基于均差估计二变量Meta-分析的随机效应模型

The Random-Effect Model of Bivariate Meta-Analysis Based on the Difference Mean

摘要: 本文给出已知协方差矩阵的基于均差估值的二变量Meta-分析随机效应模型，并通过极大似然估计法给出效应量估计并说明了其无偏性。通过计算估计效应量的协方差矩阵，给出估计效应分量的(1-α)100%置信区间。

Abstract: In this paper, we establish the random-effect model of bivariate meta-analysis based on the dif-ference mean. Under the assumption of the multilevel normal distribution of the difference mean, we obtain the maximum likelihood estimator û_MLof the difference mean by the method of max-imum likelihood estimate. We conclude that the maximum likelihood estimator û_MLof the dif-ference mean is unbiased for the difference mean effect µ. In addition, we get the corresponding covariance matrix and the (1-α)100%  confidence interval for the difference mean effect  µ.

1. 前言

Meta-分析是一种对同一问题的不同研究进行定量合并的一种方法。该方法是通过考虑每个研究内和研究间的差异性以及研究发表偏倚性得到一个综述结果，从而解决了单个研究样本量小，区域、种族差异大等的缺点。因此Meta-分析的结果对很多领域的科学研究都具有指导意义。针对具体问题的Meta-分析及其统计方法研究已持续了一个世纪之久。最早是在1904年Karl Pearson针对伤寒疫苗效果进行了合并研究 [1] ；接下来的很多年中对某些零星的问题做了类似的综合 [2] 。但是在过去的四十年中，人们逐渐意识到医学实验和临床操作需要基于整体的相关性和可靠证据，Meta-分析的影响得到极大的扩展 [3] 。仅在2018年，关于健康管理的Meta-分析发表文章将近4000篇。

虽然Meta-分析在很多领域都有很好的应用，但对Meta-分析本身的统计方法研究与其应用研究相比就少了很多。从广泛应用的意义上来讲，Meta-分析是以估计量的标准误差为权重的点估计加权平均。随机效应模型的Meta-分析在各个研究效应量不相等这一假设，具体是以估计量方差及研究间差异方差和的逆作为权重得到的加权平均效应量，并可扩展为合并研究间协变量估计 [4] 。

随着数据科学的快速发展，数据维度和数据量急速增加导致单变量Meta-分析无法满足现实应用的需求。多变量Meta-分析有许多优点：第一，可以在一个模型框架下得到所有效应量的估计量；第二，可以通过所有效应量的估计量的协方差矩阵描述多效应量之间的关系。第三，可以获得具有更好统计特性的参数估计量；第四，我们可以获得区别于单变量Meta-分析的潜在机理原因；第五，多变量分析的方法可以在一定程度上降低纳入文献的偏倚性。

由于多变量Meta-分析与单变量Meta-分析比较起来有很多的优点，因此多变量Meta-分析在各个领域的应用研究是很多的。然而，对于多变量Meta-分析本身统计方法的研究却很少。1988年，为了研究教练在SAT中的作用，Raudenbush SW等人通过广义最小二乘法建立了多效应量合并的模型 [5] ；1993年，van Houwelingen HC等人在文献 [6] 首次给出了基于比值比的二变量Meta-分析模型，并讨论了该模型的异质性检验和敏感性分析；1996年，Kalaian HA等人为了研究教练在SAT语言和数学方面的指导效果，在文献 [7] 建立了基于比值比的各研究内效应量不同情形下的多元Meta-分析模型，并通过REML方法估计了效应量的协方差矩阵；2002年，van Houw -elingen HC等人在广义多元混合线性模型的框架下，通过似然估计方法给出了多元Meta-回归模型，并将此模型扩展到了非正态分布情形 [8] 。2008年，Riley RD等人在文献 [9] 中通过极大似然估计法给出了一些特殊相关系数情形下的二变量Meta-分析的协方差矩阵估计；2008年，Ritz J等人在文献 [10] 中通过极大似然估计和估计方程给出了协方差矩阵已知情形下的多元效应量回归参数，并将此模型应用到肺癌发病率的临床推断中。2010年，Paul M等人在文献 [11] 中通过基于可积嵌套拉普拉斯近似的贝叶斯方法给出了多元Meta-分析的合并效应量估计，这种方法得到的方差估计偏移量更小且稳定。

本文通过极大似然估计法给出已知协方差矩阵的基于均差估值的二变量Meta-分析随机效应模型，并通过极大似然估计法给出效应量估计。通过计算估计效应量的协方差矩阵，给出效应分量的 $\left(1-\alpha \right)100\%$ 置信区间。

