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thesis entitled

THE APPLICATION OF BAYESIAN STATISTICS TO AUDITING:
DISCRETE VERSUS CONTINUOUS PRIOR DISTRIBUTIONS

presented by

Albert K. Francisco

has been accepted towards fulfillment
of the requirements for

Ph.D degreein Business - Accounting

 

Date May ‘5, 1972

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ABSTRACT

THE APPLICATION OF BAYESIAN STATISTICS TO
AUDITING: DISCRETE VERSUS CONTINUOUS
PRIOR DISTRIBUTIONS

BY

Albert Kenning Francisco

It has been suggested in the auditing literature
that auditors adept Bayesian statistical techniques.
Studies have shown that auditors are willing to provide
information that can be used to construct prior distri—
butions for this purpose. Practicing auditors, however,
have not widely adepted Bayesian techniques, even though
such techniques have received considerable exposure in
the literature. The articles suggesting the use of
Bayesian techniques have all demonstrated such techniques
with discrete prior distributions.

Although less difficult to work with than contin-
uous prior distributions in simple examples, discrete
prior distributions have several practical disadvantages
which may partly explain the lack of adoption of Bayesian
methods by auditors. Discrete prior distributions are
poor approximations of the continuous range of possible

rates found in most audit attributes sampling situations.

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Albert Kenning Francisco
The derivation of specific discrete prior distributions is
difficult, because a number of points must be assigned
probabilities, and these probabilities must total 1.00.
It is difficult for an auditor to perceive the meaning of
a discrete distribution because only a few of the possible
points are assigned positive probabilities. Involved com—
putations (many multiplication and division Operations) are
required for the solution of any practical problem using
discrete distributions and Bayesian statistics. This would
usually require a computer.

Bayesian methods using beta prior distributions can
overcome most of these difficulties. Such a distribution
is continuous, and fits the large number of possible popu-
lation error rates better than does the discrete model.

It is easier for an auditor to visualize continuous curves.
Sketches of representative distributions deve10ped in this
research can allow an auditor to quickly pick the specific
prior desired, and write down the numeric parameters of that
distribution. Tables of statistics (mean, variance, and
mode) of these representative distributions can provide
additional information to the auditor seeking a prior for

a given audit situation. The revision of a beta prior by

a binomial sample result is not difficult. A few addition

and subtraction Operations are required. These can be

 

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Albert Kenning Francisco
performed by an auditor with the aid of nothing more
soPhisticated than a piece of paper and a pencil. Once
this is done, the auditor can refer again to his tables for
a sketch of the posterior distribution (or a very similar
one) and for statistics he wishes to know for that distri-
bution. Another set of tables deve10ped in this study can
give confidence intervals. What would have taken consider-
able computer time with a discrete prior distribution is
quickly taken care of with this method using a few tables
and a quick addition and subtraction Operation.

Although tables and sketches of the beta distribu-
tion are not currently available, techniques exist with
which such information can be generated. The formulas for
computing statistics of beta distributions are not complex.
Tables of such statistics can be generated in seconds on
modern computers. Similarly, sketches of such distribu-
tions can be created by plotters attached to computers.
Simpson's rule for the evaluation of definite integrals
can be used as the basis for computer programs which will
generate confidence intervals based upon representative
posterior beta distributions. Examples of sketches and
tables are included in the study.

It is concluded that, although auditors have not

adOpted Bayesian statistical methods, such methods hold

Albert Kenning Francisco

promise. There are more SOphisticated statistical techni—
ques available than those demonstrated in the articles
which have prOposed Bayesian methods for use in auditing.
Such techniques, such as the beta prior distribution dis-
cussed here, may allow auditors to benefit from the advan-

tages Of Bayesian statistics that have been claimed in the

literature.

THE APPLICATION CW'BAYESIAN STATISTICS TO
AUDITING: DISCRETE VERSUS CONTINUOUS

PRIOR DISTRIBUTIONS

BY

Albert Kenning Francisco

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Accounting and

Financial Administration

1972

COpyright by
ALBERT KENNING FRANCISCO
© 1972

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ACKNOWLEDGMENTS

Thanks are due the members of my dissertation
committee, Professors Alvin Arens (chairman), George
Mead, and Robert Staudte for their review of my thesis
and constructive suggestions. Appreciation is also
expressed to Professors James Don Edwards and Gardner
Jones who, as department chairmen, ensured financial
support during my Ph.D. program. I am also indebted
to Professor James E. Sorensen of the University of
Denver for arousing my interest in the general area
in which the dissertation was done.

Appreciation is due the organizations which
provided financial support for my Ph.D. program. These
organizations include the American Accounting Associa-
tion, Ernst & Ernst, and the U. S. Government for its
National Defense Education Act fellowship.

Finally, appreciation is expressed to my parents,
for encouraging me to remain in school when other alter-
natives arose, and to my wife Barbara and daughter Eileen

for help and encouragement throughout my graduate studies.

ii

of

 

I would like to thank Mrs. Donna Ford for an excellent job

of typing the final manuscript.

iii

 

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II”

 

TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS. . . . . . . . . . . . . . . . . . . ii
LIST OF TABLES. . . . . . . . . . . . . . . . . . . . Vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . viii

Chapter

I INTRODUCTION. . . . . . . . . . . . . . . . l
A. The Auditor in Society. . . . . . . . 1
B. Audit Sampling. . . . . . . . . . . . 3
C. Classical Statistics. . . . . . . . . 6
D. Bayesian Statistics . . . . . . . . . 7
E. Purpose of this Study . . . . . . . . 8
F. Limitations of the Study. . . . . . . 11
II AUDIT EVIDENCE AND AUDIT VARIABLES. . . . . 14
A. Audit Evidence. . . . . . . . . . . . 14
B. Audit Variables . . . . . . . . . . . 16
C. Effect of Variables on Audit. . . . . 20

III ATTRIBUTES SAMPLING AND CLASSICAL

STATISTICS. . . . . . . . . . . . . . . . . 24

A. Introduction. . . . . . . . . . . . . 24
B. Classical Statistics as Used by

Auditors . . . . . . . . . . . . . . 26
C. Effect of Audit Variables . . . . . . 30

iv

Ch

 

 

VII

Chapter

IV

VI

VII

ATTRIBUTES SAMPLING USING BAYESIAN
STATISTICS: DISCRETE PRIOR
DISTRIBWIONS . O C O C C O C O O O O O O

A.
B.
C.

Bayes Theorem. . . . . . . . . . .

Discrete Prior Distributions . . .

Difficulties Found in the Use of
Discrete Prior Distributions. . .

BAYESIAN STATISTICS AND THE BETA: A
CONTINUOUS PRIOR DISTRIBUTIONS . . . . .

A.

B._

C.

D.

E.

The Beta Distribution. . . . . . .
Selection of a Prior Distribution
Revision Of a Beta Prior by a
Binomial Sample Result. . . . . .
Confidence Intervals from Beta
Posterior Distributions . . . . .
An Example of the Use of a
Continuous Prior Distribution . .

SUBJECTIVE PROBABILITIES . . . . . . . .

A.
B.

Definition . . . . . . . . . . . .
Audit Evidence and Prior
Distributions . . . . . . . . . .

CONCLUS I ON C O C C O C C O O O O O O O O

A.
B.

C.
D.

Summary. . . . . . . . . . . . . .
Relative Advantages of Various
Distributions in Applications

of Bayesian Statistics to
Auditing. . . . . . . . . . . . .
Conclusions. . . . . . . . . . . .
Suggestions for Further Research .

Page

35

35
37

43

49

49
61

66
68
73
79
79
83

89

89

93
95
98

SE

. 1- IIIUIINS

 

APPENDIX Page
A. BETA DISTRIBUTION TABLES. . . . . . . . . . . . 100
B. SKETCHES OF BETA DISTRIBUTIONS. . . . . . . . . 105

C. REVISION OF A BETA PRIOR DISTRIBUTION BY A
BINOMIAL SAMPLE RESULT . . . . . . . . . . . . 110

D. COMPUTER PROGRAMS . . . . . . . . . . . . . . . 113

E. EVALUATION OF INTEGRALS NOT SOLVABLE BY
NORML MEANS C O C O O O O O 0 O O I C O O O C 12 3

SELECTED BIBLIOGRAPHY . . . . . . . . . . . . . . . . 125

vi

LIST OF TABLES

Table
1 Statistics of Selected Beta
Distributions. . . . . . .
2 One Sided Beta Confidence
Coefficients . . . . . . .
3 One Sided Beta Confidence
Coefficients . . . . . . .
4 One Sided Beta Confidence
Coefficients . . . . . . .
5 Computer Program "Table". .
6 Computer Program "Plotit" .
7 Computer PrOgram ”Value”. .

vii

Page

101

102

103

104

115

119

121

Figure

10

11

12

13

14

15

16

Beta Distribution Graph<x=1.40 fi=8.60.

Beta Distribution Graph.d330.40 p=69.60.

Graph of

Mean and

TWO

Two

Two

Two

TWO

Two

TWO

TWO

Two

Beta

Beta

Beta

Beta

Beta

Beta

Beta

Beta

Beta

Beta

Beta

Beta

LIST OF FIGURES

Beta Curve «=2 ps8. . . .

Mode of Beta Distribution

Curves

Curves

Curves

Curves

Curves

Curves

Curves

Curves

Curves

Curves

Curves

Curves

with

with

with

with

with

with

with

with

with

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Page
51
51
54
59

106
106
106
107
107
107
108
108
108
109
109

109

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CHAPTER I

INTRODUCTION

A. The Auditor in Society

The purpose of accounting is to provide informa-
tion useful for decision making which affects the alloca-
tion of resources. Such decisions occur at all levels of
organized society from the individual level to that of

entire political entities.1

The reliability of accounting
communication is essential to the effective conduct of
economic exchange and taxation. A high degree of accept—
ability of accounting communication is needed between the
Opposite parties in such transactions. Either intentional
or unintentional errors could retard the functioning of
the entire system.

The more separated the parties (or the more infre-
quent the contact between the parties) the more the reports
become susceptible to inaccuracy and misunderstanding. The
value of an independent third party reviewing the accounting
report and vouching for its reliability is apparent. For

example, investors are separated from the use of their capi-

tal by management. Thus, the financial statements of most

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publicly owned corporations are audited by independent
certified public accountants (CPAs). Audits are imposed

as a matter of governmental regulations or can be voluntarily
adopted by the parties concerned.2

Audits are performed by a variety of organizations
in the United States. Almost everyone has heard of the tax
audit, performed by Internal Revenue Service agents on indi—
vidual or business records to substantiate claims made on
income tax returns. The Congress of the United States has
organized its own auditing agency, the General Accounting
Office, to report on the spending of public monies. Many
business organizations have their own internal auditing
group which continually checks upon the accuracy of records
and upon compliance with managerial policies.

The most common type of audit in the business world,
however, is performed by an independent firm of accountants
upon the financial statements of a business. The goal of
this type of audit is for the auditing firm to render an
Opinion upon the fairness with which the financial state-
ments issued to stockholders, creditors, various government
regulatory agencies, and the public at large reflect the
true condition Of the business entity. This written Opinion
accompanies the statements when they are issued and provides

assurance that statement users are entitled to rely upon the

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statements to a greater extent than if management had issued

them with no outside check upon their accuracy.

B. Audit Sampling

At one time, an audit involved the examination of
every transaction entered into by the firm and the resulting
record in the accounts.3 Because of the increased size of
today's business entities and the number of transactions,
auditors now restrict their tests to a sample of the records.
This reduces the time required to perform an audit and gives
an acceptable degree of reliability that major errors will
not appear in the statements. Any sampling plan, however,
increases the risk that a material error will occur and not
be found by the auditor; since he selects and examines only
a part of the available evidence. This risk must be bal-
anced against the advantages of sampling.

Until rather recently, auditors in general sampled
by a "seat of their pants" approach and took a sample of
items from some population on the basis of past experience
and intuition. This method worked reasonably well in the
majority of cases, and the judgment of an experienced audi-
tor is an indefinable quality which can not be replaced by
sophisticated mathematics. However, it can be aided and

refined by certain tools, as it has been recently with the

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4
limited adoption of classical statistics as an auditing
tool.

Several methods are used in practice to obtain
audit samples. Some Of these methods are the block, month,
haphazard, random, and statistical methods of sampling.

A block sample is Obtained by selecting a group of
items listed together in a sequence. For example, out of
checks numbered from 1 to 10,000 written during the year
under examination, the auditor may examine those numbered
2,083 through 2,336. Because the selection of one sample
item is more likely to occur if another close to it in the
sequence is also drawn, the selections of items are not
independent of one another and inferences can not be made
using statistical theory on the basis of such a non-random
sample. This is true even though a random method may have
been used to select the particular block of items examined.
With some filing systems it is so difficult to locate spe—
cific items that the use of block sampling is justified,
even though statistically based inferences can not be made
on the results.

A relatively common practice, especially on smaller
audits, has been for the auditor to select one month out of
the year and then to carefully examine entries, vouchers,

etc., for that month. Similar items relating to the rest

5

of the year are not examined. As was the case in block
sampling, statistical inferences can not be made on the
basis of such a procedure. The simplicity and convenience
of this method make it useful where specific documents are
filed by month instead of by serial number.

Another method used consists of the auditor selecting
a few items here and a few there, then making his decisions
on that basis. Appropriately, this method is known as the
haphazard method for Obtaining audit samples. As with the
other methods described above, statistical inferences can
not be made because the selection of one item is not statis—
tically independent of the selection of other items.

All three methods described above are non—random
selection methods, called judgmental methods. Any infer-
ences made on the basis of these methods are based solely
on the judgment of the individual auditor.

Statistically valid inferences are, in general,
made only on the basis of a truly random sample. A random
sample plan is any method of sampling in which the prob-
ability Of a given item being selected is not affected by
another item being selected. Samples can be either with
replacement (a given item can be drawn more than once) or
without replacement (a given item can not be drawn again

once it has been chosen). Different statistical theories

6

have been derived to deal with these two cases.

