Analysis of error-prone survival data under additive hazards models: measurement error effects and adjustments

Abstract

Covariate measurement error occurs commonly in survival analysis. Under the proportional hazards model, measurement error effects have been well studied, and various inference methods have been developed to correct for error effects under such a model. In contrast, error-contaminated survival data under the additive hazards model have received relatively less attention. In this paper, we investigate this problem by exploring measurement error effects on parameter estimation and the change of the hazard function. New insights of measurement error effects are revealed, as opposed to well-documented results for the Cox proportional hazards model. We propose a class of bias correction estimators that embraces certain existing estimators as special cases. In addition, we exploit the regression calibration method to reduce measurement error effects. Theoretical results for the developed methods are established, and numerical assessments are conducted to illustrate the finite sample performance of our methods.

Appendix

In the following, for an $m\times 1$ vector $a=(a_{(1)},a_{(2)},\cdots , a_{(m)})^T$, we use the Euclidean norm $||a||=(\sum a_{(i)}^2)^{1/2}$. For a matrix A, define $||A||=\sup _{i,j}|A_{(i)(j)}|$. When we say that the vector process $A_n(t)$ converges almost surely to A(t) uniformly in t, we mean that $\sup _{0\le t\le \tau }||A_n(t)-A(t)||\mathop {\rightarrow }\limits ^{a.s.} 0$, as $n\rightarrow \infty$.

1.1 Appendix 1: regularity conditions

• R1. $\{N_i(\cdot ),Y_i(\cdot ),Z_i(\cdot )\}, i=1,\cdots ,n$ are independent and identically distributed.

• R2. $\Pr \{Y_1(\tau )=1\}>0$.

• R3. $\Lambda _0(\tau )<\infty$.

• R4. $\sup _{t\in [0,\tau ]}||E[Z_1^{\otimes 2}(t)]||<\infty$.

• R5. Bounded variation condition: for $i=1,\cdots , n$, $j=1,\cdots , p+q$,

  $$\begin{aligned} |Z_{i(j)}(0)|+\int _0^{\tau } |dZ_{i(j)}(u)|\le K \end{aligned}$$

  holds almost surely for all the sample path, where K is a constant.

• R6. All the $n_i$, $i=1,\cdots , n$, are upper bounded by a constant $N_0$, and

  $$\begin{aligned} \lim _{n\rightarrow \infty }n^{-1}\sum _{i=1}^n I\{n_i=j\} \end{aligned}$$

  exists, where $j=1,\cdots ,N_0$.

• R7. $||E(\epsilon _{11}^{\otimes 2})||<\infty$.

• R8. $\int _0^{\tau }E\left[ Y_1(t)\{{Z}_1(t)-e(t)\}^{\otimes 2}\right] dt$, $\Sigma _{nv}$, and $\Sigma _{rc}^*$ are positive definite.

1.2 Appendix 2: Proof of Theorem 1

Since each component of the vector $n^{-1}\sum _{i=1}^n Y_i(t)Z_i(t)$ is of bounded variation by Condition R5, it can be written as the difference of two nondecreasing functions. Thus, $n^{-1}\sum _{i=1}^n Y_i(t)Z_i(t)$ is manageable (Pollard 1990, p. 38), and $n^{-1}\sum _{i=1}^n Y_i(t)\hat{Z}_i(t)$ is manageable. Together with Conditions R4, R6 and R7, the two conditions of the uniform strong law of large numbers (USLLN) (Pollard 1990, p. 41) are thus verified, and we obtain that as $n\rightarrow \infty$, $n^{-1}\sum _{i=1}^n Y_i(t)\hat{Z}_i(t)\mathop {\rightarrow }\limits ^{a.s.} E[Y_i(t)Z_i(t)]$ uniformly in t. Similarly, $n^{-1}\sum _{i=1}^n Y_i(t)\mathop {\rightarrow }\limits ^{a.s.} E[Y_i(t)]$, and $n^{-1}\sum _{i=1}^n N_i(t)\mathop {\rightarrow }\limits ^{a.s.} E[N_i(t)]$ uniformly in t. By USLLN together with Condition R2, $\sum _{i=1}^n Y_i(t)\hat{Z}_i(t)/\sum _{i=1}^n Y_i(t)\mathop {\rightarrow }\limits ^{a.s.} e(t)$ uniformly in t. Thus, $n^{-1}\sum _{i=1}^n\int _0^{\tau } \tilde{Z}(t)dN_i(t)\mathop {\rightarrow }\limits ^{a.s.}\int _0^{\tau }e(t)dE[N_i(t)]$. Similarly,

