Using Ancillary Information to Reduce Sample Size in Discovery Sampling and the Effects of Measurement Error

Axelrod, M

Lawrence Livermore National Laboratory

关键词: 99 General And Miscellaneous//Mathematics, Computing, And Information Science;    Compliance;    Sampling;    Design;    Attitudes;   

DOI  :  10.2172/877925
RP-ID  :  UCRL-TR-216206
RP-ID  :  W-7405-ENG-48
RP-ID  :  877925

美国|英语

来源: UNT Digital Library

【 摘 要 】

Discovery sampling is a tool used in a discovery auditing. The purpose of such an audit is to provide evidence that some (usually large) inventory of items complies with a defined set of criteria by inspecting (or measuring) a representative sample drawn from the inventory. If any of the items in the sample fail compliance (defective items), then the audit has discovered an impropriety, which often triggers some action. However finding defective items in a sample is an unusual event--auditors expect the inventory to be in compliance because they come to the audit with an ''innocent until proven guilty attitude''. As part of their work product, the auditors must provide a confidence statement about compliance level of the inventory. Clearly the more items they inspect, the greater their confidence, but more inspection means more cost. Audit costs can be purely economic, but in some cases, the cost is political because more inspection means more intrusion, which communicates an attitude of distrust. Thus, auditors have every incentive to minimize the number of items in the sample. Indeed, in some cases the sample size can be specifically limited by a prior agreement or an ongoing policy. Statements of confidence about the results of a discovery sample generally use the method of confidence intervals. After finding no defectives in the sample, the auditors provide a range of values that bracket the number of defective items that could credibly be in the inventory. They also state a level of confidence for the interval, usually 90% or 95%. For example, the auditors might say: ''We believe that this inventory of 1,000 items contains no more than 10 defectives with a confidence of 95%''. Frequently clients ask their auditors questions such as: How many items do you need to measure to be 95% confident that there are no more than 10 defectives in the entire inventory? Sometimes when the auditors answer with big numbers like ''300'', their clients balk. They balk because a big sample size might bust the budget, or the number seems intuitively excessive. To reduce the sample size, you can increase the tolerable number of defectives, the ''10'' in the preceding example, or back off on the confidence level, say from 95% to 90%. Auditors also frequently bump up the sample size as a safety factor. They know that something can go wrong. For example, they might find out that the measurements or inspections were subject to errors. Unless the auditors know exactly how measurement error affects sample size, they might be forced to give up the safety factor. Clients often choose to ''live dangerously'' (without a compelling argument to the contrary) to save money. Thus, sometimes the auditor finds that ''you just can't get there from here'', because the goals of the audit and the resources available are inherently in conflict. For discovery audits, there is a way out of this apparent conundrum. It turns out that the classical method of confidence intervals uses an implicit and very conservative assumption. We will see that this assumption is too pessimistic and too conservative in the context of a discovery audit. If we abandon this assumption and use ancillary information about the inventory, then we can significantly reduce the sample size required to achieve the desired confidence level. We will see exactly how the classical method ignores this ancillary information and misses the opportunity for an efficient audit. In the following sections, we first review the standard approach using confidence intervals. Then we present a method that incorporates the ancillary information about the inventory to design a very efficient discovery audit. We also provide results on how measurement errors affect the audit, and how exactly how much the sample size must be modified to compensate for these errors. Finally, we state asymptotic formulas that provide useful approximations for large inventories. It is suggested that the reader review the glossary of symbols while reading the body and the appendices as there are numerous special symbols and notations used in the text.