Multinomial Logistic Mixed Models for Clustered Categorical Data in a Complex Survey Sampling Setup

Abstract

In a finite/survey population setup, where categorical/multinomial responses are collected from individuals belonging to a cluster, in a recent study Skinner (International Statistical Review, 87, S64-S78 2019) has modeled the means of the clustered categorical responses as a function of regression parameters, and suggested a ‘working’ correlations based GEE (generalized estimating equations) approach for the estimation of the regression parameters. However, this mean model involving only regression parameters is not justified for clustered multinomial responses because of the fact that these responses share a common cluster effect which compels the clustered correlation parameter to enter into the mean function on top of the regression parameters. Consequently, the so-called GEE approach, which requires the means to be free of correlations, is not applicable for regression analysis in the clustered multinomial setup. As a remedy, in this paper we consider a multinomial mixed model which accommodates the clustered correlation parameter in the mean functions. For inferences in the present finite population setup, as the GQL (generalized quasi-likelihood) approach is known to produce consistent and more efficient estimate than the MM (method of moments) approach in an infinite population setup, we estimate the regression parameters of primary interest by using the first order response based survey weighted GQL (WGQL) approach. For the estimation of the random effects variance (also known as clustered correlation) parameter, as it is of secondary interest, we use the second order response based survey weighted MM (WMM) approach, which is simpler than the corresponding WGQL estimation approach. The estimation steps are presented clearly for the benefit to the practitioners. Also because, in practice, survey practitioners such as statistical agencies frequently deal with a large health or socio-economic data set at national or state levels, for example, we make sure for their benefit that our proposed WGQL and WMM estimators are consistent. Thus, the asymptotic properties such as asymptotic unbiasedness and consistency for both regression and clustered correlation parameters are studied in details. The asymptotic normality property, for the benefit of constructing confidence interval for the main regression parameters, is also studied.

Appendices

Appendix A: Proof for the Design Unbiased Property of the Estimating Functions (4.1) of β, and Eq. 4.4 of $\sigma ^{2}_{\gamma }$

Appendix A1: Proof for the Estimating Function (4.1) of β

Here we check the design (D) unbiasedness of $\hat {\tau }_{\beta |\sigma ^{2}_{\gamma }}(y;\cdot )$ (4.1) for $\tau _{\beta |\sigma ^{2}_{\gamma }}(y;\cdot )$ in Eq. 3.9, by taking the design expectation (E_D) as

$$ \begin{array}{@{}rcl@{}} &&{}E_{D}\left[\hat{\tau}_{\beta|\sigma^{2}_{\gamma}}(y;\cdot)\right] =E_{D}\left[{\sum}^{k}_{c=1}{\sum}^{n_{c}}_{i=1}\tilde{w}_{cs^{*}}z_{cis^{*}}(y|\beta,\sigma^{2}_{\gamma})\right] \\ &&{}=E_{D}\left[{\sum}^{k}_{c=1}{\sum}^{n_{c}}_{i=1}\frac{K}{k}\frac{N_{c}}{n_{c}}z_{cis^{*}}(y|\beta, \sigma^{2}_{\gamma})\right] = E_{C}\frac{K}{k}{\sum}^{k}_{c=1}N_{c}E_{|C}\left[\frac{1}{n_{c}} {\sum}^{n_{c}}_{i=1}z_{cis^{*}}(y|\beta,\sigma^{2}_{\gamma})\right], \end{array} $$

(a.1)

where E_C denotes the expectation over the clusters (C) and E_|C is the expectation over the individuals within a given cluster. Hence,

$$ \begin{array}{@{}rcl@{}} &&E_{D}\left[\hat{\tau}_{\beta|\sigma^{2}_{\gamma}}(y;\cdot)\right] =E_{D}\left[{\sum}^{k}_{c=1}{\sum}^{n_{c}}_{i=1}\tilde{w}_{cs^{*}}z_{cis^{*}}(y|\beta,\sigma^{2}_{\gamma})\right]\\ &=&E_{C}\frac{K}{k}{\sum}^{k}_{c=1}N_{c}\left[\frac{1}{N_{c}}{\sum}^{N_{c}}_{i=1}z_{ci}(y|\beta,\sigma^{2}_{\gamma})\right] =KE_{C}\frac{1}{k}{\sum}^{k}_{c=1}\left\{{\sum}^{N_{c}}_{i=1}z_{ci}(y|\beta,\sigma^{2}_{\gamma})\right\}\\ &=&K\frac{1}{K}{\sum}^{K}_{c=1}\left\{{\sum}^{N_{c}}_{i=1}z_{ci}(y|\beta,\sigma^{2}_{\gamma})\right\} ={\sum}^{K}_{c=1}{\sum}^{N_{c}}_{i=1}z_{ci}(y|\beta,\sigma^{2}_{\gamma}), \end{array} $$

(a.2)

which is the hypothetical estimating function (HEF) in the left hand side of the HEE in Eq. 3.9.

