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 FACULTY WORKING 
 PAPER NO. 1053 
 
 Subjective Prior Probability Distributions 
 and Audit Risk 
 
 Paul J. Beck 
 Ira Solomon 
 Lawrence A. Tomasslni 
 
 JUL 6M4 
 
 UNIVERSITY OF ILLINOIS 
 URBANA champaign 
 
 College of Commerce and Business Administration 
 Bureau of Economic and Business Research 
 University of Illinois, Uroana-Champaign 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 1053 
 College of Commerce and Business Administration 
 University of Illinois at Urbana-Champaign 
 June 1984 
 
 Subjective Prior Probability Distributions 
 and Audit Risk 
 
 Paul J. Beck, Associate Professor 
 Department of Accountancy 
 
 Ira Solomon, Associate Professor 
 Department of Accountancy 
 
 Lawrence A. Tomassini 
 University of Texas at Austin 
 
 Acknowledgment: We wish to acknowledge the suggestions of 
 the referee on earlier drafts and the comments of accounting 
 workshop participants at the University of Illinois, Indiana 
 University, Ohio State University, and the University of 
 Minnesota. Other useful comments were provided by Ed Blocher, 
 Dave Burgstahler, Dan Dhaliwal, Bill Felix, Jim Jiabalvo, D. 
 Paul Newman, Jack Robertson, Mike Shields, and Bill Waller. 
 
Digitized by the Internet Archive 
 
 in 2011 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/subjectivepriorp1053beck 
 
Subjective Prior Probability Distributions and Audit Risk 
 
 Abstract 
 
 This paper presents an analysis of the audit risk consequences of PPD ex- 
 tremeness deficiencies and miscalibration. While there is empirical evidence 
 that auditors, like many other decision makers, assess miscalibrated PPDs , the 
 attendant inferential risk consequences of such deficiencies have not been 
 addressed in the extant literature. The comparative statics analysis performed 
 in this study indicates that the risk effects of miscalibration and extremeness 
 deficiencies on the auditor's (substantive testing) evaluation decision are 
 complex and cannot be predicted from an examination of the planning (sampling 
 size) decision. 
 
1.0 Introduction 
 
 Bayesian models of the auditor's decision process (e.g., 
 Bailey and Jensen [1977], Kinney [1975], and Scott [1973, 1975]) 
 incorporate subjective beliefs formalized as prior probability 
 distributions (PPDs). Recent studies by Solomon [1982] and 
 Tomassini et al. [1982] investigated two conformance properties 
 of auditors' account balance PPDs: extremeness and calibration. 
 PPD extremeness is measured over a sequence of elicitation trials 
 by computing the average subjective probability assigned to inter- 
 vals containing the auditees' actual account values and, as such, 
 can be viewed as a measure of predictive ability (see Seaver et 
 al. [1978]). Calibration, in turn, is a measure of the ability 
 to express an appropriate degree of confidence in such subjective 
 estimates. Both the Solomon [1982] and Tomassini et al. [1982] 
 studies reported that auditors' PPDs were raiscalibrated and that 
 there was limited evidence of under confidence (see fn. 1). Other 
 probability elicitation studies in psychology also have found 
 miscalibration, but with the exception of experienced weather 
 forecasters, these other studies have reported almost universal 
 evidence of overconf idence (see Lichtenstein et al. [1982]). 
 
 Although several empirical studies have investigated PPD cali- 
 bration and extremeness, their specific decision-making consequences 
 have not been analyzed formally in either the psychology or 
 accounting literatures. Libby [1981] discussed the effect of mis- 
 calibration on the audit sample size decision in compliance 
 
-9- 
 
 testing and concluded that overconf idence impairs audit effec- 
 tiveness, and underconf idence impairs audit efficiency. Tomassini 
 et al. [1982] investigated the implications of underconf idence for 
 the sampling decision in a substantive testing context, but did 
 not consider the subsequent effect on the auditor's evaluation of 
 account book values. 
 
 Simulation studies (e.g., Blocher [1981] and Cushing [1980]) 
 have been used to investigate the sensitivity of audit planning 
 (sample size) decisions to misspecif ications of PPD parameters in 
 the compliance and substantive testing contexts, respectively. 
 While these studies indicated that planning decisions were sen- 
 sitive to misspecif ications of both PPD means and/or variances, 
 the findings permit only very limited conclusions about the adverse 
 consequences of extremeness deficiencies and miscalibration. 
 
 In our study, we identified specific audit risk consequences 
 of PPD extremeness deficiencies and miscalibration for Bayesian 
 hypothesis tests of account balances (which could be applicable to 
 quality control and cost-variance decisions). Following Kinney 
 [1975], we assumed that the auditor assesses account balance PPDs 
 which are combined with mean-per-unit (MPU) sample estimates to 
 form posterior probability distributions. The effects of extreme- 
 ness deficiencies were then analyzed for a sequence of such 
 posterior distributions by varying the PPD mean displacement rela- 
 tive to the corresponding outcome sequence. Similarly, the effects 
 
-3- 
 
 of mis calibration were determined by varying the correspondence 
 between the assessor's perceptions of PPD mean displacement and 
 the actual displacement. We subsequently identified the attendant 
 audit risk consequences by interpreting our comparative statics 
 results in the context of the Kinney [1975] audit decision model. 
 
 Our study extends previous research first by formally analyzing 
 the audit risk effects of miscalibration and extremeness deficien- 
 cies on both planning and evaluation of substantive tests. Second, 
 by allowing subjects' PPD variances to adjust in response to the 
 perceived PPD mean displacement, we were able to avoid experimen- 
 tally confounding the effects of calibration and extremeness defi- 
 ciencies. 
 
