Geometric Stratification of Accounting Data

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1 Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual in auditing, wic is muc easier to use tan te commonly used cumulative root frequency metod of Dalenius and Hodges (1957, 1959). We implement it on two audit populations, one a population of debtors in an Iris firm, and te oter a population of sales and use tax liabilities in te US. Our results sow tat te new metod compares favourably wit te cumulative root frequency metod in terms of te accuracy of te estimates. Keywords: Accounting data, statistical auditing, efficiency, sampling, stratification. 1. Introduction One of te objectives of an audit is to estimate te total amount of error in a population of interest: for example, te amount of overstated debts in te set of transactions of a firm, or underpaid taxes in te set of transactions to wic sales and use taxes apply. For large businesses, it is too costly and time consuming to examine all transactions in te population: instead a sample is selected and subjected to detailed testing in order to estimate te parameters of te wole population. Te distribution of accounting populations is usually igly skewed wit long tails to te rigt. Terefore a stratification design is often used, were te population is divided into several strata and separate random samples are drawn from eac stratum. Te objective is to decide on a stratification design to minimise te variance of te resulting estimates. Dalenius (1950) derived equations for determining boundaries for te strata so tat te variance of te resulting estimate is minimised. He pointed out tat tese equations are troublesome to solve because of dependencies among te components, and suggested a metod of approximation (Dalenius and Hodges, 1957), called te cumulative root frequency metod, wic is still today te most commonly used metod of constructing stratum boundaries. Tis metod involves first dividing te sorted frame into a fairly large number of classes, obtaining te root of te frequency in eac class, accumulating te root of te frequencies, and obtaining strata so tat tere are equal intervals on te cumulative root frequencies. Toug commonly used in practice, te cumulative root frequency metod as some arbitrariness wic makes it difficult to implement; te final stratum boundaries depend * Researcer, Scool of Computing, Dublin City University, Dublin, Ireland. Correo electrónico: ** Senior Lecturer, Scool of Computing, Dublin City University, Dublin, Ireland. Correo electrónico: *** Independent Consultant, Dallas, Texas. Correo electrónico:

2 on te initial coice of te number of classes and tere is no teory tat gives te best number of classes (Hedlin 2000). In tis paper, we suggest a new metod of stratification, suitable for use wit positively skewed populations, wic is simpler tan te solution of Dalenius and Hodges, and overcomes te arbitrariness. We examine its performance wen applied to two real accounting populations, one a population of Iris debtors, and te oter a population of US sales-and-use taxes. Te paper is structured as follows. We begin in Section 2 by overviewing stratification in statistical auditing, and go on in Section 3 to outline te procedures for constructing strata. In Section 4 te new procedure is implemented on two real accounting populations, and its efficiency is examined in Section 5. We conclude wit a summary of our results. 2. Stratification in Auditing Suppose tere are N transactions in a population. Let X 1, X 2,...,X N be te amounts stated by te debtor or taxpayer, and Y 1, Y 2,...,Y N be te true amounts to be ascertained by te auditor. Te objective is to estimate te total true amount, T Y, and compare it wit te stated amount T X. To do tis a sample of size n is cosen and used to provide an estimate of T Y. Wen a stratification design is used, te transactions are ordered in increasing size of X, and divided into L mutually exclusive strata. Te strata are intervals determined by teir endpoints k 0, k 1,...,k L. If N 1, N 2,..., N L represent te number of items witin eac stratum ten te total population is given by N= N. If n 1,n 2,...,n L represent te number of elements to be selected from eac stratum ten te total sample size is given by n= n One general strategy in coosing stratum boundaries is to coose tem in suc a way tat te variance of te resulting estimate is minimised. For example, in te case of te stratified unbiased estimate of te total: ^ L N T y, st N y 1 N were y is te mean of te sample elements in te t stratum; we coose a design in order to minimise its variance: V L 2 ^ 2 N ( y, st ) (1 T N 1 N n S ) N n were S is te standard deviation of all elements in stratum, i.e. 2 S 1 N _ Y i Y ) 2 N 1 i1 Here Y is te mean of te full set of elements (Y i, i = 1, 2,. N ) in stratum. Stratified sampling designs are distinguised by:

3 1. te number L of strata used; 2. ow te sample is allocated among te strata; 3. te construction of te stratum boundaries. As far as (1), L te number of strata is concerned, Cocran (1977) developed a model representing te approximate reduction in te variance gained over simple random sampling by stratification, and deduced tat tere is little to be gained from aving more tan five strata. Wen it comes to (2), allocating te sample elements among strata, it can be sown tat te variance of te stratified mean is minimised wen n N S n L N l l S 1 l Tis metod of sample allocation is detailed in Cocran (1977), and is referred to as Neyman allocation. We look at (3), te construction of stratum boundaries, in te next section. 3. Construction of te Stratum Boundaries Before detailing te new metod, we summarise te commonly used cumulative root frequency metod Te Cumulative Root Frequency Metod Te cumulative root frequency procedure of stratum construction is carried out for te division of a population into L strata as follows: 1. Arrange te stratification variable X in ascending order; 2. Group te X into a number of classes, J; 3. Determine te frequency in eac class f i (i=1, 2, J); 4. Determine te square root of te frequencies in eac class; 5. Cumulate te square root of te frequencies J i 1 fi 6. Divide te sum of te square root of te frequencies by te number of strata: 1 Q J i f i L 1 7. Take te upper boundaries of eac stratum to be te X values corresponding to Q, 2Q, 3Q, (L-1)Q, LQ. Notice tat tese are in an aritmetic progression.

4 To illustrate te implementation of te cumulative root frequency metod of stratum construction, we use a population of Iris debtors stated liabilities consisting of 3,369 items ranging between 40 and 28,000. Te data are divided into 30 initial classes, and given in Table 1 below; observe ow te distribution is igly skewed, te low values X ave a ig incidence of occurrence, and te incidence decreases as te X values increase. Table 1 Calculation of te Stratum Boundaries by te Cumulative Root Frequency Rule. X-value fi fi Cum f i To stratify te population into tree strata, we calculate Q = /3 = Ten we take te upper boundaries of eac stratum to be te X-values corresponding to 44.56, and Te nearest available breaks are 972, 3,768 and 28,000, giving corresponding strata in te range: ; 973-3,768; 3,769-28,000 Tis example illustrates not only te tedious nature of te calculations necessary to obtain te boundaries, but ow te coice of te initial class intervals ultimately determines te boundary values. Te final stratum boundaries depend on te coice of te number of classes. Had we cosen a different number of initial classes in Table 1, we would ave obtained different boundaries.

5 3.2. Te Metod Te new metod we are proposing to stratify skewed populations is based on an observation by Cocran (1961) tat wen te optimum boundaries of Dalenius (1950) are acieved, te coefficients of variation (CV = S x / X ) are often found to be approximately te same in all strata. i.e S1x S2x S Lx... X X X 1 Here S x is te standard deviation of te stated amounts in stratum, and X is te mean. 2 Lavallée and Hidiroglou (1988) noted tat te equality of coefficients is often asked by users of survey data. Gunning and Horgan (2004) ave derived a simple algoritm for setting te coefficients of variation approximately equal in eac stratum. Tey sow tat, if it can be assumed tat te data are approximately uniformly distributed witin eac stratum, near-equal coefficients of variation may be acieved by constructing te stratum boundaries using te geometric progression. Tis geometric stratification is implemented for dividing a population into L strata as follows: 1. Arrange te stratification variable X in ascending order; 2. Take te minimum value as te first term, and te maximum value as te last term of te geometric series wit L+1 terms; 3. Calculate te common ratio: r = (max/min) 1/L ; 4.Take te boundaries of eac stratum to be te X values corresponding to te terms in te geometric progression wit tis common ratio: Minimum k 0 = a, ar, ar 2.. ar L = maximum k L. A full proof of te validity of tis result is given in Gunning and Horgan (2004). To illustrate its simplicity we apply it to stratify te data provided in Table 1 into tree strata, i.e. L=3, k 0 =40,...,k 3 =28,000 : tus r = (28,000/40) 1/3 = 700 1/ 3 = 8.88, and k =40*8.88 (=0,1,2,3). Terefore, te strata form te ranges ; 355-3,152; 3,153-28,000. Tis is clearly a muc simpler metod of obtaining stratum breaks tan te cumulative root frequency metod. It is suitable for positively skewed populations, common in accounting data, were te low values of te variable ave a ig incidence of occurrence, and te incidence decreases as te variable values increase. It is terefore appropriate to take small intervals at te beginning and large intervals at te end. Tis is wat appens wit a geometric series of constant ratio greater tan one. In te lower range of te variable, te strata are narrow so tat an assumption of a uniform distribution is not unreasonable. As te L