2. 问题描述

在协方差矩阵的基于均差估值的二变量Meta-分析随机效应模型中，我们假设个体量 $X_{i1j}^C$ ， $X_{i1j}^T$ ， $X_{i2j}^C$ 和 $X_{i2j}^T$ 是独立的并且是正态分布，其均值分别为 ${\mu }_1^C$ ， ${\mu }_1^T$ ， ${\mu }_2^C$ 和 ${\mu }_2^T$ ，每个研究内的方差分别为 ${\sigma }_{i1}^2$ ， ${\sigma }_{i2}^2$ 。因此 ${\bar{X}}_{i1}^C$ ， ${\bar{X}}_{i1}^T$ ， ${\bar{X}}_{i2}^C$ 和 ${\bar{X}}_{i2}^T$ 是独立的且服从正态分布，其均值分别为 ${\mu }_1^C$ ， ${\mu }_1^T$ ， ${\mu }_2^C$ 和 ${\mu }_2^T$ ，方差分别为 $\frac{{\sigma }_{i1}^2}{n_{i1}^C}$ ， $\frac{{\sigma }_{i1}^2}{n_{i1}^T}$ ， $\frac{{\sigma }_{i2}^2}{n_{i2}^C}$ ， $\frac{{\sigma }_{i2}^2}{n_{i2}^T}$ 。

由上述假设，均差效应量 $D_i$ 的分布可表示为：

$D_i=\left(\begin{array}{c}{D}_{i1}\\ D_{i2}\end{array}\right)\sim N\left(\left(\begin{array}{c}{\mu }_1\\ {\mu }_2\end{array}\right),\left(\begin{array}{cc}{\sigma }_{i1}^2{w}_{i1}+{\tau }^2& {\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \\ {\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma & {\sigma }_{i2}^2{w}_{i2}+{\gamma }^2\end{array}\right)\right)$

其中 ${\mu }_1={\mu }_1^T-{\mu }_1^C$ ， ${\mu }_2={\mu }_2^T-{\mu }_2^C$ 为均值， $w_{i1}=\frac{1}{n_{i1}^T}+\frac{1}{n_{i1}^C}$ ， $w_{i2}=\frac{1}{n_{i2}^T}+\frac{1}{n_{i2}^C}$ 及 ${\rho }_0$ 是非随机的。一般地， ${\mu }_1$ ， ${\mu }_2$ 为未知参数。本文主要内容是通过极大似然估计法来估计 ${\mu }_1$ ， ${\mu }_2$ 。为了方便讨论，记 $S_i=\left(\begin{array}{cc}{\sigma }_{i1}^2{w}_{i1}& {\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}\\ {\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}& {\sigma }_{i2}^2{w}_{i2}\end{array}\right)$ 为研究内均值效应量的协方差矩阵； $\Sigma =\left(\begin{array}{cc}{\tau }^2& {\rho }_1\tau \gamma \\ {\rho }_1\tau \gamma & {\gamma }^2\end{array}\right)$ 为描述研究间关系的协方差矩阵。

3. 均值效应量的极大似然估计

根据之前的假设效应量 $D_i$ 服从正态分布，因此关于 $\mu$ 的似然函数为：

$L\left(\mu ,D_i,S_i\right)={\displaystyle \underset{i=1}{\overset{k}{\prod }}\frac{1}{2\text{π}det{\left(S_i+\Sigma \right)}^{\frac{1}{2}}}}\exp \left\{-\frac{{\left(D_i-\mu \right)}^{\text{T}}{\left(S_i+\Sigma \right)}^{-1}\left(D_i-\mu \right)}{2}\right\}$

其中 $\mu =\left(\begin{array}{c}{\mu }_1\\ {\mu }_2\end{array}\right)$ ， $S_i+\Sigma =\left(\begin{array}{cc}{\sigma }_{i1}^2{w}_{i1}+{\tau }^2& {\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \\ {\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma & {\sigma }_{i2}^2{w}_{i2}+{\gamma }^2\end{array}\right)$ 。其所对应的对数似然函数为：

$\ln L\left(\mu ,D_i,S_i\right)=-k\ln \left(2\text{π}\right)-\frac{1}{2}{\displaystyle \underset{i=1}{\overset{k}{\sum }}\ln }\left[det\left(S_i+\Sigma \right)\right]-\frac{1}{2}{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(D_i-\mu \right)}^{\text{T}}{\left(S_i+\Sigma \right)}^{-1}\left(D_i-\mu \right)}$

