A Constructive Procedure for Unbiased Controlled Rounding

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Abstract
It is often necessary to round values in a two-way statistical table, that is, replace each value by an adjacent integer multiple of an appropriately chosen integer rounding base. Such is the case when statistical data are to be presented in increments, such as personal income presented in multiples of $1,000 or retail sales in units of $1 million, or when counts must be adjusted prior to release to reduce the risk of statistical disclosure (Cox, McDonald, and Nelson 1986). It is desirable that the rounded array also be additive along rows and columns and to the grand total. A table satisfying these conditions is called a controlled rounding of the original table. If, as is often the case, entries that are already integer multiples of the base must remain unchanged, the controlled rounding is said to be zero-restricted. Controlled rounding is an important adjunct to statistical methods for adjusting contingency tables such as iterative proportional fitting (Ireland and Kullback 1968), because such methods often do not converge to additive, integer-valued tables as required. It is also desirable that the rounding procedure be unbiased. Nargundkar and Saveland (1972) developed unbiased (so-called “random”) rounding procedures, which Fellegi (1975) was able to make additive, but only for one-way tables. Unbiased controlled rounding is especially useful in sampling and estimation problems. Indeed, the controlled selection problem in two-way stratification (Goodman and Kish 1950) amounts to finding an unbiased controlled rounding (base 1) of the table of expected sample sizes. Cox and Ernst (1982) demonstrated that for any two-way table controlled roundings to any base always exist, and presented methods for finding controlled roundings optimally close to the original table. Causey, Cox, and Ernst (1985) extended these results to produce unbiased controlled roundings, thereby solving the controlled selection problem. These methods rely upon mathematical programming algorithms that, although efficient, are complex. In this article I present a constructive algorithm for achieving unbiased controlled rounding that is simple enough to be implemented by hand. Extension of this method to the larger problem of achieving balanced adjustments in tables is easily seen, thereby demonstrating that random perturbation methods for controlling statistical disclosure (Newman 1975), heretofore not additive, can be made additive. The controlled rounding problem in three dimensions is discussed and a counterexample to the existence of unbiased controlled rounding in three dimensions is given.

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