More Data or Better Data? A Statistical Decision Problem

When designing data collection, crucial questions arise regarding how much data to collect and how much effort to expend to enhance the quality of the collected data. To make choice of sample design a coherent subject of study, it is desirable to specify an explicit decision problem. We use the Wald framework of statistical decision theory to study allocation of a budget between two or more sampling processes. These processes all draw random samples from a population of interest and aim to collect data that are informative about the sample realizations of an outcome. They differ in the cost of data collection and the quality of the data obtained. One may incur lower cost per sample member but yield lower data quality than another. Increasing the allocation of budget to a low-cost process yields more data, while increasing the allocation to a high-cost process yields better data. We initially view the concept of “better data” abstractly and then fix attention on two important cases. In both cases, a high-cost sampling process accurately measures the outcome of each sample member. The cases differ in the data yielded by a low-cost process. In one, the low-cost process has non-response and in the other it provides a low-resolution interval measure of each sample member’s outcome. In these settings, we study minimax-regret sample design for prediction of a real-valued outcome under square loss; that is, design which minimizes maximum mean square error. The analysis imposes no assumptions that restrict the unobserved outcomes. Hence, the decision maker must cope with both the statistical imprecision of finite samples and the partial identification of the true state of nature.