1. Introduction
David G. Steel and Robert Graham Clark
Unequal unit costs have been reflected in sample designs by using simple linear cost models. In stratified sampling, a per-unit cost coefficient can sometimes be estimated for each stratum. The resulting allocation of sample to strata is proportional to the inverse of the square root of the stratum cost coefficients (Cochran 1977). In a multistage design the costs of including the units at the different stages of selection can be used to decide the number of units to select at each stage (Hansen, Hurwitz and Madow 1953).
While this theory is well established, unequal costs have not been used extensively in practice (Brewer and Gregoire 2009), perhaps because of a lack of good information on costs, and because of a focus on sample size rather than cost of enumeration. Groves (1989) argued that linear cost models are unrealistic, and that mathematical cost modelling can distract from more important decisions such as the mode of collection, the number of callbacks and how the survey interacts with other surveys conducted by the same organisation. Nevertheless, given the pressures on survey budgets, the final design should reflect costs and variance in a rational way, without being fixated on formal optimality.
Increasing use of computers in data collection is leading to more extensive and useful cost-related information on units on survey frames. In a programme of business surveys conducted by a national statistics institute, most medium and large enterprises will be selected in some surveys at least every year or two. This may provide information on costs for those businesses, for example some businesses may have required extensive follow-up or editing in a previous survey. Direct experience is less likely to be available for any given small business, but datasets of costs could be modelled to give predictions of likely costs.
Adaptive and responsive survey designs make use of paradata (process data) collected during a survey's operation, and auxiliary data known for the sampling frame (typically from administrative sources), to guide ongoing decisions. These may include the number of callbacks, which respondents to follow up, targeting of incentives, and choice of mode of collection for followup attempts (Groves and Heeringa 2006). In one example discussed by Groves and Heeringa (2006), interviewers designated non-respondents as having either low or high propensity to respond. The latter are less costly to convert to respondents, and a higher sampling fraction was assigned to them in a second phase of the survey. More recently, Schouten, Bethlehem, Beullens, Kleven, Loosveldt, Luiten, Rutar, Shlomo and Skinner (2012, Section 6) suggested that followup in the second phase of a survey should be designed to improve the R-indicator of non-response bias (defined in Schouten, Cobben and Bethlehem 2009; and in Schouten Shlomo and Skinner 2011). Peytchev, Riley, Rosen, Murphy and Lindblad (2010) argued that likely non-responders should be targeted with a different protocol from the very outset of a survey.
Thus, unequal unit costs can arise in practice, either for all units in advance of sampling, or for non-respondents who are to be targeted for followup. In either case, the collection and use of cost information incurs some expense and additional complexity. Moreover, effectively trading off cost and variance is only part of the picture, and response bias must also be considered. It is therefore important to understand whether the potential gains from using this information are worthwhile, particularly as any cost data is likely to be imperfect.
This paper develops relatively simple approximations to the gains arising from using unit level cost information in a model-assisted framework. Section 2 contains notation and some key expressions. Section 3 is concerned with the optimal design when cost parameters are known. Section 4 analyses the use of estimated unit costs, and Section 5 presents examples. Section 6 offers a discussion.
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