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 $\left\{{\pi }_{i}\text{ }:\text{\hspace{0.17em}}i=1,\dots ,N\right\}\text{ }.$ 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,

$\sum _{i\in s}\frac{{x}_{i}}{{\pi }_{i}}=\sum _{i=1}^{N}{x}_{i}.\text{ }\text{ }\text{ }\text{ }\text{ }\left(2.1\right)$

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 ${\sum }_{i\in s}{I}_{i}\left(\omega \right)/{\pi }_{\omega }={\sum }_{i=1}^{N}\text{\hspace{0.17em}}{I}_{i}\left(\omega \right).$ In this context the balancing equation simply requests that the number of sampled units of type $\omega ,$ ${n}_{\omega }={\sum }_{i\in s}{I}_{i}\left(\omega \right),$ is equal to its expectation,

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

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, $\stackrel{˜}{n}={\left({n}_{1},\dots ,\text{\hspace{0.17em}}{n}_{T}\right)}^{ㄒ}\text{​},$ to be equal to their expectations, $E\left(\stackrel{˜}{n}\right),$ under the sampling design. Under rejective sampling, the sample is said to be balanced if

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

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

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