 Our analysis showed that the risk consequences of miscalibra- 
 tion (overconf idence and underconf idence) were complex and depended 
 upon the actual population mean and the nature and severity of any 
 extremeness deficiencies. Furthermore, unlike Libby [1981], we 
 found that overconf idence actually can predispose auditors to com- 
 mit either effectiveness or efficiency errors. The remaining sec- 
 tions of the paper are organized as follows: Section 2 presents 
 our modeling assumptions which serve as a foundation for the com- 
 parative statics analysis in Section 3. This is followed in 
 Section 4 by a discussion of the audit risk implications of our 
 comparative statics findings. Concluding remarks, with implica- 
 tions for the design of audit training programs are provided in 
 Section 5. 
 
-4- 
 
 2.0 Audit Decision Context Assumptions 
 
 Audit verification of account balances is based upon both 
 objective and subjective information. Objective information for 
 asset accounts (such as inventory and receivables) is generally 
 obtained by sampling from populations of subsidiary book values. 
 Previous Bayesian studies in auditing have modelled such popula- 
 tions using a probability density with an uncertain mean, u. In 
 these models, a PPD is assessed for each D and then combined with 
 sample information to form a posterior probability distribution. 
 In practice, such PPDs would incorporate all of the available 
 forms of evidence typically collected prior to substantive 
 testing, including compliance tests. 
 
 2.1 Hvnothesis Testing Assumptions 
 
 We model the audit evaluation decision as a Bayesian hypothe- 
 sis test (i.e., we assume a two-point auditor loss function). 
 This simplifying assumption implies that the losses borne by the 
 auditor do not depend upon the magnitude of misstatement (see 
 Moriarity [1975]) and ensures that the optimal decision rule is 
 consistent both with the expected linear utility framework, 
 adopted herein, and with traditional definitions of audit risk in 
 the professional auditing literature. Specifically, for asset 
 accounts, we assume that the null and alternative hypotheses are: 
 
-5- 
 
 H~: U > X - M (i.e., account book value is not materially 
 overstated) 
 
 H : ji j^_ X - M (i.e., account book value is materially 
 overstated) , 
 
 where X is the auditee's reported account book value (expressed as 
 an average), and M is the materiality limit defined with respect 
 to the mean account book value. Note that the null hypothesis 
 does not consider the case of material understatements, but other- 
 wise is consistent with the hypothesis formulation initially pre- 
 sented in Elliott and Rodgers [1972] and subsequently adopted in 
 Section 4 of the recently issued Audit and Accounting Guide: 
 Audit Sampling (AICPA [1983]). A discussion of the modified risk 
 implications for cases in which the auditor employs a two-sided 
 hypothesis test is presented below (see fns. 8 and 10). 
 
 Figure 1 presents the four possible outcomes associated with 
 the audit evaluation (acceptance-rejection) decision and the two 
 hypothesized states of nature, H~ and H . When y _<_ X - M, the 
 null hypothesis (H„) should be rejected. Thus, if H» were 
 accepted, audit effectiveness would be compromised and the loss is 
 represented by L • Alternatively, when y > X - M, audit effi- 
 ciency would be compromised if H were to be rejected, and 
 L represents the associated loss. 
 
 Insert Figure 1 Here 
 
 Assuming minimization of expected losses, the optimal audit 
 decision rule (see Berger [1980]) is to reject H_ if and only if: 
 
-6- 
 
 Pr(H ) L 
 
 <T LL > (1) 
 
 1 - Pr(H Q ) Lj 
 
 2 
 where Pr(H~) represents the posterior probability that S > X - M. 
 
 The above formulation simplifies our analysis because, for a 
 given loss ratio, L T /L , the optimal decision depends solely upon 
 the posterior likelihood ratio, Pr(H Q )/(l-P (H Q )). The rela- 
 tionship between L and L affects the audit risk implications of 
 miscalibration and extremeness deficiencies and will be discussed 
 further in Section 4. 
 
 2.2 Distributional AssumDtions 
 
 Consistent with previous research (e.g., Kinney [1975] and 
 Scott [1973, 1975]), we assume that the distribution of subsidiary 
 book values and auditor's PPD are normally distributed and that 
 the ordinary MPU estimator is employed. We also assume that the 
 variance of each distribution is known. Given these assumptions, 
 the posterior distribution for each audited account value is nor- 
 mally distributed and parameterized (see Winkler [1972, p. 169]) 
 by (m",v"), where: 
 
 jm 
 m = : (2) 
 
 Ar = -V+-2- (3) 
 
 V V V 
 
 and: 
 
-7- 
 
 m = sample estimate of the population mean, p; 
 
 m 1 = PPD mean; 
 
 m" = posterior estimate of the mean; 
 
 n = quantity of sample evidence; 
 
 v/n = variance of m; 
 
 v 1 = PPD variance for the assessed m'; 
 
 v" = posterior variance; 
 
 The above assumptions are not as restrictive as they might 
 appear since, as discussed below, our analysis of the simple MPU 
 estimator can be generalized directly to the stratified MPU esti- 
 mator. Also, the assumption of population normality is defensible 
 when stratified sampling is employed since' the strata, which are 
 more internally homogeneous, are less skewed than the population 
 taken as a whole. More importantly, sample MPU estimates of the mean 
 should be normally distributed, given the central limit theorem. 
 Previous empirical results (e.g., Cushing and Romney [1981], and 
 Solomon, et al. [1982]) suggest also that auditors' PPDs assessed 
 for highly skewed populations, such as those described in Neter 
 and Loebbecke [1975], are typically symmetric and can be approxi- 
 mated by normal PPDs. Finally, relaxing the assumption of known 
 sample variance by allowing an assessment of a second PPD for the 
 variance parameter (actually a joint PPD for the mean and variance 
 parameters) as in Felix and Griralund [1977] would not change our 
 analysis in any significant manner. Calibration and extremeness, 
 
however, would have to be redefined with respect to each of the 
 unknown parameters. 
 