定理3.1：设效应量 $D_i$ 服从 $N\left(\left(\begin{array}{c}{\mu }_1\\ {\mu }_2\end{array}\right),\left(\begin{array}{cc}{\sigma }_{i1}^2{w}_{i1}+{\tau }^2& {\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \\ {\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma & {\sigma }_{i2}^2{w}_{i2}+{\gamma }^2\end{array}\right)\right)$ 的正态分布，则 $\mu$ 的极大似然估计量的分量为：

证明：因为关于 $\mu$ 的似然函数为：

$L\left(\mu ,D_i,S_i\right)={\displaystyle \underset{i=1}{\overset{k}{\prod }}\frac{1}{2\text{π}det{\left(S_i+\Sigma \right)}^{\frac{1}{2}}}\exp \left\{-\frac{{\left(D_i-\mu \right)}^{\text{T}}{\left(S_i+\Sigma \right)}^{-1}\left(D_i-\mu \right)}{2}\right\}}$

其所对应的对数似然函数为：

$\begin{array}{c}\ln L\left(\mu ,D_i,S_i\right)=-k\ln \left(2\text{π}\right)-\frac{1}{2}{\displaystyle \underset{i=1}{\overset{k}{\sum }}\ln }\left[det\left(S_i+\Sigma \right)\right]-\frac{1}{2}{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(D_i-\mu \right)}^{\text{T}}{\left(S_i+\Sigma \right)}^{-1}\left(D_i-\mu \right)}\\ =-\frac{1}{2}{\displaystyle \underset{i=1}{\overset{k}{\sum }}\ln \left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}\\ -\frac{1}{2}{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}\\ \times [\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right){\left(D_{i1}-{\mu }_1\right)}^2-2\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)\\ \times \left(D_{i1}-{\mu }_1\right)\left(D_{i2}-{\mu }_2\right)+\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right){\left(D_{i2}-{\mu }_2\right)}^2]-k\ln (2\pi )\end{array}$

上述对数似然函数对 ${\mu }_1$ ， ${\mu }_2$ 的偏导数为：

$\begin{array}{c}\frac{\partial \ln L\left(\mu ,D_i,S_i\right)}{\partial {\mu }_1}={\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}\\ \times \left[\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)\left(D_{i1}-{\mu }_1\right)-\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)\left(D_{i2}-{\mu }_2\right)\right]\end{array}$

$\begin{array}{c}\frac{\partial \ln L\left(\mu ,D_i,S_i\right)}{\partial {\mu }_2}={\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}\\ \times \left[-\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)\left(D_{i1}-{\mu }_1\right)+\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left(D_{i2}-{\mu }_2\right)\right]\end{array}$

令 $\frac{\partial \ln L\left(\mu ,D_i,S_i\right)}{\partial {\mu }_1}=0,\frac{\partial \ln L\left(\mu ,D_i,S_i\right)}{\partial {\mu }_2}=0$ 即得：

$\begin{array}{l}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}\\ \times \left[\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)\left(D_{i1}-{\mu }_1\right)-\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)\left(D_{i2}-{\mu }_2\right)\right]=0\end{array}$

$\begin{array}{l}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}\\ \times \left[-\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)\left(D_{i1}-{\mu }_1\right)+\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left(D_{i2}-{\mu }_2\right)\right]=0\end{array}$

整理可得：

$\begin{array}{l}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}\left({\sigma }_{i2}^2+{\gamma }^2\right){\omega }_{i2}{\mu }_1\\ -{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right){\mu }_2}\\ ={\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}\left({\sigma }_{i2}^2+{\gamma }^2\right){\omega }_{i2}{D}_{i1}\\ -{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)D_{i2}}\end{array}$

因为 $S_i+\Sigma =\left(\begin{array}{cc}{\sigma }_{i1}^2{w}_{i1}+{\tau }^2& {\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \\ {\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma & {\sigma }_{i2}^2{w}_{i2}+{\gamma }^2\end{array}\right)$ 可逆，且 ${\left(S_i+\Sigma \right)}^{-1}$ 为：

${\left[det\left(S_i+\Sigma \right)\right]}^{-1}\left(\begin{array}{cc}{\sigma }_{i2}^2{w}_{i2}+{\gamma }^2& -\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \right)\\ -\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \right)& {\sigma }_{i1}^2{w}_{i1}+{\tau }^2\end{array}\right)$

