A New ‘Fuzzy Logic’ Approach
Rebecca L. Rosner, Christie L. Comunale, and Thomas R. Sexton
JUNE 2006 - Two
difficulties that auditors face in assessing materiality are
the need to make a binary decision (material versus not material)
and the need to weigh certain qualitative factors in doing
so. This article shows how a “fuzzy logic” expert
system permits auditors to assess materiality on a continuous
scale from 0 to 1, and allows for explicit consideration of
important qualitative factors relevant to materiality.
practice, an auditor must make an oversimplified, binary
decision for each omission and misstatement, both individually
and in the aggregate: It is either material, or it is not.
A fuzzy logic model allows omissions and misstatements to
possess a degree of value—that is, each omission or
misstatement is material to a greater or lesser degree,
measured on a scale from 0 to 1.
tend to view materiality as a quantitative concept; the
larger the fluctuation, the more likely the auditor is to
consider it material. This is natural, because size is easier
to measure and analyze than nonquantitative factors. Both
the SEC and FASB, however, recognize that although such
thresholds and rules of thumb can be useful starting points,
exclusive reliance on numerical thresholds has no basis
in accounting literature or law. Indeed, materiality assessment
requires the consideration of many qualitative factors beyond
the size of the misstatement or omission. (The authors consider
errors, which may or may not be material, to include misstatements,
in which a numerical or textual item is reported incorrectly,
and omissions, in which a required item is absent from the
financial statement.) SEC Staff Accounting Bulletin (SAB)
99 lists qualitative factors that “render material
a quantitatively small misstatement of a financial statement
item.” These factors ask whether the misstatement
does the following:
Arises from an item capable of precise measurement, or
arises from an estimate and, if so, the degree of imprecision
inherent in the estimate;
Masks a change in earnings or other trends;
Hides a failure to meet analysts’ expectations;
Changes a loss into income, or vice versa;
Concerns a portion of the business that has been identified
as playing a significant role in the company’s operations
Affects compliance with regulatory requirements;
Affects compliance with loan covenants or other contractual
Increases management’s compensation by, for example,
satisfying requirements for the award of bonuses or other
forms of incentive compensation; or
Involves concealment of an unlawful transaction.
has also emphasized that materiality is not strictly a quantitative
concept (Statement of Financial Accounting Concepts 2).
In fact, FASB rejected a formulaic approach for determining
materiality in favor of one that incorporates all relevant
qualitative factors are more difficult to assess than the
size of a misstatement or omission. Qualitative factors
often require subjective judgment and evaluation in light
of other information that may not be readily available to
the auditor during the audit.
auditors tend to rely on quantitative evaluations and to
do so using simplistic numerical thresholds and rules of
thumb. The authors reviewed the materiality worksheets of
three national public accounting firms and found no specific
guidance regarding how to evaluate the qualitative factors
in the materiality assessment process. One worksheet made
no mention of qualitative factors. A second worksheet reminded
the auditor to consider such factors but provided only one
example, that of an illegal payment. The third worksheet
listed the qualitative factors from SAB 99 but provided
no methodology for incorporating such factors into the overall
materiality assessment. In practice, qualitative factors,
while recognized as important, are likely to be overlooked.
of Fuzzy Logic
A. Zadeh first introduced the concepts of fuzzy sets and
fuzzy logic, which rests on the conceptual premise that
an item can have partial membership in a set [“Fuzzy
Sets,” Information and Control, 8: 338–353
(1965)]. In a simple example, consider the set of all “tall”
men. The classical approach requires establishing a threshold
height (e.g., six feet) and declaring that a man is “tall”
only if he is at least six feet tall. Must we be so arbitrary?
Using the fuzzy logic approach, membership in the set of
all tall men could be equal to 0 for men who are five feet
tall, equal to 1 for men who are seven feet tall, and equal
to 0.5 for men who are six feet tall. Exhibit
1 shows one such membership function.
classical logical functions “not,” “or,”
and “and” are built on the classical theory
of sets. For example, consider the set of all men who are
“tall and wealthy.” Classical logic stipulates
that a man must be a member of both sets, the set of tall
men (at least six feet tall) and the set of wealthy men
(which may be defined as having a net personal wealth of
at least $1 million). In a parallel fashion, fuzzy logic
computes the membership of a man in the set of all men who
are both tall and wealthy using the man’s membership
in the individual sets. For example, suppose that a man
has 0.6 membership in the set of tall men and 0.2 membership
in the set of wealthy men. Then he would have 0.2 (the minimum
of 0.6 and 0.2) membership in the set of tall and wealthy
men. Similarly, he would have 0.6 (the maximum of 0.6 and
0.2) membership in the set of tall or wealthy men. Finally,
he would have 0.4 (1 minus 0.6) membership in the set of
men who are “not tall.” All of the standard
logical operations can be used with the principles of fuzzy
and Fuzzy Rule-Based Expert Systems
expert system can be thought of as an advisory board consisting
of many advisors. In this application, the auditor—the
decision maker—would ask each advisor to review a
given omission or misstatement and decide whether it is
material. Imagine that each advisor focuses on only one
aspect of the omission or misstatement in making this determination.