 Our assumptions establish a simplified environment in which 
 there is only one uncertain parameter and there are no sample esti- 
 mation problems due to skewness and/or an unknown sample variance. 
 If miscalibration can be shown to be a significant problem in the 
 simplified environment modeled herein, we can reasonably expect 
 it to be as important or more so when the probability elicitation 
 task is more complex and there are statistical estimation problems. 
 
 2 .3 Extremeness and Calibration Assumptions 
 
 Since calibration and extremeness are defined with respect to 
 a sequence of PPDs and corresponding outcomes (see Lichtenstein et 
 al. [1982]), we assumed that the probability assessment-hypothesis 
 
 testing procedures were to be repeated for a sequence of trials 
 
 3 
 
 with outcomes, (u^.; t=l, T}. This assumption is consistent with 
 
 traditional definitions of inferential risks (e.g., alpha and beta 
 risks) which are defined with respect to a sequence of estimates. 
 
 PPD extremeness, as noted above, is measured ex post by 
 imputing the probability assigned to a suitably chosen interval 
 containing each y and then averaging these probabilities over the 
 sequence (see Solomon [1982]). An achievement of high levels of 
 extremeness would then require that the auditor-assessor specify 
 distributions which are centered appropriately, relative to the 
 u sequence, and are tight (see Peters [1978]). Otherwise, the 
 
-9- 
 
 highest probabilities would be assigned to population values 
 which do not occur most frequently. We modeled extremeness 
 deficiencies in our study by PPD mean displacement (i.e., 
 m = u + A, where A denotes the PPD mean displacement). By 
 varying A in our comparative statics analysis, we manipulated the 
 severity of the assessor-auditor's extremeness deficiency. 
 
 The auditor's calibration, (see above and also fn. 1); reflects 
 the ability to express an appropriate degree of confidence in sub- 
 jective estimates of population means so it is defined with 
 respect to a given level of extremeness (i.e., magnitude of A). 
 Both overconf idence and underconf idence calibration defi- 
 ciencies were modeled. An overconfident assessor's PPDs would be 
 too tight and, therefore, over a sequence of elicitation trials, 
 too many realizations would be captured in the outer fractile 
 ranges, and too few captured in the inner fractile ranges (see 
 Lichtenstein et al. [1982]). In contrast, underconf ident 
 assessors would specify PPDs which are too diffuse, in that the 
 inner fractile ranges capture too many outcomes, while the outer 
 fractile ranges capture too few outcomes. The variance of a nor- 
 mal PPD represents the assessor's uncertainty about the population 
 mean and, thus, is related directly to the auditor's calibration 
 (see Moskowitz and Bullers [1979]). 
 
 Calibration was modeled in our comparative statics analysis by 
 assuming that the auditor would implicitly estimate the average 
 
-10- 
 
 mean displacement that would exist if the elicitation process were 
 repeated for the entire u sequence, and then would adjust the 
 assessed PPD variance on each trial to reflect the estimated (aver- 
 age) displacement. Since each trial was assumed to be exchange- 
 able, the assessor's perceptions about displacement would be the 
 same for each trial in the sequence (see fn. 3). Furthermore, 
 provided that an assessor were not completely miscalibrated, there 
 should be some correspondence between the assessor's perceptions 
 about the PPD mean displacement (A) and the actual A. Accordingly, 
 we formalized this correspondence between A and A by expressing 
 the PPD variance on each trial as a function of A (i.e., v' = 
 v'(A), where A = A(A)). Note that, within the context of our 
 model, perfect calibration would exist if A(A) = A for each pos- 
 sible level of extremeness since the auditor would be able to 
 spread the PPD variance appropriately (see fn. 2). Alternatively, 
 A(A) > A would imply underconf idence, while A(A) < A would indi- 
 cate overconf idence. The correspondence between A and A will be 
 varied in our comparative statics analysis to identify the con- 
 sequences of the two types of miscalibration. 
 
 3.0 Comparative Statics Analysis 
 
 In this section we first investigate the effects of extreme- 
 ness deficiencies and miscalibration on a sequence of posterior 
 
-11- 
 
 distributions , after which (in Section 4) the audit risk implica- 
 tions of these effects will be discussed. 
 
 3.1 Extremeness 
 
 The effect of PPD mean displacement on the posterior mean for 
 trial t can be determined by substituting u + A for m' into (2). 
 Assuming that m = u , the following simplified expression for the 
 posterior mean is obtained: 
 
 m t = \ + (1 + n v'/v) ' (4) 
 
 where n = n(m ,v'), m = u + A, and v' = v'(A(A)). 
 
 The difference between m and \i represents the magnitude of 
 
 t t F 6 
 
 posterior mean displacement on trial t. Since (1+n v'/v) > 0, 
 equation (4) predictably indicates that the posterior mean on each 
 trial will be displaced in the same direction as the PPD mean due 
 to the presence of A. However, the magnitude of the posterior 
 
 mean displacement can vary from trial to trial. Accordingly, we 
 
 4 
 take an expectation (average) with respect to {y ). 
 