Appendix A2: Proof for the Estimating Function (4.4) of $\sigma ^{2}_{\gamma }$

Notice that by similar calculations as in (a.2) for β estimation, one can show that the first term in Eq. 4.4 is an unbiased estimating function for the first term in Eq. 3.24. That is,

$$ \begin{array}{@{}rcl@{}} E_{D}\left[{\sum}^{k}_{c=1}{\sum}^{n_{c}}_{i=1}\tilde{w}_{cs^{*}}g_{1,cis^{*}} (p_{cis^{*}}|\beta,\sigma^{2}_{\gamma})\right]= {\sum}^{K}_{c=1}{\sum}^{N_{c}}_{i=1}g_{1,ci}(p_{ci}|\beta,\sigma^{2}_{\gamma}). \end{array} $$

(a.3)

However, the design expectation (E_D) of the second term in Eq. 4.4 is given by

$$ \begin{array}{@{}rcl@{}} &&E_{D}\left[{\sum}^{k}_{c=1}{\sum}^{n_{c}(n_{c}-1)/2}_{h=1}\tilde{w}_{cs^{*}} g_{2,chs^{*}}(q_{c,hs^{*}}(y)|\beta,\sigma^{2}_{\gamma})\right] \\ &=&E_{D}\left[{\sum}^{k}_{c=1}{\sum}^{n_{c}(n_{c}-1)/2}_{h=1}\frac{K}{k}\frac{N_{c}}{n_{c}} g_{2,chs^{*}}(q_{c,hs^{*}}(y)|\beta,\sigma^{2}_{\gamma})\right]\\ &=&E_{C}\frac{K}{k}{\sum}^{k}_{c=1}[N_{c}(n_{c}-1)/2]E_{|C}\\ &&\times \left[\frac{1}{n_{c}(n_{c}-1)/2}{\sum}^{n_{c}(n_{c}-1)/2}_{h=1} g_{2,chs^{*}}(q_{c,hs^{*}}(y)|\beta,\sigma^{2}_{\gamma}) \right]\\ &=&E_{C}\frac{K}{k}{\sum}^{k}_{c=1}[N_{c}(n_{c}-1)/2]\\ &&\times\left[\frac{1}{n_{c}(n_{c}-1)/2} \frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}{\sum}^{N_{c}(N_{c}-1)/2}_{h=1} g_{2,ch}(q_{c,h}(y)|\beta,\sigma^{2}_{\gamma})\right]\\ &=&E_{C}\frac{K}{k}{\sum}^{k}_{c=1}\frac{n_{c}-1}{N_{c}-1} \left[{\sum}^{N_{c}(N_{c}-1)/2}_{h=1} g_{2,ch}(q_{c,h}(y)|\beta,\sigma^{2}_{\gamma})\right]\\ &=&{\sum}^{K}_{c=1}\frac{n_{c}-1}{N_{c}-1} \left[{\sum}^{N_{c}(N_{c}-1)/2}_{h=1} g_{2,ch}(q_{c,h}(y)|\beta,\sigma^{2}_{\gamma})\right]\\ &&\neq {\sum}^{K}_{c=1}{\sum}^{N_{c}(N_{c}-1)/2}_{h=1} g_{2,ch}(q_{c,h}(y)|\beta,\sigma^{2}_{\gamma})). \end{array} $$

(a.4)

Hence, when eq. a.4 is combined with Eq. a.3, it follows that the naive estimating function $\hat {\tau }_{NMM,\sigma ^{2}_{\gamma }|\beta }(y;\cdot )$ in (4.4) is not an unbiased estimating function for the HEF, in the left hand side of (3.24). However, by a bias correction adjustment, one can construct an unbiased function as done in (4.5).

Appendix B: Asymptotic Unbiasedness and Variance of the Survey Weighted GQL Estimator $\hat {\beta }_{WGQL}$

Asymptotic Unbiasedness:

By Eq. 5.2 we write

$$ \begin{array}{@{}rcl@{}} &&E\left[\hat{\beta}_{WGQL}-\beta\right] \simeq -\left[E_{D}{\sum}^{k}_{c=1}{\sum}^{n_{c}}_{i=1} \frac{K}{k}\frac{N_{c}}{n_{c}} \frac{\partial {z}_{cis^{*}}(\beta)}{\partial \beta'} \right]^{-1} \\ &\times & E_{Y}E_{D}\left[{\sum}^{k}_{c=1}{\sum}^{n_{c}}_{i=1} \frac{K}{k}\frac{N_{c}}{n_{c}} {z}_{cis^{*}}\right]+o_{p}(1/\sqrt{n}). \end{array} $$

(b.1)

To compute the expectations involved in Eq. b.1, first we write from Eq. b.2 that

$$ \begin{array}{@{}rcl@{}} &&E_{D}\left[{\sum}^{k}_{c=1}{\sum}^{n_{c}}_{i=1} \frac{K}{k}\frac{N_{c}}{n_{c}} {z}_{cis^{*}}\right]={\sum}^{K}_{c=1}{\sum}^{N_{c}}_{i=1} z_{ci}, \end{array} $$

