}(s; {\varvec{\beta }}).$

[3]    $n^{\frac{1}{2}} \left\{ \widehat{{\varLambda }}^\#_{ijk}(t)-{\varLambda }^\#_{ijk}(t) \right\}$

We start with the following decomposition,

$$\begin{aligned}{}[3]= & {} n^{\frac{1}{2}} \left\{ \int _0^t Y _{ij}(s) \exp (\widehat{{\varvec{\beta }}}_k^T {\varvec{Z}}_i) d\widehat{{\varLambda }}^\#_{0jk} (s; \widehat{{\varvec{\beta }}}_k) \right. \nonumber \\&\quad \left. - \int _0^t Y _{ij}(s) \exp ({\varvec{\beta }}_k^T {\varvec{Z}}_i) d\widehat{{\varLambda }}^\#_{0jk} (s; \widehat{{\varvec{\beta }}}_k) \right\} \end{aligned}$$

(8)

$$\begin{aligned}&+\, n^{\frac{1}{2}} \left\{ \int _0^t Y _{ij}(s) \exp ({\varvec{\beta }}_k^T {\varvec{Z}}_i) d\widehat{{\varLambda }}^\#_{0jk} (s; \widehat{{\varvec{\beta }}}_k) \right. \nonumber \\&\quad \left. - \int _0^t Y _{ij}(s) \exp ({\varvec{\beta }}_k^T {\varvec{Z}}_i) d{\varLambda }^\#_{0jk} (s; {\varvec{\beta }}_k) \right\} . \end{aligned}$$

(9)

By a Taylor expansion, then applying Result [1], we can write

$$\begin{aligned} (8)= & {} \int _0^t {\varvec{Z}}_i^T Y_{ij}(s) d{\varLambda }^\#_{ijk} (s; {\varvec{\beta }}_k) {\varvec{I}}_k({\varvec{\beta }}_k)^{-1}n^{-\frac{1}{2}} \sum _{\ell =1}^n {\varvec{U}}_{\ell k}({\varvec{\beta }}_k) + o_p(1) , \end{aligned}$$

where the last equality holds by the convergence in probability of $\widehat{{\varLambda }}^\#_{ijk}(t)$ to ${\varLambda }^\#_{ijk}(t)$. We re-express (9) as

$$\begin{aligned} (9)= & {} \int _0^t Y _{ij}(s) \exp ({\varvec{\beta }}_k^T {\varvec{Z}}_i) \left\{ n^{-\frac{1}{2}} \sum _{\ell =1}^n d{\varPhi }_{\ell jk} (t;{\varvec{\beta }}_k)\right\} + o_p(1), \end{aligned}$$

by incorporating Result [2], where we define

$$\begin{aligned} d{\varPhi }_{ijk}(t;{\varvec{\beta }}_k)= & {} \left[ -\overline{{\varvec{z}}}_j(s;{\varvec{\beta }}_k) d{\varLambda }_{0jk} (s;{\varvec{\beta }}_k) \right] ^T {\varvec{I}}_k({\varvec{\beta }}_k)^{-1}{\varvec{U}}_{ik}({\varvec{\beta }}_k) \\&+\, r_{jk}^{(0)} (s;{\varvec{\beta }}_k) ^{-1} dM_{ijk}(s; {\varvec{\beta }}_k). \end{aligned}$$

