# Using balanced sampling in creel surveys Section 5. Discussion

In the context of creel surveys, balanced sampling techniques such as the cube method or the rejective algorithm are used to ensure a predetermined sample size in small domains of the survey population. The cube method is very effective at doing so especially in complex survey designs with several stages of sampling. It does not change the selection probabilities and it yields domain sample sizes that are very close their target values. The rejective method, on the other hand, changes the selection probabilities slightly and produce domain sample sizes that are more variable. With a large number of constraints, Fuller’s rejective sampling scheme is not really applicable as it requires the evaluation and the inversion of a large covariance matrix in (2.3); alternative acceptation criteria for a sample need to be investigated.

## Acknowledgements

We thank the Associate Editor and the referees for their constructive comments on the first version of this manuscript. The assistance and the suggestions of Valérie Bujold, Michel Legault and of Hélène Crépeau who participated to the initial phase of this project is gratefully acknowledged. This project benefitted from the financial assistance of the Canada Research Chair in Statistical Sampling and Data Analysis and form a discovery grant (5244/2012) from the Natural Sciences and Engineering Research Council of Canada.

## Appendix

### Calculation of the joint selection probabilities when $N=3$

Consider a population of size 3 and let ${\pi }_{1},$ ${\pi }_{2},$ and ${\pi }_{3}$ be the marginal selection probabilities when drawing a sample of size $n=2.$ The joint selection probabilities ${\pi }_{ij},$ $i\ne j=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3$ satisfy

$\left(\begin{array}{lll}1\hfill & 1\hfill & 0\hfill \\ 1\hfill & 0\hfill & 1\hfill \\ 0\hfill & 1\hfill & 1\hfill \end{array}\right)\left(\begin{array}{l}{\pi }_{12}\hfill \\ {\pi }_{13}\hfill \\ {\pi }_{23}\hfill \end{array}\right)=\left(\begin{array}{l}{\pi }_{1}\hfill \\ {\pi }_{2}\hfill \\ {\pi }_{3}\hfill \end{array}\right).$

Thus

$\left(\begin{array}{l}{\pi }_{12}\hfill \\ {\pi }_{13}\hfill \\ {\pi }_{23}\hfill \end{array}\right)={\left(\begin{array}{lll}1\hfill & 1\hfill & 0\hfill \\ 1\hfill & 0\hfill & 1\hfill \\ 0\hfill & 1\hfill & 1\hfill \end{array}\right)}^{-1}\left(\begin{array}{l}{\pi }_{1}\hfill \\ {\pi }_{2}\hfill \\ {\pi }_{3}\hfill \end{array}\right)=\frac{1}{2}\left(\begin{array}{ccc}1& 1& -1\\ 1& -1& 1\\ -1& 1& 1\end{array}\right)\left(\begin{array}{l}{\pi }_{1}\hfill \\ {\pi }_{2}\hfill \\ {\pi }_{3}\hfill \end{array}\right).$

Using these equations, the entries of the covariance matrix (4.1) can be evaluated using the stage 1 and the stage 2 selection probabilities.

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