 E[nT] = E[u t J + E[A/(l+n t v'/v)]. (5) 
 
 The consequences of altering PPD extremeness deficiencies for 
 the sequence are determined by differentiating (5) totally with 
 respect to A using the quotient and chain rules, 
 
-12- 
 
 dn 
 
 ._, „, (1 + n v'/v) [A[(n /v>^-+ ( v '/v>-f] 
 aE [m J _ r t L t dA dA J 1 
 
 "dA~" = E[ T~. 7772" T~ 7772 ; 
 
 (1 + n t v'/v) (1 + n t v'/v)' 
 Simplifying (6) algebraically, 
 
 dE[m"] f 1 V dv' v'A dn t i m 
 
 dA " Wl + n v'/v) n . ,, ,2 ,. n . ,. ,2 dA '* U) 
 t v(l + n v'/v) dA v(l + n v'/v) 
 
 Equation (7) shows that the total effect of PPD mean displace- 
 ment on the expected value of the posterior mean sequence can be 
 decomposed into three terms. The first term on the right side of 
 (7) represents the expected (upward) posterior mean displacement 
 which would occur if ^v /<ja = ^ n t/dA = 0, whereas the second and 
 third terms reflect the sensitivity of the PPD variance and sample 
 size, respectively,' to changes in the magnitude of PPD mean dis- 
 placement. The second term can be further decomposed as follows: 
 
 dv' dv' dA „.. . . . . ,. 
 
 = — . This decomposition indicates that the expected mag- 
 
 dA dA dA 
 
 nitude of posterior mean displacement in (7) is influenced by the 
 
 auditor's calibration (sensitivity) to PPD mean displacement, as 
 
 <?i , , dA - . dv' _ , . , 
 
 reflected by the -rr component of the — — terra. In particular, if 
 * da dA 
 
 -t— < (which implies very severe overconf idence caused by the 
 assessor's believing that the PPD mean displacement has been reduced 
 when it actually has increased), the expected posterior mean 
 displacement in (7) would be increased due to the minus sign pre- 
 ceding the second term. 
 
-13- 
 
 Another possibility is that <_ -7— < 1. In this situation, 
 the auditor would still be overconfident, because the PPD variance 
 would not change commensurately with the magnitude of the PPD mean 
 displacement. Hence, the auditor would fail to recognize the 
 actual level of PPD mean displacement. Note that such overcon- 
 fidence was effectively incorporated in Cushing's [1980] study in 
 
 which the PPD mean and variance were perturbed independently 
 
 c dv ' d * m 
 (i.e. , — - = 0;. 
 
 dA dA 
 
 Alternatively, if -rr = 1.0, the auditor would exhibit an 
 appropriate calibration sensitivity to the PPD mean displacement 
 by increasing the PPD dispersion, thereby, effectively eliminating 
 posterior mean displacement. Finally, if -rr- > 1.0 (i.e., auditor 
 is underconf ident) , the assessed PPD variances would be too large 
 given the actual level of PPD mean displacement. While (8) indi- 
 cates that severe underconf idence can effectively drive the 
 expected posterior mean displacement to zero, unnecessary sampling 
 would still be performed. 
 
 Further insight regarding the sample size effect is obtained 
 
 by decomposing the dn /dA term in (7) which will show the depen- 
 
 dence of n on A through the A and v' terms: 
 
 dn 3n , , 3n , , * 
 t_ = t dm' t dv' dA , g \ 
 
 dA 8m' dA 3v* d £ dA * 
 
 3n 
 
 The -r— T-J7— term in (8) can be either positive or negative, 
 dm a/i 
 
 depending on the auditor's loss function. Hence, general conclu- 
 
-14- 
 
 sions about the sample size effect and the magnitude of the 
 posterior mean displacement are not possible. Nevertheless, the 
 decomposition in (8) indicates that there is a "calibration com- 
 ponent" embedded in the ^ n t/dA term in (7). Since the sign of the 
 terms multiplying c^dA i- n (7) is negative, the expected magni- 
 tude of the posterior mean displacement will be reduced by the 
 
 , • v • L dv' d'A x . . . dv' dA . _ 
 calibration component when — z > 0, but increased when — s < 0. 
 
 dA dA dA dA 
 
 3.2 Calibration 
 
 This subsection extends the previous analysis by investigating 
 the effects of PPD miscalibration on the posterior variance 
 sequence. This is done by holding A constant, and varying the 
 inaccuracy of the assessor's perception of the actual displacement. 
 More specifically, we model the effects of miscalibration by 
 assuming that A(A) ■ (1+<J>)A, where <J> represents the inaccuracy of 
 the assessor's perception of A. In this framework, $ = would 
 imply that the assessor's perception of the actual level of PPD 
 mean displacement across the sequence of trials is accurate, so 
 that the PPD variance can be adjusted appropriately. Alterna- 
 tively, $ < and $ > would indicate overconf idence and under- 
 confidence, respectively. 
 
 As a first step here, we solve (3) for v". Upon simplifica- 
 
 tion, the posterior variance for the t trial can be expressed 
 
 as : 
 
 v t = v/(n t + v/v'), (9) 
 
-15- 
 
 where: n = n(m t ,v f ), m = u + A, v' = v'(A), and A = (l+<j>)A. 
 Taking an expectation with respect to the u sequence, we 
 obtain the average variance of the posterior sequence: 
 
 E[v t ] = vE[l/(n t + v/v')]. (10) 
 
 Differentiating (10) with respect to <j>, 
 
 dE[v"] " dn 
 
 #T7 f r , fl -2 r t dv' dA dv' dA, f 2i ,,,. 
 
