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Balanced Sampling

BalancedSampling

Balanced sampling is a random method of selection of units from a population that provides a sample such that the Horvitz-Thompson estimators of the totals are the same or almost the same as the true population totals for a set of control variables.

More precisely, let $U=\{1,\dots, k,\dots, N\}$ be a finite population and $s\subset U$ a sample or a subset of . A sampling design is a probability distribution on the set of all the subsets $s\subset U,$ i.e. $p(s)\ge 0$ and

$\displaystyle \sum_{s\subset U} p(s)=1.$

The inclusion probability $\pi_k=pr(k\in s)$ of a unit

is its probability of being selected in the sample

Consider a variable of interest that takes the value on unit . Let also be the total of the values taken by on the units of the population

$\displaystyle Y =\sum_{k\in U} y_k.$

If $\pi_k>0,$ for all $k\in U,$ the Horvitz-Thompson (1952) estimator

$\displaystyle \widehat{Y} =\sum_{k\in S} \frac{y_k}{\pi_k}$

is unbiased for

Let also $x_1, \dots,x_j, \dots, x_J$ be a sequence of auxiliary variables whose the values $x_{k1}, \dots,x_{kj}, \dots, x_{kJ}$ are known for each unit of the population. According to the definition given in Tillé (2006), a sampling design is said to be balanced on the variables if

$\displaystyle \sum_{k\in S} \frac{x_{kj}}{\pi_k}\approx \sum_{k\in U} x_{kj},$ for $\displaystyle j=1,\dots,J.$

Balanced sampling involves sampling with fixed sample size and stratification. Indeed, a stratified design is balanced on the indicator variables of the strata, because the Horvitz-Thompson estimators of the sizes of the strata are equal to the population sizes of the strata. In the design-based inference, balanced sampling allows a strong improvement of the efficiency of the Horvitz-Thompson estimator when the auxiliary variables are correlated with the interest variable (see Deville and Tillé, 2004). In the model-based inference, balanced samples are advocated to protect under miss-specification of the model (see Valliant et al., 2000).

Balanced sampling must not be confused with a representative sample. Representativity is a vague concept that usually means that some groups have the same proportions in the sample and in the population. This definition is fallacious because some groups can be over or under-represent in a sample to obtain a more accurate unbiased estimator. Moreover, balanced sampling implies that the sample is randomly selected and that predefined inclusion probabilities, that can be unequal, are satisfied at least approximately.

There exists a large family of methods for selecting balanced samples. The first one was probably proposed by Yates (1949) and consists of selecting a sample by a simple random sampling and next eventually changing some units to get a more balanced sample. Other methods, called rejective, consist of selecting several samples with an initial sampling design until obtaining a sample that is well balanced. Rejective methods however have the drawback that the inclusion probabilities of the rejective design are not the same as the ones of the initial design and are generally impossible to compute.

The cube method proposed by Deville and Tillé (2004) is a general multivariate algorithm for selecting balanced samples that can use several decades of auxiliary variables with equal of unequal inclusion probabilities. The cube method exactly satisfies the predefined inclusion probabilities and provides a sample that is balanced as well as possible. SAS and R language implementations are available.

Reprinted with permission from Lovric, Miodrag (2011), International Encyclopedia of Statistical Science. Heidelberg: Springer Science +Business Media, LLC

Bibliography

Deville and Tillé, 2004: Deville, J.-C. and Tillé, Y. (2004).
Efficient balanced sampling: The cube method.
Biometrika, 91, 893-912.
Horvitz and Thompson, 1952: Horvitz, D.G. and Thompson, D.J. (1952).
A generalization of sampling without replacement from a finite universe.
Journal of the American Statistical Association, 47, 663-685.
Tillé, 2006: Tillé, Y. (2006).
Sampling algorithms.
New York: Springer-Verlag.
Valliant et al., 2000: Valliant, R., Dorfman, A.H. and Royall, R.M. (2000).
Finite Population Sampling and Inference: A Prediction Approach.
New York: Wiley.
Yates, 1949: Yates, F. (1949).
Sampling Methods for Censuses and Surveys.
London: Griffin.

Balanced Sampling is owned by Yves Till\'e.

Note

NOTE ON THE PROPER CITATION OF THIS ARTICLE: The proper citation for this article is Yves Tille, "Balanced Sampling." In: Lovric, Miodrag (Editor), (2010). International Encyclopedia of Statistical Science. Heidelberg: Springer Science +Business Media, LLC, reprinted and freely available at StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies.

How to Cite This Entry

Yves Till\'e. "Balanced Sampling" (version 4). StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. Freely available at http://statprob.com/encyclopedia/BalancedSampling.html

Classification

AMS MSC:

62D05 (Statistics :: Sampling theory, sample surveys)

Pending Errata and Addenda

None.

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