Using balanced sampling in creel surveys
Section 2. Balanced sampling

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Suppose that $U$ is a finite population of size $N$ that is sampled with a design having selection probabilities given by $\left\{{\pi }_i:i\mathrm{<mo>=</mo>\; 1,}\dots ,N\right\}.$ 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,

$${\displaystyle \sum _{i\in s}\frac{x_i}{{\pi }_i}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{x}_i}\mathrm{.}(2.1)$$

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 ${\displaystyle {\sum }_{i\in s}{I}_i\left(\omega \right)}/{\pi }_{\omega }\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{I}_i\left(\omega \right)}.$ In this context the balancing equation simply requests that the number of sampled units of type $\omega ,$ $n_{\omega }\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s}{I}_i\left(\omega \right),}$ is equal to its expectation,

$$n_{\omega }\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{I}_i\left(\omega \right){\pi }_{\omega }}\mathrm{.}(2.2)$$

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, $\tilde{n}={\left(n_1\mathrm{,}\dots \mathrm{,}{n}_T\right)}^{ㄒ}\text{},$ 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

$$Q_{T\text{}\mathrm{,}n}\mathrm{<mo>=</mo>}{\left(\tilde{n}-E\left(\tilde{n}\right)\right)}^{ㄒ}{\left[\text{Var}\left(\tilde{n}\right)\right]}^{-1}\left(\tilde{n}-E\left(\tilde{n}\right)\right)\mathrm{<mo><</mo>}{\gamma }^2(2.3)$$

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 $Q_{T\text{}\mathrm{,}n}\mathrm{<mo><</mo>}{\gamma }^2$ are simply rejected.

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