 = -vE [n + v/v f ] i-T-r—^--rr- v— s— -rr/v f } . (11) 
 d<t> t d v dA d $ dA d< J> 
 
 Given that -rr = A and — s— > 0, (11) indicates that 
 d * dA 
 dE[v"]/ d 4> >(<) o when 
 
 dn t 2 
 
 -7-r<(>) v/v' Z . (12) 
 
 dv 
 
 Since the direction of the inequality in (12) depends upon the 
 specific parameter values, general conclusions about the effects 
 either of underconf idence or overconf idence are not possible. An 
 inspection of (12) indicates, however, that misspecif ication of 
 the PPD variance will have the greatest impact on the posterior 
 variance when the PPD variance is small in relation to the sample 
 variance — i.e., the auditor has substantial subjective knowledge — 
 or sampling costs are high so that t^dv' ^ s small. Hence, in 
 situations in which the formal incorporation of subjective audit 
 knowledge potentially would be most beneficial, underconf idence 
 will cause the posterior variance to be biased upward, while over- 
 confidence will cause the posterior variance to be biased downward. 
 
-16- 
 
 3. 3 Stratification 
 
 So far our analysis of extremeness deficiencies and miscali- 
 bration has assumed unstratified MPU sampling. In practice, 
 however, population skewness often necessitates stratification. 
 Application of Bayesian stratified methods requires the assessment 
 of a vector of means and k(k+l)/2 elements of the variance- 
 covariance matrix (see Ericson [1965]). But, in the special case 
 in which the stratum means are independent (i.e., the variance- 
 covariance matrix is diagonal), extremeness and calibration can be 
 computed with respect to a PPD sequence for each stratum. This 
 case represents the direct Bayesian analogue of the classical 
 stratified sampling model in which the sample mean estimates are 
 assumed to be distributed independently. 
 
 In such Bayesian stratified sampling models, posterior esti- 
 mates of both the mean and variance are computed for each stratum 
 and then aggregated linearly to obtain a stratified posterior 
 estimate for the population as a whole. As a result, if our 
 assumed hypothesis test (decision rule) were employed to evaluate 
 the stratified posterior estimates, we would obtain comparative 
 statics results similar to those reported above. For example, if 
 the stratum PPD mean sequences are displaced in the same direc- 
 tion, the posterior means and stratified mean estimate for the 
 population also will be displaced and in the same direction. If, 
 in addition, the auditor is overconfident (underconf ident ) in 
 
-17- 
 
 assessing the PPDs for one or more strata, the expected magnitude 
 of posterior mean displacement will be increased (decreased) and 
 the expected posterior variance also will be increased (decreased) 
 Unfortunately we are unable to provide general results when the 
 strata means are correlated or if calibration deficiencies differ 
 across strata. 
 
 4.0 Audit Risk Implications 
 
 We now identify the risk implications of extremeness deficien- 
 cies and miscalibration for the audit evaluation decision. An ex 
 post perspective is implicitly adopted in which we focus on a 
 representative trial (t) and separately analyze Case I in which 
 \i <_ X - M and Case II in which u > X - M. Separate risk analy- 
 ses are required, because audit efficiency and effectiveness 
 errors depend jointly upon the audit decision and whether the 
 account balance actually contains a material (overstatement) 
 error. 
 
 4.1 Case I: Book Value Materially Overstated 
 
 Since the account book value is assumed to be materially 
 overstated (i.e., u^ _<_ X - M) , we focus on the auditor's risk of 
 committing an effectiveness error in the four subcases below. 
 
 ^.1.1 No Mean Displacement 
 
 We initially simplify by considering the risk effects of mis- 
 
 6 - 
 calibration in the absence of PPD mean displacement. Figure 2 
 
-18- 
 
 presents two normal posterior probability distributions which, for 
 expositional purposes, are assumed to be based upon PPDs assessed 
 by auditors A and B, respectively. Since the PPDs and posterior 
 distributions correspond to the same u outcome, we henceforth 
 replace the t subscript on all PPD parameters with the letters A 
 and B, so that each distribution can be associated with a par- 
 ticular auditor-assessor. While both auditors are assumed to 
 assess PPDs which are properly centered vis-a-vis the actual 
 account balance (i.e., m = ol = u ) , Auditor A is more under- 
 confident than Auditor B (i.e., <j). > <f>„ > 0). Therefore, con- 
 
 A a 
 
 av r "i 
 sistent with our assumption that — > 0, distribution A is 
 
 more diffuse than distribution B and, thus, can be interpreted as 
 
 reflecting greater underconfidence. 
 
 Insert Figure 2 Here 
 
 The area under each distribution to the right of X-M repre- 
 sents the probability that the actual account balance mean (u ) 
 exceeds X-M and is denoted by Pr(H»), while that to the left of 
 X-M represent Pr(H ) = 1 - Pr(H ). Given the symmetry property of 
 normal distributions and the further assumption that m ■ V , the 
 posterior means, m , nL _<_ X - M, so the computed Pr(H~) _< .5 for 
 both distributions A and B. 
 