(b.2)

implying that

$$ \begin{array}{@{}rcl@{}} &&E_{D}\left[{\sum}^{k}_{c=1}{\sum}^{n_{c}}_{i=1}\frac{K}{k}\frac{N_{c}}{n_{c}}\frac{\partial z_{cis^{*}}}{\partial \beta'}\right] {\rightarrow}_{p} E_{D}\left[{\sum}^{k}_{c=1}{\sum}^{n_{c}}_{i=1}\frac{K}{k}\frac{N_{c}}{n_{c}}\frac{\partial z_{cis^{*}}}{\partial \beta'}\right] \\ &=&{\sum}^{K}_{c=1}{\sum}^{N_{c}}_{i=1}\frac{\partial z_{ci}} {\partial \beta'}. \end{array} $$

(b.3)

Next because

$${\sum}^{N_{c}}_{i=1}z_{ci}(\cdot)={\sum}^{N_{c}}_{i=1}a'_{ci}(y_{ci}-\pi_{ci}(\cdot)) =\frac{\partial \pi'_{c}(\beta,\sigma^{2}_{\gamma})}{\partial \beta}{\Sigma}^{-1}_{c} (\beta,{\sigma^{2}_{\gamma}})(y_{c}-\pi_{c}(\beta,\sigma^{2}_{\gamma}))$$

by Eq. 3.10, it follows from Eqs. b.2 and b.3 that

$$ \begin{array}{@{}rcl@{}} E_{Y}E_{D}\left[{\sum}^{k}_{c=1}{\sum}^{n_{c}}_{i=1} \frac{K}{k}\frac{N_{c}}{n_{c}} {z}_{cis^{*}}\right]&=&0, \text{and} \end{array} $$

(b.4)

$$ \begin{array}{@{}rcl@{}} E_{D}\left[{\sum}^{k}_{c=1}{\sum}^{n_{c}}_{i=1}\frac{K}{k}\frac{N_{c}}{n_{c}}\frac{\partial z_{cis^{*}}}{\partial \beta'}\right]&=&{\sum}^{K}_{c=1} \frac{\partial \pi'_{c}(\beta,\sigma^{2}_{\gamma})}{\partial \beta}{\Sigma}^{-1}_{c} (\beta,{\sigma^{2}_{\gamma}})\frac{\partial \pi_{c}(\beta,\sigma^{2}_{\gamma})}{\partial \beta'} \\ &=&\tilde{G}(\beta,\sigma^{2}_{\gamma}), \text{as in (\ref{Equ63})}, \end{array} $$

(b.5)

respectively.

Consequently, by applying (b.4) and (b.5), it follows from Eq. b.1 that

$$ \begin{array}{@{}rcl@{}} &&E\left[\hat{\beta}_{WGQL}-\beta\right] \simeq o_{p}(1/\sqrt{n})=0, \end{array} $$

(b.6)

as $n \rightarrow N \rightarrow \infty ,$ showing that $\hat {\beta }_{WGQL}$ is asymptotically unbiased for β.

Computation of

$\text {cov}\left ({\sum }^{k}_{c=1}{\sum }^{n_{c}}_{i=1}\tilde {w}_{cs^{*}}z_{cis^{*}} \right )$ Notice that as the sample of size $n={\sum }^{k}_{c=1}n_{c}$ is selected from the FP of size $N={\sum }^{K}_{c=1}N_{c}$ by using the TSCS, it then follows that

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

(b.7)

Now computing the within and between clustered expectations and covariances, one simplifies (b.7) and obtains $V^{*}(\beta ,\sigma ^{2}_{\gamma }) = \text {cov}\left ({\sum }^{k}_{c=1}{\sum }^{n_{c}}_{i=1}\tilde {w}_{cs^{*}}z_{cis^{*}} \right )$ as in Eq. 5.8 under Section 5.1.

Asymptotic Variance of the Regression Estimator $\hat {\beta }_{WGQL}:$

Because $\hat {\beta }_{WGQL},$ by Eq. b.6, is asymptotically unbiased for β, by applying (b.5) and (b.7), it follows from (5.2) (see also Eq. b.1) that

$$ \begin{array}{@{}rcl@{}} &&\lim_{{\sum}^{k}_{c=1}n_{c}=n\rightarrow N \rightarrow \infty}\text{cov}[\hat{\beta}_{WGQL}]\\ &=&\left[ {\sum}^{K}_{c=1} \frac{\partial \pi'_{c}(\beta,\sigma^{2}_{\gamma})}{\partial \beta}{\Sigma}^{-1}_{c} (\beta,{\sigma^{2}_{\gamma}})\frac{\partial \pi_{c}(\beta,\sigma^{2}_{\gamma})}{\partial \beta'} \right]^{-1} \text{cov}\left( {\sum}^{k}_{c=1}{\sum}^{n_{c}}_{i=1}\tilde{w}_{cs^{*}}z_{cis^{*}} \right) \\ &&\times \left[ {\sum}^{K}_{c=1} \frac{\partial \pi'_{c}(\beta,\sigma^{2}_{\gamma})}{\partial \beta}{\Sigma}^{-1}_{c} (\beta,{\sigma^{2}_{\gamma}})\frac{\partial \pi_{c}(\beta,\sigma^{2}_{\gamma})}{\partial \beta'} \right]^{-1}\\ &=&\tilde{G}^{-1}(\beta,\sigma^{2}_{\gamma})V^{*}_{n}(\beta,\sigma^{2}_{\gamma}) \tilde{G}^{-1}(\beta,\sigma^{2}_{\gamma}), \end{array} $$

(b.8)

which, as expected, is the same as the covariance matrix of the normal distribution given by (5.26).