其中 $det\left(S_i+\Sigma \right)={\left[\left({\sigma }_{i1}^2{w}_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{w}_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}$ ，

因此上述关于 ${\mu }_1$ ， ${\mu }_2$ 的线性方程组的解为：

注：由 ${\left(S_i+\Sigma \right)}^{-1}$ 为： ${\left[det\left(S_i+\Sigma \right)\right]}^{-1}\left(\begin{array}{cc}{\sigma }_{i2}^2{w}_{i2}+{\gamma }^2& -\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \right)\\ -\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \right)& {\sigma }_{i1}^2{w}_{i1}+{\tau }^2\end{array}\right)$ ，

其中 $det\left(S_i+\Sigma \right)={\left[\left({\sigma }_{i1}^2{w}_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{w}_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}$ ，

可以给出效应量 $D_i$ 服从正态分布的合并效应量的矩阵形式为：

${\hat{\mu }}_{ML}={\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}{D}_i$ 。

由上述效应量 $D_i$ 服从正态分布的合并效应量均值估计量分量的具体形式，我们可以得到合并效应量均值估计量的如下性质：

性质2.1：设效应量设效应量 $D_i$ 服从 $N\left(\left(\begin{array}{c}{\mu }_1\\ {\mu }_2\end{array}\right),\left(\begin{array}{cc}{\sigma }_{i1}^2{w}_{i1}+{\tau }^2& {\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \\ {\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma & {\sigma }_{i2}^2{w}_{i2}+{\gamma }^2\end{array}\right)\right)$ 的正态分布，其极大似然估计量为： ${\hat{\mu }}_{ML}={\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}{D}_i}$ ，则 ${\hat{\mu }}_{ML}$ 对于 $\mu$ 是无偏的。

证明：要证明估计量的无偏性，需证等式 $E\left({\hat{\mu }}_{ML}\right)=\mu$ 成立。根据我们给出的效应量 $D_i$ 服从正态分布的合并效应量均值估计量分量的具体形式，我们需要证明 $E\left({\hat{\mu }}_1\right)={\mu }_1$ ， $E\left({\hat{\mu }}_2\right)={\mu }_2$ 。因为 $E\left(D_{i1}\right)={\mu }_1$ ， $E\left(D_{i1}\right)={\mu }_2$ ，所以

$\begin{array}{l}=\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}}[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)\\ -{{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2]}^{-1}[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)\\ {\begin{array}{c}\\ \end{array}\times \left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)]\}}^{-1}\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}}\\ \times {\left[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}\{[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)\\ -\left({\rho }_1\tau \gamma +{\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}\right)\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)]E\left(D_{j1}\right)\\ \begin{array}{c}\\ \end{array}-\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)-\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)\left({\sigma }_{j1}^2{\omega }_{i1}+{\tau }^2\right)\right]E\left(D_{j2}\right)\}\}\end{array}$

$\begin{array}{l}=\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}}[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)\\ -{{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2]}^{-1}[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)\\ {\begin{array}{c}\\ \end{array}\times \left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)]\}}^{-1}\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}}\\ \times {\left[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}\{[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)\\ -\left({\rho }_1\tau \gamma +{\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}\right)\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)]{\mu }_1\\ \begin{array}{c}\\ \end{array}-\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)-\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)\left({\sigma }_{j1}^2{\omega }_{i1}+{\tau }^2\right)\right]{\mu }_2\}\}\end{array}$

$\begin{array}{l}=\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}}[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)\\ -{{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2]}^{-1}[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)\\ {\begin{array}{c}\\ \end{array}\times \left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)]\}}^{-1}\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}}\\ \times {\left[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}\{[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)\\ -\left({\rho }_1\tau \gamma +{\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}\right)\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)]{\mu }_1\\ \begin{array}{c}\\ \end{array}-\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)-\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\right]{\mu }_2\}\}\end{array}$

$\begin{array}{l}=\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}}[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)\\ -{{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2]}^{-1}[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)\\ {\begin{array}{c}\\ \end{array}\times \left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)]\}}^{-1}\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}}\\ \times {\left[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)\\ \begin{array}{c}\\ \end{array}-\left({\rho }_1\tau \gamma +{\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}\right)\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)]\}{\mu }_1\\ ={\mu }_1\end{array}$

$\begin{array}{l}=\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)}}\\ -{{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2]}^{-1}[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)\\ {\begin{array}{c}\\ \end{array}\times \left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)]\}}^{-1}\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}}\\ \times {\left[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}\{[\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)\\ \times \left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)]E\left(D_{j1}\right)\\ \begin{array}{c}\\ \end{array}-\left[\left({\rho }_1\tau \gamma +{\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}\right)\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)-\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\right]E\left(D_{J2}\right)\}\}\end{array}$