 The optimal decision, for either auditor, is to reject H~ when 
 the computed likelihood ratio, Pr(H )/(l-Pr(H )) < L /L . Since 
 
-19- 
 
 the costs of effectiveness and efficiency errors depend upon fac- 
 tors specific to the decision-making context, general conclusions 
 cannot be drawn about the loss ratio (L /L ) . Typically, however, 
 the assumption is that losses from audit effectiveness errors are 
 larger than those from audit efficiency errors, implying that 
 L /L > 1.0 (see Elliott and Rodgers [1972] and AICPA [1981, 
 1983]). This means that, in the present subcase in which 
 Pr(H )/(l-Pr(H )) _<_ 1.0, H Q would be (correctly) rejected by 
 either Auditor A or B. In fact, varying the magnitude of the pos- 
 terior variance would not alter the rejection decision, because 
 the computed Pr(H_) < .5 when m , til = u <_ X - M. Therefore, in 
 the absence of mean displacement, underconf idence does not affect 
 audit risk exposure (see Table 1). 
 
 But the audit planning decision would be adversely affected by 
 underconf idence. That is, ceteris paribus , an underconf ident 
 auditor would choose a larger sample size than a properly cali- 
 brated auditor when ^ n /dv' > 0* This would result in an oppor- 
 tunity cost due to insufficient reliance upon subjective 
 knowledge. The next two subcases extend the preceding analysis by 
 incorporating PPD mean displacement. 
 
 Insert Table 1 Here 
 
 4.1.2 Moderate Upward Mean Displacement 
 
 We now assume that both Auditors A and B assess PPDs whose 
 means are equally displaced upward (i.e., m = hl > M._ )> but 
 
-20- 
 
 Auditor A is more underconf ident than B. From our previous analy- 
 sis, it is apparent that both posterior means will be displaced 
 upward (i.e., m , itl > \i ), so that the computed Pr(H~) and like- 
 lihood ratio will be increased. Here, we consider the risk 
 effects when u < ra = nu < X - M, but in the next subsection we 
 consider more serious upward mean displacements wherein 
 \i <X-M<m=nL. 
 
 When u < m = nL < X - M, the posterior means, m , nu < X - M, 
 so the computed Pr(H~) < .5 for both distributions. Therefore, 
 given that L > L , both auditors will correctly reject H„, so 
 changes in the PPD variance due to miscalibration would not alter 
 the rejection decision. Therefore, upward mean displacement 
 and/or miscacalibration do not affect risks provided that 
 = nu < X - M (see Table 1). 
 
 » i 
 m 
 
 4.1.3 Serious Upward Mean Displacement 
 
 In practice, the magnitude of the upward PPD mean displacement 
 may be so great that ra = hl > X - M. If so, both likelihood 
 ratios could exceed the loss ratio, which would result in the com- 
 mission of effectiveness errors (see Table 1). The predisposi- 
 tion to commit an effectiveness error, however, is influenced by 
 
 Q 
 
 the severity and nature of the auditor's miscalibration. Figure 
 3 presents two posterior probability distributions for Auditors 
 A and B. As in 4.1.1, m = nu, but the posterior mean displace- 
 ment is less for distribution A, because the (displaced) PPD mean 
 
-21- 
 
 is weighted less heavily vis-a-vis the sample mean due to more 
 serious underconf idence. 
 
 Insert Figure 3 Here 
 
 Reducing the upward posterior mean displacement, ceteris pari- 
 bus , increases the computed Pr(H ) and reduces Pr(H ). Since the 
 area under the lower tail of the distribution also is increased by 
 underconf idence, the computed Pr(H ) is further increased, thereby 
 
 3. 
 
 reducing the computed likelihood ratio. Thus, underconf idence 
 reduces the auditor's predisposition to commit an effectiveness 
 error due to upward mean displacement (see Table 1), but at the 
 cost of suboptimal (larger) sample sizes. 
 
 The risk effects of overconf idence can also be determined from 
 Figure 3 by reinterpreting distribution B as exhibiting greater 
 overconf idence than A. Predictably, the effects of overconf idence 
 on the posterior distribution are contrary to those described 
 above for underconf idence, because the weight accorded the PPD 
 mean is increased and the audit sample size is simultaneously 
 reduced. As a result, the posterior mean displacement is 
 increased and the posterior variance is decreased. Both effects 
 reduce the area to the left of the point X-M thereby increasing 
 the computed Pr(H ). Hence, when ra = iil > X - M, overconf idence 
 
 increases the auditor's predisposition to commit an effectiveness 
 
 9 
 
 error resulting from upward PPD mean displacement (see Table 1). 
 
-22- 
 
 4.1.4 Downward Mean Displacement 
 
 We now assume that the PPD means are displaced downward (i.e., 
 
 » i 
 
 . < u ) . Since by assumption, ra = u <_ X - M, our previous 
 
 analysis indicates that the posterior means, m , itl < X - M. 
 Therefore, the computed Pr(H ) < .5 for both distributions which 
 ensures that both auditors will correctly reject H„. Furthermore, 
 as discussed above, changes in the PPD variance due to miscalibra- 
 tion would not increase the computed Pr(H n ) above .5. Hence, no 
 risk consequences are associated with downward PPD mean displace- 
 ment and/or miscalibration (see Table 1). 
 
 4.2 Case II; Account Book Value NOT Materially Overstated 
 
 We now assume that the account book value is not materially 
 overstated (i.e., u > X - M) . This allows us to focus on the 
 risk of committing an efficiency error due to PPD extremeness 
 deficiencies and miscalibration. 
 
