6 Concluding remarks

J.N.K. Rao, F. Verret and M.A. Hidiroglou

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In this paper, we have proposed a unified weighted composite likelihood (WCL) approach for two-level models to make inferences from complex survey data. The proposed WCL methods are asymptotically valid even when the sample sizes within sampled clusters (level 1 units) are small, unlike some of the existing methods, but knowledge of the joint inclusion probabilities within sampled clusters is required. Often it may be possible to treat the sample within clusters as drawn with replacement because of small sampling fractions within clusters. Also, excellent approximations to joint inclusion probabilities, depending only on the marginal inclusion probabilities, are also available when the sampling fractions are not small (Haziza et al., 2008). We plan to study the accuracy of such approximations in a future study. Simulation studies on the performance of the WCL estimators (4.5) and (4.6) for two-level models (2.3), based on the pairwise method, will also be conducted.

Composite likelihood methods are mostly used when the full likelihood is complex. Our development in the survey sampling context demonstrates that the full likelihood method with weights is not feasible for multi-level models whereas the weighted composite likelihood method facilitates valid inferences even when the cluster sample sizes are small.

Acknowledgments

We thank two referees and the associate editor for constructive comments and suggestions.

Appendix

Weighted score equations: nested error linear regression model

For the nested error linear regression model (2.3), an explicit form for the census full log-likelihood is obtained using the explicit form for the covariance matrix $V_i$ of $y_i={\left(y_{i1},\mathrm{...},y_{i{M}_i}\right)}^T.$ We have $V_i^{-1}={\sigma }_e^{-2}\left[I_i-{\sigma }_v^2/{\lambda }_i{1}_i{1}_i^T\right],$ where ${\lambda }_i={\sigma }_e^2+M_i{\sigma }_v^2$ , $I_i$ is the $M_i\times M_i$ identity matrix and $1_i$ is the $M_i\times 1$ unit vector. Using the expression for $V_i^{-1}$ , the census score equations are obtained as

$$\beta :\left[{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^{M_i}{x}_{ij}{y}_{ij}}}-{\sigma }_v^2{\displaystyle \sum _{i=1}^N{\lambda }_i^{-1}\left({\displaystyle \sum _{j=1}^{M_i}{\displaystyle \sum _{k=1}^{M_i}{x}_{ij}{y}_{ik}}}\right)}\right]-\left[{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^{M_i}{x}_{ij}{x}_{ij}^T}}-{\sigma }_v^2{\displaystyle \sum _{i=1}^N{\lambda }_i^{-1}\left({\displaystyle \sum _{j=1}^{M_i}{\displaystyle \sum _{k=1}^{M_i}{x}_{ij}{x}_{ik}^T}}\right)}\right]\beta =0\left(A.1\right)$$

$${\sigma }_v^2:{\displaystyle \sum _{i=1}^N{\lambda }_i^{-2}\left[{\displaystyle \sum _{j=1}^{M_i}{\displaystyle \sum _{k=1}^{M_i}\left(y_{ij}-x_{ij}^T\beta \right)\left(y_{ik}-x_{ik}^T\beta \right)}}\right]}-{\displaystyle \sum _{i=1}^N{\lambda }_i^{-1}{M}_i}=0\left(A.2\right)$$

$${\sigma }_e^2:{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^{M_i}{\left(y_{ij}-x_{ij}^T\beta \right)}^2}}+{\displaystyle \sum _{i=1}^N\left(M_i{\sigma }_v^4{\lambda }_i^{-2}-2{\sigma }_v^2{\lambda }_i^{-1}\right){\displaystyle \sum _{j,k=1}^{M_i}\left(y_{ij}-x_{ij}^T\beta \right)\left(y_{ik}-x_{ik}^T\beta \right)}}-{\sigma }_e^2{\displaystyle \sum _{i=1}^N\left(1-{\sigma }_v^2{\lambda }_i^{-1}\right)M_i}=0\left(A.3\right)$$

From (A.1), we obtain weighted score equations

$$\begin{array}{l}\beta :{\displaystyle \sum _{i\in s}{w}_i{\displaystyle \sum _{j\in s(i)}{w}_{j|i}{x}_{ij}{y}_{ij}}}-{\sigma }_v^2{\displaystyle \sum _{i\in s}{w}_i{\lambda }_i^{-1}\left({\displaystyle \sum _{j\in s(i)}{\displaystyle \sum _{k\in s(i)}{w}_{jk|i}{x}_{ij}{y}_{ik}}}\right)}\left(A.4\right)\\ -\left[{\displaystyle \sum _{i\in s}{w}_i{\displaystyle \sum _{j\in s(i)}{w}_{j|i}{x}_{ij}{x}_{ij}^T}}-{\sigma }_v^2{\displaystyle \sum _{i\in s}{w}_i{\lambda }_i^{-1}\left({\displaystyle \sum _{j\in s(i)}{\displaystyle \sum _{k\in s(i)}{w}_{jk|i}{x}_{ij}{x}_{ik}}}\right)}\right]\beta =0\end{array}$$

where $w_{jj|i}=w_{j|i}$ . Note that the cluster sizes $M_i$ for $i\in s$ are assumed to be known. One should not replace $M_i$ by its estimate ${\displaystyle {\sum }_{j\in s(i)}{w}_{j|i}}$ because it includes ratio bias due to small within cluster sample sizes. The estimating equation (A.4) is design-unbiased for the census equation (A.1).

Turning to the weighted score equation for ${\sigma }_v^2$ , we obtain from (A.2)

$${\sigma }_v^2:{\displaystyle \sum _{i\in s}{w}_i{\lambda }_i^{-2}\left[{\displaystyle \sum _{j\in s(i)}{\displaystyle \sum _{k\in s(i)}{w}_{jk|i}\left(y_{ij}-x_{ij}^T\beta \right)\left(y_{ik}-x_{ik}^T\beta \right)}}\right]}-{\displaystyle \sum _{i\in s}{w}_i{\lambda }_i^{-1}{\displaystyle \sum _{j\in s(i)}{w}_{j|i}}}=0\left(A.5\right)$$

The estimating equation (A.5) is unbiased for (A.2). Finally, the weighted score equation for ${\sigma }_e^2$ is obtained from (A.3) as

$$\begin{array}{l}{\sigma }_e^2:{\displaystyle \sum _{i\in s}{w}_i{\displaystyle \sum _{j\in s(i)}{w}_{j|i}{\left(y_{ij}-x_{ij}^T\beta \right)}^2}}+{\displaystyle \sum _{i\in s}{w}_i\left(M_i{\sigma }_v^4{\lambda }_i^{-2}-2{\sigma }_v^2{\lambda }_i^{-1}\right){\displaystyle \sum _{j,k\in s(i)}{w}_{jk|i}\left(y_{ij}-x_{ij}^T\beta \right)\left(y_{ik}-x_{ik}^T\beta \right)}}\left(A.6\right)\\ -{\sigma }_e^2{\displaystyle \sum _{i\in s}{w}_i\left(1-{\sigma }_v^2{\lambda }_i^{-1}\right){\displaystyle \sum _{j\in s(i)}{w}_{j|i}}}=0\end{array}$$

It follows from (A.4)-(A.6) that the weighted score equations depend only on the first order weights $w_i$ and $w_{j|i}$ and the second order weights $w_{jk|i}$ in the special case of a nested error linear regression model.

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