$n^{-1}\sum _{i=1}^n\int _0^{\tau } \hat{Z}_i(t)dN_i(t)\mathop {\rightarrow }\limits ^{a.s.}\int _0^{\tau }E\left[ Z_i(t)dN_i(t)\right] .$ Note that $E[\hat{Z}_i^{\otimes 2}(t)]=E[Z_i^{\otimes 2}(t)]+\Sigma _1/n_i$. Thus,

$$\begin{aligned} n^{-1}\sum _{i=1}^n\int _0^{\tau }Y_i(t)\hat{Z}_i^{\otimes 2}(t)\beta dt\mathop {\longrightarrow }\limits ^{a.s.} \int _0^{\tau }E\left[ Y_i(t){Z}_i^{\otimes 2}(t)\beta \right] dt+\rho _0\Sigma _1\beta \int _0^{\tau }E\left[ Y_i(t)\right] dt. \end{aligned}$$

After some algebra, these results lead to that $n^{-1} U_{nv}(\beta )\mathop {\longrightarrow }\limits ^{a.s.}\mathcal {U}_{nv}(\beta )$, as $n\rightarrow \infty$. Thus, $\hat{\beta }_{nv}\mathop {\rightarrow }\limits ^{a.s.}\beta _{nv}^*$, as $n\rightarrow \infty$.

Now we show that

$$\begin{aligned} n^{-1/2}U_{nv}(\beta _{nv}^*)=&\,n^{-1/2}\sum _{i=1}^n\biggl [\int _0^{\tau }\left\{ \hat{Z}_i(t)-e(t)\right\} d \tilde{M}_i(t;\beta )\nonumber \\&- \int _0^{\tau }Y_i(t)\left\{ \hat{Z}_i(t)-e(t)\right\} ^{\otimes 2}\left( \beta _{nv}^*-\beta \right) dt\biggr ]+o_p(1). \end{aligned}$$

(13)

Note that $U_{nv}(\beta _{nv}^*)=U_1^*-U_2^*$, where

$$\begin{aligned} U_1^*= & {} \sum _{i=1}^n\int _0^{\tau }\left\{ \hat{Z_i}(t)-\tilde{Z}(t)\right\} d\tilde{M}_i(t;\beta ),\\ \text {and}\ \ U_2^*= & {} \sum _{i=1}^n\int _0^{\tau }\left\{ \hat{Z_i}(t)-\tilde{Z}(t)\right\} Y_i(t)\hat{Z}_i^T(t)\left( \beta _{nv}^*-\beta \right) dt. \end{aligned}$$

Similar to the proof of Theorem 1 of Kulich and Lin (2000), we obtain that

$$\begin{aligned} n^{-1/2}U_1^*=n^{-1/2}\sum _{i=1}^n\int _0^{\tau }\left\{ \hat{Z_i}(t)-e(t)\right\} d\tilde{M}_i(t;\beta )+o_p(1). \end{aligned}$$

To prove (13), it remains to prove that

$$\begin{aligned} n^{-1/2}U_2^*\equiv & {} n^{-1/2}\sum _{i=1}^n\int _0^{\tau }Y_i(t)\left\{ \hat{Z}_i(t)-\tilde{Z}(t)\right\} ^{\otimes 2}\left( \beta _{nv}^*-\beta \right) dt\nonumber \\= & {} n^{-1/2}\sum _{i=1}^n\int _0^{\tau }Y_i(t)\left\{ \hat{Z}_i(t)-e(t)\right\} ^{\otimes 2}\left( \beta _{nv}^*-\beta \right) dt+o_p(1). \end{aligned}$$

(14)

