FORMULAS FOR SAMPLE SIZE CALCULATION AND EXTRAPOLATION OF ERRORS
ANNEX IISupplementary provisions
ANNEX II FORMULAS FOR SAMPLE SIZE CALCULATION AND EXTRAPOLATION OF ERRORS 1. MUS STANDARD APPROACH 1.1. MUS standard approach – one period NON-STRATIFIED STRATIFIED Sample size calculation where - weighted mean of the variances of the error rates for the whole set of strata, with the weight for each stratum equal to the ratio between the stratum book value ( BV h ) and the book value for the whole population ( BV ) and is the variance of error rates in each stratum where BV - population book value (total declared expenditure) z - coefficient z from a normal distribution TE - tolerable error (maximum 2 % of the total expenditure) AE - anticipated error σ r - the standard deviation of the error rates Extrapolation of errors Projected/extrapolated error (MUS standard approach/PPS): For the exhaustive stratum, the projected error is the sum of the errors found in the units belonging to the stratum: For the non-exhaustive stratum, i.e. the stratum containing the sampling units with book value smaller than the interval, the projected error is The projected error at the level of population is the sum of the two components above: Projected/extrapolated error (MUS standard approach/PPS): For the exhaustive groups, the projected error is the sum of the errors found in the units belonging to those groups: For the non-exhaustive groups, i.e. the groups containing the sampling units with book value smaller than the interval, , the projected error is The projected error at the level of population is just the sum of these two components above: Sampling precision: where is the standard-deviation of error rates in the sample of the non-exhaustive stratum (calculated from the same sample used to extrapolate the errors to the population) Sampling precision: where is the standard-deviation of error rates in the sample of the non-exhaustive group of stratum h (calculated from the same sample used to extrapolate the errors to the population) 1.2. MUS standard approach – two periods NON-STRATIFIED STRATIFIED Sample size calculation First period where First period where Second period Second period where Notes: Whenever different approximations for the standard-deviations of each period cannot be obtained/are not applicable, the same value of standard deviation may be applied to all periods. In such a case is just equal to the single standard-deviation of error rates . The parameter σ refers to the standard-deviation obtained from auxiliary data (e.g. historical data) and s refers to the standard-deviation obtained from the audited sample. In the formulas, whenever s is not available, it may be substituted by σ. Formulas under the heading “First period” are used to calculate the sample size after the first sampling period of the accounting year in the case of a standard recalculation of the sample size referred to in Article 5(6), point (a). In the case of the global recalculation of the sample size referred to in Article 5(6), point (b), these formulas are used after the first sampling period and if needed also after the second sampling period in order to adjust to updated sampling parameters. Formulas under the heading “Second period” are applicable only in the case of a standard recalculation of the sample size referred to in Article 5(6), point (a). They are used to recalculate the sample size of the second period in order to adjust to updated sampling parameters. If the formula results in a negative number, the formula and consequently the standard approach to recalculation of the sample size cannot be applied based on the established set of the updated parameters. Extrapolation of errors Projected/extrapolated error (MUS standard approach/PPS): For the exhaustive strata, the projected error is the sum of the errors found in the units belonging to the strata: For the non-exhaustive strata, i.e. the strata containing the sampling units with book value smaller than the interval, the projected error is The projected error at the level of population is the sum of the two components above: Projected/extrapolated error (MUS standard approach/PPS): For the exhaustive strata, the projected error is the sum of the errors found in the units belonging to the strata: For the non-exhaustive strata, i.e. the strata containing the sampling units with book value smaller than the interval, the projected error is The projected error at the level of population is the sum of the two components above: Sampling precision: where is the standard-deviation of error rates in the sample of the non-exhaustive strata of period t (calculated from the same sample used to extrapolate the errors to the population) Sampling precision: where is the standard-deviation of error rates in the sample of the non-exhaustive group of stratum h in period t (calculated from the same sample used to extrapolate the errors to the population) 1.3. MUS standard approach – three periods ( 1 ) NON-STRATIFIED STRATIFIED Sample size calculation First period where First period where Second period where Second period where Third period Third period Notes: Whenever different approximations for the standard-deviations of each period cannot be obtained/are not applicable, the same value of standard deviation may be applied to all periods. In such a case is just equal to the single standard-deviation of error rates . The parameter σ refers to the standard-deviation obtained from auxiliary data (e.g. historical data) and s refers to the standard-deviation obtained from the audited sample. In the