C. Classical Statistigs

If a random sample is drawn, inferences can be made
on the basis of statistical theory. There are many models
for different situations in classical statistics, but all
share one common trait--probabilities can be mathematically
derived from a random sample result. These probabilities,
given a specific pOpulation and sample therefrom, should
be the same no matter which statistician computes them.
Judgmental inference, on the other hand, provides no means
for different auditors to come up with the same answers in
complex situations. Not only are statistical methods more
consistent from one user to the next, but a body of mathe-
matical theory is available to support statistical results.
No similar theory backs the user of judgmental methods.

In the last several years classical statistics has
become popular as an auditing tool. The larger CPA firms
engaged in auditing have statistical sampling worksheets,
tables, etc., which use classical statistical methods to
arrive at inferences based on audit samples. This increased
popularity is probably due to several factors. Among them
are the inherent applicability of classical statistics to

many audit sampling situations, the better education in

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statistics of recent college graduates entering the firms.
and the leadership of various groups such as the American
Institute of Certified Public Accountants in promoting the
use Of new methods.

Recent research suggests that Bayesian statistics
may be even more applicable to certain audit situations
than classical statistics. Bayesian methods are discussed

in the following section.

D. Bayesian Statistics

Bayesian statistics is similar to classical methods,
but uses a prior probability distribution as well as the
current sample result. The combination of the two distribu-
tions is called a pgsterior distribution and is a sort of
mathematical average of the two Obtained with the use of
Bayes theorem.

Bayesian methods recognize the subjective nature of
some types of audit evidence. Instead of ignoring these
sources as classical statistics models do, Bayesian statis-
tics attempts to accept them as part of the analysis. In
so doing, the auditor, based on his past experience, is
forced to put down on paper what he expects of various parts
of the audit. This can provide more evidence that a suit-

able audit will be conducted than do alternative methods

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8
which ignore these sources of information altogether. With
the increasing frequency of lawsuits naming auditing firms
as defendants for not living up to expectations in the work
done, adequate defense is more and more important to the

accountant engaged in auditing.

E. Purpose of this Study

The purpose of this study is to eXplore more deep-
ly the Bayesian method, which has been proposed and shown
to be of at least limited applicability in auditing.4
Classical statistics, as useful as it is, has certain
limitations, which will be explored. In addition, certain
practical difficulties inherent in the Bayesian methods
which have been prOposed in the literature will be discussed
and an alternative prOposed.

Any audit involves the gathering of evidence. The
amount and kinds of evidence which are obtained should be
determined by the audit variables. Whether judgmental
methods, classical statistics, or Bayesian statistics is
used, it is necessary that the audit variables be considered.
Because the subject is so important to any audit, Chapter

II is devoted to a discussion of audit evidence and audit

variables.

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Auditors have used classical statistics for several
years. It has proven to be a valuable audit tool. But it
has certain limitations, which advocates of Bayesian statis-
tics have claimed their method can overcome. In addition,
the method which auditors call classical statistics is not
the method used by statisticians who consider themselves
classicists. A discussion of this distinction, as well as
a description of "classical auditing" statistics is present-
ed in Chapter III.

The accounting literature has included several
suggestions that Bayesian statistics be used by auditors.
Examples have shown how Bayesian statistics with discrete
prior distributions can be applied to selected auditing
situations. Although both the binomial and the hyper—
geometric distributions have been used as prior distribu-
tions in those examples, no explanation of the difference
between the two or the advantages of either has appeared.
Chapter IV of this study shows how the discrete binomial
and hypergeometric distributions can be used as prior
distributions in auditing situations and explains the differ-
ence between the two.

Although continuous distributions are used in many

statistical applications, such distributions have not

10
appeared in the literature as prior distributions in appli-
cations Of Bayesian statistics to auditing. One continuous
distribution which has potential in those applications is
the beta. To be used in auditing, however, it is necessary
to have tables and graphs of the beta. The use of the beta
as a prior distribution in auditing, as well as the develop-
ment of the necessary tables and graphs, is discussed in
Chapter V.

The application of Bayesian statistics to auditing,
with either discrete or continuous prior distributions,
requires that the relationship between the audit variables
and prior distributions be considered. Additionally, an
auditor attempting to use Bayesian methods needs informa-
tion about subjective probabilities and the criticism
which the use of such probabilities has brought from stat-
isticians who feel statistical methods should not be applied
to situations which can not be repeated. Others have
held that improved decision making capability is suffi—
cient justification for the use of subjective probabilities.

These tOpics are explored in Chapter VI of this study.

11

F. Limitations of the Study

In audit variables sampling, classical statistics
gives almost the same results as those obtained through the
use of Bayesian statistics. This is because the prior dis—
tribution provided by the auditor in variables sampling
situations has a large variance. Internal control evalua-
tions, work on other accounts, and prior audit eXperience
gives little evidence to indicate whether the cash account
should be $100,000 or $150,000 for example. In attributes
sampling, on the other hand, these sources of information
do give the auditor a good basis from which to predict
likely error rates, so that the prior distribution provided
by the auditor can contribute significant information. For
this reason, the study is concerned only with attributes
sampling.

A Bayesian method using a continuous prior distri-
bution is prOposed in this study. Illustrative tables and
sketches, not complete enough for general use, are shown.
Extended tables and sketches would require computer time and
other resources beyond those available.

The general subject of Bayesian statistics in audit-
ing includes many subsets which have not yet been research—

ed. Although the exploration of most of these areas is

12
beyond the sc0pe of this study, some are mentioned in the

last section of Chapter VII.

 

:lll Ill-Ill I III! |

 

13

CHAPTER I--FOOTNOTES

1The Study Group on Introductory Accounting, A New
Introduction 29 Accounting (Seattle, Washington: Price
Waterhouse Foundation, 1971), p. 11.

2

 

 

Ibid., p. 18.

3Bookkeepers' Handy Guide, (New York: The Ronald
Press Company, 1936), p. 518.

 

 

4For example, see John A. Tracy, "Bayesian Statisti-
cal Methods in Auditing, The Accounting Review, January,
1969, p. 90.

CHAPTER II

AUDIT EVIDENCE AND AUDIT VARIABLES

A. Audit Evidence
"The objective of the ordinary examination of
financial statements by the independent auditor is the
expression of an Opinion on the fairness with which they
present financial position and results of Operations."1
The auditor's Opinion, however, must be based upon evi-
dence obtained by the auditor in the course of his examin—
ation.
Sufficient competent evidential matter is to
be Obtained through inspection, observation,
inquires and confirmations to afford a reason-
able basis for an Opinion regarding the finan-
cial statements under examination.2
Audit evidence is a general term which includes all
those factors in a given audit upon which the Opinion is
based. In addition to written material, such as reconcilia-
tions and confirmations, the term evidence covers such

intangible factors as the competence of personnel in the

client's accounting system.

14

15

The auditing profession provides few specific

guidelines for auditors to use in their determination of

what to require as evidence or how much evidence is neces-

sary in a given situation.

The amount and kinds of evidential matter re-
quired to support an informed opinion are
matters for the auditor to determine in the
exercise of his professional judgment after

a careful study of the circumstances in a
particular case.3

There are many kinds of audit evidence, some more

important to the auditor than others. Mautz and Sharaf

categorized audit evidence into nine general types in The

 

 

Philosophngf Auditing:

1.

Physical examination by the auditor of the
thing represented in the accounts

Statements by independent third parties
written
oral

Authoritative documents
prepared outside the enterprise under
examination
prepared inside the enterprise under
examination

Statements by officers and employees of the
company under examination

formal

informal

Calculations performed by the auditor

Satisfactory internal control procedures

l6

7. Subsequent actions by the company under
examination and others

8. Subsidiary or detailed records with no
significant indications of irregularity

9. Interrelationships with other data4

The overall result of combining the individual items
of evidence is used as the basis for the auditor's opinion.
This involves an evaluation of the evidence gathered in
light of the materiality of the several accounting prOposi-
tions which the Opinion is to cover. In this evaluation,
several factors, or "audit variables", are considered by
the auditor in determining whether sufficient evidence has

been accumulated, as outlined in the following sections.

B. Audit Variables

An audit is an investigation, the extent of which
depends upon specific circumstances encountered. Even under
the most ideal conditions a certain minimum amount of evi-
dence must be gathered. For any given audit there will be
a minimum audit program which will be applied if all circum-
stances and the results of all tests meet the auditor's best
- 5
expectations.

The variables of the audit are those factors or
circumstances of a particular audit situation
which should affect an auditor's decision on

which procedures to apply and the extent of
their application.

17
Some audit variables determine the minimum audit program,
while others result in procedures beyond the minimum audit
program.

The variety of situations encountered by an auditor
is quite large. To set forth here some of the more impor-
tant variables which the auditor may encounter, let us
summarize briefly two recent items from accounting litera—
ture. Extended discussions about audit variables appear in
a 1970 article by Anderson, Giese, and Booker? and in a 1970
thesis by Arens.8

Audit variables may be broken down into several cata-
gories. Here three general categories will be considered:
1) the auditor's environment, 2) the client's environment,
3) internal factors peculiar to that client.

The auditor’s environment includes several factors
which affect the evidence required for an audit. The rela-
tive sizes of the auditing firm and the client are an impor-
tant variable.9 If the auditing fee paid by that one client
is a substantial portion of the auditing firm's billings,
questions may arise as to the auditor's appearance of inde-
pendence. In such circumstances the auditor should be
extremely cautious to avoid arousing suspicion about his

work and therefore should take relatively large samples.

18
Another variable from the auditor's environment is the
auditor's background. If he has had excellent experience
with this client and similar clients, he may be willing to
reduce sample sizes somewhat because his risk appears to be
lower than with other clients. In addition, previous years'
results of specific audit tests are important to the audi-
tor, when the results are available. If tests in the area
have produced few errors in past years, the auditor will
have some confidence that the area will again produce few
errors.

Another category of audit variables relates to the
client's environment. The absolute size of the client's
organization is of interest to the auditor.10 The auditor
of a small client which has little public exposure faces
less risk than the auditor of a larger client. This is
simply because there are fewer parties that can be damaged.
In the audit of a smaller client the only directly interest-
ed party may be a banker, who knows nearly as much about
the client's affairs as does the auditor. Consequently,
the auditor is less likely to have his work attacked.
Another variable of interest to the auditor is the client's
industry.11 Some industries are noted for steady growth

and others for wide fluctuations in sales and earnings and

19

resulting management difficulties. An auditor would require
more evidence in the audit of a client in an unstable indus-
try than for a similar client in a stable one.12 Another
variable is the type of legal form used by the client.
Different legal obligations are placed upon the auditor
depending on the client's legal form, e.g. a corporation or
a partnership. If the client is listed on a major stock
exchange, requirements are often imposed which may increase
the risk assumed by the auditor. Similarly, if securities
issued by the client are subject to the registration re—
quirements Of the Securities and Exchange Commission, Spe-
cial rules are imposed by law upon the auditor. The mode
of financing chosen by the client is a variable of interest
to the auditor, as is the profitability of the client.13
Finally, general economic conditions can change the audi-
tor's risk, and these should be considered. A sharp
downturn in the economy can have adverse effects upon many
organizations, and difficulties caused by such a slump can
result in close scrutiny of the auditor's work.

The final category of audit variables is composed
of items from within the client's organization. Internal

auditing results are often an important variable to the

outside auditor, as are other items related to internal

20

control.14 Another factor is the applicability of the
client's records to auditing and the condition of those
records. Each test made in the course of an audit has
variables associated with it, e.g. the pOpulation size and
the materiality of the item subject to the test.15

Many more variables could be listed. These should,
however, be sufficient for the purposes at hand. In an
audit the auditor weighs all these and more factors in his
mind and then produces an audit program, complete with
sample sizes and specific items to be examined. In the
past much of this process has been based upon arbitrary
judgments and previous practice. Audit variables should
have an effect upon the audit, whether judgmental or stat-
istical methods are used. This effect is discussed in the

following section.

C. Effect of Variables on Audit

When the auditor discovers variables which indicate
that more than the minimum audit program is necessary,
there are several routes he can take. If he feels the tests
are adequate for the circumstances, he can increase sample
sizes, thus increasing the probability that the sample re—
sult will reflect the actual situation. In some audits

he may wish to change the procedures performed in order to

21

anticipate a greater variety of possible discrepancies.
When the situation is highly abnormal, new personnel may be
assigned to the audit to bring in skills especially appli-
cable to the areas of concern. Finally, the timing of the
tests may be changed.
These actions are, however, not without cost to
the auditor. Increased sample sizes, additional procedures,
timing variations, and more experienced personnel are all
actions which increase the cost of an audit. Situations
exist where all or a good portion of these costs can be
passed on to the client. In other cases, the auditor him—
self must absorb them. This limits the extent to which
the auditor can increase his costs without incurring losses.
Once the auditor understands the variables in a
particular engagement, he must have an algorithm to deter—
mine the evidence required. Because this study is about
statistical sampling in auditing, it is concerned only with
the size of the samples taken. The specific procedures
used in the audit, the timing of the tests, and the person—
nel are important items, but are not within the scope of
this study. However, the audit variables do influence
the selection of the population, the definition of the

attribute to be tested, and the precision limits and

22
confidence level deemed necessary by the auditor. In the
following chapters the variables will be considered in the

context of Bayesian and classical statistics.

23

CHAPTER II--FOOTNOTES

1Committee on Auditing Procedure of the American
Institute of Certified Public Accountants, Auditing
Standards and Procedures (New York: American Institute
of Certified Public Accountants, 1963), p. 9.

 

21bid., p. 16.
3 .
Ibid., p. 36.
4R. K. Mautz and Hussein A. Sharaf. The Philosthy

_g£ Auditing (American Accounting Association, 1961),
p. 86.

 

 

5R. K. Mautz and Donald L. Mini, ”Internal Control
Evaluation and Audit Program Modification," The Accounting
Review, April, 1966, p 284.

 

6Alvin A. Arens, The Adeguacy 9; Audit Evidence
Accumulation ig_Public Accounting (unpublished Ph.D.
thesis, University of Minnesota, 1970), p. 17.

 

 

 

 

7H. M. Anderson, J. W. Giese and Jon Booker,
"Some PrOpositions About Auditing," The Accounting Review,
July, 1970, p. 524.

8Arens, p. 23-76.

9Anderson, Giese and Booker, p. 528.

loIbid., p. 528.