Observe that

$$\begin{aligned}&n^{-1/2}\int _0^{\tau }\sum _{i=1}^nY_i(t)\tilde{Z}^{\otimes 2}(t)dt\nonumber \\= & {} n^{-1/2}\int _0^{\tau }\left\{ \tilde{Z}(t)-e(t)\right\} \sum _{i=1}^nY_i(t)\hat{Z}_i^T(t)dt+n^{-1/2}\int _0^{\tau }e(t) \sum _{i=1}^nY_i(t)\hat{Z}_i^T(t)dt\nonumber \\= & {} n^{1/2}\int _0^{\tau }\left\{ \tilde{Z}(t)-e(t)\right\} E\left\{ Y_i(t)\hat{Z}_i^T(t)\right\} dt+n^{-1/2}\int _0^{\tau }e(t) \sum _{i=1}^nY_i(t)\hat{Z}_i^T(t)dt+o_p(1)\nonumber \\= & {} n^{-1/2}\int _0^{\tau }\frac{\sum _{i=1}^nY_i(t)\hat{Z}_i(t)}{E\{Y_i(t)\}} E\left\{ Y_i(t)\hat{Z}_i^T(t)\right\} dt\nonumber \\&-\int _0^{\tau }\frac{E\left\{ Y_i(t)\hat{Z}_i(t)\right\} }{\left[ E\{Y_i(t)\}\right] ^2}{E\left\{ Y_i(t)\hat{Z}_i^T(t)\right\} }\sqrt{n} \left[ \frac{1}{n}\sum _{i=1}^nY_i(t)-E\{Y_i(t)\}\right] dt\nonumber \\&-\,n^{1/2}\int _0^{\tau }e(t) E\left\{ Y_i(t)\hat{Z}_i^T(t)\right\} dt+n^{-1/2}\int _0^{\tau }e(t) \sum _{i=1}^nY_i(t)\hat{Z}_i^T(t)dt+o_p(1)\nonumber \\= & {} n^{-1/2}\int _0^{\tau }\sum _{i=1}^nY_i(t)\left\{ \hat{Z}_i(t)e^T(t)+e(t)\hat{Z}_i^T(t)-e^{\otimes 2}(t)\right\} dt+o_p(1). \end{aligned}$$

(15)

Plugging (15) into $n^{-1/2}U_2^*$, we obtain (14), and thus (13) is proved.

By the Taylor series expansion, $0=n^{-1/2}U_{nv}(\hat{\beta }_{nv})= n^{-1/2}U_{nv}(\beta _{nv}^*)+ \left[ n^{-1}\frac{\partial U_{nv}(\beta )}{\partial \beta }\right] n^{1/2}(\hat{\beta }_{nv}-\beta _{nv}^*)$, leading to

$$\begin{aligned} n^{1/2}\left( \hat{\beta }_{nv}-\beta _{nv}^*\right) =-\left[ n^{-1}\frac{\partial U_{nv}(\beta )}{\partial \beta }\right] ^{-1}n^{-1/2}U_{nv}(\beta _{nv}^*). \end{aligned}$$

(16)

It is straightforward that

$$\begin{aligned} -n^{-1}\frac{\partial U_{nv}(\beta )}{\partial \beta }\mathop {\rightarrow }\limits ^{a.s.}\lim _{n\rightarrow \infty }n^{-1}\int _0^{\tau }\sum _{i=1}^nE\left[ Y_i(t)\left\{ \hat{Z}_i(t)-e(t)\right\} ^{\otimes 2}\right] dt= \mathcal {D}_{nv}. \end{aligned}$$

Let $U_{nv,i}=\int _0^{\tau }\left\{ \hat{Z}_i(t)-e(t)\right\} d \tilde{M}_i(t;\beta )- \int _0^{\tau }Y_i(t)\left\{ \hat{Z}_i(t)-e(t)\right\} ^{\otimes 2}(\beta _{nv}^*-\beta )dt,i=1,\cdots ,n$. By Condition R6, $E[||n^{-1/2}U_{nv,i}||^2I\{||n^{-1/2}U_{nv,i}||>\epsilon \}]$ can only take at most $N_0$ possible values for fixed $\epsilon >0$. Without loss of generality, suppose when $i=1$, it achieves the maximum value. It follows from the Markov inequality that $\Pr \{||n^{-1/2}U_{nv,1}||>\epsilon \}\le n^{-1}E[||U_{nv,1}||^2]/\epsilon ^2\rightarrow 0$ as $n\rightarrow \infty$, and thus

$$\begin{aligned}&\sum _{i=1}^nE[||n^{-1/2}U_{nv,i}||^2I\{||n^{-1/2}U_{nv,i}||>\epsilon \}]\\&\quad \le E[||U_{nv,1}||^2I\{||n^{-1/2}U_{nv,1}||>\epsilon \}]\rightarrow 0 \ \ \text {as}\ \ n\rightarrow \infty . \end{aligned}$$

