Cluster Correlations and Complexity in Binary Regression Analysis Using Two-stage Cluster Samples

Abstract

In a two-stage cluster sampling setup for binary data, a sample of clusters such as hospitals is chosen at the first stage from a large number of clusters belonging to a finite population, and in the second stage a random sample of individuals such as nurses is chosen from the selected cluster and the binary responses along with covariates are collected from the selected individuals. Because the hypothetical binary responses from the individuals in a given cluster/hospital under the first stage sample are correlated (as they share a common cluster effect), this correlation plays a complex role in developing the second stage sample based estimating equations for the underlying regression parameters. Moreover, the correlation parameters have to be consistently estimated too. In this paper, unlike the existing studies, we demonstrate how to accommodate (1) the so-called inverse correlation weights arising from a finite population based generalized quasi-likelihood (GQL) estimating function, on top of (2) the sampling weights, to develop a survey sample based doubly weighted (SSDW) estimation approach, for consistent estimation of both regression and correlation parameters. For simplicity, we refer to this GQL cum SSDW approach as the SSDW approach only. The method of moments (MM) cum SSDW approach will be simpler but less efficient, which is not included in the paper. The estimating function involved in the proposed SSDW estimating equation has the form of a sample total, which unbiasedly estimate the corresponding finite population total that arises from the aforementioned generalized quasi-likelihood function for the targeted finite population parameter. The resulting SSDW estimators, thus, become consistent for the respective parameters. This consistency property for the SSDW estimator for both regression and cluster correlation parameters is studied in details.

Appendices

Appendix A: Computation of the Mixed Effects Based Marginal Mean (1.7), and Covariance Matrix (1.8)

Computation of unconditional marginal mean

Use $\gamma ^{*}_{c}=\gamma _{c}/\sigma _{\gamma }$ in (1.2), and re-express the conditional mean as

$$ \begin{array}{@{}rcl@{}} \pi^{*}_{ci}(\beta,\sigma^{2}_{\gamma}, \gamma^{*}_{c})&=& =\frac{\exp(x^{\prime}_{ci}\beta+\sigma_{\gamma}\gamma^{*}_{c})}{[1+\exp(x^{\prime}_{ci}\beta+ \sigma_{\gamma}\gamma^{*}_{c})]}, \end{array} $$

(a.1)

where $\gamma ^{*}_{c} {\stackrel {iid}{\sim }} N(0,1)$. One may them compute the unconditional mean as

$$ \begin{array}{@{}rcl@{}} E_{M}[Y_{ci}]&=&\pi_{ci}(\beta,\sigma^{2}_{\gamma})=\int \pi^{*}_{ci}(\beta,\sigma^{2}_{\gamma}, \gamma^{*}_{c})g_{N}(\gamma^{*}_{c})d\gamma^{*}_{c}, \end{array} $$

(a.2)

where $g_{N}(\gamma ^{*}_{c})$ denotes the standard normal density.

Computation of unconditional covariance matrix

First, because y_ci is a binary response, the formula for its variance is written as

$$ \text{var}_{M}[Y_{ci}|x_{ci}]=\sigma_{c,ii}(\beta,\sigma^{2}_{\gamma}) =\pi_{ci}(\beta,\sigma^{2}_{\gamma})(1-\pi_{ci}(\beta,\sigma^{2}_{\gamma})), $$

(a.3)

where $\pi _{ci}(\beta ,\sigma ^{2}_{\gamma })$ is the unconditional mean, given by (a.2).

