1 Introduction
Marco Ballin and Giulio Barcaroli
The optimality of a sample can be defined in terms of costs (associated to fieldwork, especially in terms of the number of units to be interviewed) and accuracy (related to the sampling variance of target estimates). Stratified sampling is a widely adopted design that may ensure savings in costs and gains in accuracy of estimates, when stratification variables are available in the sampling frame.
Many studies have been published on the problem of the optimization of stratified sample design. We can classify them accordingly to the object of the optimization:
- the allocation has to be optimized, while stratification is considered as given;
- stratification has to be optimized, while the allocation issue is postponed to a later stage;
- stratification and allocation are optimized in a single step.
In the first group we can include Cochran (1977), Bethel (1985, 1989), Chromy (1987), Huddleston, Claypool and Hocking (1970), Kish (1976), Stokes and Plummer (2004), Day (2006, 2010), Díaz-García and Cortez (2008), Kozak, Zieliński and Singh (2008), Khan, Maiti and Ahsan (2010), Kozak and Wang (2010). Bethel (1985, 1989) and Chromy (1987) propose similar algorithms for the extension of the Neyman allocation to the multivariate case by using Convex Programming methods. Stokes and Plummer (2004) show how to make use of the Non Linear Programming tool available in Excel spreadsheets as a solver for the same problem. In Day (2006, 2010) the evolutionary algorithm approach is proposed to solve the multivariate allocation problem under the same setting indicated by Bethel and Chromy. In Díaz-García and Cortez (2008), the optimum multivariate allocation problem is solved as a problem of multi-objective optimization of integers. Kozak et al. (2008) investigate the case of stratified two-stage sampling.
In the second group, we can consider Dalenius and Hodges (1959), Singh (1971), Hidiroglou (1986), Lavallée and Hidiroglou (1988), Gunning and Horgan (2004), Khan, Nand and Ahmad (2008). In general, the problem dealt with is related to the optimization of the stratification obtainable by one or more continuous variables, correlated to one or more target variables.
A number of papers deal with both problems (stratification and allocation) jointly. Kozak, Verma and Zieliński (2007) propose a method to obtain multivariate stratification while minimising the overall sample size. The method is defined only on a theoretical base, and the claim is that in the univariate case the optimization is not difficult, while in the multivariate case more research is needed. Keskintürk and Er (2007) make use of the genetic algorithm to solve jointly the allocation and strata boundaries problems, in the case of only one continuous stratification variable, and considering as given both the number of strata and the total sample size. The proposal of Benedetti, Espa and Lafratta (2008) is based on the use of a tree-based approach: their procedure defines a path from the null stratification towards the so called atomic stratification (characterised by the maximum number of strata, obtained by using all auxiliary variables, with the most detailed classifications), generally without reaching it, given that a number of stopping rules are used. Baillargeon and Rivest (2009, 2011) propose a method that can jointly optimise stratum boundaries and sample size, using an iterative algorithm: stratum boundaries (related to only one stratification variable) are obtained by minimising the anticipated sample size required for estimating the population total of only one survey variable (so this approach is univariate with respect to both stratification and target variables). In conclusion, most contributions in this group are dedicated to solving the problem of finding best strata boundaries for only one, continuous, auxiliary variable: only Benedetti et al. (2008) deal with the multivariate stratification case.
In case of categorical stratification variables, we could consider the stratification given by their Cartesian product; but when the number of produced strata is high, this could determine a huge sample size, far beyond the one affordable, or the one necessary to ensure the required precision levels. So, a crucial task is to choose the "best� auxiliary variables cross product, i.e., the best partition of the frame, that takes the maximum advantage of the auxiliary information, but at the same time does not lead to an explosion of the number of the strata.
This paper proposes a solution to the problem of jointly determining the optimal stratification of a sampling frame together with the optimal sample size and allocation, in the full multivariate case (i.e., with regard to both stratification and target variables). The only restriction is on the nature of the stratification variables that must be categorical (but we give indications on a suitable way to transform continuous ones into categorical ones). The proposed solution is based on the use of the genetic algorithm. The general procedure has been implemented in an R package, named SamplingStrata, available on the CRAN (Barcaroli, Pagliuca and Willighagen 2013a). This package makes use of a modified version of some functions of another R package, genalg (Willighagen 2012).
The paper is structured as follows: Section 2 contains a formalization of the optimization problem. Section 3 details how the genetic algorithm is employed in order to optimally solve the problem of finding the best stratification that allows the minimal cost of the required sample. To better illustrate this, Section 4 contains an example based on a well known dataset (the "iris flowers� data). Section 5 reports and analyses the results of the application of the algorithm to a real survey, the Italian Farm Structure Survey, and these results are compared to the practical solution adopted by survey statisticians. A further application, to the Monthly Survey on milk and milk products, is reported in Section 6. Final conclusions are contained in Section 7.
- Date modified: