Fixed-width confidence interval for covariate-adjusted response-adaptive designs

Abstract

In this paper, we obtain fixed-width confidence interval for covariate-adjusted response-adaptive designs. Specifically, we consider logistic regression model and the normal regression model for binary and continuous responses, respectively, both in the situations for presence and absence of treatment–covariate interactions. Simulation study and real-data analysis are carried out.

Appendix A

Result 7

As $m\rightarrow \infty$,

$$\begin{aligned} \frac{1}{m}\sum _{i=1}^m\delta _{ki}\tau _k(\theta _k,Z_i)Z_iZ_i^\top \rightarrow I_{Zk} \end{aligned}$$

almost surely, $k=A,B$.

Proof

It is not difficult to observe that

$$\begin{aligned} \sum _{i=1}^m[\delta _{ki}\tau _k(\theta _k,Z_i)Z_iZ_i^\top -E(\pi _k({\widehat{\theta }}_{i-1},Z_i)\tau _k(\theta _k,Z_i)Z_iZ_i^\top )], \end{aligned}$$

where ${\widehat{\theta }}_i$ is the ML estimator based on $\{\delta _{kj},Y_j,Z_j; j=1,2,\ldots ,i;k=A,B\}$, is a martingale sequence. Hence, applying the same technique as in (A.14) under Corollary 3.1 of Zhang et al. (2007), the required result follows. $\square$

Note 3

Note A.1. If $\tau _k(\theta _k,Z)=1$, the above result implies

$$\begin{aligned} {\widehat{I}}_{mk}=\frac{1}{m}\sum _{i=1}^m\delta _{ki}Z_iZ_i^\top \rightarrow E[\pi _k(\theta ,Z)ZZ^\top ] \end{aligned}$$

(17)

almost surely as $m\rightarrow \infty$.

If $\tau _k(\theta _k,Z)=\frac{\mathrm{e}^{\theta _k^\top Z}}{( 1+\mathrm{e}^{\theta _k^\top Z})^2}$, then the mean-value theorem yields

$$\begin{aligned} |\tau _k({\widehat{\theta }}_{km},Z)-\tau _k(\theta _k,Z)|<\frac{M}{2}||{\widehat{\theta }}_{km}-\theta _k||, \end{aligned}$$

(18)

where M is described in Sect. 2. Consequently, (17) and (18) imply

$$\begin{aligned} {\widehat{I}}_{mk}= & {} \frac{1}{m}\sum _{i=1}^m\delta _{ki}\left[ \tau _k(\theta _k,Z_i)+O\left( ||{\widehat{\theta }}_{km}-\theta _k||\right) \right] Z_iZ_i^\top \\= & {} \frac{1}{m}\sum _{i=1}^m\delta _{ki}\tau _k(\theta _k,Z_i)Z_iZ_i^\top +O( ||{\widehat{\theta }}_{km}-\theta _k||).O(1)\nonumber \end{aligned}$$

almost surely. Hence, by the above result, it follows that

$$\begin{aligned} {\widehat{I}}_{mk}\rightarrow E[\pi _k(\theta ,Z)p_k(Z)(1-p_k(Z))ZZ^\top ] \end{aligned}$$

almost surely as $m\rightarrow \infty$, $k=A,B$.

Result 8

As $\nu \rightarrow \infty$,

$$\begin{aligned} \frac{1}{\sqrt{\nu }}S_{k\nu }\rightarrow N_r(0,\Sigma _k) \end{aligned}$$

in distribution with $\Sigma _k$ given in Result 1, $k=A,B$.

Proof

Let $\ell _k$, $k=A,B$, be two r-component fixed vectors. Then, setting $a_i=\ell _A^\top I_{ZA}^{-1}Z_i$ and $b_i=\ell _B^\top I_{ZB}^{-1}Z_i$, consider

$$\begin{aligned} V_{\nu i}=\frac{1}{\sqrt{\nu }}\left[ \delta _{Ai}a_i(X_{Ai}-\mu _A(Z_i))+\delta _{Bi}b_i(X_{Bi}-\mu _B(Z_i)) \right] , \end{aligned}$$

where $\mu _k(Z_i)=p_k(Z_i)$ and $\theta _k^\top Z_i$ for models (a) and (b), $i=1,2,\ldots ,\nu$, $k=A,B$. The sequences $\left\{ V_{\nu i}, ~1\le i\le \nu , ~\nu \ge 1\right\}$ represent differences corresponding to the martingale sequence $T_{\nu }=\sum _{i=1}^{\nu }V_{\nu i}$, $\nu \ge 1$. Then, we get

$$\begin{aligned} E(T_{\nu }^2)=\sum _{i=1}^{\nu }E(V_{\nu i}^2)\le \sum _{k=A,B}\ell ^\top I_{Zk}^{-1}E[\tau _k(\theta _k,Z)ZZ^\top ] I_{Zk}^{-1}\ell _k<\infty . \end{aligned}$$

Hence, by martingale central limit theorem (see, for example. Theorem 3.2 of Hall and Heyde 1980, p. 58), it will follow that

$$\begin{aligned} T_{\nu }\rightarrow N(0,\eta ^2) \end{aligned}$$

in distribution as $\nu \rightarrow \infty$, where $\eta ^2=\sum _{k=A,B}(\ell _k^\top \Sigma _k\ell _k)$, provided all the conditions of Theorem 3.2 in Hall and Heyde (1980) are satisfied. Here, all the conditions, except (3.19) of this theorem, are trivially satisfied. Now, to prove this non-trivial condition, write

$$\begin{aligned} \sum _{i=1}^{\nu }W_i=\sum _{i=1}^{\nu }[\nu V_{\nu i}^2-a_i^2\tau _A(\theta ,Z_i)\delta _{Ai}-b_i^2\tau _B(\theta ,Z_i)\delta _{Bi}] , \end{aligned}$$

where

$$\begin{aligned} W_i=\delta _{Ai}a_i^2[\left( X_{Ai}-\mu _A(Z_i)\right) ^2-\tau _A(\theta ,Z_i)]+\delta _{Bi}b_i^2[\left( X_{Bi}-\mu _B(Z_i)\right) ^2-\tau _B(\theta ,Z_i)] . \end{aligned}$$

Using Theorem 2.13 (iii) of Hall and Heyde (1980), we get, as $\nu \rightarrow \infty$,

$$\begin{aligned} \frac{1}{\nu }\sum _{i=1}^{\nu }W_i\rightarrow 0 \end{aligned}$$

in probability, which implies

$$\begin{aligned} \sum _{i=1}^{\nu }V_{\nu i}^2\rightarrow \eta ^2 \end{aligned}$$

in probability as $\nu \rightarrow \infty$. Hence, by the Cramer–Wold device, we get the required result. $\square$