Appendix C: Proof for Consistency of the Survey Weighted MM Estimator $\hat {\sigma }^{2}_{\gamma ,WMM}$

Because $\hat {\sigma }^{2}_{\gamma , WMM}$ is the solution of the WMM estimating equation (4.5), a first order Taylor series expansion of the estimating function in the left hand side of (4.5) about $\sigma ^{2}_{\gamma }$ provides

$$ \begin{array}{@{}rcl@{}} &&\hat{\sigma}^{2}_{\gamma,WMM} - \sigma^{2}_{\gamma} \simeq - \left[{\sum}^{k}_{c=1}\tilde{w}_{cs^{*}}\left\{\frac{\partial \lambda'_{p_{cs^{*}}}}{\partial \sigma^{2}_{\gamma}}\frac{\partial \lambda_{p_{cs^{*}}}}{\partial \sigma^{2}_{\gamma}} +\frac{N_{c}-1}{n_{c}-1} \frac{\partial \lambda'_{q_{cs^{*}}}}{\partial \sigma^{2}_{\gamma}} \frac{\partial \lambda_{q_{cs^{*}}}}{\partial \sigma^{2}_{\gamma}}\right\}\right]^{-1} \\ &\times & \left[{\sum}^{k}_{c=1}\tilde{w}_{cs^{*}}\left\{\frac{\partial \lambda'_{p_{cs^{*}}}}{\partial \sigma^{2}_{\gamma}}(p_{cs^{*}}-\lambda_{p_{cs^{*}}}) +\frac{N_{c}-1}{n_{c}-1}\frac{\partial \lambda'_{q_{cs^{*}}}}{\partial \sigma^{2}_{\gamma}} (q_{cs^{*}}-\lambda_{q_{cs^{*}}})\right\}\right] \\ &+&o_{p}(1/\sqrt{n}), \end{array} $$

(c.1)

where $n={\sum }^{k}_{c=1}n_{c}$ (see also (4.6)). For convenience of further calculations, we re-express the equation in Eq. c.1 as

$$ \begin{array}{@{}rcl@{}} \hat{\sigma}^{2}_{\gamma,WMM}-\sigma^{2}_{\gamma} \simeq -S^{-1}_{1}S_{2,y}+o_{p}(1/\sqrt{n}). \end{array} $$

(c.2)

Notice that S₁ is free of responses, whereas S_{2, y} contains the responses {y} because $p_{cs^{*}}$ and $q_{cs^{*}}$ in Eq. c.1 are functions of second order responses. More specifically, by (4.4) (see also (3.20)), $p_{cs^{*}}$ is constructed by using the second order responses from the matrix {y_ciyci′} for c = 1, … , k; i = 1, … , n_c, where by (1.7), y_ci is the (J − 1)-dimensional multinomial response vector. Similarly, $q_{cs^{*}}$ is constructed by using the elements of second order response matrix {y_ciycm′} given in (3.12), for i ≠ m; i, m = 1, … , n_c. Consequently, one can show that

$$ \begin{array}{@{}rcl@{}} S_{1} &\rightarrow & E_{D}[S_{1}] \\ &=& {\sum}^{K}_{c=1}\left\{\frac{\partial \lambda'_{p_{c}}}{\partial \sigma^{2}_{\gamma}}\frac{\partial \lambda_{p_{c}}}{\partial \sigma^{2}_{\gamma}} +\frac{\partial \lambda'_{q_{c}}}{\partial \sigma^{2}_{\gamma}} \frac{\partial \lambda_{q_{c}}}{\partial \sigma^{2}_{\gamma}}\right\}. \end{array} $$

(c.3)

Suppose that this survey/finite population based quantity in Eq. c.3 is finite and bounded. More specifically, we assume that the following regularity condition holds:

C1.

For $N={\sum }^{K}_{c=1}N_{c},$ let f_N increases as N gets larger but it is a finite quantity. Also suppose that the scalar quantity in Eq. c.3 satisfies

$$ \begin{array}{@{}rcl@{}} \frac{1}{N} {\sum}^{K}_{c=1} \left\{\frac{\partial \lambda'_{p_{c}}}{\partial \sigma^{2}_{\gamma}}\frac{\partial \lambda_{p_{c}}}{\partial \sigma^{2}_{\gamma}} +\frac{\partial \lambda'_{q_{c}}}{\partial \sigma^{2}_{\gamma}} \frac{\partial \lambda_{q_{c}}}{\partial \sigma^{2}_{\gamma}}\right\}\leq f_{N}. \end{array} $$

(c.4)

By applying (c.4) to (c.3), one obtains the order of $S^{-1}_{1}$ in Eq. c.2 as

$$ S^{-1}_{1} = O(N^{-1}f^{-1}_{N}). $$

(c.5)