 4.2.1 No Mean Displacement 
 
 We first consider a subcase in which there is no PPD or 
 sample mean displacement (i.e., m = nu = m,= M ), but the auditor 
 is underconf ident . Figure 4 presents two posterior probability 
 distributions which are based upon the PPDs assessed by Auditors A 
 and B and are analogous to those of Figure 2, except we now assume 
 that U > X - M. As in 4.1.1, the computed Pr(H ) is higher for 
 the more diffuse distribution (A) which reduces the likelihood 
 
-23- 
 
 ratio. However, in contrast with the earlier effects, the more 
 underconf ident auditor has a greater predisposition to commit an 
 efficiency error by incorrectly rejecting H_. 
 
 Insert Figure 4 Here 
 
 4.2.2 Upward Mean Displacement 
 
 Upward PPD mean displacement (i.e., m. = m_ > u ), ceteris 
 paribus results in upward posterior mean displacement which, as in 
 cases 4.1.2 and 4.1.3, increases the computed Pr(H n ). Here, 
 however, increasing the likelihood ratio reduces the risk of effi- 
 ciency errors (see Table 1). Figure 5 presents two posterior 
 distributions depicting the joint effects of upward PPD mean 
 displacement and miscalibration. Posterior distribution A is cen- 
 tered to the right of distribution B and is less diffuse to 
 reflect greater overconf idence. As in subsection 4.1.3, both the 
 displacement and dispersion effects upon the posterior distribu- 
 tion increase the computed Pr(H ). However, the effect of 
 increasing the computed Pr(H_) now is to reduce the probability of 
 efficiency errors. 
 
 Insert Figure 5 Here 
 
 Unfortunately, when there is underconf idence, the risk effects 
 are more difficult to discern. Figure 5 can be reinterpreted to 
 facilitate an analysis of underconf idence in conjunction with 
 
-24- 
 
 upward PPD mean displacement. The more diffuse distribution B now 
 represents greater underconf idence than distribution A. Note that 
 greater PPD underconf idence reduces the upward posterior mean 
 displacement while simultaneously increasing the posterior vari- 
 ance. This effect tends to mitigate the effect of upward PPD mean 
 displacement. As a result, the ultimate risk consequences are 
 indeterminate unless specific assumptions are made both about the 
 magnitude of the PPD mean displacement and the severity of the 
 auditor's underconf idence (see Table 1). 
 
 4.2.3 Downward Mean Displacement 
 
 We now analyze the case of downward PPD mean displacement in 
 which m = nu < y . Since the posterior mean is displaced in the 
 same direction as the PPD mean, the computed Pr(H_) is decreased 
 which, ceteris paribus , predisposes the auditor to commit effi- 
 ciency errors (see Table 1). Such predisposition, however, is 
 also influenced by the auditor's miscalibration. Figure 6 pre- 
 sents two posterior distributions which depict the joint effects 
 of downward PPD mean displacement and miscalibration. Auditor A's 
 posterior distribution is displaced further than B's and also is 
 tighter than distribution B due to greater overconf idence. Note 
 that the areas to the left of X - M are approximately the same for 
 both distributions, given the opposing mean and variance effects 
 on the computed Pr(H ). Hence, the efficiency risk consequences 
 cannot be determined. Finally, note that, while we have assumed 
 
-25- 
 
 implicitly that X - M < m. * m_ < U , the same indeterminate risk 
 effects also would arise when m=m_<X-M<u. 
 
 Insert Figure 6 Here 
 
 The risk consequences of underconf idence also can be deter— 
 mined from Figure 6 by reinterpreting the relationship between the 
 two posterior distributions. Auditor B is now assumed to be more 
 underconf ident than A, so distribution B exhibits less downward 
 mean displacement and also is more diffuse than distribution A. 
 However, since the joint effects of underconf idence and downward 
 mean displacement are in opposition to one another, the audit 
 efficiency risk exposure cannot be determined again without speci- 
 fic assumptions both about the magnitude of PPD mean displacement 
 and severity of the underconf idence (see Table 1). 
 
 5.0 Concluding Remarks 
 
 In this paper, we have investigated the audit risk consequences 
 of PPD extremeness deficiencies and miscalibration in a Bayesian 
 hypothesis testing context. These risk consequences, as sum- 
 marized in Table 1, are complex and their identification requires 
 an analysis of both the auditor's planning and evaluation deci- 
 sions. In contrast with the existing literature, our analysis 
 showed that overconf idence could predispose the auditor toward 
 either efficiency or effectiveness errors. Furthermore, the risk 
 consequences of each type of miscalibration were found to depend 
 
-26- 
 
 jointly upon the actual population mean and the nature and sever- 
 ity of the extremeness deficiencies. 
 
 Public accounting firms devote significant resources to 
 training programs designed to impart audit knowledge which can 
 improve the subjective judgment accuracy component portrayed here 
 by PPD extremeness. Our analysis suggests the need to give addi- 
 tional consideration to training directed specifically at 
 improving auditor calibration. 
 
-27- 
 FOOTNOTES 
 
 1. A sequence of continuous PPDs would exhibit proper calibra- 
 tion (and the assessor's confidence would be appropriate) if 
 the fractiles of the PPDs correspond with the relative out- 
 come frequencies. For example, only 10% of the outcomes 
 should fall to the right of the .90 fractiles of the PPD 
 sequence. If more (less) than 10% of the outcomes should 
 occur in the right tails of the PPDs, overconf idence (under- 
 confidence) is indicated. 
 