For example, one advisor may consider only the size of the
omission or misstatement, whereas a second advisor may consider
only the extent to which it increases management compensation.
Thus, each advisor represents one of the rules in the expert
system. Finally, the decision maker would assimilate the
various opinions of the advisors and reach a conclusion
on the materiality of the omission or misstatement. In doing
so, the decision maker would also incorporate her assessment
of the aptitude of the advisors in making materiality judgments.
a classical system, each advisor would be required to make
a binary decision: the omission or misstatement is material
or it is not material. For example, the first advisor may
respond that, based on the size of the omission or misstatement,
the omission is not material; the second advisor may say
that, because it increases management compensation, it is
a fuzzy system, each advisor would be allowed to express
his materiality judgment as a number anywhere between 0
and 1, with 1 meaning the omission or misstatement is material,
and 0 meaning it is not material. Their responses represent
initial materiality assessments that would be modified later
based upon the aptitude of the advisor.
a classical system, every rule is assumed valid, meaning
that every advisor’s viewpoint is assumed equally
relevant to the materiality assessment. In a fuzzy system,
not every rule is equally relevant to the materiality assessment.
The decision maker must assign to each rule a validity value
that corresponds to the auditor’s assessment of the
aptitude of the advisor. For example, the auditor may assign
a modest 0.35 validity value to the rule:
(Size of Misstatement is Medium), then (Misstatement
while assigning a greater 0.85 validity value to the rule:
If (Misstatement Increases Management Compensation),
(Misstatement is Material).
the fuzzy system, the auditor then multiplies the initial
materiality membership produced by each rule by its validity
value, to produce the misstatement’s final value.
Therefore, the final materiality assessment involving the
medium size of the omission or misstatement would be 0.14
(0.4 x 0.35). The management compensation issue would yield
a result of 0.765 (0.9 x 0.85). No such computations are
required in the classical system, because the classical
system simply produces a collection of binary (material
or not material) assessments, one for each rule.
last step is to assimilate the final assessments produced
by the rules into an ultimate materiality assessment. The
standard approach is to select the maximum final materiality
value produced by any rule:
Materiality Value = Max (Final Materiality Value of All
the classical system, the misstatement would be assessed
as material if one or more of the rules indicated that the
misstatement was material. In the fuzzy system, the auditor
would use the two formulas above to determine a value indicating
the level of materiality.
Fuzzy Materiality Assessments
auditor might decide on a policy of how to convert fuzzy
materiality values into audit actions. One approach is to
interpret materiality assessments as follows:
the example, the auditor would conclude that the ultimate
materiality assessment is high on the materiality scale;
indeed, close to very high. The firm’s policy would
then dictate the appropriate actions for the auditor to
follow, depending upon the materiality assessment category.
effective fuzzy logic expert system to assess materiality
must incorporate rules that capture both the quantitative
and the qualitative aspects of a misstatement. The quantitative
aspects should include both the size of the misstatement
and the precision with which it is measured. The qualitative
aspects should include the SAB 99 factors listed above.
They could also be defined as fuzzy sets, as shown in Exhibit
Insight for Auditors
routinely encounter misstatements during the course of an
audit. By assigning a fuzzy degree of materiality anywhere
between 0 and 1, the auditor has more flexibility and precision
in materiality assessment, and greater insight regarding
subsequent testing and investigation.
providing a formal model structure, the fuzzy system formalizes
and documents the materiality assessment process. It requires
an auditor to evaluate each qualitative and quantitative
factor explicitly. It also requires that the system designers
state each rule specifically and assign validities to each
rule unambiguously. This facilitates better communication
within the audit team and with the client, and enhances
process consistency across auditors, engagements, and years.
build a valid and reliable fuzzy expert system to assess
materiality, auditing experts need to carefully extract
the quantitative and qualitative factors that auditors should
apply, the fuzzy rules they should use, and the validities
of these rules. Once the initial design is complete, a formal
feedback system may be incorporated to improve future performance.
If materiality assessments later prove to be inaccurate,
the system can modify itself to improve future assessments
of similar situations.
the era of SOX and increased auditor responsibility, where
auditors are expected to do a more thorough job in assessing
the risk of material misstatements, there may be value in
considering the fuzzy logic approach.
L. Rosner, PhD, CPA, CISA, and Christie L.
Comunale, PhD, CPA, are both associate professors
in the school of professional accountancy at Long Island University—C.W.
Post Campus, Brookville, N.Y.
Thomas R. Sexton, PhD, is a professor in
the college of business at Stony Brook University, Stony Brook,