6 value of te variable increases, te stratum widt increases geometrically. Tis coincides wit te decreased rate of cange of te incidence of te positively skewed variable, so ere also te assumption of uniformity is reasonable. We expect te best results, in terms of equality of coefficients of variation, wen te distribution is igly positively skewed and te upper part contains a small percentage of te total frequency, and a large percentage of te monetary amount. Again tis can be said to be an apt description of accounting data. In te next section, we examine ow successful tis algoritm is in equalising te coefficients of variation in te strata. 4. Comparison of te Metods of Stratum Construction To test our algoritm, we implement it on two specific accounting populations, bot of wic are positively skewed Te Accounting Data Our first population (Population 1) is tat already given in Table 1 and consists of debtors stated liabilities in an Iris firm; it is furter detailed in Horgan (2003). Our second population (Population 2) is a population of stated liabilities of taxpayers in a US firm. Te audit often consists of a take-all stratum of very large items tat are audited on a 100% basis, and a take-none stratum of very small items tat are not audited at all. In Population 1, te take-all stratum consisted of all items over 28,000, and te take-none stratum consisted of all items under 40. In Population 2, te take-all stratum consisted of all items over $50,000, and te take-none stratum consisted of all items under $100. In eac population, te remainder of te population is sampled. Te trimmed populations tat are sampled are summarised in Table 2 Table 2 Summary Statistics Population N Range Skewness Mean Variance Population 1( ) 3, , ,511,827 Population 2($) 249, , ,258 24,352,340 It is interesting to note from Table 2 tat, altoug te two populations represent different types of accounting situations in different countries, tey are bot igly positively skewed. Te US population contains a substantially greater number of items, and its skewness coefficient is smaller Te Stratum Boundaries Te two populations described in Table 2 were divided into L = 3, 4 and 5 strata using te cumulative root frequency metod (cumroot), and te new geometric metod, and te coefficient of variation was calculated in eac stratum. Tables 3, 4 and 5 give te results for L = 3, 4 and 5 strata respectively.

7 Table 3 Strata Construction: 3 Strata Population Metod Range % of Pop 56% 38% 6% CV Range % of Pop. 69% 22% 9% CV Range % of Pop. 55% 36% 9% CV Range % of Pop. 72% 17% 11% CV Table 4: Strata Construction: 4 Strata Population Metod Range % of Pop 42% 41% 14% 3% CV Range % of Pop. 69% 14% 10% 7% CV Range % of Pop % 16% 5% CV Range % of Pop. 53% 31% 12% 4% CV

8 Table 5 Strata Construction: 5 Strata Population Metod Range % of Pop 31% 37% 22% 8% 2% CV Range % of Pop. 49% 30% 10 7% 4% CV Range ,425 14,426-50,000 % of Pop..33%.34%.21%.9%.3% CV Range % of Pop..61%.16%.10%.9%.4% CV A cursory examination of te coefficients of variation in Tables 3, 4, and 5 suggests tat te geometric metod is more successful tan te cumulative root frequency metod in obtaining near-equal CV : te CV differ substantially more from eac oter wen te cumulative root frequency metod is used to make te breaks tan wen te geometric metod is used. A more detailed analysis of te variability of te CV between strata is given in Table 6, were te standard deviation of te CV is calculated for eac design. Table 6 Te Variability of te CV for Eac Design Strata Metod Population 1 Population We see from Table 6 tat te standard deviations of te CV are substantially lower wit te geometric metod of stratum construction tan wit te cumulative root metod. Te new algoritm appears to be successful in breaking te strata in a way tat te CV are close in value. Tose obtained wit te cumulative root frequency metod are substantially more variable. On te basis of te observation of Cocran (1961) tis suggests tat te variance of te mean is close to minimal wit geometric stratification. We investigate tis in te next section.