 2. The auditor has two choices: 1) conclude that the account 
 book value is materially misstated and possibly extend 
 substantive tests or 2) conclude that the account book value 
 is fairly stated and discontinue testing. If the former 
 choice is made, the expected loss is: 
 
 P r (H ) • + (1 - P r (HQ))L II . Alternatively, if Hq is 
 rejected, the expected loss is: P r (H())Li + (1 - P r (H()))0. 
 The rejection decision is optimal when: P r (Ho)L]; < (1 - 
 P r (HQ))Lxi which is equivalent to the condition that: 
 P r (H )/<l - P r (H )) < Ln/L!. 
 
 3. Calibration must be measured with respect to a sequence of 
 PPDs assessed for exchangeable events (i.e., the assessor 
 perceives the events as equally uncertain). This implies 
 that, for normally distributed PPDs, the same v' would be 
 specified for each trial in the sequence. For purposes of 
 
-28- 
 
 identifying audit risk consequences, however, we must further 
 assume that, for all uncertain account balances in the 
 sequence, the auditor's loss ratio is the same. Such an 
 assumption might be satisfied if the PPDs are assessed for 
 the same account and the auditees are in similar circumstances . 
 We also assume that the PPD sequence is assessed by a single 
 auditor. Multiple auditor— assessors could be introduced (as 
 • in most calibration studies, see Lichtenstein et al. [1982]), 
 provided that they are homogeneous with respect to their 
 substantive knowledge, calibration deficiencies, and per- 
 ceived uncertainty. 
 
 4. One could assume that the {ptl are drawn from a super- 
 population (see Godfrey and Andrews [1982]) modelled by a 
 probability density function, g(u t ). Given such an assump- 
 tion, the expectation in (5) would be taken with respect to 
 g(»). Alternatively, one could view the expectation as a 
 simple average over {y t }. 
 
 5. Raiffa and Schlaiffer [1961] have shown that z is 
 
 3v' dA dA 
 
 positive for most commonly employed loss functions. The 
 
 3n dm , , , , , c , . _ 
 
 - — . — — - term is related to the expected value of sample mtor- 
 dm dA 
 
 nation and depends, therefore, on the location and dispersion 
 of the auditor's PPD. In our model, the term would be posi- 
 tive if the likelihood ratio, P r (Ho)/(l - P r (Ho)), initially 
 were close to the critical ratio (Lti/Li) and the PPD mean 
 
-29- 
 
 displacement changed the likelihood ratio, so that the 
 auditor's optimal evaluation decision changed. 
 
 6. Our benchmark case deals with underconf idence which would 
 occur if the auditor failed to recognize that A = 0. When 
 A = 0, we would not expect to observe overconf idence on a 
 representative trial from the sequence. 
 
 7. The audit risk consequences of PPD mean displacement and 
 miscalibration (discussion below) are the same when the audi- 
 tor adopts the Statement on Auditing Procedure (SAP) No. 54 
 [AICPA, 1972] decision rule (i.e., interchanges the null and 
 alternate hypotheses), provided that the PPD sequence is 
 assessed for account balances. However, if the PPD sequence 
 were assessed for the monetary error in the account rather 
 than the account value itself, the risk consequences asso- 
 ciated with upward PPD mean displacement would be the same as 
 downward displacement in our model and vice versa. 
 
 8. Given a one-sided hypothesis test, upward displacement of the 
 posterior mean increases P r (Ho). If a two-sided hypothesis 
 test were employed, P r (Ho) would not increase monotonically 
 with the posterior mean. Hence, sufficiently large posterior 
 mean displacement could result in rejection of Hq. Note 
 that, while rejection of Hq would be correct, the auditor's 
 conclusion about the direction of the material error would be 
 incorrect. This is considered in more detail below. 
 
-30- 
 
 9. In the unlikely event that dE[v"]/d<f> < 0, distribution A 
 would be more diffuse than distribution B, so that the 
 effects of overconf idence would be analogous to those 
 described above for underconf idence. Similarly, the effects 
 of underconf idence when dE[v"]/d<}> < 0, would be similar to 
 those for overconf idence when dE[v"]/d<£ > 0. 
 10. As discussed above (see fn. 8), if a two sided hypothesis test 
 were employed, upward posterior mean displacement could 
 decrease P r (H()) and create a predisposition toward effi- 
 ciency errors. 
 
-31- 
 
 Decisions: 
 
 States : 
 H > X - M y < X - M 
 
 Accept 
 
 H o ° L n 
 
 Reject 
 
 H o L i 
 
 y = actual population mean 
 
 X = average value of subsidiary account reported by the 
 client 
 
 L = loss from audit inefficiency 
 
 L TT = loss from audit ineffectiveness 
 
 Auditor's Loss Matrix 
 
 Figure 1 
 
-32- 
 
 Frequency 
 
 m A , m B 
 
 Monetary Value 
 
 Fig. 2-;Underconfidence-No Mean Displacement 
 
-33- 
 
 Frequency 
 
 " 1 T ' il il 
 
 fi t x-m m A m B 
 
 Monetary Value 
 
 Fig. 3 - Miscalibration , Upward Mean Displacement 
 
-34- 
 
 Frequency 
 
 TW'B 
 Monetary Value 
 
 Fig.4- Underconfidence - No Mean Displacement 
 
-JD- 
 
 Frequency 
 
 x-m m B fj. t m A 
 
 Monetary Value 
 
 Fig.5 - Miscalibration, Upward Mean Displacemef 
 
-36- 
 
 Frequency 
 
 ii ii 
 
 x-m m A m B fi t 
 
 Monetary Value 
 
 Fig. 6 - Miscalibration - Downward Mean Displacerr 
 
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-38- 
 
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