Combining (8) and (9), then further reorganizing, we obtain

$$\begin{aligned}{}[3]= & {} {\varvec{D}}_{ijk}^T(t;{\varvec{\beta }}_k) {\varvec{I}}_{k}({\varvec{\beta }}_k)^{-1} n^{-\frac{1}{2}} \sum _{\ell =1}^n {\varvec{U}}_{\ell k}({\varvec{\beta }}_k) + n^{-\frac{1}{2}} \sum _{\ell =1}^n J_{i\ell jk} (t;{\varvec{\beta }}_k), \end{aligned}$$

where we let

$$\begin{aligned} {\varvec{D}}_{ijk}(t;{\varvec{\beta }})= & {} \int _0^t \left\{ {\varvec{Z}}_i -\overline{{\varvec{z}}}_j(s;{\varvec{\beta }}) \right\} d{\varLambda }^\#_{ijk}(s;{\varvec{\beta }}) \\ J_{i\ell jk} (t;{\varvec{\beta }})= & {} \int _0^t Y _{ij}(s) \cdot \exp ({\varvec{\beta }}_k^T {\varvec{Z}}_i) r_{jk}^{(0)} (s;{\varvec{\beta }}) ^{-1} dM_{\ell jk}(s; {\varvec{\beta }}). \end{aligned}$$

[4]   $n^{\frac{1}{2}}\left\{ \widehat{S}_{ij}(t)-S_{ij}(t) \right\}$

We decompose [4] as follows,

$$\begin{aligned}{}[4]= & {} - \sum _{m=1}^K S_{ij}(t) n^{-1/2} \left\{ \widehat{{\varLambda }}^\#_{ijm}(t) - {\varLambda }^\#_{ijm}(t) \right\} , \end{aligned}$$

due to the Functional Delta Method, combined with the convergence, $e^{-\widehat{{\varLambda }}^\#_{ijm}(s;\widehat{{\varvec{\beta }}}_m)} \rightarrow e^{-{\varLambda }^\#_{ijm}(s;{\varvec{\beta }}_m)}$, by continuity. Using Result [3], we then obtain

$$\begin{aligned}{}[4]= & {} -S_{ij}(t) \sum _{m=1}^K \left\{ {\varvec{D}}_{ijm}^T(t;{\varvec{\beta }}_m) {\varvec{I}}_{m}({\varvec{\beta }}_m)^{-1} n^{-\frac{1}{2}} \sum _{\ell =1}^n {\varvec{U}}_{\ell m}({\varvec{\beta }}_m) \right. \\&\left. +\, n^{-\frac{1}{2}} \sum _{\ell =1}^n J_{i\ell jm} (t;{\varvec{\beta }}_m) \right\} . \end{aligned}$$

[5]    $n^{\frac{1}{2}} \left\{ \widehat{F}_{ijk}(t) - F_{ijk}(t)\right\}$

We use the following decomposition,

$$\begin{aligned}{}[5]= & {} n^{\frac{1}{2}} \left\{ \int _0^t \widehat{S}_{ij}(s; \widehat{{\varvec{\beta }}}) d\widehat{{\varLambda }}^\#_{ijk}(s;\widehat{{\varvec{\beta }}}_k) - \int _0^t S_{ij}(s; {\varvec{\beta }}) d\widehat{{\varLambda }}^\#_{ijk}(s;\widehat{{\varvec{\beta }}}_k) \right\} \end{aligned}$$

(10)

$$\begin{aligned}&+\, n^{\frac{1}{2}} \left\{ \int _0^t S_{ij}(s; {\varvec{\beta }}) d\widehat{{\varLambda }}^\#_{ijk}(s;\widehat{{\varvec{\beta }}}_k) - \int _0^t S_{ij}(s; {\varvec{\beta }}) d{\varLambda }^\#_{ijk}(s;{\varvec{\beta }}_k) \right\} , \end{aligned}$$