Next, it is clear from Eq. c.1 that

$$ S_{2,y} \rightarrow_{p} E_{Y}E_{D}[S_{2,y}]=0, $$

(c.6)

in order of $[\text {var}(S_{2,y})]^{\frac {1}{2}}.$ To obtain this order we compute the variance var(S_{2, y}) as follows:

$$ \begin{array}{@{}rcl@{}} &&\text{var}(S_{2,y})=\text{var} \left[{\sum}^{k}_{c=1}\tilde{w}_{cs^{*}}\left\{\frac{\partial \lambda'_{p_{cs^{*}}}}{\partial \sigma^{2}_{\gamma}}(p_{cs^{*}}-\lambda_{p_{cs^{*}}}) \right.\right.\\&&\left.\left.+\frac{N_{c}-1}{n_{c}-1}\frac{\partial \lambda'_{q_{cs^{*}}}}{\partial \sigma^{2}_{\gamma}} (q_{cs^{*}}-\lambda_{q_{cs^{*}}})\right\}\right]\\ &=&\text{var}\left[{\sum}^{k}_{c=1}\frac{K}{k} \frac{N_{c}}{n_{c}}\left\{{\sum}^{n_{c}}_{i=1}b'_{ci}(p_{ci}-\lambda_{p_{ci}}) \right. \right. \\ &+& \left. \left. \{(N_{c}-1)/(n_{c}-1)\} {\sum}^{n_{c}(n_{c}-1)/2}_{i=1}d'_{ci}(q_{ci}-\lambda_{q_{ci}})\right\} \right], \\ &&\text{by Eq.~\ref{Equ52} (see also Eqs.~\ref{Equ47}-\ref{Equ48})} \\ &=& (\frac{K}{k})^{2}{\sum}^{k}_{c=1}{N^{2}_{c}} \text{var}_{C}\left\{\frac{1}{n_{c}} {\sum}^{n_{c}}_{i=1}b'_{ci}(p_{ci}-\lambda_{p_{ci}}) \right. \\ &+& \left. \{(N_{c}-1)/2\}\frac{1}{n_{c}(n_{c}-1)/2} {\sum}^{n_{c}}_{i<j}d'_{c,ij}(q_{c,ij}-\lambda_{q_{c,ij}})\right\}\\ &=&(\frac{K}{k})^{2}{\sum}^{k}_{c=1}{N^{2}_{c}} \left\{\frac{1}{{n^{2}_{c}}}\text{var}\left( {\sum}^{n_{c}}_{i=1}b'_{ci}(p_{ci}-\lambda_{p_{ci}})\right) \right. \\ &+&\frac{(N_{c}-1)^{2}}{[n_{c}(n_{c}-1)]^{2}}\text{var}\left( {\sum}^{n_{c}}_{i<j}d'_{c,ij}(q_{c,ij}-\lambda_{q_{c,ij}})\right)\\ &+& \left. \frac{(N_{c}-1)}{[{n^{2}_{c}}(n_{c}-1)]}\text{cov} \left( {\sum}^{n_{c}}_{i=1}b'_{ci}(p_{ci} - \lambda_{p_{ci}}), {\sum}^{n_{c}}_{j<k}d'_{c,jk}(q_{c,jk} - \lambda_{q_{c,jk}})\right) \right\}, \end{array} $$

(c.7)

where the variances and covariances can be computed as follows.

$$ \begin{array}{@{}rcl@{}} &&V_{c,1}(\cdot)=\text{var}\left( {\sum}^{n_{c}}_{i=1}b'_{ci}(p_{ci}-\lambda_{p_{ci}})\right)\\ &=&\text{var}\left( {\sum}^{N_{c}}_{i=1}s_{i}b'_{ci}(p_{ci}-\lambda_{p_{ci}})\right) \\ &=&{\sum}^{N_{c}}_{i=1}b'_{ci}(p_{ci}-\lambda_{p_{ci}}) (p_{ci}-\lambda_{p_{ci}})'b_{ci}\text{var}(s_{i})\\ &+&2{\sum}^{N_{c}}_{i<j}b'_{ci}(p_{ci}-\lambda_{p_{ci}}) (p_{cj}-\lambda_{p_{cj}})'b_{cj}\text{cov}(s_{i},s_{j}), \end{array} $$

(c.8)

where s_i is a random indicator variable such that

$$ \begin{array}{@{}rcl@{}} s_{i}&=& \left\{ \begin{array}{ll} 1 & \text{if the \textit{i}th unit from the \textit{c}-th cluster is in the sample} \\ [2ex] 0 & \text{otherwise}\\ [2ex] \end{array} \right. \end{array} $$

(c.9)

[Cochran (1977, Section 2.9)]. It then follows that

$$ \begin{array}{@{}rcl@{}} E[s_{i}]=\frac{n_{c}}{N_{c}}, \text{var}(s_{i})=\frac{n_{c}}{N_{c}}(1-\frac{n_{c}}{N_{c}}), \end{array} $$

(c.10)

and

$$ \begin{array}{@{}rcl@{}} \text{cov}(s_{i},s_{j})&=&E(s_{i}s_{j})-E(s_{i})E(s_{j}) \\ &=&\frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}-\left( \frac{n_{c}}{N_{c}}\right)^{2} =-\frac{n_{c}}{N_{c}(N_{c}-1)}(1-\frac{n_{c}}{N_{c}}). \end{array} $$

(c.11)

