Estimation of intra-cluster correlation coefficient via the failure of Bartlett’s second identity

Abstract

A new means of estimating the correlation coefficient for cluster binary data in the regression settings is introduced. The creation of this method is founded upon the violation of Bartlett’s second identity when adopting the binomial distributions to model binary data that are correlated. The new methodology applies to any sensible link functions that connect the success probability and covariates. One can easily implement the procedure by using any statistical software providing the naïve and the sandwich covariance matrices for regression parameter estimates. Simulations and real data analyses are used to demonstrate the efficacy of our new procedure.

Appendix

As remarked by Presnell and Boos (2004), the asymptotic variance of $trace(\widehat{I}^{-1}\widehat{V})$ is equal to the bottom right element of $C^{-1}D(C^{-1})^{t}/m$, where C and D are $6\times 6$ matrices, with

$$\begin{aligned} C=\left({{\begin{array}{ccc} {-I}&\quad 0&\quad 0 \\ {E(\partial vech({l}^{\prime \prime })/\partial \beta ^{t})}&\quad {-H_q (I\otimes I)G_q }&\quad 0 \\ {2E({l}^{\prime t}I^{-1}{l}^{\prime \prime })}&\quad {(vech[2E({l}^{\prime }{l}^{\prime t})-diag\{E({l}^{\prime }{l}^{\prime t})\}])^{t}}&\quad {-1} \\ \end{array} }} \right) \end{aligned}$$

and D having elements denoted by $D_{ij} ,i,j=1,2,3$ introduced later.

Consider a simple regression model with $\eta _i =\beta _0 + x_i \beta _1$. The log likelihood function is

$$\begin{aligned} l(\beta )=\sum _{i=1}^m {l_i (\beta )=} \sum _{i=1}^m {\left[{y_i \log \left({\frac{p_i}{1-p_i}}\right)+n_i \log \left({1-p_i} \right)}\right]} +\sum _{i=1}^m {\log } \left({{\begin{array}{c} {n_i } \\ {y_i } \\ \end{array}}} \right). \end{aligned}$$

For simplicity, we will use $l_i$ to denote $l_i(\beta )$.

We first let ${p}^{\prime }_i$ and $p_i^{\prime \prime }$ denote the first and second derivatives of$p_i$with respect to $\beta$. Then $l_i^{\prime }=\partial l_i /\partial \beta =w_i [1,x_i]^{t}$, where $w_i =(y_i -n_i p_i){p}^{\prime }_i /\left({p_i (1-p_i)} \right)$ and

$$\begin{aligned} l_i^{\prime \prime } =\partial ^{2}l_i / \partial \beta \partial \beta ^{t}= w_i^{\prime } \left[{{\begin{array}{cc} 1&{x_i} \\ {x_i}&{x_i^2} \\ \end{array}}} \right], \end{aligned}$$

where

$$\begin{aligned} w_i^{\prime } = \frac{{p}^{\prime \prime }_i (y_i -n_i p_i)-n_i {p}_i ^{\prime 2}}{p_i (1-p_i)}-\frac{{p}_i^{\prime 2}(y_i -n_i p_i )(1 -2p_i)}{[p_i (1-p_i)]^{2}}, \end{aligned}$$

which is the derivative of $w_i$ with respect to $p_i$.

Note that if the logistic function is employed, then $p_i =e^{\eta _i}/(1+e^{\eta _i})$ giving ${p}^{\prime }_i = p_i\left({1-p_i} \right)$ and ${p}^{\prime \prime }_i = p_i\left({1-p_i}\right)(1-2p_i)$.