(11)

where ${\varvec{\beta }}^T = [{\varvec{\beta }}_1^T, \ldots , {\varvec{\beta }}_K^T]$. Note that Eq. (10) will eventually give rise to $\phi _{i\ell jk}^1(t, {\varvec{\beta }}), \phi _{i\ell jk}^2(t, {\varvec{\beta }})$ as defined in Theorem 2, while (11) will give rise to $\phi _{i\ell jk}^3(t, {\varvec{\beta }})$ and $\phi _{i\ell jk}^4(t, {\varvec{\beta }})$. We can write (10) as

$$\begin{aligned} (10)= & {} \int _0^t n^{\frac{1}{2}} \left[ \widehat{S}_{ij}(s; \widehat{{\varvec{\beta }}})-S_{ij}(s; {\varvec{\beta }})\right] d\widehat{{\varLambda }}^\#_{ijk}(s;\widehat{{\varvec{\beta }}}_k) \nonumber \\= & {} - \sum _{m=1}^K \int _0^t S_{ij}(s) {\varvec{D}}_{ijm}^T (s;{\varvec{\beta }}_m) d\widehat{{\varLambda }}^\#_{ijk}(s;\widehat{{\varvec{\beta }}}_k) {\varvec{I}}_{m}({\varvec{\beta }}_m)^{-1} n^{-\frac{1}{2}} \sum _{\ell =1}^n {\varvec{U}}_{\ell m}({\varvec{\beta }}_m) \nonumber \\\end{aligned}$$

(12)

$$\begin{aligned}&- \quad n^{-\frac{1}{2}} \sum _{m=1}^K \int _0^t S_{ij}(s) \sum _{\ell =1}^n J_{i\ell jm} (s;{\varvec{\beta }}_m) d\widehat{{\varLambda }}^\#_{ijk}(s;\widehat{{\varvec{\beta }}}_k), \end{aligned}$$

(13)

where we have used Result [4]. Focusing on (12), we have,

$$\begin{aligned} (12) = - \sum _{m=1}^K \int _0^t {\varvec{D}}_{ijm}^T (s;{\varvec{\beta }}_m) dF_{ijk}(s) \ {\varvec{I}}_{m}({\varvec{\beta }}_m)^{-1}n^{-\frac{1}{2}} \sum _{\ell =1}^n {\varvec{U}}_{\ell m}({\varvec{\beta }}_m) + o_p(1), \end{aligned}$$

by the fact that $\widehat{{\varLambda }}^\#_{ijk}(s;{\varvec{\beta }}_k) {\rightarrow } {\varLambda }^\#_{ijk}(s;{\varvec{\beta }}_k)$, and $\widehat{{\varvec{\beta }}}_k {\rightarrow } {\varvec{\beta }}_k$ and so, by the CMT, $\widehat{{\varLambda }}^\#_{ijk}(s;\widehat{{\varvec{\beta }}}_k) {\rightarrow } {\varLambda }^\#_{ijk}(s;{\varvec{\beta }}_k).$ Therefore, from (12), define

$$\begin{aligned} \phi _{i\ell jk}^1(t, {\varvec{\beta }})= & {} \sum _{m=1}^K \left\{ \int _0^t \left\{ {\varvec{Z}}_i -\overline{{\varvec{z}}}_j(u;{\varvec{\beta }}_k) \right\} ^T F_{ijk}(u) \right. \\&\left. - F_{ijk}(t) \int _0^t \left\{ {\varvec{Z}}_i -\overline{{\varvec{z}}}_j(u;{\varvec{\beta }}_k) \right\} ^T d{\varLambda }^\#_{ijk}(u;{\varvec{\beta }}_k) \right\} {\varvec{I}}_{m}({\varvec{\beta }}_m)^{-1} {\varvec{U}}_{\ell m}({\varvec{\beta }}_m). \end{aligned}$$

Using analogous arguments, we can write

$$\begin{aligned} (13)= & {} -n^{-\frac{1}{2}} \sum _{m=1}^K \sum _{\ell =1}^n \left\{ F_{ijk}(t) \int _0^t Y _{ij}(u) \exp ({\varvec{\beta }}_m^T {\varvec{Z}}_i) r_{jk}^{(0)} (u;{\varvec{\beta }}_m) ^{-1} dM_{\ell jm}(u; {\varvec{\beta }}_m) \right. \\&\left. - \int _0^t F_{ijk}(u) Y _{ij}(u) \exp ({\varvec{\beta }}_m^T {\varvec{Z}}_i) r_{jk}^{(0)} (u;{\varvec{\beta }}_m) ^{-1} dM_{\ell jm}(u; {\varvec{\beta }}_m) \right\} , \end{aligned}$$