$\begin{array}{l}=\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)}}\\ -{{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2]}^{-1}[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)\\ {\begin{array}{c}\\ \end{array}\times \left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)]\}}^{-1}\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}}\\ \times {\left[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}\{[\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)\\ \times \left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)]{\mu }_1\\ \begin{array}{c}\\ \end{array}-\left[\left({\rho }_1\tau \gamma +{\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}\right)\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)-\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\right]{\mu }_2\}\}\end{array}$

$\begin{array}{l}=\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)}}\\ -{{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2]}^{-1}[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)\\ {\begin{array}{c}\\ \end{array}\times \left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)]\}}^{-1}\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}}\\ \times {\left[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}\{[\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)\\ \times \left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)]{\mu }_1\\ \begin{array}{c}\\ \end{array}-\left[\left({\rho }_1\tau \gamma +{\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}\right)\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)-\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\right]{\mu }_2\}\}\end{array}$

$\begin{array}{l}=\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)}}\\ -{{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2]}^{-1}[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)\\ {\begin{array}{c}\\ \end{array}\times \left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)]\}}^{-1}\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}}\\ \times {\left[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}\\ \begin{array}{c}\\ \end{array}\times \left[\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)-\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)\right]{\mu }_2\}\\ ={\mu }_2\end{array}$

综上所述， ${\hat{\mu }}_{ML}$ 对于 $\mu$ 是无偏的。

4. 均值效应量极大似然估计量的协方差矩阵

假设所有的研究都提供所有的均值效应。由多变量统计学的大数定理，合并效应量均值估计量可以近似为一个多元正态分布，其对应的协方差矩阵可由下面的定理给出：

定理4.1：设效应量设效应量 $D_i$ 服从 $N\left(\left(\begin{array}{c}{\mu }_1\\ {\mu }_2\end{array}\right),\left(\begin{array}{cc}{\sigma }_{i1}^2{w}_{i1}+{\tau }^2& {\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \\ {\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma & {\sigma }_{i2}^2{w}_{i2}+{\gamma }^2\end{array}\right)\right)$ 的正态分布，其极大似然估计量为： ${\hat{\mu }}_{ML}={\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}{D}_i$ ，则 ${\hat{\mu }}_{ML}$ 所对应的协方差矩阵 $Var\left({\hat{\mu }}_{ML}\right)$ 为：

$Var{\left({\hat{\mu }}_{ML}\right)}_{\left(1,1\right)}={\left\{det\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]\right\}}^{-1}\left\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}det{\left[\left(S_i+\Sigma \right)\right]}^{-1}}\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\right\}$ ，

$Var{\left({\hat{\mu }}_{ML}\right)}_{\left(1,2\right)}={\left\{det\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]\right\}}^{-1}\left\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}det{\left[\left(S_i+\Sigma \right)\right]}^{-1}}\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)\right\}$ ，

$Var{\left({\hat{\mu }}_{ML}\right)}_{\left(2,1\right)}={\left\{det\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]\right\}}^{-1}\left\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}det{\left[\left(S_i+\Sigma \right)\right]}^{-1}}\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)\right\}$ ，

$Var{\left({\hat{\mu }}_{ML}\right)}_{\left(2,2\right)}={\left\{det\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]\right\}}^{-1}\left\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}det{\left[\left(S_i+\Sigma \right)\right]}^{-1}}\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)\right\}$ 。

其中 $Var{\left({\hat{\mu }}_{ML}\right)}_{\left(i,j\right)}$ 表示矩阵 $Var\left({\hat{\mu }}_{ML}\right)$ 第i行第j列的分量，

$\begin{array}{l}det\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]\\ ={\displaystyle \underset{i=1}{\overset{k}{\sum }}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}}}\\ \times {\left[\left({\sigma }_{j1}^2{\omega }_{j1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)}^2\right]}^{-1}\\ \times \left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{j2}^2{\omega }_{j2}+{\gamma }^2\right)-\left({\rho }_0{\sigma }_{j1}{\sigma }_{j2}\sqrt{{\omega }_{j1}{\omega }_{j2}}+{\rho }_1\tau \gamma \right)\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)\right]\end{array}$