Let $I^{-1}$ denote the inverse of the matrix $I=-E(l^{\prime \prime })=\frac{1}{m}\sum _{i=1}^m {\frac{n_i {p}^{\prime 2}_i}{p_i (1-p_i)} \left[{{\begin{array}{cc} 1&{x_i} \\ {x_i}&{x_i^2} \\ \end{array}}} \right]}$. One can show that

$$\begin{aligned} E\left[{\frac{\partial }{\partial \beta ^{t}}vech(l^{\prime \prime })} \right]&= \frac{1}{m}\sum _{i=1}^m {\left({\frac{2n_i {p}_i^{\prime 3}(1-2p_i)}{[p_i (1-p_i)]^{2}}-\frac{3n_i {p}^{\prime }_i {p}^{\prime \prime }_i}{p_i (1-p_i)}} \right) \left[{{\begin{array}{cc} 1&{x_i} \\ {x_i}&{x_i^2} \\ {x_i^2}&{x_i^3} \\ \end{array}}} \right]},\\ -H_q (I\otimes I)G_q&= \left[{{\begin{array}{cccc} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&0&1 \\ \end{array}}}\right] \left[{{\begin{array}{cc} {{\begin{array}{cc} {I_{11}^{2}}&{I_{11} I_{12}} \\ {I_{11} I_{12}}&{I_{22}^{2}} \\ \end{array}}}&{{\begin{array}{cc} {I_{11} I_{12}}&{I_{12}^{2}} \\ {I_{12}^{2}}&{I_{12} I_{22}} \\ \end{array}}} \\ {{\begin{array}{cc} {I_{11} I_{12}}&{I_{12} ^{2}} \\ {I_{12}^{2}}&{I_{12} I_{22}} \\ \end{array}}}&{{\begin{array}{cc} {I_{11} I_{22}}&{I_{12} I_{22}} \\ {I_{12} I_{22}}&{I_{22}^{2}} \\ \end{array}}} \\ \end{array}}}\right] \left[{{\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&1&0 \\ 0&0&1 \\ \end{array}}} \right], \end{aligned}$$

where $I_{ij}$ denotes the $i\text{-}j\text{ th}$ element of $I$, and

$$\begin{aligned} E(l^{\prime t}I^{-1}l^{\prime \prime })&= \frac{1}{m}\sum _{i=1}^m \frac{Var(Y_i)\left[{{p}^{\prime }_i {p}^{\prime \prime }_i p_i (1-p_i)-{p}_i ^{\prime 3}(1-2p_i)} \right]}{[p_i (1-p_i)]^{3}}\\&\quad \left\{ {\left[{1,x_i}\right]I^{-1} \left[{{\begin{array}{cc} 1&{x_i} \\ {x_i}&{x_i^2} \\ \end{array}}}\right]}\right\} . \end{aligned}$$