due to previously described properties of $\widehat{{\varLambda }}^\#_{ijk}(s;{\varvec{\beta }}_k)$ and $\widehat{{\varvec{\beta }}}_k$. Thus, from (13) define:

$$\begin{aligned} \phi _{i\ell jk}^2(t, {\varvec{\beta }}_1)&= - \sum _{m=1}^K \int _0^t \left\{ F_{ijk}(t) - F_{ijk}(u) \right\} \ Y _{ij}(u) \exp ({\varvec{\beta }}_m^T {\varvec{Z}}_i) r_{jk}^{(0)} (u;{\varvec{\beta }}_m) ^{-1}\\&\quad d M_{\ell jm}(u;{\varvec{\beta }}_m). \end{aligned}$$

Now shifting attention to (11), we obtain

$$\begin{aligned} (11)= & {} \left\{ \int _0^t S_{ij}(s; {\varvec{\beta }}) d{\varvec{D}}_{ijk}^T (s;{\varvec{\beta }}_k) \right\} \ {\varvec{I}}_{k}({\varvec{\beta }}_k)^{-1} n^{-\frac{1}{2}} \sum _{\ell =1}^n {\varvec{U}}_{\ell k}({\varvec{\beta }}_k) \end{aligned}$$

(14)

$$\begin{aligned}&\quad + \,n^{-\frac{1}{2}} \int _0^t S_{ij}(s; {\varvec{\beta }}) \sum _{\ell =1}^n dJ_{i\ell jk} (s;{\varvec{\beta }}_k), \end{aligned}$$

(15)

through Result [3]. We can then write

$$\begin{aligned} (14)= \left\{ \int _0^t S_{ij}(s; {\varvec{\beta }}) \left[ {\varvec{Z}}_i-\overline{{\varvec{z}}}_j(s;{\varvec{\beta }}_k)\right] ^T d{\varLambda }^\#_{ijk} (s;{\varvec{\beta }}_k) \right\} {\varvec{I}}_{k}({\varvec{\beta }}_k)^{-1} n^{-\frac{1}{2}} \sum _{\ell =1}^n {\varvec{U}}_{\ell k}({\varvec{\beta }}_k) \end{aligned}$$

and, correspondingly, we define

$$\begin{aligned} \phi _{i\ell jk}^3(t, {\varvec{\beta }}) = \left\{ \int _0^t S_{ij}(s; {\varvec{\beta }}) \left[ {\varvec{Z}}_i-\overline{{\varvec{z}}}_j(s;{\varvec{\beta }}_k)\right] ^T d{\varLambda }^\#_{ijk} (s;{\varvec{\beta }}_k) \right\} {\varvec{I}}_{k}({\varvec{\beta }}_k)^{-1} {\varvec{U}}_{\ell k}({\varvec{\beta }}_k). \end{aligned}$$

We can re-express (15) as follows,

$$\begin{aligned} (15)= n^{-\frac{1}{2}} \sum _{\ell =1}^n \int _0^t S_{ij}(s; {\varvec{\beta }}) Y _{ij}(s) \exp ({\varvec{\beta }}_k^T {\varvec{Z}}_i) r_{jk}^{(0)} (s;{\varvec{\beta }}_k) ^{-1} dM_{ljk}(s; {\varvec{\beta }}_k), \end{aligned}$$

and, correspondingly redefine

$$\begin{aligned} \phi _{i\ell jk}^4(t, {\varvec{\beta }}) = \int _0^t S_{ij}(s; {\varvec{\beta }}_1) Y _{ij}(s) \exp ({\varvec{\beta }}_k^T {\varvec{Z}}_i) r_{jk}^{(0)} (s;{\varvec{\beta }}_k) ^{-1} dM_{\ell jk}(s; {\varvec{\beta }}_k). \end{aligned}$$