Next, because given the cluster effect, the individuals within a cluster must be pair-wise independent, we write

$$ \begin{array}{@{}rcl@{}} &&\text{cov}_{M}[\{Y_{ci},Y_{cj}\}|x_{ci},x_{cj},\gamma_{c}]=0, \end{array} $$

(a.4)

implying that

$$ \begin{array}{@{}rcl@{}} E_{M}[[\{Y_{ci},Y_{cj}\}|\gamma_{c}]&=&E_{M}[Y_{ci}|\gamma_{c}]E_{M}[Y_{cj}|\gamma_{c}] \\ &=&\pi^{*}_{ci}(\beta,\sigma^{2}_{\gamma}, \gamma^{*}_{c})\pi^{*}_{cj}(\beta,\sigma^{2}_{\gamma}, \gamma^{*}_{c}). \end{array} $$

(a.5)

Hence, the unconditional covariance between y_ci and y_cj, is given by

$$ \begin{array}{@{}rcl@{}} \text{cov}_{M}[\{Y_{ci},Y_{cj}\}|x_{ci},x_{cj}]&=&\sigma_{c,ij}(\beta, \sigma^{2}_{\gamma}) \\ &=&\lambda_{c,ij}(\beta,\sigma^{2}_{\gamma})-\pi_{ci}(\beta,\sigma^{2}_{\gamma}) \pi_{cj}(\beta,\sigma^{2}_{\gamma}), \end{array} $$

(a.6)

where

$$ \begin{array}{@{}rcl@{}} \lambda_{c,ij}(\beta,\sigma^{2}_{\gamma}) &=&E_{M}[Y_{ci}Y_{cj}]=E_{\gamma_{c}}E[\{Y_{ci}Y_{cj}\}|\gamma_{c}] \\ &=& E_{\gamma^{*}}[\pi^{*}_{ci}(\beta,\sigma^{2}_{\gamma}, \gamma^{*}_{c})\pi^{*}_{cj}(\beta,\sigma^{2}_{\gamma}, \gamma^{*}_{c})] \\ &=&\int \frac{\exp[(x_{ci}+x_{cj})'\beta+2\sigma_{\gamma} \gamma^{*}_{c}]}{[1+\exp(x^{\prime}_{ci}\beta+\sigma_{\gamma}\gamma^{*}_{c})] [1+\exp(x^{\prime}_{cj}\beta+\sigma_{\gamma}\gamma^{*}_{c})]}g_{N}(\gamma^{*}_{c})d\gamma^{*}_{c} \\ &=&\int \pi^{*}_{ci}(\beta,\sigma^{2}_{\gamma},\gamma^{*}_{c})\pi^{*}_{cj}(\beta,\sigma^{2}_{\gamma}, \gamma^{*}_{c}) g_{N}(\gamma^{*}_{c})d\gamma^{*}_{c}. \end{array} $$

Appendix B: Computation of the Covariance Matrix $V^{*}_{n}(\beta ,\sigma ^{2}_{\gamma })$ in (3.33)

By applying the indicator variables from (3.21)-(3.22), we express the formula of this matrix from (3.33), as

$$ \begin{array}{@{}rcl@{}} &&V^{*}_{n}(\beta,\sigma^{2}_{\gamma}) =\text{cov}_{p_{1}}\left[\frac{K}{k}{\sum}^{K}_{c=1}\frac{N_{c}}{n_{c}} \delta_{1,c}E_{p_{2c}}\left\{{\sum}^{N_{c}}_{i=1} \delta_{2,i|c}z_{ci}|p_{1}\right\}\right] \\ &+&E_{p_{1}}\left[(K^{2}/k^{2}){\sum}^{K}_{c=1}\frac{{N^{2}_{c}}}{{n^{2}_{c}}}\delta_{1,c} \text{cov}_{p_{2c}}\left\{{\sum}^{N_{c}}_{i=1}\delta_{2,i|c}z_{ci}|p_{1}\right\} \right]. \end{array} $$

(b.1)

Computational formula for the first term in (b1)

Notice that for hypothetically known z_ci under the FP, the expectation with respect to the sampling design p_2c, in the first term, may be computed as

$$ \begin{array}{@{}rcl@{}} E_{p_{2c}}\left\{{\sum}^{N_{c}}_{i=1} \delta_{2,i|c}z_{ci}|p_{1}\right\} &=&{\sum}^{N_{c}}_{i=1} E_{p_{2c}}[\delta_{2,i|c}]z_{ci}|p_{1} \\ &=&\frac{n_{c}}{N_{c}}{\sum}^{N_{c}}_{i=1}z_{ci}=\frac{n_{c}}{N_{c}}Z_{c}, \text{(say).} \end{array} $$