Consequently, the Lindeberg condition (van der Vaart 1998, p. 20) is verified. By the multivariate Lindeberg–Feller central limit theorem (van der Vaart 1998, p. 20), we obtain that $n^{-1/2}U_{nv}(\beta _{nv}^*)$ is asymptotically normal with mean 0 and covariance matrix $\Sigma _{nv}=\lim _{n\rightarrow \infty }n^{-1}\sum _{i=1}^nE( U_{nv,i})^{\otimes 2}$. It follows from (16) that $\hat{\beta }_{nv}$ is asymptotically normal with mean $\beta _{nv}^*$ and covariance matrix $\mathcal {D}_{nv}^{-1}\Sigma _{nv}\mathcal {D}_{nv}^{-1}$. The matrices $\Sigma _{nv}$ and $\mathcal {D}_{nv}$ can be consistently estimated by their empirical counterpart.

1.3 Appendix 3: asymptotic equivalence of $\hat{\beta }_{bc}$ and $\hat{\beta }_{szs}$

Note that

$$\begin{aligned} \sqrt{n}\left\{ \tilde{Z}_{r}(t)-e(t)\right\} \left\{ \tilde{Z}_{s}(t)-e(t)\right\} ^T=o_p(1). \end{aligned}$$

Thus,

$$\begin{aligned} {\sqrt{n}\hat{B}_{bc,1}}= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^{\tau }\frac{Y_i(t)}{n_i(n_i-1)}\sum _{1\le r\ne s\le n_i}\left\{ \hat{Z}_{ir}(t)- \tilde{Z}_{r}(t)\right\} \left\{ \hat{Z}_{is}(t)-\tilde{Z}_{s}(t)\right\} ^Tdt\\= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^{\tau }\frac{Y_i(t)}{n_i(n_i-1)}\sum _{1\le r\ne s\le n_i}\hat{Z}_{ir}(t)\hat{Z}_{is}(t)^T dt\\&-\frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^{\tau }{Y_i(t)} \tilde{Z}(t)e(t)^T dt\\&-\frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^{\tau }{Y_i(t)} e(t)\tilde{Z}(t)^T dt+\frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^{\tau }{Y_i(t)} e^{\otimes 2}(t) dt+o_p(1)\\= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^{\tau }\frac{Y_i(t)}{n_i(n_i-1)}\sum _{1\le r\ne s\le n_i}\hat{Z}_{ir}(t)\hat{Z}_{is}(t)^T dt\\&-\frac{1}{\sqrt{n}}\int _0^{\tau }\sum _{i=1}^nY_i(t)\tilde{Z}^{\otimes 2}(t)dt+o_p(1)\\= & {} {\sqrt{n}\hat{B}_{szs,1}+o_p(1)}, \end{aligned}$$

where the third equation comes from (15). Together with error bounds for the inverse matrices (Horn and Johnson 1985), we obtain $\sqrt{n}(\hat{B}_{bc,1}^{-1}-\hat{B}_{szs,1}^{-1})=o_p(1)$. Note also that $\hat{B}_{bc,1}+\hat{B}_{bc,2}=\hat{B}_{szs,1}+\hat{B}_{szs,2}$. It then follows that

$$\begin{aligned} \sqrt{n}\left( \hat{\beta }_{bc}-\hat{\beta }_{szs}\right)= & {} \sqrt{n}\left\{ \hat{B}_{bc,1}^{-1}\left( \hat{B}_{bc,1}+\hat{B}_{bc,2}\right) \hat{\beta }_{nv}-\hat{B}_{szs,1}^{-1}\left( \hat{B}_{szs,1}+\hat{B}_{szs,2}\right) \hat{\beta }_{nv}\right\} \\= & {} \sqrt{n}\left( \hat{B}_{bc,1}^{-1}-\hat{B}_{szs,1}^{-1}\right) \left( \hat{B}_{bc,1}+\hat{B}_{bc,2}\right) \hat{\beta }_{nv}\\= & {} o_p(1). \end{aligned}$$

Thus, $\hat{\beta }_{bc}$ and $\hat{\beta }_{szs}$ are asymptotically equivalent, and they have the same asymptotic distribution.