Hence by putting (c.11) and (c.10) in (c.8), we obtain the variance

$$ \begin{array}{@{}rcl@{}} V_{c,1}(\beta,\sigma^{2}_{\gamma})&=&\frac{n_{c}}{N_{c}}(1-\frac{n_{c}}{N_{c}}) {\sum}^{N_{c}}_{i=1}b'_{ci}(p_{ci}-\lambda_{p_{ci}}) (p_{ci}-\lambda_{p_{ci}})'b_{ci} \\ &-&2\frac{n_{c}}{N_{c}(N_{c}-1)}(1-\frac{n_{c}}{N_{c}}){\sum}^{N_{c}}_{i<j} b'_{ci}(p_{ci}-\lambda_{p_{ci}}) (p_{cj}-\lambda_{p_{cj}})'b_{cj}. \\ &=&\frac{n_{c}}{N_{c}}(1-\frac{n_{c}}{N_{c}})S^{*}_{1,N_{c}}(y) \\ &-&2\frac{n_{c}}{N_{c}(N_{c}-1)}(1-\frac{n_{c}}{N_{c}})S^{*}{2,N_{c}}(y), \text{(say)}. \end{array} $$

(c.12)

Notice that the computation of the remaining variance and the covariance is slightly more complicated because it will require the fourth order product moments of indicator random variables. Specifically, we write

$$ \begin{array}{@{}rcl@{}} &&V_{c,2}(\beta,\sigma^{2}_{\gamma}) =\text{var}\left( {\sum}^{n_{c}}_{i<j}d'_{c,ij}(q_{c,ij}-\lambda_{q_{c,ij}})\right)\\ &=&\text{var}\left( {\sum}^{N_{c}}_{i<j}s_{i}s_{j}d'_{c,ij}(q_{c,ij}-\lambda_{q_{c,ij}})\right)\\ &=&{\sum}^{N_{c}}_{i<j}\text{var}(s_{i}s_{j})d'_{c,ij} (q_{c,ij}-\lambda_{q_{c,ij}})(q_{c,ij}-\lambda_{q_{c,ij}})'d_{c,ij} \\ &+&{\sum}^{N_{c}}_{i<j,k<\ell}\text{cov}[s_{i}s_{j},s_{k}s_{\ell}] d'_{c,ij}(q_{c,ij}-\lambda_{q_{c,ij}})(q_{c,k\ell}- \lambda_{q_{c,k\ell}})'d_{c,k\ell}, \end{array} $$

(c.13)

where

$$ \begin{array}{@{}rcl@{}} \text{var}(s_{i}s_{j})&=&E[{s^{2}_{i}}{s^{2}_{j}}]-\left( E(s_{i}s_{j})\right)^{2} =E[s_{i}s_{j}]-\left( E(s_{i}s_{j})\right)^{2}\\ &=&E[s_{i}s_{j}]\left( 1-E[s_{i}s_{j}]\right) \\ &=&\frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\left[1- \frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\right], \end{array} $$

(c.14)

by Eq. c.11. The covariance in Eq. c.13 is computed as

$$ \begin{array}{@{}rcl@{}} \text{cov}[s_{i}s_{j},s_{k}s_{\ell}] &=& \left\{ \begin{array}{ll} E[s_{i}s_{j}s_{k}s_{\ell}]-E[s_{i}s_{j}]E[s_{k}s_{\ell}] & \text{for} i\neq j \neq k \neq \ell \\ [2ex] E[s_{i}s_{j}s_{k}]-E[s_{i}s_{j}]E[s_{k}s_{\ell}] & \text{otherwise}\\ [2ex] \end{array} \right. \\ &=& \left\{ \begin{array}{ll} \frac{n_{c}(n_{c}-1)(n_{c}-2)(n_{c}-3)}{N_{c}(N_{c}-1) (N_{c}-2)(N_{c}-3)} - \left( \frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\right)^{2} & \text{for} i\neq j \neq k \neq \ell \\ [2ex] \frac{n_{c}(n_{c}-1)(n_{c}-2)}{N_{c}(N_{c}-1) (N_{c}-2)}-\left( \frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\right)^{2} & \text{otherwise}\\ [2ex] \end{array} \right. \\ &=& \left\{ \begin{array}{ll} \frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\left[ \frac{(n_{c}-2)(n_{c}-3)}{ (N_{c}-2)(N_{c}-3)}- \frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\right] & \text{for} i\neq j \neq k \neq \ell \\ [2ex] \frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\left[ \frac{(n_{c}-2)}{(N_{c}-2)}- \frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\right] & \text{otherwise}\\ [2ex] \end{array} \right. \\ &=&\left\{\begin{array}{ll} \frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\left[ \delta_{1}(n_{c},N_{c})\right] & \text{for} i\neq j \neq k \neq \ell \\ [2ex] \frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\left[\delta_{2}(n_{c},N_{c})\right] & \text{otherwise}\\ [2ex] \end{array} \right. \end{array} $$

(c.15)

