# Attribute Sampling: A Belief-Function Approach to Statistical Audit Evidence

## Article excerpt

SUMMARY

The Dempster-Shafer belief function framework has been used to model the aggregation of audit evidence based on subjectively assessed beliefs. This paper shows how statistical evidence obtained by means of attribute sampling may be represented as belief functions, so that it can be incorporated into such models. In particular, the article shows: (1) how to determine the sample size in attribute sampling to obtain a desired level of belief that the true attribute occurrence rate of the population lies in a given interval; (2) what level of belief is obtained for a specified interval, given the sample result. As intuitively expected, we find that the sample size increases as the desired level of belief in the interval increases. In evaluating the sample results, our findings are again intuitively appealing. For example, provided the sample occurrence rate falls in the interval B for a given number of occurrences of the attribute, we find that the belief in B, Bel(B), increases as the sample size increases. However, if the sample occurrence rate falls outside of the interval, then Bel(B) is zero. Note that, in general, both Bel(B) and Bel(notB) are zero when the sample occurrence rate falls at the end points of the interval. These results extend similar results already available for variables sampling. However, the auditor faces an additional problem for attribute sampling: how to convert belief in an interval for control exceptions into belief in an interval for material misstatements in the financial statements, so that it can be combined with evidence from other sources in implementations of the Audit Risk Model.

INTRODUCTION

Several researchers in recent years have argued that belief functions can provide a better framework for representing uncertainties in audit evidence than probability theory (see, e.g., Akresh et al. 1988; Gillett 1993, 1996; Sharer and Srivastava 1990; Srivastava and Shafer 1992; Srivastava 1993). Whereas in probability theory, probability mass is assigned to individual elementary outcomes in the sample space, in belief functions mass is assigned to subsets of the sample space, which need not be singleton subsets. This increased generality (called nonspecificity) allows for a more natural representation of ignorance on the belief-function framework. In general, the structure of audit evidence forms a network of variables: the accounts in the financial statements, the audit objectives for the accounts, and the financial statements as a whole. Aggregating all the evidence in an audit becomes a problem of propagating beliefs in a network (Shafer et al. 1988; Srivastava 1995; Srivastava et al. 1996).

It often seems that the belief-function approach to modeling the aggregation of audit evidence is limited to subjectively assessed beliefs, and that only probabilistic models are able to incorporate objective assessments based on statistical evidence. In contrast, Srivastava and Shafer (1994) discuss how to incorporate statistical evidence from mean per-unit variable sampling in belief-function models. Some audit procedures, however, require different forms of sampling, such as attribute sampling (e.g., when testing the effectiveness of internal controls). The purpose of the present paper is to show how statistical evidence obtained via attribute sampling may be represented as a belief function so that it, too, may be incorporated in belief-function models of auditing.

The remainder of the paper is divided into four sections, dealing respectively with: the standard statistical approach to attribute sampling, consonant belief functions and their relationship with statistical evidence, the belief-function approach to attribute sampling, and finally a summary and conclusion with potential research problems.

THE STANDARD STATISTICAL APPROACH

Suppose an auditor wants to test, with a certain level of confidence, that the control "customers are not invoiced without supporting documents" is effective. …