1.4 Appendix 4

Following Carroll et al. (2006), we estimate ${\Sigma }_{xx}$, ${\Sigma }_{xv}$, ${\Sigma }_{vv}$, $\mu _x$, and $\mu _{v}$ by $\hat{\Sigma }_{xx}$, $\hat{\Sigma }_{xv}$, $\hat{\Sigma }_{vv}$, $\bar{W}_{\cdot \cdot }={\sum _{i=1}^n\sum _{j=1}^{n_i}W_{ij}}/{\sum _{i=1}^n n_i}$ and $\bar{V}_{\cdot }(t)$, respectively, where

$$\begin{aligned} \hat{\Sigma }_{vv}= & {} ({n-1})^{-1}\sum _{i=1}^n\left( V_i(t)-\bar{V}_{\cdot }(t))(V_i(t)-\bar{V}_{\cdot }(t)\right) ^T,\\ \hat{\Sigma }_{xv}= & {} \frac{\sum _{i=1}^n n_i}{\left( \sum _{i=1}^n n_i\right) ^2-\sum _{i=1}^n n_i^2}\sum _{i=1}^n n_i \left( \bar{W}_{i \cdot }-\bar{W}_{\cdot \cdot }\right) \left( V_i(t)-\bar{V}_{\cdot }(t)\right) ^T,\\ \text {and} \ \ \hat{\Sigma }_{xx}= & {} \frac{\sum _{i=1}^n n_i}{\left( \sum _{i=1}^n n_i\right) ^2-\sum _{i=1}^n n_i^2}\left[ \sum _{i=1}^n n_i \left( \bar{W}_{i \cdot }-\bar{W}_{\cdot \cdot }\right) \left( \bar{W}_{i \cdot }-\bar{W}_{\cdot \cdot }\right) ^T-(n-1)\hat{\Sigma }_0\right] . \end{aligned}$$

1.5 Appendix 5: Proof of Theorem 2

First, we consider the case where all $n_i$ are equal. Note that

$$\begin{aligned} \hat{\beta }_{rc}= & {} \left[ \sum _{i=1}^n\int _0^{\tau } Y_i(t)\hat{A}_{rc,i}\left\{ \hat{Z}_{i}(t)-\tilde{Z}(t)\right\} ^{\otimes 2}\hat{A}_{rc,i}^Tdt\right] ^{-1}\\&\left[ \sum _{i=1}^n\int _0^{\tau }\hat{A}_{rc,i}\left\{ \hat{Z}_i(t)-\tilde{Z}(t)\right\} dN_i(t)\right] \\= & {} \hat{A}_{rc,1}^{-1T}\left[ \sum _{i=1}^n\int _0^{\tau } Y_i(t)\left\{ \hat{Z}_{i}(t)-\tilde{Z}(t)\right\} ^{\otimes 2}dt\right] ^{-1}\hat{A}_{rc,1}^{-1}\hat{A}_{rc,1}\\&\left[ \sum _{i=1}^n\int _0^{\tau }\left\{ \hat{Z}_i(t)-\tilde{Z}(t)\right\} dN_i(t)\right] =\left( \hat{A}_{rc,1}^{-1}\right) ^T\hat{\beta }_{nv}. \end{aligned}$$

It follows that $\hat{\beta }_{rc}\mathop {\longrightarrow }\limits ^{a.s.}\beta _{rc}^*$, as $n\rightarrow \infty$. Together with the fact that

$$\begin{aligned} \sqrt{n}\left\{ \left( \hat{A}_{rc,1}^{-1}\right) ^T-\left( {A}_{rc,1}^{-1}\right) ^T\right\} \left( \hat{\beta }_{nv}-\beta _{nv}^*\right) =o_p(1), \end{aligned}$$

