## Internal Revenue Bulletin: 2004-20 |

## May 17, 2004 |

The statistical sampling must be conducted in accordance with the following methodology.

1. Statistical (probability) sampling methodology may not include the use of judgment sampling.

2. Taxpayers may apply the results of a statistical sample only to the taxable years included in the sample.

3. A statistical sample may include data from no more than three consecutive taxable years.

4. Data from a taxable year may be included in only one statistical sample.

5. The estimated amount of expenses not subject to the § 274(n)(1) limitation must be based on a statistical (probability) sample, in which each sampling unit has a known (non-zero) chance of selection, using either a simple random sampling method or stratified random sampling method.

6. In general, the computation of the estimated amount of expenses not subject to the § 274(n)(1) limitation must be at the least advantageous 95% one-sided confidence limit. The “least advantageous” confidence limit is either the upper or lower limit that results in the least benefit to the taxpayer. However, if the precision of the change in the estimated deductible amount of expenses not subject to the § 274(n)(1) limitation (see paragraph 9 below) divided by the change in the estimated deductible amount of expenses not subject to the § 274(n)(1) limitation does not exceed 10%, the point estimate may be used in place of the least advantageous confidence limit. All strata for which “substantially all” of the population sampling units are sampled will be treated as 100% strata. That is, the overall point estimate and its precision will be estimated by treating all 100% strata appropriately for the sample design used. Also, the calculation of the denominator for the relative precision will exclude all 100% strata. For this revenue procedure, “substantially all” is defined as 80% or more.

7. Recognizing that many methods exist to estimate population values from the sample data, the Service will consider acceptable only the following estimators. Variable estimators permitted include the mean (also known as the direct projection method), difference (using “paired variables”), (combined) ratio (using a variable of interest and a “correlated” variable), and (combined) regression (using a variable of interest and a “correlated” variable). The first variable used for the difference, ratio and regression estimators must be the variable used in the mean estimator. The second variable used for the difference, ratio and regression estimators must be a variable that can be paired with the first variable and should be related to the first variable. For example, in a typical audit-sampling situation, the first variable would be the audited value of a transaction and the second variable would be the originally reported value of the same transaction. Since the latter two variable methods are statistically biased, there must be a demonstration that the bias is negligible before the Service will accept the method.

8. Variable sampling plans must use the qualifying final estimate with the smallest overall standard error as an absolute value (for example, the size of the estimate is irrelevant in the determination of the reported value).

9. Variable sampling plans must calculate confidence limits by addition and subtraction of the precision of the estimate from
the point estimate in which the determination of precision proceeds by multiplication of the standard error by (i) the 95%
one-sided confidence coefficient based on the Student’s *t*-distribution with the appropriate degrees of freedom, or (ii) 1.645 (the normal distribution), assuming the sample size is
at least 100 in each non-100% stratum.

10. For either the (combined) ratio or regression methods, to demonstrate little statistical bias exists, the following applies after excluding all strata tested on 100% basis (the entire population of a stratum is selected for evaluation).

a. The total sample size of all strata must be at least 100 units.

b. Each stratum for a population estimate should contain at least 30 sample units.

c. The coefficient of variation of the paired variable must be 15% or less. The coefficient of variation of the paired variable (y) is defined as the standard error of the total “y” variables divided by point estimate of the total “y” variables when the “y” variables are commonly the reported values in accounting situations.

d. The coefficient of variation of the primary variable of interest, represented by either the corrected value or the difference between the reported and corrected values in common accounting situations, must be 15% or less. The coefficient of variation for the corrected value (x) is defined as the standard error of the total “x” variables divided by point estimate of the total “x” variables when the “x” variables are commonly the corrected values in accounting situations. The coefficient of variation for the difference (d) between the reported and corrected values (x-y) is defined as the smaller of the standard error of the total “x-y” or total “d” variables divided by the amount equaling total population value represented by “Y” plus point estimate of the total “x-y” or total “d” variables or the standard error of the total “x-y” or total “d” variables divided by the total “x-y” or total “d” variables when the “x-y” variables are commonly the difference (“d”) between the reported (“y”) and corrected (“x”) values in accounting situations.

e. For only the (combined) ratio method, the reported values of units must be of the same sign.

11. When sampling the same expense accounts for multiple taxable years, if a single projection does not materially affect other computations that are more appropriately made on a yearly basis, it is permissible to combine the accounts into one population. There should be allocation of the combined result by a reasonable method determined prior to the selection of the sampling units.

12. A written sampling plan is required prior to the execution of a sample. A plan must include the following:

a. The objective of the plan including a description of the value for estimation and the applicable taxable year(s);

b. Population definition and reconciliation of the population to the tax return;

c. Definition of the sampling frame;

d. Definition of the sampling unit;

e. Source of the random numbers, the starting point or seed, and the method of selection;

f. Sample size, along with supporting factors in the determination;

g. Method to associate random numbers to the frame;

h. Steps to ensure that the serialization of the frame is independent of the drawing of random numbers;

i. Steps for evaluating the sampling unit; and

j. The estimator that was used for appraising the sample.

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