11Arens, p. 41.

12Anderson, Giese and Booker, p. 529.

l3Ibid., p. 529.
14Ibid., p. 528.

15Arens, p. 47.

 

bution
random
one or
ment.1
"A Ber.
0UtC0mI
Stlldy ;
ing rec
example
One auc
bank CC

cash ac-

 

variabl

 

CHAPTER III

ATTRIBUTES SAMPLING AND

CLASSICAL STATISTICS

A. Introduction

Statistics involves the use of probability distri-
butions to make inferences about some population using
random samples taken from that pOpulation. The taking of
one or more samples from a pOpulation is called an experi-

ment.1

One execution of an experiment is called a trial.

"A Bernoulli trial is an experiment which has two possible
outcomes, generally called success and failure."2 In this
study a trial will either produce an "error" in the account-
ing records, or will not produce an error. There are many
examples of Bernoulli trials in auditing: the result of
one audit is either an unqualified Opinion or it is not: a
bank confirmation either reconciles satisfactorily with the
cash account or fails to; an accounting student either
passes a course or fails it.

When one or more trials of a Bernoulli random

variable occur, the probability distribution of the number

24

25

of errors found in those trials in called a binomial.

 

The probability of any possible result occurring can be
computed if two things are known
n the number of trials

p the probability of success on any one
trial.

The computation is made with the following formula in which
x is the number of errors for which the probability is being

found:

p (X) =' (2) pan‘x

where (2)8 (mi): 'x'

 

n18 n'(n-l)‘(n-2)°°'---°(2)-(l)

q =l-p

Because there are tables and approximation methods
available, the formula can be bypassed. It is significant
‘with the binomial that the probability of a success must
be constant from one trial to the next, implying either
replacement of each item sampled in the pOpulation before
another is taken or an infinitely large pOpulation.

The hypergeometric distribution is similar to the
‘binomial except that it is drawn from a pOpulation of
finite size, resulting in a changing probability of success

after each trial. Computations are somewhat more involved

than

Assun

 

whict
follc

the

For 5
geome
probe
is re
more

to a;

stati

in th

 

 

 

26
than with the binomial because of this added variable.
Assume the existance of a finite pOpulation of m items of
which w are successes. If a sample of size n is drawn the
following formula gives the probability of z successes in
the sample:
(1‘1) (:23)

10(2) = (‘3) ,

k= 0,1,2, ..... n

 

........using the convention (2)30 for a)b.3
For small samples taken from large populations the hyper-
geometric closely approximates the binomial because the
probability of a success changes very little when one item
is removed from the large pOpulation. Because tables are
more readily available for the binomial, it is Often used
to approximate the hypergeometric for large populations.

Statisticians and auditors have used classical

statistics in different manners. The distinction is covered

in the following section.

B. Classical Statistics as Used by Auditors

One way in which classical statistics is used in
auditing is the creation of confidence intervals. Confi-
dence intervals are ranges within which the actual value of
lsome variable is likely to be. For example, the result of

a sample may allow an auditor to state that he is 80%

 

 

 

conf:
subj]
reser
fider

to st

 

0.057

sariI
fideI
abou‘
resu
fixel
of i
an e.
the I
samp;
ical
mator
90% c

be co

 

27
confident that the rate of errors made on payroll checks
subject to audit is less than 0.4%. Such a statement rep-
resents a one sided confidence interval. A two sided con—
fidence interval would result if a sample allowed an auditor
to state that the error rate on payroll checks was between
0.05% and 0.10% with 80% certainty.

Auditors using classical statistics do not neces-
sarily understand the meaning of such statements about con-
fidence intervals. Classical statistics makes statements
about an unknown population parameter on the basis of sample
results. The unknown pOpulation parameter is considered
fixed.4 Suppose an auditor takes a sample from a population
of items. Each sample item taken is either an error or not
an error. The auditor will want a confidence interval about
the percentage of errors (p) in the pOpulation. If the
sample consists Of 100 items, 20 of which are errors, class-
ical statistics would say that the maximum likelihood esti-

A

mator of the population error rate is p = 0.20. A two sided

90% confidence interval for the pOpulation error rate would
be computed in the following manner:
Confidence Limits = 3 5; 55% r—fig = 0.20 ;_I-_ l.645(0.04)
= 0.20 1 0.0658.
Classical statistics would interpret this to say that p

lies in the interval 0.1342 to 0.2658 with confidence

28
coefficient 0.90.5 This means that if a very large number
of samples were taken from this same pOpulation and a
confidence interval constructed from each in the above
manner, 90% of those intervals would contain the actual
population error rate p.
This is not the manner in which auditors have typi-

cally used classical statistics. They have gone through
the computations shown above and then made probabilistic
statements about the population parameter, as though it
were the random variable instead of the sample result. This
is shown by statements made about the pOpulation parameter,
such as the following from Arkin.6

....if a random sample of 100 contains 50%

black balls, the universe from which it was

drawn probably (99% probability) would not

have contained less than about 37.1% or more

than 62.9% black balls....
This statement was made about a large binomial pOpulation,
from which a sample was found to contain 50% black balls.
It is clear that the person making this statement considers
the population parameter p a variable, while the sample
parameter‘S’is fixed at 0.50. The following equation'can
be constructed from Arkin's statement:

Probability (.371(p(.629|8 0.5) all0.99.

The only variable in this equation is the pOpulation para—

meter p. Classical statistics would not interpret the

29

sample result in this manner, because it implies that the
population parameter p is not fixed, rather, that it was
drawn from some large number of possible populations. These
pOpulations would have some distribution of p among them-
selves. This distribution would be the prior. Thus, the
"classical auditing statistics" is not the classical stat-
tics, but is some combination of it and Bayesian statistics.
In this study the term "classical statistics" will refer to
the way such methods have been applied to auditing.

The following example illustrates the application
of classical statistics to auditing. Assume that an auditor
takes a sample of 169 from a large population and observes

55 errors in the sample. He would compute

 

91—21% (——-—55 Mil—4.)
0;" n ‘3 169 159 = 0.0361
169
A one—sided confidence interval would be computed from the
sample mean (i) of 55/169, or 32.6%, and the product of<F%
and a factor taken from a table of the normal distribution.
If the auditor desired a 97.8% confidence level, this nor-
mal factor would be 2.03.7

Confidence Limit 3'; + 2.03«$%)

. 0.326 + 2.03(0.0361)

ll

0.326 + 0.0735

3 0.400.

 

 

 

 

 

of

U0

Th:

30
Thus, the auditor using classical statistical methods would
have a 97.8% confidence level that the error rate in the
population would be less than 40%. If his maximum accept-
able error rate is 40% or greater, he will accept the sample
results and go no further. On the other hand, if his maxi—
mum acceptable error rate is smaller than 40%, he can either
reject the entire population as containing such a large per—
centage Of errors as to be useless, or expand his sample by
taking additional sample units in an attempt to produce an
acceptable sample result. This decision depends upon the
audit variables. The relationship between classical stat-
istics and audit variables is treated in the following

section.

C. Effect of Audit Variables

 

As discussed in Chapter II, audit variables are
the relationships which exist between circumstances en-
countered by the auditor and the appropriate sample sizes.
If the circumstances are good (excellent internal control,
few errors in the records, prOper valuation methods, low
risk of the client having difficulty in meeting the claims
of its creditors, etc.), the auditor is able to form an
unqualified Opinion with relatively small sample sizes.

This is due to few problem areas being encountered in the

31

audit and the low risk of sanctions being applied to the
auditor as a result of that engagement. In a less favor-
able situation, sample sizes must be increased in order
that the auditor can enjoy a similar degree of risk, be-
cause more questionable items are likely to arise during
the audit. These warn the auditor that he must be more
prepared to defend an unqualified opinion rendered as a
result of that engagement.

In the classical statistics model, the auditor
can keep the same small sample size under worsening audit
variables only by increasing the error rate he is willing
to accept or by decreasing the confidence level he has
that the actual error rate is within bounds acceptable
to him. Classical statistics allows the auditor to con-
sider the audit variables in calculating confidence inter-
vals, but only indirectly. One method of incorporating
the variables is to change the degree of confidence
required. For example, assume a test in the audit of one
client has produced no errors in each of the last six
audits. The same test in another audit has produced high
error rates in every one of the last six years. The audi-
tor may require an 80%,confidence level in the "errorless"
engagement, and a 90% level in the other. As a result of

a change in one of the audit variables, the auditor wishes

E\

 

Jam .1. ytlerl .
i

5.! .r

 

32
to have a greater level of certainty in the more risky
engagement. If the auditor does not change the confidence
level, but instead decides to allow an increased acceptable
error rate, the risk of sanctions being imposed upon him
increases.

One audit variable which the classical statistics
model does explicitly consider is the size of the population
from which the sample is taken. On page 29, an example
shows how an auditor might construct a confidence interval
using classical statistical methods. In that example, a
large population (meaning that the population is consider-
ably larger than the sample size) was assumed. If this had
not been the case, it would have been necessary to correct

the result with a "finite correction factor." The formula

N - n
N - l

where the pOpulation size is referred to as "N" and the

for this factor is

sample size as "n". It should be noted that this factor

is very close to 1.00 when N is large relative to n, re-
sulting in the correction factor being passed by, as in the
example mentioned above. If n is a relatively large part
of the population, this factor must be referred to, how-

ever. It is used by multiplying it by the product of the

33

value obtained from the normal table andfl'; in the confi-
dence limit computation.

In summary, classical statistics does not provide
a way in which subjective information can be directly in—
serted into the model. Adjustments can be made to sample
size by way of the acceptable error rate and the confidence
level, but it is not a simple task to determine which of
these to change and by how much it should be changed to
reflect a given change in the audit variables. This diffi-
culty is illustrated by the prevalence of 90%, 95%, and
99% confidence levels in academic studies of widely vary-
ing subjects, to the almost total exclusion of other con-
fidence levels.8

In contrast, Bayesian statistics directly incor-

porates this subjective information, as set forth in the

following chapters.

34

CHAPTER III--FOOTNOTES

1Harold J. Larson, Introduction £9 Probability
Theory and Statistical Inference (John Wiley and Sons,
1969), p. 17.

21bid., p. 106.

 

31bid., p. 116.

4William Mendenhall, Introduction £9 Probability
and Statistics (Wadsworth, 1967), p. 151.

 

51bid., p. 163, 4.

6Herbert Arkin, Handbodk‘gf Sampling For Auditing
and Accounting, Volume I (McGraw-Hill, 1963), p. 74.

7See any introductory statistics text, under
"normal probability distribution."

8William H. Kraft Jr., "Statistical Sampling for

Auditors: A New Look", Journal 9; Accountancy, August,
1968, p. 49.

 

CHAPTER IV

ATTRIBUTES SAMPLING USING BAYESIAN STATISTICS:

DISCRETE PRIOR DISTRIBUTIONS

A. Bayes Theorem

Bayes theorem is the basis of a method which allows
inferences to be made about a pOpulation on the basis of
more than one probability distribution derived from that
pOpulation. Classical statistics, discussed in the pre-
vious chapter, does not allow more than one input distri—
bution in an analysis. Bayes theorem is named for the
Reverend Thomas Bayes, one of the early writers on prob-
ability theory. Recently his theorem has been applied to
a wide variety of situations and is especially important
in many branches of applied statistics.1

Bayes theorem itself is a method for combining two
probabilitydistributions. The first distribution is
called a prior: the second is derived from the results of
a current sample, and the result of the combination is
known as the posterior distribution. conceptually the

method is not difficult, but in practice the decision as

35

36
to which distributions to combine and the meaning of the
result present difficulties.
In the discrete form, Bayes theorem takes the
following structure. "Suppose that we are given k events

A A p 000000; Ak SUCh that:

1' 2
l. A]_\JA2LJ....UA.k : S
2. AinAj=¢ , for all i 3‘ j

(these events form a partition of S): then for any event

 

ECS.

PA- PEA.
P(Aj|E) k( 3) (‘3) , j: 1,2,....,k."

ZP(Ai)P(E|Ai)
i=1

 

The notation above may be read U (union),r\ (intersection),
#(not equal), S (the entire possible sample space), ¢
(the empty set--no possible result), ‘ (given), Z:(summa-
tion from the point under the symbol by intervals of one
to the point above the symbol), andC: (an element of).

To use Bayesian statistics in an auditing attributes
sampling situation, several steps must be completed. First,
a suitable prior distribution must be provided which ex—
presses the expectations of the auditor regarding a parti-
cular sample outcome. Secondly, an actual sample must be
taken from the population and the results (sample size and

number of errors found in the sample) recorded. That

37
information is used with Bayes formula to revise the prior
distribution and obtain the posterior distribution. Fin—
ally, it is necessary that the posterior distribution be
used to create a confidence interval in a manner similar
to that used with the normal distribution in classical
statistics.

The prior distribution provided by the auditor can
be either a discrete or a continuous distribution. The
remainder of this chapter discusses the discrete case,
while a continuous prior distribution is the subject of

Chapter V.

B. Discrete Prior Distributions

The following examples showlunv Bayesian statis—
tics might treat one auditing situation with the use of two
discrete models that have been proposed in the literature.
The auditor would have to provide information about the
expected population error rate under consideration before
the sample was taken. This is in contrast to the classi-
cal approach, which would base the entire analysis on the
results of the sample.

Suppose the auditor felt, based on the results of
other samples taken during the current audit and on pre—

vious years' results, that only an 80%.population error

38
rate (A) or a 20% rate (B) were possible and he felt that
A was twice as likely to occur as B. Since the total
probability (that of A and B) must equal 1.00, the auditor
must believe that
P(A):: 2/3
P(B) : 1/3.
The Bayesian statistician could use the formula for the
binomial distribution which was discussed in Chapter 3.
P(x) = (§)pxq“’x
If a sample of 3 were taken and no errors were found then
P(xzolA) = (3) (0.8)0(0.2)3= 0.008
P(x=0IB) :2 (g) (0.2)°(0.8)3 : 0.512
would be calculated. Using Bayes theorem, the following

computations would be made

Ptx=olA)P(A)

 

 

 

p(A\x=0) = P(x=0[ A)p(A)+P(x=0\B)P(B)
_ .008(2/3L ____ 0.03
.008(2/3)+.512(1/3)
P(BIX=0) _ P(X:0|B)P(B)

— P (x=0l A) P (A)+P (x=0| B)P (B)

= .512(L/3) :_0.97
.008(2/3r+.512(l/3)
He would check his results, knowing that the total proba-

bility of A or B must equal 1.00.