The sub-matrices $D_{ij},i,j=1,2,3$ constituting $D$ are, respectively,

$$\begin{aligned} D_{11}&= E(l^{\prime }l^{\prime t})=\frac{1}{m}\sum _{i=1}^m {\frac{Var(Y_i){p}_i^{\prime 2}}{[p_i (1-p_i)]^{2}} \left[{{\begin{array}{cc} 1&{x_i} \\ {x_i}&{x_i^2} \\ \end{array}}} \right]},\\ D_{12}&= D_{21}^{t}=E\{{l}^{\prime }(vech{l}^{\prime \prime })^{t}\} \text{=}\frac{\text{1}}{\text{ m}}\sum _{i=1}^m \frac{Var(Y_i)[{p}^{\prime }_i {p}^{\prime \prime }_i p_i(1-p_i)-{p}_i^{\prime 3}(1-2p_i)]}{[p_i(1-p_i)]^{3}}\\&\qquad \qquad \qquad \qquad \qquad \qquad \left[{{\begin{array}{ccc} 1&{x_i}&{x_i^2} \\ {x_i}&{x_i^2}&{x_i^3} \\ \end{array}}} \right],\\ D_{13}&= D_{31}^{t} = E(l^{\prime }l^{\prime t}I^{-1}l^{\prime }) = \frac{1}{m}\sum _{i=1}^{m} \frac{E(Y_i -n_i p_i)^{3}{p}_i^{\prime 3}}{[p_i (1-p_i)]^{3}} \\&\qquad \qquad \qquad \qquad \qquad \qquad \times \left\{ {\left[{{\begin{array}{cc} 1&x_i \\ x_i&x_i^2 \\ \end{array}}}\right] I^{-1} \left[{{\begin{array}{c} 1 \\ x_i \\ \end{array}}}\right]}\right\} ,\\ D_{22}&= E\{(vechl^{\prime \prime })(vechl^{\prime \prime })^{t}\}-(vech(I)(vech(I))^{t} \\&= \frac{1}{m}\sum _{i=1}^m {E({w}_i^{\prime 2}) \left[{{\begin{array}{ccc} 1&{x_i}&{x_i^2} \\ {x_i}&{x_i^2}&{x_i^3} \\ {x_i^2}&{x_i^3}&{x_i^4} \\ \end{array}}}\right]- \left[{{\begin{array}{ccc} {I_{11}^{2}}&{I_{11} I_{12}}&{I_{11} I_{22}} \\ {I_{11} I_{12}}&{I_{12}^{2}}&{I_{12} I_{22}} \\ {I_{11} I_{22}}&{I_{12} I_{22}}&{I_{22}^{2}} \\ \end{array}}} \right]}, \end{aligned}$$

where $E({w}_i^{\prime 2})=Var(Y_i)\left[{\frac{{p}^{\prime \prime 2} +n_i^{2}{p}_i^{\prime 4}}{[p_i (1-p_i)]^{2}}-\frac{2{p}^{\prime \prime }_i ({p}_i ^{\prime 2}-2p_i {p}_i^{\prime 2})}{[p_i (1-p_i)]^{3}}+\frac{({p}_i ^{\prime 2}-2p_i {p}_i^{\prime 2})^{2}}{[p_i (1-p_i)]^{4}}} \right]$,

$$\begin{aligned} D_{23}&= D_{32} ^{t}=E\{l^{\prime t}I^{-1}l^{\prime }(vech(l^{\prime \prime }))\}+trace(I^{-1}V)(vech(I))\\&= \frac{1}{m}\sum _{i=1}^m {E(w_i^2 {w}^{\prime }_i)\left\{ {\left[{1,x_i} \right]I^{-1} \left[{{\begin{array}{c} 1 \\ x_i \\ \end{array}}}\right] \left[{{\begin{array}{c} 1 \\ x_i\\ x_i^2\\ \end{array}}}\right]}\right\} +trace(I^{-1}V) \left[{{\begin{array}{c} {I_{11}} \\ {I_{12}} \\ {I_{22}} \\ \end{array}}} \right]}, \\ \end{aligned}$$

where $E(w_i^2 {w}^{\prime }_i)=\frac{E(Y_i -n_i p_i)^{3}{p}_i ^{\prime 2}{p}^{\prime \prime }_i}{[p_i (1-p_i)]^{3}}-\frac{Var(Y_i)n_i {p}_i^{\prime 4}}{[p_i (1-p_i)]^{3}}-\frac{E(Y_i -n_i p_i)^{3}{p}_i ^{\prime 4}(1-2p_i)}{[p_i (1-p_i)]^{4}}$, and

$$\begin{aligned} D_{33}&= E(l^{\prime t}I^{-1}l^{\prime })^{2}-[trace(I^{-1}V)]^{2} =\frac{1}{m}\sum _{i=1}^m \frac{E(Y_i -n_i p_i )^{4}{p}^{\prime 4}_i}{[p_i (1-p_i)]^{4}}\\&\left\{ {\left[{1,x_i} \right]I^{-1} \left[{{\begin{array}{c} 1 \\ {x_i} \\ \end{array}}}\right]}\right\} ^{2} -[trace(I^{-1}V)]^{2}. \end{aligned}$$

Note that $D_{33}$ is a scalar.