Using Eqs. c.15 and c.14 in Eq. c.13, we obtain the formula for V_c,2(⋅) as

$$ \begin{array}{@{}rcl@{}} &&V_{c,2}(\beta,\sigma^{2}_{\gamma}) =\text{var}\left( {\sum}^{n_{c}}_{i<j}d'_{c,ij}(q_{c,ij}-\lambda_{q_{c,ij}})\right)\\ &=&\frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\left[1- \frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\right] {\sum}^{N_{c}}_{i<j}d'_{c,ij}(q_{c,ij}-\lambda_{q_{c,ij}}) (q_{c,ij}-\lambda_{q_{c,ij}})'d_{c,ij} \\ &+&\frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\left[ \delta_{1}(n_{c},N_{c})\right] {\sum}^{N_{c}}_{i \neq j \neq k \neq \ell} d'_{c,ij}(q_{c,ij}-\lambda_{q_{c,ij}})(q_{c,k\ell}- \lambda_{q_{c,k\ell}})'d_{c,k\ell} \\ &+&\frac{n_{c}(n_{c} - 1)}{N_{c}(N_{c} - 1)}\left[ \delta_{2}(n_{c},N_{c})\right] \left\{{\sum}^{N_{c}}_{i <j, k <\ell, i=k} d'_{c,ij}(q_{c,ij} - \lambda_{q_{c,ij}})(q_{c,k\ell} - \lambda_{q_{c,k\ell}})'d_{c,k\ell} \right. \\ &+&{\sum}^{N_{c}}_{i <j, k <\ell, i=\ell} d'_{c,ij}(q_{c,ij}-\lambda_{q_{c,ij}})(q_{c,k\ell}- \lambda_{q_{c,k\ell}})'d_{c,k\ell} \\ &+& {\sum}^{N_{c}}_{i <j, k <\ell, j=k} d'_{c,ij}(q_{c,ij}-\lambda_{q_{c,ij}})(q_{c,k\ell}- \lambda_{q_{c,k\ell}})'d_{c,k\ell} \\ &+& \left. {\sum}^{N_{c}}_{i <j, k <\ell, j=\ell} d'_{c,ij}(q_{c,ij}-\lambda_{q_{c,ij}})(q_{c,k\ell}- \lambda_{q_{c,k\ell}})'d_{c,k\ell} \right\}\\ &=&\frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\left[1- \frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\right]S^{*}_{3,N_{c}}(y) \\ &+&\frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\left[ \delta_{1}(n_{c},N_{c})S^{*}_{4,N_{c}}(y) + \delta_{2}(n_{c},N_{c})S^{*}_{5,N_{c}}(y) \right], \text{(say)}. \end{array} $$

(c.16)

Next we compute

$$ \begin{array}{@{}rcl@{}} &&V_{c,12}(\beta,\sigma^{2}_{\gamma})=\text{cov} \left( {\sum}^{n_{c}}_{i=1}b'_{ci}(p_{ci}-\lambda_{p_{ci}}), {\sum}^{n_{c}}_{j<k}d'_{c,jk}(q_{c,jk}-\lambda_{q_{c,jk}})\right) \\ &=& \text{cov} \left( {\sum}^{N_{c}}_{i=1}s_{i}b'_{ci}(p_{ci}-\lambda_{p_{ci}}), {\sum}^{N_{c}}_{j<k}s_{j}s_{k}d'_{c,jk}(q_{c,jk}-\lambda_{q_{c,jk}})\right) \\ &=&{\sum}^{N_{c}}_{i \neq j, i \neq k, j <k}\text{cov}[s_{i},s_{j}s_{k}] b'_{ci}(p_{ci}-\lambda_{p_{ci}})(q_{c,jk}-\lambda_{q_{c,jk}})'d_{c,jk}\\ &&+{\sum}^{N_{c}}_{i =j, j<k}\text{cov}[s_{i},s_{i}s_{k}] b'_{ci}(p_{ci}-\lambda_{p_{ci}})(q_{c,jk}-\lambda_{q_{c,jk}})'d_{c,jk}\\ &&+{\sum}^{N_{c}}_{i =k, j<k}\text{cov}[s_{i},s_{j}s_{i}] b'_{ci}(p_{ci}-\lambda_{p_{ci}})(q_{c,jk}-\lambda_{q_{c,jk}})'d_{hc,jk}\\ &=&\left[\frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)} \left( \frac{n_{c}-2}{N_{c}-2}-\frac{n_{c}}{N_{c}}\right)\right] {\sum}^{N_{c}}_{i =j, j<k} b'_{ci}(p_{ci} - \lambda_{p_{ci}})(q_{c,jk} - \lambda_{q_{c,jk}})' d_{c,jk}\\ &&+\left[\frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\left( 1-\frac{n_{c}}{N_{c}} \right)\right] \left[{\sum}^{N_{c}}_{i =j, j<k} b'_{ci}(p_{ci}-\lambda_{p_{ci}})(q_{c,jk}-\lambda_{q_{c,jk}})' d_{c,jk}\right.\\ &&+\left. {\sum}^{N_{c}}_{i =k, j<k} b'_{ci}(p_{ci}-\lambda_{p_{ci}})(q_{c,jk}-\lambda_{q_{c,jk}})'d_{c,jk}\right] \\ &=&\left[\frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)} \left( \frac{n_{c}-2}{N_{c}-2}-\frac{n_{c}}{N_{c}}\right)\right]S^{*}_{6,N_{c}}(y) \\ &&+\left[\frac{n_{c}(n_{c}-1)}{N_{c}(N_{c}-1)}\left( 1-\frac{n_{c}}{N_{c}} \right)\right]S^{*}_{7,N_{c}}(y),\text{(say)}. \end{array} $$