P(A|x=0) + P(B|x=o) = 0.03 + 0.97 = 1.00

39

The auditor is 97% certain that the population error rate
equals 0.20. This example is, of course, highly simpli-
fied, but techniques are discussed in the following sec-
tions which are capable of incorporating much more complex
distributions that might better reflect reality

The Bayesian approach generally requires a smaller
sample size to reach the same confidence level as the
classical approach. This is not free information, however.
The auditor must have some reason for the prior distribu—
tion he provides, whether it is the result of his internal
control inquiries, other audit tests, or previous years'
audit results. Bayesian statistics simply provides a
method for formalizing the information available. Chapter
VI discusses the problems inherent in deriving a realistic
prior distribution.

In a 1968 Journal_g£ Accountancy article,3 William

 

H. Kraft, Jr., proposed using Bayesian statistics to solve
a problem similar to the preceding. He detailed a method
where the prior distribution (with six possible error rates)
can be specified from information the auditor has gained
previously. The binomial formula, used in the preceding
example and in Kraft's article, was meant for a situation
in which the pOpulation size is infinite or in which each

item sampled is replaced before another is taken. This

40

implies a constant probability of obtaining a given result
from one sample item selection to the next. More infor-
mation can be obtained from the same size sample, however,
if each item is Egg replaced as it is sampled, making the
remaining population smaller with each successive item
drawn, and thus increasing the probability that any one
remaining item will be selected on one of the remaining
draws. A more complicated formula is necessary to gain
this advantage because the binomial does not consider the
decrease in pOpulation size which occurs as sample items
are removed.

The distribution which permits the use of this add-
ed information is known as the hypergeometric and was dis-
cussed with the binomial in Chapter III. In two 1969

4’5 prOposed the use of the hyper—

articles, John A. Tracy
geometric distribution in auditing situations and gave
examples which depicted results obtained using the hyper-
geometric with Bayesian statistics in a payroll check
examination.

The following example depicts the same situation
as that in the binomial example above, except that the

population size is only 20. Using the hypergeometric

formula from Chapter III.

41

(1‘2) (‘32)!)

 

 

 

 

 

 

P(X) = ———————— for k ==0,l,2,....
(R)
the following results are obtained.
MM) ”3:.
P(szB) =-. 0 3 = 13.3. :2-14
(20) 20: 3-19
3 17:3:
16 4 ._4_1_
P(X=~'OIA) :- ( 0H3) = 3:1: ____ 1
(20) 20: 15-19
3 17:3:

Using Bayes formula, the following results are obtained.

 

 

 

 

 

 

 

= -_- P(X=0|BIPIBI
P(B‘X 0’ P(X=0|A)P(A)+'P(X=O\B)P(B)
2‘14 (2.)
_, 3 19 3 __ g_8__9_ __
.. 2.14 (3 2)+ (i) - 281 _ 0.9964
3'19 15119 3
P(A|X=0) =-_ P£X=0 ‘ A) PUB)
P (x=olA) P (A)+P (x=0 IB) P (B)
1 (l)
.. 15° 19 3 _. __1_ =
. (2)+ (l) - 281 9J-9——-__9_3_§
3 19 15119 3

Since P(A|X=0)-t P(B\X=0)'= 140000, the results check.

It should be noted that the revised probability of B under
the hypergeometric assumption and a pOpulation of 20 is
99.64%. This is considerably more than the similar prob-
ability of 97% under the binomial distribution and the
infinite pOpulation size assumption. The greater proba-

bility obtained with the hypergeometric distribution is the

42

result of the additional information gained by sampling
from a relatively small population without replacement of
items selected for the sample. As the population size
increases, less and less advantage is gained until finally
the cost of using the more complex hypergeometric formula
is greater than the value of the additional information
gained. Thus, where the sample size is large relative to
the pOpulation size and drawing is without replacement, the
hypergeometric would be used. In situations where the pOp-
ulation is much larger than the sample the binomial would
be used whether or not sampling is with replacement.

In this section the use of Bayesian statistics and
discrete prior distributions in auditing was discussed.
Most of what has been written by others about the applica—
tion of Bayesian statistics to auditing has relied upon
discrete distributions. The exception is a thesis written

6 which is con-

at the University of Minnesota by Corless,
cerned with the ability of auditors to quantify expecta-
tions into both discrete and continuous prior distributions.
However, it does not concern itself with the use that might
be made of the distributions. Some implications of Corless'
study are considered in Chapter V. Important practical

difficulties found in the use of discrete prior distribu-

tions are discussed in the next section.

43

C. Difficulties Found in the Use of Discrete
Prior Distributions

Discrete prior distributions have been the basis of
Bayesian methods prOposed to date for use in auditing situ-
ations. Several important difficulties are found, however,
in practical applications of such methods. These diffi-
culties include poor approximation of continuous phenomenon,
inconvenience of deriving discrete prior distributions, and
the complex computations which would require a computer in
practical situations.

The phenomenon being represented by the prior
distribution is continuous. Not only is the actual number
of errors in the population unknown (or a sample would be
unnecessary), but often the auditor does not know the pre-
cise number of items in the pOpulation being sampled. He
is interested in rates instead of absolute numbers. In
this situation, the actual error rate could take a vast
number of specific values because both the numerator and
denominator of the fraction which determines the rate are
unknown. Any listing of a workable number of possible
error rates making up a discrete distribution must neces-
sarily be an approximation of the actual continuous dis-
tribution. Even a relatively large number of possible

rates may be an inaccurate approximation because most of

44
the probability could easily be concentrated between just
two or three of the listed rates with probabilities close
to zero assigned to the majority of the listed possible
rates.

An auditor who wishes to use Bayesian statistics
can approximate his feelings about a given attribute with
a relatively large number of possible rates in such a
fashion that the total of the probabilities is 1.00. This
would give a discrete prior usable with Bayes theorem and
the results of a current sample to produce a meaningful
posterior distribution. Tracy used this approach in his
Accounting Review article.7 He took 100 possible error
rates (0.002, 0.003, 0.004,....,0.099, 0.100, 0.101) and
assigned a probability to each of these which reflected
the auditor's prior thinking about that particular error
rate. First he took an easy approach by assigning an equal
probability to each of the 100 error rates so that the
prior distribution included p(rate=0.002)== 0.01, p(rate=
0.003)== 0.01, p(rate=0.004)== 0.01, etc. Such a prior
is called a uniform distribution. This uniform prior did
not, however, contribute much information to the posterior
distribution. The result was similar to a classical
statistics analysis of the same problem. He then decreased

the variance of the prior by lowering the probability

45
associated with higher or lower error rates and by increas-
ing the probabilities associated with error rates near 2.2%
to 3.1%. As would be expected, this second prior had a
greater effect upon the posterior than did the original.
The difficulty with this approach is that listing 100 dif-
ferent probabilities is a considerable task. Furthermore,
making certain they sum to 1.00 while simultaneously re-
flecting the information desired makes this method of de-
riving prior distributions so time consuming as to be im-
practical. It is important, also, to note that error rates
other than the 100 listed are usually possible (anywhere
between 0.02 and 0.03 for instance) but all of these
possibilities are assigned a probability of zero with such
a discrete prior distribution. This can only contribute
error to the result.

To decrease the difficulty of listing 100 different
probability figures, the number of points considered can
be reduced. In the Kraft article8 only six points were
assigned probabilities. These represented error rates of
0.001, 0.01, 0.02, 0.03, 0.04, and 0.05. It would be a
rare audit situation indeed which could be accurately rep-
resented by so few possible error rates, although the
accuracy could be improved by increasing the number of

points as Tracy did. By increasing the number of error

46
rates assigned probabilities, it becomes increasingly
obvious that this method of deriving prior distributions
produces only an approximation to some continuous distri-
bution which admits a range of possible rates instead of a
finite number, in most practical situations. Statistical
techniques exist which deal directly with continuous dis—
tributions instead of approximating them with various
discrete points. One of these techniques is considered in
the next chapter.

A final difficulty with the use of discrete distri-
butions is that the revision of a discrete prior distribu-
tion by a discrete sample result requires a large number of
multiplication and division Operations, even for relatively
simple problems. Most practical auditing problems would
require a computer for solution. This is an important
drawback, because computers and a means of translating in-
formation about the prior distribution and current sample
result into a computer-readable form without error are not
often available to the auditor in the field. Even with the
availability of computers, discrete distributions would
require significant input and processing costs.

The introduction of Bayesian statistics to auditors
by the use of discrete prior distributions has been an

excellent way to provide information about Bayesian

47
statistics. Discrete distributions are easier to work
with in simple examples when tables are not available than
continuous distributions. But the practical limitations
of discrete prior distributions in all but the most limited
audit applications of Bayesian statistics are too important
to ignore. Discrete prior distributions provide a poor
approximation to continuous phenomenon, are inconvenient
for auditors to derive, are difficult to perceive, and re—
quire involved computations when used with Bayes theorem.
The use of Bayesian methods with discrete prior distribu-
tions which have been proposed in the auditing literature
is not feasible in practice. An alternative, the use of
Bayesian statistics with continuous prior distributions,

is discussed in the following chapter.

48

CHAPTER IV--FOOTNOTES

1Harold J. Larson, Introduction £9 Probability
Theory and Statistical Inference (John Wiley & Sons, Inc.,
1969), p. 46.

 

2Ibid., p. 47.

3William H. Kraft, Jr., "Statistical Sampling
For Auditors: A New Look," Journal 9£_Accountancy, August,
1968, p. 49.

4John A. Tracy, "Bayesian Statistical Methods in
Auditing," The Accounting Review, January, 1969, p. 90.

 

 

5John A. Tracy, "Bayesian Statistical Confidence
Intervals For Auditors," Journal gf Accountancy, July,
1969, p. 41.

 

6John C. Corless, "The Assessment of Prior Dis-
tributions For Applying Bayesian Statistics in Auditing
Situations," Unpublished Ph.D. Dissertation, University of
Minnesota, 1971.

7Tracy, The Accounting Review, p. 90.

 

8Kraft, p. 49.

CHAPTER V

BAYESIAN STATISTICS AND THE BETA:

A CONTINUOUS PRIOR DISTRIBUTION

A. The Beta Distribution

Continuous prior distributions, as well as the
discrete ones discussed in the previous chapter, can be
used with Bayesian statistics in auditing situations. In
attempts to apply Bayesian statistics to auditing which
have appeared in the literature to date, discrete prior
distributions have been almost exclusively employed. How-
ever, many other fields have found continuous distributions
or a mixture of the two more useful than discrete distri-
butions alone. For example, the normal distributibn is
widely employed in natural science studies. The normal
is not well suited to application as a prior distribution
in auditing situations because the revision of a normal
prior distribution by Bayes theorem can lead to widely
varying posterior distributions which can be difficult to

evaluate.

49

50

One continuous prior distribution which produces
a more usable posterior distribution when revised by the
results of an audit attributes sample is the beta. Math-
ematically, a beta function is one which relates two unknown
variable terms by the use of three constants. The unknown
variables will be referred to as "x" and as "p(x)". The
value of p(x) is a function of the value of x. The three
constants will be called "mfl (alpha), ”9" (beta), and "c".

The relationship between these terms takes the form1

p(X) = cx“'1(1-x)‘3-1 (beta function).

A beta function can be graphed with x represented by the
horizontal axis and p(x) represented by the vertical axis.
When a graph of a beta function is drawn in this

study, it will be limited to the region between x=0 and
x=l. Auditors are concerned with rates of occurrence in
attributes sampling situations. In this discussion, these
rates will be referred to as error rates to make visualiza-
tion easier, although it is recognized that other rates

may sometimes be of interest. Two facts about rates are

of interest here. First, rates of occurrence such as those
found in auditing situation are never negative. Secondly,
such rates are never greater than 100%. The "x" from the
discussion of the beta function above will represent an

audit error rate. Therefore, it will be considered only

51

P(X) MODE : 0.05

 

N

0 0.25 0.50
x(error rate)

FIGURE 1. BETA DISTRIBUTION GRAPH

p(X)

MODE : 0.30

fiwvwr—

 

 

0 0.25 0050
x(error rate)

FIGURE 2. BETA DISTRIBUTION GRAPH

0.75

o<=30.40 [5:69.60

to the extent that its value lies between x=0 and x=1,

inclusive.

The beta function is quite flexible, and by prOper

choice of the constants,oa, fl, and c, its graph can be made

to assume a wide variety of shapes. The graphs of two beta

distributions are shown in Figures 1 and 2.

52

Auditors will, in general, be interested only in
beta distributions which have an "upside down bowl" shape
such as those in Figures 1 and 2. It can be shown that
any beta which exhibits such a shape will be the result of
a function with constants okand fl both greater than one.2

To make the beta function into a probability
distribution function (such as a prior distribution for use
by auditors), it is necessary to insure that the area be-
tween the curve and the x axis (see Figures 1 and 2) be
made equal to 1.00. OnceIX and 5 have been chosen, this
can be done by selecting the constant c in the beta func-
tion above so that this condition holds.

Under the conditions described, p(x) can be computed
at any point x along the horizontal axis, and plotted on a
graph. It will never have a negative value.

To illustrate the use of a beta function, an
example will be considered with 0k=2 and. 3:8. First, the
constant c must be found so that the area between the curve
and the x axis between XaO and x=l is equal to 1.00. The

function takes the form
p(x)'= cx2—1(l-x)8-l = cx(1-x)7.
The area under a beta curve between x=0 and x=1 can be

computed with gamma functions if both (X and B are integers.

53

Since cc and fl are integers in this case, gamma functions
will be used to compute the area under

p00 = x(l-x)7
and the reciprocal of the area used in place of c in the
original formula to make the area between the curve and the
x axis between x=0 and x=l equal to 1.00. The value of a
beta integral over x=0 to x=l with 0‘ and 3 both integers

can be computed from the following formula.3

(Efl)1(9-lli

 

 

In the case at hand:
(2-1)1(8-l)1 1:7: 1 , 1
A = : ———— =-——— ~-—-—
rea (2+8-1): 9: 9.8 72 .