Combining results derived in this subsection, we obtain,

$$\begin{aligned} n^{\frac{1}{2}} \left\{ \widehat{F}_{ijk}(t) - F_{ijk}(t)\right\} = n^{-\frac{1}{2}} \sum _{\ell =1}^n \left\{ \sum _{d=1}^4 \phi _{i\ell jk}^d(t, {\varvec{\beta }}) \right\} + o_p(1), \end{aligned}$$

where the $\sum _{d=1}^4 \phi _{i\ell jk}^d(t, {\varvec{\beta }})$ are asymptotically independent and identically distributed variates with mean 0.

[6]    $n^{\frac{1}{2}} \left\{ \widehat{\delta }_{jk}(t) - \delta _{jk}(t) \right\}$

We now complete the proof by averaging over $i = 1, \ldots , n$ to obtain the limiting distribution of the proposed estimator. Setting $n_j = \sum _{i=1}^n A_{ij}$ and $p_j = E[A_{ij}]$, we have

$$\begin{aligned}{}[6]= & {} \frac{1}{n_j}\sum _{i=1}^{n} A_{ij} \left\{ n^{-\frac{1}{2}} \sum _{\ell =1}^n \left[ \sum _{d=1}^4 \phi _{i\ell jk}^d(t, {\varvec{\beta }}) - \frac{1}{n} \sum _{r=1}^J \left\{ \sum _{d=1}^4 \phi _{i\ell rk}^d(t, {\varvec{\beta }}) \right\} n_r \right] \right\} \nonumber \\= & {} n^{-\frac{1}{2}} \sum _{\ell =1}^n \left\{ \sum _{d=1}^4 \left[ \frac{1}{n_j}\sum _{i=1}^{n} A_{ij} \phi _{i\ell jk}^d(t, {\varvec{\beta }})\,-\,\frac{1}{n_j}\sum _{i=1}^{n} A_{ij} \frac{1}{n} \sum _{r=1}^J \phi _{i\ell rk}^d(t, {\varvec{\beta }}) n_r \right] \right\} .\nonumber \\ \end{aligned}$$

(16)

Focusing on each component in (16), we have the following for the expression involving $\phi ^1_{i\ell jk}(t;{\varvec{\beta }})$,

$$\begin{aligned}&\frac{1}{n_j}\sum _{i=1}^{n} A_{ij} \phi ^1_{i\ell jk}(t;{\varvec{\beta }}) \\&\quad = - \frac{1}{p_j} E \left[ \sum _{m=1}^K A_{ij} \int _0^t {\varvec{D}}_{ijm}^T (s;{\varvec{\beta }}_m) dF_{ijk}(s;{\varvec{\beta }}_k) \right] {\varvec{I}}_{m}({\varvec{\beta }}_m)^{-1} {\varvec{U}}_{\ell m}({\varvec{\beta }}_m) \\&\quad = \phi _{\ell jk}^1 (t;{\varvec{\beta }}), \end{aligned}$$

since $n_j/n {\rightarrow } p_j$ by the Weak Law of Large Numbers (WLLN), continuity, and Slutsky’s Theorem. The simplification for the term involving $\phi ^3_{i\ell jk}(t;{\varvec{\beta }})$ unfold in a similar way. The term involving $\phi ^2_{i\ell jk}(t;{\varvec{\beta }})$ can be written as the following:

$$\begin{aligned}&\frac{1}{n_j}\sum _{i=1}^{n} A_{ij} \phi ^2_{i\ell jk}(t;{\varvec{\beta }}) \\&\quad = - \sum _{m=1}^K \int _0^t \left[ \frac{1}{p_j} \frac{1}{n} \sum _{i=1}^{n} A_{ij} \left\{ F_{ijk}(t) \right. \right. \\&\qquad \left. \left. -\,F_{ijk}(u) \right\} Y _{ij}(u) \exp ({\varvec{\beta }}_m^T {\varvec{Z}}_i) r_{jk}^{(0)} (u;{\varvec{\beta }}_m) ^{-1} \right] dM_{\ell jm}(u; {\varvec{\beta }}_m) \\&\quad = \phi _{\ell jk}^2(t;{\varvec{\beta }}), \end{aligned}$$

by the WLLN, continuity, and Slutsky’s Theorem. The term involving $\phi ^4_{i\ell jk}(t;{\varvec{\beta }})$ unfold in a similar way. The term involving $\phi ^1_{i\ell rk}(t;{\varvec{\beta }})$ can be written as the following:

$$\begin{aligned}&\frac{1}{n_j}\sum _{i=1}^{n} \sum _{m=1}^K A_{ij} \left\{ \frac{1}{n} \sum _{r=1}^J \phi ^1_{i\ell rk}(t;{\varvec{\beta }}) n_r \right\} \\&\quad = - \frac{1}{p_j} \sum _{r=1}^J p_r E \left[ \sum _{m=1}^K A_{ij} \int _0^t {\varvec{D}}_{irm}^T (s;{\varvec{\beta }}_m) dF_{irk}(s;{\varvec{\beta }}_k) \right] {\varvec{I}}_{m}({\varvec{\beta }}_m)^{-1} {\varvec{U}}_{\ell m}({\varvec{\beta }}_m) \\&\quad = \phi _{\ell k}^1(t;{\varvec{\beta }}). \end{aligned}$$

The term involving $\phi ^3_{i\ell rk}(t;{\varvec{\beta }})$ can be expressed in a similar way. The term involving $\phi ^2_{i\ell rk}(t;{\varvec{\beta }})$ can be written as,

$$\begin{aligned}&\frac{1}{n_j}\sum _{i=1}^{n} \sum _{m=1}^K A_{ij} \left\{ \frac{1}{n} \sum _{r=1}^J \phi ^3_{i\ell rk}(t;{\varvec{\beta }}) \cdot n_r \right\} \\&\quad = \sum _{m=1}^K \int _0^t \sum _{r=1}^J \frac{1}{p_j} p_r E \left[ A_{ij} \left\{ F_{irk}(t) -F_{irk}(u) \right\} Y _{ir}(u) \right. \\&\quad \left. \times \exp ({\varvec{\beta }}_m^T {\varvec{Z}}_i) r_{rk}^{(0)} (u;{\varvec{\beta }}_m) ^{-1} \right] dM_{\ell rm}(u; {\varvec{\beta }}_m) \\&\quad = \phi _{\ell k}^2(t;{\varvec{\beta }}). \end{aligned}$$

The term involving $\phi ^4_{i\ell rk}(t;{\varvec{\beta }})$ can be expressed analogously. Therefore, we can write

$$\begin{aligned}{}[6] = n^{-\frac{1}{2}} \sum _{\ell =1}^n \left\{ \sum _{d=1}^4 \left( \phi _{\ell jk}^d(t;{\varvec{\beta }}) + \phi _{\ell k}^d (t;{\varvec{\beta }}) \right) \right\} . \end{aligned}$$

(17)

All summands across $l$ have mean 0 since the $\phi$’s have mean 0. If we apply the Functional Central Limit Theorem to [6], where each component is independent across $l$, we have

$$\begin{aligned} V \left( n^{\frac{1}{2}} \left\{ \widehat{\delta }_{jk}(t) - \delta _{jk}(t) \right\} \right) = E \left[ \left\{ \sum _{d=1}^4 \left( \phi _{\ell jk}^d (t;{\varvec{\beta }})- \phi _{\ell k}^d(t;{\varvec{\beta }}) \right) \right\} ^2 \right] . \end{aligned}$$

(18)