(c.17)

Finally, by applying (c.12), (c.16) and (c.17), the desired variance follows from Eq. c.7 as

$$ \begin{array}{@{}rcl@{}} &&\text{var}[S_{2,y}(\cdot)]\\ &=&\left[(\frac{K^{2}}{k})\frac{1}{K}{\sum}^{K}_{c=1}{N^{2}_{c}} \left\{\frac{1}{{n^{2}_{c}}}V_{c,1}(\cdot) +\frac{(N_{c}-1)^{2}}{[n_{c}(n_{c}-1)]^{2}}V_{c,2}(\cdot) \right. \right. \\ &&+\left. \left. \frac{(N_{c}-1)}{[{n^{2}_{c}}(n_{c}-1)]}V_{c,12}(\cdot) \right\}\right]. \end{array} $$

(c.18)

For simplicity, we assume that the sampling fractions are negligible. It then follows from Eqs. c.12, c.16 and c.17 that

$$ \begin{array}{@{}rcl@{}} V_{c,1}(\beta,\sigma^{2}_{\gamma}) &\simeq & n_{c}\left[\frac{1}{N_{c}}S^{*}_{1,N_{c}}(y)\right]-2n_{c}\left[\frac{1}{N_{c}(N_{c}-1)}S^{*}_{2,N_{c}}(y)\right] \end{array} $$

(c.19)

$$ \begin{array}{@{}rcl@{}} V_{c,2}(\beta,\sigma^{2}_{\gamma}) &\simeq & n_{c}(n_{c}-1)\left[\frac{1}{N_{c}(N_{c}-1)}\left\{S^{*}_{3,N_{c}}(y) \right. \right. \\ &+&\left. \left.\delta_{1}(n_{c},N_{c})S^{*}_{4,N_{c}}(y) +\delta_{2}(n_{c},N_{c})S^{*}_{5,N_{c}}(y) \right\}\right] \end{array} $$

(c.20)

$$ \begin{array}{@{}rcl@{}} V_{c,12}(\beta,\sigma^{2}_{\gamma}) &\simeq & n_{c}(n_{c}-1)\left[\frac{1}{N_{c}(N_{c}-1)}S^{*}_{7,N_{c}}(y)\right], \end{array} $$

(c.21)

respectively. Consequently, we can simplify the desired variance in Eq. c.18 as

$$ \begin{array}{@{}rcl@{}} &&\text{var}[S_{2,y}(\cdot)]\\ &=&\left[(\frac{K}{k}){\sum}^{K}_{c=1}\frac{{N^{2}_{c}}}{n_{c}} \left\{\left( \frac{1}{N_{c}}S^{*}_{1,N_{c}}(y)-\frac{2} {N_{c}(N_{c}-1)}S^{*}_{2,N_{c}}(y)\right) \right. \right. \\ &&+\frac{1}{n_{c}-1} \left( S^{*}_{3,N_{c}}(y)+\delta_{1}(n_{c},N_{c})S^{*}_{4,N_{c}}(y) + \delta_{2}(n_{c},N_{c})S^{*}_{5,N_{c}}(y) \right) \\ &&+\left. \left. \frac{1}{N_{c}}S^{*}_{7,N_{c}}(y) \right\}\right]. \end{array} $$

(c.22)

Next we assume that the following regularity conditions hold with regard to the finite population:

C2

For $N={\sum }^{K}_{c=1}N_{c},$ and for N dependent finite and bounded quantity r_N, the summed quantities in Eq. c.22 satisfy

$$ \begin{array}{@{}rcl@{}} &&\max_{c}\left[\frac{K{N^{2}_{c}}}{N} \left\{\left( \frac{1}{N_{c}}S^{*}_{1,N_{c}}(y)-\frac{2} {N_{c}(N_{c}-1)}S^{*}_{2,N_{c}}(y)\right) \right. \right. \\ &+&\frac{1}{n_{c}-1} \left( S^{*}_{3,N_{c}}(y)+\delta_{1}(n_{c},N_{c})S^{*}_{4,N_{c}}(y) + \delta_{2}(n_{c},N_{c})S^{*}_{5,N_{c}}(y) \right) \\ &+&\left. \left. \frac{1}{N_{c}}S^{*}_{7,N_{c}}(y) \right\}\right] \leq r_{N}. \end{array} $$

(c.23)

It then follows that the variance in Eq. c.22 satisfies the condition

$$ \begin{array}{@{}rcl@{}} &&\text{var}[S_{2,y}(\cdot)] \leq N\left[ r_{N}{\sum}^{K}_{c=1}\frac{1}{kn_{c}}\right]. \end{array} $$

(c.24)

Finally by applying (c.24), (c.6), and (c.5), the convergence result stated in (6.1)-(6.2) under Section 6 follows from Eq. c.2.