The reciprocal of this area will be substituted for c
in the original formula. The value of c is therefore equal
to 72 and the formula becomes:

p(X) = 72x(l-x) 7
A rough graph of that function can be drawn by computing
several points from the formula, plotting them, and
connecting the points. Such a graph for the function
above is shown in Figure 3. The points used are shown
below. Later in this chapter, it will be shown how the
highest point on a beta curve can be computed. In this
case that point occurs at x =:l/8. This information is

useful in plotting the curve shown in Figure 3.

54

P(x)

 

 

 

 

G! l 1 l 1
0 1/4 1/2 /4
X

FIGURE 3. GRAPH OF BETA CURVE oar-2 3:8

 

x P(X)
0 0
1/8 3.53
1/4 2.40

3/8 1.005
3/4 0.0033

1 0

 

The function used to find the area under the curve

between x=0 and any point x = x1 is the beta integral. For

the example at hand, this integral takes the form

55
Area = cjxle—x) 7dx = 72jX1x(l-x) 7dx.
0 0
This formula will be used to create confidence intervals
based upon the beta distribution determined by 0‘ = 2 and
fl: . It is necessary to integrate by parts to evaluate

this function. The formula for integration by parts is

x x x
l
fludv = uv - g 1vdu.
0 0 0

In this instance u and v are defined as follows:

 

u z: x dv :: (l-x)7dx
x1 x

v: (l-x)7dx ‘: — %(l—x)8
0 0

 

X1 X
S udv = x(--;;_) (1—x)8 l - fl(—)é(l-x) 8dx
0 0

0

 

 

X1 X1
:: —lx(1—x) 8 + 1 (l-x) 8dx
8 B'
O 0
8 X 9 X
-.-.-_1_x(1-x) - _l(1-x)

 

 

To check this substitution the area under the curve from

x=0 to x=l will be evaluated.

56

1

Area ==-?13.,x(l-x)8

 

 

-.__l_(l-x)9
72

0 0

= (0 - 0) —.__1_(0) - (-) _1(1-0)9
72 72

= '(‘)—l.:-—l
72 72 °

This is the same answer as that obtained through the eval-
uation of the beta by gamma functions earlier. It should
be noted that the constant c, equal to the reciprocal of
the area computed above (c = 72), must be multiplied by
the function to insure that the area under the curve be-
tween x=0 and x=l equals 1.00. The formula for deriving
confidence intervals from this distribution becomes
Area under curve from x=0 to x=x

l

X X

= 72 -%x(l-x) 8

 

- _1(1-x) 9
72

0 0

 

The area under the curve from x=0 to x=% can be computed

with the use of this formula.

Area under curve from x=0 to x:%

=72’-lav 8-10 —__1_ 9 .11-09
8(2)05) 8( ) 72(32) -+ 72 ( )

 

=72L-1-(———1)-_l(———)+—l
. 8 512 72 12 72
=1 _ 9 _ 1 512 - 10 502 =0 981

 

57
When this value is compared to the graph of the function
(Figure 3), the answer appears to be reasonable. In a
similar fashion, the area under the beta curve with °£= 2
and p= 8 between x=0 and any other desired value of x can
be computed. If the above computation were made for the
purpose of obtaining a confidence interval, it would yield
a 98.1% confidence coefficient that the error rate of the
pOpulation did not exceed 50%. The confidence coefficient
is the probability that the computed confidence interval
contains the actual population error rate.

The methods demonstrated above are useful only
when the constants ok and )8 of the beta distribution under
consideration are integers. If either constant is not
an integer other computation techniques must be employed,
one of which is discussed in section D of this chapter.

The Beta $5.3 Prior Distribution. A prior distri-
bution is required before Bayes theorem can be employed to
produce a posterior. Prior distributions can be viewed
as providing an "additional sample" from the population
of interest to the auditor. The parameters N and {3 of the
prior distribution can be looked upon as being representa-
tive of this additional sample. The sum.°b+fi would rough—

ly correspond to the size of the additional sample, while

58

utwould roughly represent the number of errors found in
that sample. Viewing the prior in this manner may aid
an auditor who is attempting to determine the parameters
Otand B in a specific audit situation. Other information
which the auditor may find useful in this task includes
various statistics computed for possible prior distribu-
tions. Some of these statistics are the mean, variance,
and mode. The formulas used to compute these statistics
from the O< and ,6 parameters of a specific beta distribu-
tion will now be discussed.

The mean of any function is the expected value of
that function. It can be shown that the mean of a beta
distribution occurs at an error rate of 4

._£§__
o<+p-

As an example, the mean of the beta distribution shown in
Figure 1, page 51, will be computed. The constants<x
and p of that distribution are ok=1.40 and p=8.60.

(x __ 1.40 __. 1.40

__.—__.. —- 2.0.14
°<+fl 1.40+8.60 10.00

Mean 1'

The variance of a distribution is an indication of
the relative density with which the values would be expect-
ed to occur about the mean. The variance of a beta

distribution is5

59

mode

mean
’/’

  

 

 

—-—_—-—-—_————

.2

FIGURE 4. MEAN AND MODE OF BETA DISTRIBUTION

 

one
(cup) I (owed)

The variance of the beta distribution shown in Figure 1

will now be computed.

. - up __ (1.4) (8.6)
Variance -— (OH-)6)7(°‘+p+1) "' (10,0)1(10.O+l.0)

= 12.04
1100.0

= 0.10945 .......
Another measure which is sometimes used is the

mode of a distribution. On a graph of a continuous func-

tion, the mode appears as the highest point on the curve,

as shown in Figure 4.

60
The vertical distance of the beta curve from the x axis
is computed from the formula:
p (x) :: cxm"l (l-x)p—l
To find the maximum value of p(x) the first derivative is

found and set equal to zero.

dp(x) _ djcx“_ljl-x)p’l)
dx ‘ dx

=c(«-1>x°“2(1-x)P'1 + (-1)cx 403-1) (1905‘2
:2 (car-1)x°"2(1--x)'3'1 - x“‘1(p-1)(1-x)9‘2 = 0.
(d-1)Xq—2(l-X)B’l = (p-1)§'1(1-x>P'2

E :; x(l-x)-l

dram-1+x == fix-x
d'l ‘3 (6‘2”)x
The mode of the beta distribution occurs at

X: 00-1
“+fi-2

To illustrate the computation of the mode of a beta dis-
tribution, the mode of the distribution shown in Figure 1

will be computed.

o( -1 = 1
«+5-2 1.4 +

Mode:
= 0.05

The mean, variance, and mode discussed here are

useful to an auditor attempting to fit a prior beta

61

distribution to evidence available to him. Once he has
determined the specific constants (0(and p ) which deter-
mine the prior distribution, it is necessary to revise the
prior by the results of a current audit sample. Determina—
tion of the prior distribution is discussed in Section B,
while the revision of the prior by the results of a

current audit sample is explained in Section C, and the
procedure used to make inferences from the result is

covered in Section D.

B. (Selection of a Prior Distribution

Before Bayes formula or a posterior confidence
interval can be used, a meaningful prior distribution must
be selected by the auditor. This prior is based upon evi-
dence actually available to the auditor: evidence which
relates to the error rate of the pOpulation under con—
sideration.

It is important that an auditor interested in
the use of Bayesian statistics understand the meaning of
the prior distribution he selects. To aid in this,
various statistics can be computed about representative
prior distributions. The mean, mode, and variance have
been discussed in this study, and such statistics can be

of value to an auditor who has had even limited

62
statistical training. A short table of such statistics
computed for selected beta distributions appears in
Appendix A (Table l).

A useful aid to the auditor faced with the problem
of deriving a prior distribution might be a book of
sketches of representative beta prior distributions.

These might be arranged in a fashion similar to a police
department's books of "mug shots" which include a number
to identify each portrait. As the victim of a criminal
act might thumb through the "mug shots" and note the
identifying number of a picture which bears a resemblance
to the offender, an auditor attempting to fit a prior
distribution to specific circumstances might thumb through
the sketches and note the appr0priate o( and ,3 parameters
for the distribution which best fits his situation. These
sketches might be ordered in sections by mean or by mode
to provide the auditor with a starting point. A short
example of such a "mug boo " of sketches of Specific

beta distributions appears in Appendix B.

The use of such sketches would save time. In-
stead of listing a number of probability points (the
Inethod used by those who have prOposed the use of discrete

Iprior distributions in auditing applications of Bayesian

63
statistics), the auditor would merely look through the
sketches until the appr0priate prior was located. Marked
on that sketch would be the parameters 6! and 5 which deter-
mine the function. After noting those parameters, the
auditor would be ready to take the current sample and
revise the prior with Bayes theorem.

Graphs of beta distributions, in the form of the
sketches discussed above, can be produced by a plotter
attached to a digital computer. The sketches shown in
Appendix B were derived in that fashion. The computer
programs which resulted in those sketches are shown in
Appendix D. Also shown in that appendix are the programs
used to produce the beta distribution tables which appear
in Appendix A.

Several statistics can be used by the auditor in
locating the apprOpriate prior distribution for his use.
One mark with which he might begin is the mean of his
prior expectations. This is not the single rate he feels
is most likely to occur, based on his prior expectations,
but rather is the expected value of the function. The
mean is a useful measure and it is widely employed.
However, when faced with the problem of specifying a

distribution, the typical auditor might well have

64
difficulty identifying the expected value of the result.
He could specify the single error rate he feels is most
likely to occur. This is the definition of the mode,
another statistic discussed with reference to the beta
distribution earlier in this chapter. Because it appears
that the mode is less difficult for an auditor to use,
it is used to identify the distributions shown in Appendix
A and Appendix B of this study, although the mean is given
as supplementary information.

One factor the user of Bayesian statitics would
consider is the relative weights given to the prior dis-
tribution and the current sample in the derivation of
the posterior. As the sum (Xi-Sis increased, with the
mode of the prior held constant, the variance of the
prior decreases, implying greater certainty. An auditor
who has considerable evidence that an error rate is near
5% may use a prior distribution with a relatively large
(when compared to the current sample size):K4-fi3. Another
auditor with some smaller amount of evidence which also
indicates an error rate near 5% is likely to select a
prior in which the sum cup is somewhat smaller than did
the first auditor. If each draws the same current sample

result, the prior selected by the first auditor will

65

have a greater relative effect upon the posterior than
that selected by the second auditor.

An auditor with no statistical training could
not be expected to derive meaningful prior distributions
for use with Bayes theorem. One study has shown that
auditors are "willing to quantify their conclusions as
probability distributions."6 The same study7 found that
different auditors produce different prior distributions
from the same data. This implies either a lack of agree-
ment between the auditors as to the audit variables under-
lying the situation, or a difference in how each auditor
subjectively interprets the audit variables. Either
alternative suggests that more than a quick introductory
lesson would be necessary before several auditors would
reach similar independent conclusions from the same data.

Once an auditor has selected a prior distribution
from a set Of sketches or tables of statistics, he has
the constants c: and (3 which are required for the revision
of a prior beta distribution by Bayes theorem. These
constants are combined with the results of the current
audit sample to find the posterior distribution upon
which statistical inferences will be based. The Operation
of revising a beta prior distribution by a binomial sample

result is discussed in the following section.

66

C. Revision of a Beta Prior by a Binomial
Sample Result

 

After an auditor has selected a prior distribution
and taken the current sample, only two steps remain in
the use of Bayesian statistics with beta prior distribu-
tions in audit attributes sampling situations: The revi-
sion of the prior with Bayes formula and the creation of
confidence intervals based upon the posterior beta distri-
bution. This section discusses the revision Of the prior
distribution, while section D eXplains how a confidence
interval can be obtained from a beta posterior distribu-
tion. An example illustrating the entire process appears
in Section E of this chapter.

The revision of a beta prior distribution by a
binomial sample result is unexpectedly simple. A beta
distribution is determined by only two parameters, re-
ferred to as o( and f3 . The result of an audit sample can
be specified by n (the sample size) and x (the number of
errors found in the sample). If the prior is a beta
distribution with parameters at and p , and the sample a
binomial of size n with x errors, the posterior will be
a beta with parameters

a" z o<+ x - 1 and

fl'2 P-F n - X - l.

67
The derivation of this result is shown in Appendix C.
Here it is important only to note that revision of a
beta prior by a binomial sample is a simple arithmetic
manipulation requiring no higher mathematics than addi-
tion and subtraction, no computer, and no tables. The
resulting posterior distribution is another beta, which
may be further revised by additional binomial samples, in
the same manner. The result will always be another beta
distribution.

The Bayesian method discussed in the preceding
paragraph would be relatively easy for an auditor in the
field to use. If he had sketches of a number of possible
beta prior distributions and tables of relevant statistics,
he could select the prior which best represents the evi-
dence available about a given error rate before a current
sample is taken. He would then write down the appr0priate
constantscx and f for that distribution. Once the current
sample has been completed, its outcome can be summarized
by two figures, n (the sample size), and x (the number
of errors found in the sample). (Note that this x is not
the same as the x used to represent the error rate in
the beta function. Statisticians use the same term in

two different ways.) The determination of the posterior

68
distribution can be quickly accomplished with the aid of
a pencil, a piece of paper, and the two formulas discussed
previously in this section. The final step in the analysis
involves drawing inferences from the posterior distribu-
tion thus Obtained. This requires the derivation of
confidence intervals from posterior beta distributions.
The following section contains an example of the revision
of a beta prior distribution by Bayes theorem and explains
how confidence intervals can be obtained from beta
posterior distributions.
D. Confidence Intervals from Beta
Posterior Distributions

The procedure used to select a prior distribution,
summarize the results of the current audit sample, revise
the prior by those results, and obtain the posterior beta
distribution has been discussed in the three preceding
sections. Once this procedure has been followed in an
actual audit situation, there remains only the problem of
determining confidence intervals from the posterior
distribution.

In the first section of this chapter, an example
of the calculation of a confidence interval from a beta

posterior distribution was shown. The method used in

69
that example involved the evaluation of a beta integral
through the calculus technique of integrating by parts.
That method is not generally applicable to beta integrals
in which the constants‘d.and p are not integers. Since
auditors would use beta distributions with non-integer
constants, an alternative computation technique is
required.