(b.2)

Because the first stage sample of clusters is chosen based on the SRS without replacement, by substituting (b.2) in the first term in (b.159), we can compute the covariance over the sampling design p₁, as

$$ \begin{array}{@{}rcl@{}} &&\text{cov}_{p_{1}}\!\left[\frac{K}{k}{\sum}^{K}_{c=1} \delta_{1,c}Z_{c}\right] = \frac{K^{2}}{k^{2}}\!\left[{\sum}^{K}_{c=1}Z_{c}Z^{\prime}_{c}\text{var}[\delta_{1,c}] + {\sum}^{K}_{c \neq d}Z_{c}Z^{\prime}_{d} \text{cov}[\delta_{1,c},\delta_{1,d}]\right]\!. \end{array} $$

(b.3)

Now because δ_1,c is the indicator variable as defined by (3.21) under the sampling design p₁ (SRS without replacement), we have

$$ \begin{array}{@{}rcl@{}} \text{var}(\delta_{1,c})&=&\frac{k}{K}(1-\frac{k}{K})\\ \text{cov}(\delta_{1,c},\delta_{1,d})&=&E(\delta_{1,c}\delta_{1,d}) -E(\delta_{1,c})E(\delta_{1,d}) \\ &=&\frac{k(k-1)}{K(K-1)}-\left( \frac{k}{K}\right)^{2} =-\frac{k}{K(K-1)}(1-\frac{k}{K}). \end{array} $$

(b.4)

Substitute (b.4) in (b.5), and write

$$ \begin{array}{@{}rcl@{}} &&\text{cov}_{p_{1}}\left[\frac{K}{k}{\sum}^{K}_{c=1} \delta_{1,c}Z_{c}\right]=\frac{K^{2}}{k^{2}}\frac{k}{K}(1-\frac{k}{K})\left[ {\sum}^{K}_{c=1}Z_{c}Z^{\prime}_{c}-\frac{1}{K-1}{\sum}^{K}_{c \neq d}Z_{c}Z^{\prime}_{d}\right] \\ &=&\frac{K}{K-1}\frac{K}{k}(1-\frac{k}{K})\left[\frac{(K-1)}{K} {\sum}^{K}_{c=1}Z_{c}Z^{\prime}_{c}-\frac{1}{K}{\sum}^{K}_{c \neq d}Z_{c}Z^{\prime}_{d}\right] \\ &=&\frac{1}{k}\frac{K^{2}}{K-1}(1-\frac{k}{K})\left[{\sum}^{K}_{c=1}Z_{c}Z^{\prime}_{c} -\frac{1}{K}\left\{{\sum}^{K}_{c=1}Z_{c}Z^{\prime}_{c}+{\sum}^{K}_{c \neq d}Z_{c}Z^{\prime}_{d}\right\}\right] \\ &=&\frac{1}{k}\frac{K^{2}}{K-1}(1-\frac{k}{K})\left[{\sum}^{K}_{c=1}Z_{c}Z^{\prime}_{c} -\frac{1}{K}\left\{{\sum}^{K}_{c=1}Z_{c} {\sum}^{K}_{c=1}Z^{\prime}_{c}\right\}\right] \\ &=&\frac{1}{k}\frac{K^{2}}{K-1}(1-\frac{k}{K})\left[{\sum}^{K}_{c=1}(Z_{c}-\bar{Z}) (Z_{c}-\bar{Z})'\right] \\ &=&K^{2}\left( \frac{K-k}{K}\right)\frac{1}{k}V_{1\cdot}(\beta,\sigma^{2}_{\gamma}), \end{array} $$

(b.5)

where, we have used

$$\bar{Z}=\frac{1}{K}{\sum}^{K}_{c=1}Z_{c}, \text{and} V_{1\cdot}(\beta,\sigma^{2}_{\gamma})=\frac{1}{K-1} {\sum}^{K}_{c=1}(Z_{c}-\bar{Z})(Z_{c}-\bar{Z})'.$$