we obtain

$$\begin{aligned}&\sqrt{n}\left( \hat{\beta }_{rc}-{\beta }_{rc}^*\right) = \sqrt{n}\left\{ \left( \hat{A}_{rc,1}^{-1}\right) ^T\hat{\beta }_{nv}-\left( {A}_{rc,1}^{-1}\right) ^T{\beta }_{nv}^*\right\} \nonumber \\&\qquad =\sqrt{n}\left( {A}_{rc,1}^{-1}\right) ^T\left( \hat{\beta }_{nv}-{\beta }_{nv}^*\right) - \sqrt{n}\left\{ \left( \hat{A}_{rc,1}^{-1}\right) ^T-\left( {A}_{rc,1}^{-1}\right) ^T\right\} {\beta }_{nv}^*+o_p(1). \nonumber \\ \end{aligned}$$

(17)

Since the asymptotic expansion of $\sqrt{n}({A}_{rc,1}^{-1})^T(\hat{\beta }_{nv}-{\beta }_{nv}^*)$ is obtained in Theorem 1, thus, to obtain the asymptotic expansion of $\sqrt{n}(\hat{\beta }_{rc}-{\beta }_{rc}^*)$, by (17) we need only to examine $\sqrt{n}\{(\hat{A}_{rc,1}^{-1})^T-({A}_{rc,1}^{-1})^T\}{\beta }_{nv}^*$. This can be done, in principle, by a Taylor series expansion. In the following, we first study the asymptotic expansion for the univariate case, and then the multivariate case.

By considering a Taylor series expansion, we obtain

$$\begin{aligned} \sqrt{n}\left\{ \left( \hat{A}_{rc,1}^{-1}\right) ^T-\left( {A}_{rc,1}^{-1}\right) ^T\right\}= & {} \sqrt{n}\left( \frac{\hat{\Sigma }_{xx}+\hat{\Sigma }_0/n_1}{\hat{\Sigma }_{xx}} -\frac{{\Sigma }_{xx}+{\Sigma }_0/n_1}{{\Sigma }_{xx}}\right) \\= & {} \frac{\sqrt{n}\hat{\Sigma }_0/n_1}{{\Sigma }_{xx}}- \frac{{\Sigma }_0/n_1}{{\Sigma }_{xx}^2}\sqrt{n}\hat{\Sigma }_{xx}+o_p(1). \end{aligned}$$

Noting that

$$\begin{aligned} \sqrt{n}\hat{\Sigma }_{xx}= & {} \sqrt{n} \left\{ \frac{\sum _{i=1}^n\left( \bar{W}_{i\cdot }-\bar{W}_{\cdot \cdot }\right) ^2}{n-1}\right\} -\sqrt{n}\hat{\Sigma }_0/n_1\\= & {} \sqrt{n} \left\{ \frac{\sum _{i=1}^n\left( \bar{W}_{i\cdot }-\mu _x\right) ^2}{n}\right\} -\sqrt{n}\left\{ \frac{\sum _{i=1}^n\sum _{r=1}^{n_i}\left( W_{ir}-\bar{W}_{i\cdot }\right) ^{\otimes 2}}{n_1\sum _{i=1}^n(n_i-1)}\right\} +o_p(1), \end{aligned}$$

together with (17), we obtain that $\sqrt{n}(\hat{\beta }_{rc}-{\beta }_{rc}^*)$ is a sum of independent terms asymptotically, and thus, $\hat{\beta }_{rc}$ is asymptotic normal with mean ${\beta }_{rc}^*$ and variance $\Sigma _{rc}^*$, which can be obtained easily by (17).

For the multivariate case, $\hat{\beta }_{rc}$ is still asymptotic normal with mean ${\beta }_{rc}^*$ and covariance matrix $\Sigma _{rc}^*$ whose form is complicated. We suggest to use the first term of (17) only to obtain an approximate variance of $\hat{\beta }_{rc}$: $\Sigma _{rc}^*\approx [{A}_{rc,1}^{-1}]^T\mathcal {D}_{nv}^{-1}\Sigma _{nv}\mathcal {D}_{nv}^{-1}[{A}_{rc,1}^{-1}]$, which can be consistently estimated by the empirical counterpart.

For the general case where the $n_i$ are not necessarily equal, it can be shown that $\hat{\beta }_{rc}$ is still asymptotically normal, with a complicated form for the asymptotic covariance.