Mathematicians have deve10ped approximation
techniques for the evaluation of functions which can not
be computed directly. One of these techniques is
Simpson's rule for the approximation of definite integrals.
It approximates the area under a given curve by summing
the areas Of a number of segments which approximate the
area of portions of the desired area. A further discussion
of Simpson's rule can be found in Appendix E.

Simpson's rule can be used to create tables of
confidence intervals for the beta posterior distribution.
If a set of representative posterior distributions were
used as the basis for a group of confidence interval
tables, the auditor with access to such tables could
quickly determine the limits of the desired confidence
interval, once he had completed the tasks leading to the

posterior distribution. An abbreviated set of such

70
tables, which were computed with the use of Simpson's
rule, is shown in Appendix A (Table 2, 3, and 4). Those
tables are shown as an example. They are not complete
enough for use in practical situations. The computer
programs which were used to obtain those tables appear in
Appendix D.

The entire procedure followed by the auditor,
from selecting a prior distribution through looking up
the limits Of the confidence interval in the tables,
would take only a few minutes (aside from the time required
to draw and examine sample documents, etc.). The auditor
would not require the use of a computer, or even an
adding machine.

To illustrate the use of a beta prior distribution
and a binomial sample result in an application Of Bayesian
statistics to auditing, it will be assumed that an auditor
has evidence regarding errors in some pOpulation which he
feels is represented by a beta distribution with para-
meters D(=1.4 and p=8.6. He takes a sample of 92 (n)
items and finds 30 (x) errors. What are the parameters
of the posterior beta distribution?

a' «+X-l. fl :fi+n-x-l

II

= 1.4'+'30.0 - 1.0 ::8.6 +~92.0 - 30.0 - 1.0

=- 30.4 =69.6

71
Illustrations of the shapes of both the prior and
posterior distributions in this example can be found on
page 51. It should be noted that the mode of the prior
is 0.05, while the information contained in the sample
results in the mode of the posterior equaling 0.30,
certainly a significant shift for most auditors. The
prior in this example has relatively less influence upon
the posterior than does the sample result.

Confidence intervals can be constructed from the
posterior beta distribution by referring to apprOpriate
tables, such as those in Appendix A (Tables 2, 3, and 4).
To use the confidence interval tables, the appropriate
at and p parameters are located along the left side of
the tables, while the confidence coefficient desired is
located in the body of the table on that same line. The
number at the tOp of the column in which the desired
confidence coefficient appears is the upper limit of a
one sided confidence interval. If the auditor in the
example discussed in the preceding paragraph wished to
construct a one sided confidence interval with a 97.8%
confidence coefficient, the tables would show him that
97.8% of the area under a beta curve with parameters

°( = 30.4 and flz69.6 would lie below an error rate of

72

40%. In Chapter III, it was shown that the classical
statistics approach resulted in a similar conclusion of

a 97.8% confidence coefficient that the pOpulation error
rate was below 40%. That result, however, was based upon
a sample of 169, while the Bayesian example shown here
requires a sample of only 92 to reach the same confidence
interval. In each instance the sample error rate was
32.5%. Thus, the additional information contributed by
the auditor's prior distribution reduced the sample size
necessary to obtain the same assurance as that gained from
the classical model.

The relative effect upon the posterior distribution
by the prior and current sample result is determined by
both the "gentleness" of the prior curve and the size of
the sample taken. If a prior which is rather noncommittal
(has a large variance) is revised by a large sample, the
result will be largely determined by the sample. On the
other hand, if a prior with a small variance is revised
by a small sample, the result will be similar to the
prior.8 This matter is further considered in Chapter VI.

In summary, the use of a beta prior distribution
by auditors in applications of Bayesian statistics to

auditing requires several steps. In the first section Of

73
this chapter, beta distributions were discussed and it
was demonstrated how computations can be made which yield
information about specific beta curves. Several ways
of aiding an auditor who is faced with the problem of
determining a beta prior distribution for specific circum-
stances were discussed in the second section Of this
chapter. That was followed by an explanation of the
process used to revise a beta prior by the results of a
current audit attributes sample, while the confidence
intervals necessary to the making of inferences based upon
beta posterior distributions were the subject of this
section. The last section of this chapter contains an
example of the use of this method in a specific auditing
situation.
E. An Example of the Use of a Continuous
Prior Distribution

The preceding sections have explained the pro-
cedures and computations necessary to arrive at a posterior
distribution from a beta prior distribution and a binomial
audit sample result. The following example illustrates
the application of a beta prior distribution to a common
audit situation in which classical methods result in an

illogical conclusion.

74

Assume an auditor has been doing the field work
on a certain engagement for the past six years. Each year
he has sampled 40 of the approximately 6000 sales invoices,
and each year has found no errors. There have been no
significant changes in the internal control system over
sales or in the accounting staff. From the tables in
Arking, he determined that there was a 99% confidence
coefficient that the actual error rate was less than 10.8%.

Bayesian methods allow the auditor to create a
prior distribution which includes such information, when
desired. In this situation, the auditor would be likely
to use a prior with a very small mean and mode, probably
in the neighborhood of 1% to 2%, because he feels strongly
that the actual rate is at least that small, based upon
past experience. Assume that this auditor is attempting
to be very cautious in his prior and picks a beta prior
with «x: 2.98 and B = 59.02. This means that he feels

that "most likely“ error rate is the mode Of

«-1 __,_ 2.98 - 1.00 ____
“+fl-2 2.98 + 59.02 - 2.00

 

3 . 30%:

and that the "average" expected value is the mean of

, 2.98 2.98

0‘
«+13 " 2.98 + 59.02 60.00

 

 

:1 4.83%.

75
These values are both high (conservativex compared to the
auditor's expectation of the actual rate.
As stated above, the actual sample yielded no
errors (x=0) out of a sample of 40 sales invoices (n=40).
To determine the posterior distribution, the following

computations would be made:

at. on—X - 1 fl' ‘: fi-f- n - X - 1
:2.98+ 0 - 1 :59.02+ 40 - 0 - 1
=1.98. 298.02.

A one-sided classical confidence limit was found above to
yield a 99% probability that the actual rate was less than
10.8%. By referring to the beta table, a similar confi-
dence interval can be constructed from the posterior
Bayesian distribution (Table II in Appendix A). Such a
confidence interval would give a 99% confidence coefficient
that the actual error rate lies below 7.0%, instead of the
10.8% which the classical method resulted in. Thus, even

a very ”conservative" (in the sense of having mean and mode
greater than the actual expectation of the auditor) prior
distribution results in a more realistic posterior distri-
bution and resulting confidence interval than did the
classical methods. A prior which more accurately reflected

the expectations of the auditor would result in an even

76
more reasonable confidence interval.

It should be noted that an auditor using classical
statistical methods in the example discussed above would
probably approach the situation by reducing the confidence
level he requires in the current year, knowing this will
reduce the sample size. But he must make this decision
by "feel", with no formal guide as to the effect of this
change upon the analysis. The Bayesian method, on the
other hand, allows the auditor to be as specific as he
desires in making up a prior distribution based upon the
results of previous years' audits.

The preceding situation is one to which Bayesian
methods seem naturally adOpted. Not all auditing situa-
tions are so constructed as to fit the Bayesian model in
this manner, however. For example, the audit variables
which affect the risk of the auditor being subject to
sanctions because of the bankruptcy of the client can be
considered. It would be difficult for an auditor, even
one with considerable training and experience in statis-
tical methods, to construct a prior distribution which
meaningfully reflected this possibility. It appears that
there is a sort of scale of audit variables, with those

obviously applicable to Bayesian methods at one end, and

77
others, such as the bankruptcy example mentioned here,
at the opposite end.
Subjective probabilities, such as those used as
prior beta distributions, have been the subject of some

controversy. This is the subject of the following chapter.

78

CHAPTER V--FOOTNOTES

1Harold J. Larson, Introduction £9_Probability
Theory and Statistical Inference (Belmont, California:
John Wiley & Sons, Inc., 1969), p. 358.

 

 

2Irvin H. LaValle, Ag Introduction £9 Probability,
Decision, and Igference (New York: Holt, Rinehart and
Winston, 1970), p. 256.

 

 

 

3Larson, p. 358.

4Ibid., p. 305.

5Ibid., p. 305.

6John C. Corless, "The Assessment of Prior Distri-
butions for Applying Bayesian Statistics in Auditing Situ-
ations," unpublished Ph.D. dissertation, University of
Minnesota, 1971, p. 127.

7Ibid., p. 127.

8Robert Schlaifer, Introduction Egyggatistigg for
Business Decisions (New York: McGraw—Hill, 1961), p. 217.

9Herbert Arkin, HandboOk gf Sampling for Auditing
and Accounting Volume I Methods (New York: McGraw-Hill,
1963), p. 506.

 

CHAPTER VI

SUBJECTIVE PROBABILITIES

A. Definition

An auditor who has decided to use Bayesian stat—
istics must express in a prior distribution the evidence
available to him. The mathematics associated with the
beta distribution was discussed in Chapter V. But the
auditor attempting to use Bayesian statistics needs more
than formulas, or tables, or other such aids. He needs
an understanding of what lies behind Bayes theorem. This
chapter attempts to provide some of that information.

Probability distributions are pOpularly thought
of as being based upon objective, verifiable evidence.
Thus, in the usual sense of the term, a probability dis-
tribution would not vary even when derived by different
auditors from the same sample. But a qualified auditor
may have Opinions as to the distributions underlying the
sample taken, in addition to the sample result. Often
these Opinions are based upon facts just as real as the

sample result, but not directly expressible as a

79

80

probability distribution. These Opinions may give rise
to a "personalistic" or "subjectivist" probability dis-
tribution based upon the Opinion of the qualified auditor
instead of directly upon a sample. Since different audi—
tors have varying perceptions and experiences, these sub—
jective distributions are not necessarily constant. This
makes the information contained in them no less real, how-
ever. It simply reflects the differing information con-
tained in the Opinions of the various auditors.

Information of this type is sometimes criticized
because it is not based directly on solid, verifiable
evidence. It is said that this type of information is
so subjective as to be worthless. But if the auditor
generating the information is qualified and has relevant
experience, some part of the data will be ignored if this
entire class of information is not used.1

If the Bayesian method were employed, the audi—
tor would be forced to put down his expectations on
paper. At the very least he would be forced to pay
attention to this element, and if he were trained to use
an input distribution such as the beta which lends it-
self to simple adjustments for various subjective distri-

butions, it would be possible for him to approximate the

81

sum total of his own experience which is relevant to the
decisions being made.

Some who criticize the subjective school contend
that unlimited repeatability is a prerequisite for an
experiment to be associated with a probability distribu-
tion. They would limit probabilistic inference to games
of chance, problems of social mass phenomena such as life
insurance, and mechanical phenomena such as the movement

Of molecules in a gas.2

A rebuttal against such criti-
cisms is that the formal inclusion of subjective factors
in the decision process is better than informal considera-
tion outside the statistical sampling model. As a minimum,
it provides written evidence of where information arose in
the event of a lawsuit or inquiry into the audit after the
fact. Another defense is that of usefulness. If one can
Obtain better results (in the sense Of more information
for the same cost or the same information for a lower
cost) through the application Of statistical principles
to one time only events, then one is in a more desirable
position using such tools than not using them.

When an auditor attempts to derive a prior dis-
tribution, it is imperative that he understand the

variance statistic and use it or a similar measure to

insure that his prior is not so clustered about a single

82

point that other possibilities are excluded. In this
the traditional conservatism of an auditor may be an
asset because the use of a distribution with too small a
variance has the result of putting more information in
the prior than is justified by the supporting evidence.
This may expose the auditor to sanctions for not being
cautious enough in his work, while too large a variance in
the prior allows a margin of safety for the auditor. As
an auditor learns more about the use of statistics, he may
approach a smaller variance with less fear when he has
reason to justify the greater certainty implicit in such
a distribution. The auditor just beginning to use
Bayesian methods should be cautious to insure that any
errors made will tend toward larger than necessary sample
sizes.

Once it has been determined that Bayesian stat-
istics is to be used, there comes the problem of driving a
prior distribution that expresses the information avail-
able to the auditor before a current sample is taken.
Several approaches have been suggested. In one the audi-
tor lists error rates 0.00, 0.01, 0.02, 0.03,......, 0.98,
0.99, 1.00 and determines his "feelings" about the infor-
mation available to him by associating a probability

with each error rate in such a fashion that the sum of

83

these 101 probabilities totals 1.00. Other approaches

use more or fewer possible error rates, and still others
utilize continuous distributions to reflect the prior
information available. Both continuous and discrete prior
distributions must be based upon available evidence. The
relationship between the audit variables and the selection

of priors is touched upon in the following section.

B. Audit Evidence and Prior Distributions

It has been shown3 that auditors are willing to
provide information about their feelings in an audit case
which can be used to construct probability distributions
adaptable to priors. One of these prior distributions can
be combined with the results of an audit sample using
Bayes theorem to produce a posterior probability distribu-
tion from which confidence intervals can be constructed
or tests of hypotheses can be conducted. A problem that
has plagued those considering the use of Bayesian stat-
istics in auditing has been the difficulty of translating
an auditor's feelings about the audit situation with which
he is faced into a statistical distribution usable in the
model.

The previously cited work with discrete distribu-

tions has required the precise listing of a number of

84
sample points and the related probability of each. To
avoid a sacrifice of accuracy in the approximation, the
number of sample points must be large. This listing is not
an easy task for an auditor to undertake, and he is further
frustrated by the necessity that the sum of all the prob-
abilities totals l.00. Consistency is difficult to obtain
with such a method.

If continuous distributions are also considered,
however, the problem of creating a prior distribution from
the auditor's feelings is simplified considerably. Because
of the nature of continuous distributions, they are smooth
curves which may be plotted so that the auditor can com-
pare a specific curve with his feelings in a given situa—
tion. Once he has found a "picture" of the curve which
best fits his perceptions, the mathematical specifications
Of that curve may be noted and used as input to a computer
program or tables which have already solved the mathe-
matics involved.