Computational formula for the second term in (b.1)

First we obtain the covariance matrix over the second stage sampling design p_2c, as

$$ \begin{array}{@{}rcl@{}} &&\text{cov}_{p_{2c}}\left\{{\sum}^{N_{c}}_{i=1}\delta_{2,i|c}z_{ci}|p_{1}\right\} \\ &=&{\sum}^{N_{c}}_{i=1}\text{var}[\delta_{2,i|c}]z_{ci}z^{\prime}_{ci} + {\sum}^{N_{c}}_{i \neq j}\text{cov}[\delta_{2,i|c},\delta_{2,j|c}]z_{ci}z^{\prime}_{cj} \\ &=&\frac{n_{c}}{N_{c}}(1-\frac{n_{c}}{N_{c}}){\sum}^{N_{c}}_{i=1}z_{ci}z^{\prime}_{ci} -\frac{n_{c}}{N_{c}(N_{c}-1)}(1-\frac{n_{c}}{N_{c}}) {\sum}^{N_{c}}_{i \neq j}z_{ci}z^{\prime}_{cj}, \end{array} $$

(b.6)

by using the similar formula as in (b.4). Furthermore, by similar algebras as in (b.5), (b.6) reduces to

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!&&\text{cov}_{p_{2c}}\left\{{\sum}^{N_{c}}_{i=1}\delta_{2,i|c}z_{ci}|p_{1}\right\} \\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!&=&\!\!\!\frac{n_{c}}{N_{c}}(1-\frac{n_{c}}{N_{c}})\frac{N_{c}}{N_{c}-1} \left[\frac{(N_{c}-1)}{N_{c}}{\sum}^{N_{c}}_{i=1}z_{ci}z^{\prime}_{ci}-\frac{1}{N_{c}} {\sum}^{N_{c}}_{i \neq j}z_{ci}z^{\prime}_{cj}\right] \\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!&=&\!\!\!(1-\frac{n_{c}}{N_{c}})\frac{n_{c}}{N_{c}-1}{\sum}^{N_{c}}_{i=1}(z_{ci} - \bar{Z}_{c}) (z_{ci} - \bar{Z}_{c})' = n_{c}(1 - \frac{n_{c}}{N_{c}})V^{*}_{c}(\beta,\sigma^{2}_{\gamma}), \text{(say),} \end{array} $$

(b.7)

where we have used $\bar {Z}_{c}=\frac {Z_{c}}{N_{c}}=\frac {1}{N_{c}}{\sum }^{N_{c}}_{i=1}z_{ci}$. After putting (b.7) in the second term in (b.1), we take the desired expectation over the first stage sampling design p₁, which yields the formula for the second term, as

$$ \begin{array}{@{}rcl@{}} &&E_{p_{1}}\left[(K^{2}/k^{2}){\sum}^{K}_{c=1}\frac{{N^{2}_{c}}}{{n^{2}_{c}}}\delta_{1,c} \text{cov}_{p_{2c}}\left\{{\sum}^{N_{c}}_{i=1}\delta_{2,i|c}z_{ci}|p_{1}\right\} \right] \\ &=&\left[(K^{2}/k^{2}){\sum}^{K}_{c=1}\frac{{N^{2}_{c}}}{{n^{2}_{c}}}E_{p_{1}}[\delta_{1,c}] n_{c}(1-\frac{n_{c}}{N_{c}})V^{*}_{c}(\beta,\sigma^{2}_{\gamma}) \right] \\ &=&\frac{K}{k}{\sum}^{K}_{c=1}{N^{2}_{c}}\frac{N_{c}-n_{c}}{N_{c}}\frac{1}{n_{c}} V^{*}_{c}(\beta,\sigma^{2}_{\gamma}). \end{array} $$

(b.8)

Finally by combining (b.5) and (b.8), we obtain the covariance matrix $V^{*}_{n}(\beta ,\sigma ^{2}_{\gamma }),$ in (b.1), which is reported in (3.34), under Section 3.2.2.