9 5. Te Efficiency of Stratification Te geometric metod is compared wit te cumulative root frequency in terms of te relative efficiency (Eff), te ratio of te variances of te means obtained wen te strata are constructed using eac metod. Eff V geom V cum ( x ( x st ) st ) In eac case te sample elements are allocated optimally among te strata. In sample size planning, te relative efficiency represents te proportionate increase or decrease in te sample size wit te cumulative root frequency metod to obtain te same precision as tat of te geometric metod. Te variance calculations are based on X, te stated values of te debts (Population 1), and te stated taxpayers liabilities (Population 2). Since X is igly correlated wit te true values Y, we can assume tat te relative efficiency Eff, given above, is a reasonable approximation of te relative efficiency of Y. Te relative efficiencies are given in Tables 7 and 8 for Population 1 and Population 2 respectively. Te sample size n is 100 in Population 1, and 1000 in te larger Population 2. Table 7 Population 1: Relative Efficiency wen n =100 Strata Metod Variance Efficiency 3 2, , , , Table 8 Population 2: Relative Efficiency wen n =1000 Strata Metod Variance Efficiency 3 1, ,

10 We see from Tables 7 and 8 tat te relative efficiency is near 1 in most cases, indicating tat te precision of te new metod is not substantially different from tat of te cumulative root frequency metod. In all except one case te efficiency is witin.06 of 1. Te exception is in Population 1, wit L =4 strata were a gain of nearly 20% is observed for te geometric metod of stratification; in tis case a sample of size 80 would suffice wit te geometric metod to obtain te same precision as te cumulative root frequency metod wit n = Summary Tis paper gives a simple algoritm for te construction of stratum boundaries in positively skewed populations. It is based on an observation by Cocran (1961) tat te coefficients of variation witin strata are equal wen te optimum boundaries ave been acieved. We ave sown tat, wit positively skewed populations, te stratum breaks may profitably be obtained using te geometric progression. Tis metod is easier to implement tan te commonly used cumulative root frequency of Dalenius and Hodges (1957). Comparisons of te two metods, carried out wit two positively skewed real accounting populations divided into tree, four and five strata, sow tat te new metod is as precise as te cumulative root frequency metod in most cases. ACKNOWLEDGEMENTS Tis work was supported by a grant from te Iris Researc Council for Science, Engineering and Tecnology. We would like to tank te referees for teir constructive comments, wic elped to improve te paper. REFERENCES Cocran, W.G. (1961), Comparison of Metods for Determining Stratum Boundaries, Bulletin of te International Statistical Institute, 32, 2, pp Cocran, W.G. (1977), Sampling Tecniques, 3rd Edition, New York: Wiley. Dalenius, T. (1950), Te Problem of Optimum Stratification, Skandinavisk Aktuarietidskrift, pp Dalenius, T and J.L. Hodges (1957), Te Coice of Stratification Points, Skandinavisk Aktuarietidskrift, pp (1959), Minimum Variance Stratification, Journal of te American Statistical Association, 54, pp Gunning, P. and J. M. Horgan (2004), A New Algoritm for te Construction of Stratum Boundaries in Skewed Populations, Survey Metodology, 2, p. 30.

11 Hedlin, D. (2000), A Procedure for Stratification by an Extended Ekman Rule, Journal of Official Statistics, 16, pp Horgan, J.M. (2003), A List Sequential Sampling Sceme wit Applications in Financial Auditing, IMA Journal of Management Matematics, 14, pp Lavallée, P and M.A. Hidiroglou (1988), On te Stratification of Skewed Populations, Survey Metodology, 14, pp

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