An auditor attempting to construct prior distri-
butions must utilize information from his environment,
from the client's environment, and from within the client's
organization. Several audit variables from each of these
three categories were discussed in Chapter II. They vary

in applicability to Bayesian statistics.

85

Information from the auditor's environment is
important in the derivation of meaningful prior distri-
butions. The auditor's general experience with this and
with similar clients aids in determining the variance
associated with prior distributions. Specific sample
results from other years' work papers provide an excellent
source of information about the likely rate in the current
year. Such information, it must be remembered, is only a
supplement to current test results, even though the infor-
mation contained therein is important.4 There are numerous
other audit variables from the auditor's environment, such
as the relative sizes of the auditor and client, which
provide some information useful in the determination of
prior distributions.

The client's environment is important to the
auditor. General economic conditions can determine whether
the client succeeds or fails. If the national economy
appears to be headed toward a depression, the auditor
may be less likely to rely upon information from prior
distributions derived in different economic circumstances.
There are many other factors, such as the absolute size
of the client, the industry, the legal form chosen by the
client, listing on stock exchanges, registration with the

Securities and Exchange Commission, and the mode of

86
financing chosen by the client, which have some bearing
upon the risk faced by the client's owners and creditors,
and thus upon the risk that the audit results will be
challenged. Such information can be used in determining
the variance Of prior distributions and the resulting
weight contributed to the posterior by the prior distri-
bution and current sample results.

Information from within the client's organization
is especially important to the auditor attempting to derive
a specific prior distribution. If the client has an in—
ternal auditing staff, this fact, as well as the results
Of tests by the internal audit staff, can be very helpful
in determining rates which will probably be found in the
samples taken, and the variability to be associated with
the prior distributions. Other internal control factors
also contribute information useful in the derivation of
prior distributions. Finally, the auditor is interested
in facts about a specific area when making inferences
about that area. In deriving a prior distribution, he
needs information about such things as the materiality
of the items on the financial statements to which the
specific sample relates. Many more items of information
from the audit variables can be useful to the auditor in

choosing prior distributions for the audit, but is is not

87
the purpose of this study to explore such items in depth.

In addition to being reflected in the prior, the
audit variables can be used in Bayesian statistics to
determine the confidence coefficient and interval limits
just as in classical statistics. This gives an additional
input to the model for such information. In general, a
smaller current sample size will be required with Bayesian
statistics to produce a given confidence coefficient and
interval limit as compared to classical statistics.

The ultimate decision as to whether Bayesian
statistics or classical statistics will be used in auditing
probably depends upon the ability of auditors to develOp
prior distributions. It was attempted, in this study, to
deve10p more feasible approaches to Bayesian statistics in
auditing than had before been prOposed. Although the pro-
cess of choosing a prior distribution in a Specific audit
situation is beyond the scope of this thesis, methods have
been suggested which may aid in that process. This area
is important to the development of Bayesian statistics in
auditing, and requires study beyond that which has been

devoted to the subject at this point in time.

88

CHAPTER VI--FOOTNOTES

lSir Ronald Fisher, "The Nature of Probability,"
The Centennial Review, Summer, 1958, p. 272.

2Richard Von Ness, "Probability: An Objectivist
View," reprinted in Edwin Mansfield, Elementary Statis-
tics for Economics and Business (New York: Norton, 1970),
p. 61.

 

3John C. Corless, "The Assessment of Prior Dis—
tributions for Applying Bayesian Statistics in Auditing
Situations," unpublished Ph.D. disseration, University
Of Minnesota, 1971, p. 127.

4James E. Sorensen, "Bayesian Analysis in Auditing,"
The Accounting Review, July, 1969, p. 561.

CHAPTER VII

CONCLUSION

A. Summary

The objective of an audit by an independent
accountant is to render an Opinion on the fairness of
financial statements prepared from the records being
audited. The Opinion must be based upon evidence ob-
tained by the application of auditing procedures. Audit
variables are the elements of an audit situation which
should be considered in the decision about Which proce-
dures to apply and how extensively they should be applied.
Some variables determine the minimum audit program and
others govern modifications to that program.

The auditor's objective is to obtain sufficient
evidence to justify an Opinion at the least cost. Non-
statistical methods, referred to as judgment sampling
methods, have been used since the early 1900's when audit-
ing changed from a check upon every transaction entered
in the accounting records to a sampling process. More

recently, classical statistics has also become commonly

89

90

used in auditing to determine sample sizes and to construct
confidence intervals about attributes, among other uses.
Classical statistics as used by auditors is not
necessarily classical statistics. The difference lies in
the interpretation of confidence intervals. Classical
statistics considers pOpulation parameters, even though
unknown, as fixed. Auditors, however, typically make
statements about confidence intervals which refer to the
population parameter as an unknown, implying that it is
drawn from a group of possible values. This indicates
that a distribution of pOpulation parameters must exist.
Information about this distribution can be considered in
the Bayesian formula as a prior distribution and can be
used to make inferences about pOpulation parameters.
Bayesian statistics differs from classical methods
in that information from two sources (the prior distribu-
tion and the current audit sample) is combined into a
single distribution. Classical auditing statistics admits
only information from the current sample into the model.
The inferences made from classical auditing statistical
models approximate those arising from Bayesian methods
in which uniform prior distributions are used. Auditors

typically have information from prior audits, similar

91
engagements, other audit tests, internal control evalua—
tions, and other sources which provide prior evidence about
attributes tests. This prior information would seldom
imply a uniform prior distribution, but instead would
indicate certain rates that are more likely to occur than
others. Auditors are taking larger samples than necessary
to the extent that they rely upon classical auditing
statistical methods when prior evidence indicates that
some possible rates are more likely than others.

Several articles have shown how Bayesian statisti-
cal methods utilizing discrete distributions can be applied
to auditing situations. Both the binomial and hypergeome—
tric distributions have been discussed. Little has been
said in the literature, however, about applications of
continuous probability functions to Bayesian audit
statistics.

Other fields of study, especially the natural
sciences, have found the normal distribution to be of value
in a wide variety of situations. When the normal distri—
bution is revised with Bayes formula, the posterior distri-
bution is sometimes very difficult to evaluate. There-
fore, it is not well suited to applications of Bayesian

statistics to auditing. A distribution which is adaptable

92
to such uses is the beta. When a beta prior distribution
(continuous) is revised by Bayes formula and a binomial
sample (discrete), the result is always another beta dis—
tribution. This may be further revised by another binomial
distribution, if desired. The combination of continuous
and discrete distributions in this manner models the
typical attributes auditing sample.

Revision of a beta prior distribution by a bi-
nomial sample result with Bayes theorem requires only a
few addition and subtraction Operations. NO computer
or tables are necessary. This makes the technique feasible
for use by auditors who have had the necessary training,
and who have tables and graphs Of the prior and posterior
distributions. Although these are not currently available
to auditors, methods for producing the necessary tables
and sketches do exist.

To produce the tables required for the derivation
of confidence intervals from posterior beta distributions,
it is necessary to evaluate beta integrals. Such integrals
cannot be evaluated with the methods of calculus. They
can, however, be approximated to any desired degree of
accuracy with Simpson's rule for the approximation of

definite integrals. Even though this approximation is

93
not practical if done by hand, modern digital computers
can perform the necessary calculations in seconds, at a
very low cost.

Some applications of Bayesian statistics have been
criticized because subjective probabilities were used. It
is argued that statistics should be applied only to situa-
tions that can be repeated a large number of times. Suffi—
cient justification for the use of such methods in the
analysis of one time only events exists, however, if the
application of these methods provides a basis for better
decisions at a reasonable cost. In addition, the formal
consideration of subjective factors in a statistical model
is preferable to informal consideration.

B. Relative Advantages of Various Distributions In
Applications of Bayesian Statistics to Auditing

If Bayesian statistics is to be used by auditors,
a decision must be reached as to which of the many stat-
istical distributions available best fits the auditing
situation and is most efficient in actual use. Discrete
distributions have been found less difficult for those
not trained in higher mathematics to use than continuous
distributions. But this study has shown that the actual

revision of a prior distribution by a sample result is

94

less complex if a continuous prior distribution, the beta,
is used. Continuous distributions are also more exact
representations of the prior information available to
auditors.

In situations where discrete prior distributions
are used, it was discovered that the hypergeometric pro-
vides more information from samples which are large rela—
tive to the size of the pOpulation than the binomial
distribution. But this advantage was found to be out-
weighed if the sample was small compared to the pOpula-
tion size because of the additional complexity of
computations involving the hypergeometric.

The combination of a beta (continuous) prior
distribution and a binomial (discrete) sample result with
Bayes formula was found to closely fit the situation
found in attributes audit sampling. Once the prior has
been determined by the auditor and the sample taken, the
actual Operation of obtaining the posterior beta distri-
bution is easily accomplished in a few minutes by simple
addition and subtraction. This was not found to be the
case with situations in which discrete prior distributions
are used. If discrete priors are used, lengthy computa—

tions, generally involving a computer, are required to

95

obtain the posterior in practical auditing situations.
Evaluation of the beta prior and posterior distributions

is the only difficult part of a Bayesian analysis using
beta prior and posterior distributions and a binomial
sample result. Graphs and tables of representative distri—
butions can be prepared in advance, which would make the
use of this technique feasible in actual auditing situa-

tions, even if no computer were available.

C. Conclusions

 

Judgment sampling was once relied upon almost
exclusively in auditing, but has been supplemented by,
and in some cases, replaced by, classical statistical
methods. Statistical methods provide more consistent
answers and are more readily defended in the event of a
challenge to the results of the audit than judgmental
methods. Some audit situations do not lend themselves
to statistical methods, due to the small size Of the
audit, or the organization of the records, and in such
cases judgmental methods should continue to be used.

It has been suggested that Bayesian statistics
may be used in some auditing situations. Discrete
distributions (the binomial and hypergeometric) have been

used in examples where Bayesian statistics was suggested

96

for use in attributes audit sampling. Discrete distri-
butions are, however, unusable in practical auditing
situations. They are poor approximations of the contin-
uous range of possible values found in most audit attri-
butes samples. The derivation of specific discrete prior
distributions is difficult, because a number of points
must be assigned probabilities, and these probabilities
must total 1.00. Involved computations (many multiplica-
tion and division Operations) are required for the solu-
tion of any practical problem using discrete prior dis-
tributions and Bayesian statistics. This would require

a computer and a means for getting information about the
prior and current sample result into the computer without
error. These are usually not available in the field. At
this time auditors have not adOpted these methods, even
though such methods have received rather wide exposure

in the literature.

Bayesian methods using a beta prior distribution
can overcome most of these difficulties. Such a distri—
bution is continuous, and therefore fits the large number
of possible pOpulation error rates better than does the
discrete model. Visualization of the priors is easier

because continuous curves can be drawn (plotted on a

 

l u ‘t'... 1...
Jill. ul. [1»).—
I

1“:

97
computer). Sketches of representative distributions can
allow an auditor to quickly pick the specific prior he
wants and write down the numeric parameters of that
distribution. Tables of statistics (mean, mode, variance,
etc.) can be prepared for these representative distribu-
tions.

The revision Of a beta prior by a binomial sample
result is not difficult. Once this has been done, the
auditor can refer again to his tables for a sketch of the
posterior distribution (or a very similar one) and for
statistics he wishes to know for that distribution.
Another set of tables can give confidence intervals from
that distribution (one sided or two sided) at several
confidence levels. What would have taken considerable
computer time with a discrete prior distribution is
quickly taken care of with this method by using a few
tables in a book and a quick addition and subtraction
Operation.

This study has attempted to help convert an
infeasible technique with a great deal of potential into
a method feasible for use by auditors. It has been
limited in several respects, however. NO attempt was

made to discuss variables sampling in auditing. A

98
proposal was made suggesting the use of a continuous
prior distribution in auditing applications of Bayesian
statistics. For this it was necessary that tables and
sketches of representative beta distributions be deve10ped.
These were restricted to illustrative models. Additional
resources would be necessary to the develOpment of complete

sets of these tables and sketches.

D. Suggestions for Further Research

More questions arose during this study than could
possibly be answered by this project. One area yet Open
for investigation is the application of Bayesian statisti-
cal methods to variables sampling. In addition, there
are numerous statistical distributions and combinations
of distributions which may prove applicable to auditing
applications and have not yet been investigated.

Some studies have been done which used Bayesian
statistics with discrete prior distributions in "real
world" audit situations. Much can be learned by more
studies of this type, especially where applications of
classical statistics and Bayesian statistics utilizing

various distributions are compared.

 

99

The derivation of prior distributions from evidence
available to the auditor is an area about which little is
known. Studies drawing On the behavioral sciences are
needed to make prior distributions in Bayesian statistical
applications more meaningful to auditors. Comparisons can
also be made between the various types of discrete and
continuous distributions used as priors to determine the
relative efficiency of each in arriving at correct deci—
sions from the information available to auditors.

Finally, it would be helpful to have information
as to how some Of the many audit variables fit into the
Bayesian model. This could provide insight as to the
types of audit situations in which judgmental sampling,
classical statistics, and Bayesian statistics are most

applicable.

 

APPENDICES

APPENDIX A

BETA DISTRIBUTION TABLES

Table l Of this Appendix gives statistics of
selected beta distributions. The mode, mean, and vari-
ance of each distribution are shown. If explanation is
necessary, any introductory statistics discusses these
measures.

Also shown are the parameters a and p , and
their sum. The sum ov+p roughly corresponds to the sample
size that would be required if the beta distribution were
derived from a sample, while the value of<x roughly corres-
ponds to the number of errors found in that sample.

Tables 1, 2 and 3 of this appendix give approx—
imations of the value of selected beta integrals eval—
uated from 0 to the point shown along the tOp of the
table. These were found using Simpson's rule for the
approximation of definite integrals. The computer programs
are shown in Appendix D, while Simpson's rule is discussed

in Appendix E.

100

 

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a

APPENDIX B

SKETCHES OF BETA DISTRIBUTIONS

Plots of selected beta distributions are shown
in this Appendix. Certain statistics related to each
of the curves shown here were given in Appendix A.

Two curves, with the same mode, are shown on each
plate. The error rate corresponding to the highest point
on each of the two curves is the same. indicating they
share the same mode.

