# Using balanced sampling in creel surveys

Section 2. Balanced sampling

Suppose that $U$ is a finite population of size $N$ that is sampled with a design having selection probabilities given by $\left\{{\pi}_{i}\text{\hspace{0.05em}}:\text{\hspace{0.17em}}i\mathrm{=1,}\dots ,N\right\}\text{\hspace{0.05em}}.$ If $x$ is an auxiliary variable known for all population units, then the sample is balanced on $x$ if the Horvitz-Thompson estimator for the total of $x$ is equal to the known total of $x.$ In other words, for any balanced sample $s,$ the following equation has to be satisfied,

$$\sum _{i\in s}\frac{{x}_{i}}{{\pi}_{i}}}\mathrm{=}{\displaystyle \sum _{i\mathrm{=1}}^{N}{x}_{i}}\mathrm{.}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}(2.1)$$

For the surveys considered here, we balance on indicator variables ${I}_{i}\left(\omega \right)$ equal to 1 if unit $i$ is of type $\omega $ and 0 otherwise. If all the units $i$ for which ${I}_{i}\left(\omega \right)$ is equal to 1 have the same selection probability ${\pi}_{\omega},$ then equation (2.1) reduces to ${\sum}_{i\in s}{I}_{i}\left(\omega \right)}/{\pi}_{\omega}\mathrm{=}{\displaystyle {\sum}_{i\mathrm{=1}}^{N}\text{\hspace{0.17em}}{I}_{i}\left(\omega \right)}.$ In this context the balancing equation simply requests that the number of sampled units of type $\omega ,$ ${n}_{\omega}\mathrm{=}{\displaystyle {\sum}_{i\in s}{I}_{i}\left(\omega \right),}$ is equal to its expectation,

$${n}_{\omega}\mathrm{=}{\displaystyle \sum _{i\mathrm{=1}}^{N}{I}_{i}\left(\omega \right){\pi}_{\omega}}\mathrm{.}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}(2.2)$$

To implement balanced sampling we use the cube method of Deville and Tillé (2004), and the extension of Hasler and Tillé (2014) to cope with highly stratified populations. In Section 4 this method is compared with the implementation of the rejection method proposed by Fuller (2009). In the context of this study, we are balancing on $T$ types of units; we want the sampled numbers of units for the $T$ types, $\tilde{n}={\left({n}_{1}\mathrm{,}\dots \mathrm{,}\text{\hspace{0.17em}}{n}_{T}\right)}^{\u3112}\text{},$ to be equal to their expectations, $E\left(\tilde{n}\right),$ under the sampling design. Under rejective sampling, the sample is said to be balanced if

$${Q}_{T\text{}\mathrm{,}\text{\hspace{0.17em}}n}\mathrm{=}{\left(\tilde{n}-E\left(\tilde{n}\right)\right)}^{\u3112}{\left[\text{Var}\left(\tilde{n}\right)\right]}^{-1}\left(\tilde{n}-E\left(\tilde{n}\right)\right)\mathrm{<}{\gamma}^{2}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}(2.3)$$

where $\text{Var}\left(\tilde{n}\right)$ represents the design based covariance matrix of $\tilde{n}$ and ${\gamma}^{2}$ is a tolerance value that determines the balancing condition. Samples that do not meet the balancing equation ${Q}_{T\text{}\mathrm{,}\text{\hspace{0.17em}}n}\mathrm{<}{\gamma}^{2}$ are simply rejected.

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