$det\left[{\left(S_i+\Sigma \right)}^{-1}\right]={\displaystyle \underset{j=1}{\overset{k}{\sum }}\left[\left({\sigma }_{i1}^2{\omega }_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{\omega }_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{{\omega }_{i1}{\omega }_{i2}}+{\rho }_1\tau \gamma \right)}^2\right]}$ 。

证明：根据多元随机变量的知识可知：如果随机变量 $D~N\left(\mu ,\hat{\Sigma }\right)$ ，则 $Var\left(MD\right)=M\hat{\Sigma }{M}^{\text{T}}$ ，其中T表示矩阵的转置。由 $S_i=S_i^{\text{T}}$ 且 ${\Sigma }^{\text{T}}=\Sigma$ ，通过计算可得： ${\left(S_i+\Sigma \right)}^{-1}={\left[{\left(S_i+\Sigma \right)}^{-1}\right]}^{\text{T}}$ ， ${\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}={\left\{{\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}\right\}}^{\text{T}}$ 。则

$\begin{array}{c}Var\left({\hat{\mu }}_{ML}\right)=Var\left\{{\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}{D}_i}\right\}\\ =Var\left\{{\displaystyle \underset{i=1}{\overset{k}{\sum }}\left\{{\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}{\left(S_i+\Sigma \right)}^{-1}{D}_i\right\}}\right\}\\ ={\displaystyle \underset{i=1}{\overset{k}{\sum }}Var\left\{{\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}{\left(S_i+\Sigma \right)}^{-1}{D}_i\right\}}\\ ={\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left\{{\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}{\left(S_i+\Sigma \right)}^{-1}\left(S_i+\Sigma \right){\left(S_i+\Sigma \right)}^{-1}\right\}}^{\text{T}}{\left\{{\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}\right\}}^{\text{T}}}\end{array}$

$\begin{array}{l}={\displaystyle \underset{i=1}{\overset{k}{\sum }}\left\{{\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}{\left(S_i+\Sigma \right)}^{-1}\left(S_i+\Sigma \right){\left(S_i+\Sigma \right)}^{-1}{\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}\right\}}\\ ={\displaystyle \underset{i=1}{\overset{k}{\sum }}\left\{{\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}{\left(S_i+\Sigma \right)}^{-1}{\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}\right\}}\\ ={\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]{\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}\\ ={\left[{\displaystyle \underset{i=1}{\overset{k}{\sum }}{\left(S_i+\Sigma \right)}^{-1}}\right]}^{-1}\end{array}$

因为 ${\left(S_i+\Sigma \right)}^{-1}={\left[det\left(S_i+\Sigma \right)\right]}^{-1}\left(\begin{array}{cc}{\sigma }_{i2}^2{w}_{i2}+{\gamma }^2& -\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \right)\\ -\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \right)& {\sigma }_{i1}^2{w}_{i1}+{\tau }^2\end{array}\right)$ ，

其中 $det\left(S_i+\Sigma \right)=\left({\sigma }_{i1}^2{w}_{i1}+{\tau }^2\right)\left({\sigma }_{i2}^2{w}_{i2}+{\gamma }^2\right)-{\left({\rho }_0{\sigma }_{i1}{\sigma }_{i2}\sqrt{w_{i1}{w}_{i2}}+{\rho }_1\tau \gamma \right)}^2$ ，可得 ${\hat{\mu }}_{ML}$ 所对应的协方差矩阵为：

，

，

，

。

其中

。

通过所对应的协方差矩阵可以给出单变量和联合变量的置信区域，其具体形式为：

性质3.1：设效应量设效应量服从的正态分布，其极大似然估计量为：，则合并均值效应量所对应各个分量的置信区间为：

和，其中表示正态分布的分位数， 表示矩阵第i行第j列的分量。

5. 结束语

本文给出已知协方差矩阵的基于均差估值的二变量Meta-分析随机效应模型，并通过极大似然估计法给出效应量估计。通过计算估计效应量的协方差矩阵，给出效应分量的置信区间。这些结论对临床医学，特别是流行病学会有较好的统计学和现实意义。

基金项目

本项目由如下基金支持：国家自然科学基金(61871475, 61471133, 61571444, 61473331)，广东省科技计划(2017B010126001,2017A070712019, 2016A040402043, 2015A070709015, 2015A020209171, 2016B010125004, 2014B040404070,2015A040405014, 2016A070712020)，广东省教育厅科技计划(2017GCZX001, 2016KQNCX075, 2016GCZX001, 2017KTSCX094, 2017KTSCX095, 2017KQNCX098)，广州市科技计划(201707010221)。

NOTES

^*通讯作者。