These plots were done by the CDC 3600 computer in
the Michigan State University Computer Center. The
actual program, written in the FORTRAN IV language, is

shown in Appendix D.

105

106

L

J,

g4/0(=1.98 6:98.02
}.

I
. \
' 1

 

 

P (X) ,
7“: \_ /0(=l.08 (5:8.92
k \W '1‘ “‘T“‘--~— 1 J
0 0.25 0.50 0.75

Error Rate

FIGURE 5. TWO BETA CURVES WITH MODE OF 0.01

b

A

)1
\/ O(=2.96 5:97.04

1

A

r" (\‘(cleue {3=8.84

.-
“'0-”

M-
l n...-

0.25

P (x)

 

0 0.50
Error Rate

 

FIGURE 6. TWO BETA CURVES WITH MODE OF 0.02
' \

P(x) "/o(=3.94 {5:96.06
1-1. .. o<=1.24 =8.76
/ \“‘\’:» fl

0 I 0.25“

 

0.50 0.75
Error Rate

FIGURE 7. TWO BETA CURVES WITH MODE OF 0.03

107

“l‘
P(x) / 004.92 fi=95.08

 

I} 'N"\‘if*<.:%1_°32 p=B.68

‘ ‘A—

0 0.25 0.50 0.75
Error Rate

FIGURE 8. TWO BETA CURVES WITH MODE OF 0.04

P(x) /0(=5.90 5:94.10

.. «=1.40 =8.60

\. ‘7“*-~—— .
0 0.25 0.50 0.75
Error Rate

 

FIGURE 9. TWO BETA CURVES WITH MODE OF 0.05

P(x)

 

. A =10.80 fi=89.20
{ < 001.80 ,6:8.20
/ 6““:1 +— 1

0 0.25 0.50 0.75
Error Rate

FIGURE 10. TWO BETA CURVES WITH MODE OF 0.10

108

 

 

P(x) + /9<=15.70 8:84.30
7 \
7 . =2.20 =7.8
',,,/./~«—--—~e..-.-....-“__.L {ff 4 °
0 0.25 0.50 0.75

Error Rate

FIGURE 11. TWO BETA CURVES WITH MODE OF 0.15

 

 

 

P(x) (0:20.60 [3:79.40
L.
/°(=2.60 ’627.40
4 __M‘ L
0 0.25 0.50 0.75

Error Rate

FIGURE 12. TWO BETA CURVES WITH MODE OF 0.20

 

 

P(X) /o(=25.50 8:74.50
///\\\‘ x/<><=3.00 p:7.00
A’ L ‘ ‘~~._:~‘"“"T--—~— l
0 0.25 0.50 0.75

Error Rate

FIGURE 13. TWO BETA CURVES WITH MODE OF 0.25

P(x)

P (x)

P (X)

 

.__,4-vf- ., w---h_.
p..." _/ 1 - . l

 

109

[0:30 .40 3:69.60

l/z/‘7~~,\\ /d:3.40 fl36.60

4"..- ’—_—---“ _ -
_f’v-‘I,’ "".--\—...-.~‘.-.

0.25 0.50 0.75
Error Rate

FIGURE 14. TWO BETA CURVES WITH MODE OF 0.30

/6<=40.20 [3:59.80

 

 

. = .2 = .
.//\_§. /o( 4 0 )3 5 80
,be»**::37”."_fig Ah?“"‘“*-w-— - so
0.25 0.50 0.75

Error Rate

FIGURE 15. TWO BETA CURVES WITH MODE OF 0.40

«=50 .00 3:50.00
/~<. f/o<==5.00 /3=5.00
.Ivrrofir::7*I——n I\\——““**1 504

0.25 0.50 0.75
Error Rate

FIGURE 16. TWO BETA CURVES WITH MODE OF 0.50

 

APPENDIX C

REVISION OF A BETA PRIOR DISTRIBUTION BY
A BINOMIAL SAMPLE RESULT
Given the prior distribution with mean, mode,
variance, etc. chosen to reflect a belief about the under-

lying pOpulation parameter 0, and expressed as

_ “-1 B-1 001
339(0) _. C19 (1-0) on

where C1 is chosen so that

1 (K-1 -1
Cl J6, 0 (1—0)8 d0 = 1

then the prior distribution can be revised using the re-

sults of a sample of n items, x of which were errors.

_' n x _ n-x
fX|9(X19) —-(X)9 (l 9)

The joint (unconditional) distribution of X and 8

is simply the product

fX 9(x,0) = fx|9(X\9)f9(9)

I

__ OK-l _ fl—l n x _ n-X
fX,Q(X'9) _. C18 (1 8) (X)8 (l 8)
__ n OH-X-l B-tn-x-l

The margional density of X can be found by inte-
grating the joint density fX e(x,8) over the range of 9

(0 to l).

110

111

fx(X):: /fX 9(X.8) d0

Range
of 9

—_- l n 0H-X-1 _ 3+n-x-l
A cl(x)0 (1 8) d0

:0 (n) 10“+X’1(1-8)B+n_x’ 100
l x O

The posterior function f8|x(9lx) is found by divid-
ing the joint distribution by the margional distribution.
f .
x 9(X 9)

fg‘XWIX): fx(x)

 

C1 (:2) e«H-x-l (1_9)A+n-x-1
r1 71;q+x—1 B+n—x41
C1(X)/O 8 (1-0) d9

901+X-1(1_9)5+n-X—1
‘/019“+X-1(l_9)fl+n-X‘ld9

However, the denominator is a definite integral, equal to

a constant, C2.

02 :.-. [018“+X'1(1-8)3*“’X'lde

._ F (OH-x) Fig +n—XL
I r“ (006”!)

Since we wish the integral (cumulative distribution

function) of the revised distribution to be equal to 1

over the range (0,1)

1 _ _ _
[1 —— 0"”X 1(1-0)’Mn X 180 2 1
0 02

then C3 may be set equal to %— and
2

112

ottx-l __ B+n—x-—l
f9|X(9|x) -_-. c 8 (1 8)

3
which is another beta distribution, with parameters

o('= d-i-x-l

(3': fl-tn-x-l.

APPENDIX D

COMPUTER PROGRAMS

This appendix contains the computer programs
used to produce the beta distribution tables in Appendix
A and the plots in Appendix B. All programs were written
in FORTRAN IV. The plotting program was run on the
Michigan State University CDC 3600 machine, while the
other routines were run on the CDC 6500. These programs
are given here for reference only since certain software
features used are not commonly available on other machines
and changes would be necessary before they could be run
on most other computers.

This set Of programs was written as a group.
Program TABLE computes statistics for a beta distribution
with various modes and with A+B (map) of either 10 or 100.
It punches a deck of cards, one card for each mode and
A+B combination, which gives A, B, and the constant c for
the basic beta distribution formula discussed in Chapter V.
In addition, TABLE prints a table of statistics giving
the mean, mode,and variance of each distribution. The main
program determines A, B, and the mode. Subroutine WORK

computes the mean and variance of that function and prints

113

114

the table. Subroutine SIMPSON uses the Simpson rule for
the approximation of definite integrals to find the con-
stant Of integration, c, which will make the value of the
integral over the range X30 to X=l equal to 1.00. Sub-
routine ONE finds the Y value Of the beta function at each
specific X value called for by SIMPSON.

Program TABLE and its subroutines follow on the

next three pages.

115

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118

Program PLOTIT uses the cards punched by TABLE
as input data and directs a routine on file in the Michi-
gan State University Computer Program Library to plot the
curves in Appendix B. One plot is produced for each run
of the program and several curves may be plotted on the
same set of X and Y axes by running the apprOpriate in—
put data cards together in one run. PLOTIT first initial-
izes the plotter pointer, then directs the drawing Of the
X and Y axes, and finally computes and plots the apprOp-
riate X and Y values for the beta curve determined by the
A, B,and C values found on the input card being read. The
last call to PLOT prepares the plotter mechanism for the
call of the next user.

Program PLOTIT appears on the following page.

119

TABLE 6. COMPUTER PROGRAM ”PLOTIT"

FDCTFQ -LOTIT
9“ FORMAT (?F10050F40020)
1? ”FA” PSOAORQC
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120
Program VALUE computes an approximation of the
definite integral of the beta function discussed in Chapter
V. The value of A, B,and C are gathered from punched cards
previously prepared by program TABLE. The main program in

VALUE reads this data, converts C from its lOgarithmic

 

value and prints the values determined by the subroutines. f
Subroutine SIMPSON uses the Simpson rule for the approxi- ;
mation of definite integrals to find the integral value.

Subroutine ONE computes the Y value for a given X value b

for the parameters A, B,and C read in by the main program.
The approximations are not perfect. They could be made
better by dividng the X axis into more than the 100 parts
which SIMPSON uses. This would, however, require addi-
tional computer time which should be weighed against the
added accuracy. The effect of this approximation may be
noted as xl approaches 1.0 under the smaller modes in the
last table in Appendix A. It can be seen that the approx—
imation to the value of the integral reaches a peak and
then decreases slightly before xl=l.O. If two place
accuracy were sufficient for the purposes at hand the
accuracy here obtained should suffice; however, more

computer time would be required for greater accuracy.

The following two pages contain program VALUE.

121

 

 

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APPENDIX E

EVALUATION OF INTEGRALS NOT SOLVABLE BY NORMAL MEANS

In order that statements may be made about the
probability of the result of an experiment represented
by a beta distribution it is necessary to evaluate defin-

ite integrals of the form
f _. _
p(d<x<f): ,j; ex“ l(l-x)g ldx.

Unfortunately, the calculus does not provide a general
method for handling such functions. However, any definite
integral can be considered as a series of areas, and the
value of the integral approximated by any method which
approximates the area. Simpson's rule for the approxima-
tion of definite integrals takes advantage of this fact and
approximates the value of the integral by summing the areas
of many small slices which themselves approximate small
parts of the integral. It divides the width of the
integral into n equal parts, each of width h::.%. The area
is bounded on the left by x0 and on the right by Xn' with
the ”inside" broken up by x1, x2, ...... Xn—Z' Xn—l' At each
x point the apprOpriate vertical distance from the x axis
to the curve is computed; yo, yl, ..., Yn-l'yn' The approx-
imate value of the integral is found by the formula

123

 

124

_ h 1
As" 3 [y6r4yI+2y2+4y3+2y4+ ..... +2yn_2+4yn_ffyn].

This formula can be used to advantage in finding the approx-
imate value of beta integrals which can not be directly
computed by the usual techniques. In general, the larger
the number of sections into which the area is divided, the
better the approximation will be. For smooth curves such

as the beta which do not have sharp peaks the approxima-
tion is quite good with a rather small number of divisions
(say 20) and approaches the actual value rapidly as the
number of divisions gets larger. With the digital com—
puter, division of the integral width into 100 or 1000 parts
for computation is easily accomplished and provides accuracy

far beyond the needs of general audit use.

 

1George B. Thomas, Jr., Calculus and Analytic
Geometry (Addison-Wesley, Reading, Massachusetts, 1968),
p. 309.

 

SELECTED BIBLIOGRAPHY

 

 

 

 

lull-hull. -
. .“1 I41:

 

SELECTED BIBLIOGRAPHY

Reference Works

 

Arens, Alvin A. "The Adequacy of Audit Evidence Accumula-
tion in Public Accounting." Unpublished Ph.D.
dissertation, Graduate School of Business, Univer-
sity of Minnesota, 1970.

Arkin, Herbert. Handbook 9f_Samplinq for Auditing and
Accounting, Volume I, Methods. New York: McGraw-
Hill Book Company, Inc., 1963.

 

 

 

Bookkeepers' Handy Guide. New York: The Ronald Press
Company, 1936.

 

 

Committee on Auditing Procedure of the American Institute
of Certified Public Accountants. Auditing Stan-
dards and Procedures. New York: AICPA, 1963.

 

 

Corless, John C. "The Assessment of Prior Distributions
for Applying Bayesian Statistics in Auditing Situa-
tions." Unpublished Ph.D. dissertation, Graduate
School of Business, University of Minnesota.

Larson, Harold J. Introduction 32 Probability Theory and
Statistical Inference. Monterey, California:
John Wiley and Sons, 1969.

 

 

LaValle, Irving H. ‘Afl Introduction £9 Probability, Deci-
sion, and Inference. New York: Holt, Rinehart
and Winston, 1970.

 

 

 

 

Mautz, R. K. and Sharaf, Hussein A. The Philosophy 9f
Auditing. Evanston, Illinois: American Account-
ing Association, 1961.

 

 

Mendenhall, William. Introduction £9 Probability and Stat-
istics. Belmont, California; Wadsworth Publishing
Company, Inc., 1967.

 

125

 

126

Schlaifer, Robert. Introduction :2 Statistics for Business
Decisions. New York: McGraw—Hill Book Company,
Inc., 1961.

 

 

 

The Study Group On Introductory Accounting. A_New Intro—
duction 32 Accounting. Seattle, Washington:
Price Waterhouse Foundation, 1971.

 

 

Thomas, Jr., George B. Calculus and Analytic Geometry.
Reading, Massachusetts: Addison—Wesley Publishing
Company, 1968.

 

 

Von Ness, Richard. "Probability: An Objectivist View,"
Elementary Statistics for Economics and Business.
Edited by Edwin Mansfield. New York: Norton, 1970.

 

Periodical Literature

 

Anderson, H. M., Giese, J. W., Booker, Jon. "Some PrOpo-
sitions About Auditing", The Accounting Review,
July, 1970.

 

 

Fisher, Sir Ronald. "The Nature of Probability,” The
Centennial Review, Summer, 1958.

 

Kraft, William H. "Statistical Sampling For Auditors: A
New Look," Journal gf Accountanqy, August, 1968.

 

Mautz, R. K., and Mini, Donald L., "Internal Control
Evaluation and Audit Program Modification," The
Accounting Review, April, 1966.

 

Sorensen, James E. ”Bayesian Analysis In Auditing", The
Accounting Review, July 1969.

 

Tracy, John A. "Bayesian Statistical Confidence Intervals
For Auditors," Journal 2: Accountancy, July, 1969.

 

Tracy, John A. "Bayesian Statistical Methods in Auditing,"
The Accounting Review, January, 1969.

 

 

 

 

          

   

    

 

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