formulas, whenever s is not available, it may be substituted by σ. See also notes above for the two-period MUS standard approach as regards the use of the standard approach to the recalculation of the sample size and the global approach referred to in Article 5(6). Extrapolation of errors Projected/extrapolated error (MUS standard approach/PPS): For the exhaustive strata, the projected error is the sum of the errors found in the units belonging to the strata: For the non-exhaustive strata, i.e. the strata containing the sampling units with book value smaller than the interval, the projected error is The projected error at the level of population is the sum of the two components above: Projected/extrapolated error (MUS standard approach/PPS): For the exhaustive strata, the projected error is the sum of the errors found in the units belonging to the strata: For the non-exhaustive strata, i.e. the strata containing the sampling units with book value smaller than the interval, the projected error is The projected error at the level of population is the sum of the two components above: Sampling precision: where is the standard-deviation of error rates in the sample of the non-exhaustive strata of period t (calculated from the same sample used to extrapolate the errors to the population) Sampling precision: where is the standard-deviation of error rates in the sample of the non-exhaustive group of stratum h in period t (calculated from the same sample used to extrapolate the errors to the population) 2. SIMPLE RANDOM SAMPLING 2.1. Simple random sampling – one period NON-STRATIFIED STRATIFIED Sample size calculation where is the standard-deviation of errors in the population where - the weighted mean of the variances of the errors for the whole set of strata: and is the variance of errors in each stratum where N - population size z - coefficient z from a normal distribution TE - tolerable error (maximum 2 % of the total expenditure) AE - anticipated error σ e - the standard deviation of the errors Extrapolation of errors In the framework of application of the off-the-shelf methodologies laid down in this Delegated Regulation, a single extrapolation method, ratio estimation, applies for SRS referred to in Article 5(1), point (b), and equal probability selection referred to in Article 6(1), point (b), for the purpose of simplification and legal certainty. This does not limit the application of other extrapolation methods by the audit authorities under Article 79 of Regulation (EU) 2021/1060. Projected/extrapolated error (SRS/equal probability selection): If an exhaustive stratum is used, the projected error in this group is the sum of the errors found in the units belonging to the stratum: For the random stratum the projected error is The projected error at the level of population is the sum of the two components above: Projected/extrapolated error (SRS/equal probability selection): If an exhaustive stratum is used, the projected error in this group is the sum of the errors found in the units belonging to those groups: For the random strata the projected error is The projected error at the level of population is just the sum of these two components above: Sampling precision: where s q is the sample standard deviation of the variable q : Precision is exclusively calculated with data pertaining to the non-exhaustive strata. Sampling precision: where is a weighted mean of the sample variances of the variable q h , with Precision is exclusively calculated with data pertaining to the non-exhaustive strata. 2.2. Simple random sampling – two periods NON-STRATIFIED STRATIFIED Sample size calculation First period where First period where Second period Second period Notes: Whenever different approximations for the standard-deviations of each period cannot be obtained/are not applicable, the same value of standard deviation may be applied to all periods. In such a case is just equal to the single standard-deviation of errors . The parameter σ refers to the standard-deviation obtained from auxiliary data (e.g. historical data) and s refers to the standard-deviation obtained from the audited sample. In the formulas, whenever s is not available, it may be substituted by σ. Formulas under the heading “First period” are used to calculate the sample size after the first sampling period of the accounting year in the case of a standard recalculation of the sample size referred to in Article 5(6), point (a). In the case of the global recalculation of the sample size referred to in Article 5(6), point (b), these formulas are used after the first sampling period and if needed also after the second sampling period in order to adjust to updated sampling parameters. Formulas under the heading “Second period” are applicable only in the case of a standard recalculation of the sample size referred to in Article 5(6), point (a). They are used to recalculate the sample size of the second period in order to adjust to updated sampling parameters. If the formula results in a negative number, the formula and consequently the standard approach to re-calculation of the sample size cannot be applied based on the established set of the updated parameters. Extrapolation of errors In the framework of application of the off-the-shelf methodologies laid down in this Delegated Regulation, a single extrapolation method, ratio estimation, applies for SRS referred to in Article 5(1), point (b) and equal probability selection referred to in Article 6(1), point (b), for the purpose of simplification and legal certainty. This does not limit the application of other extrapolation methods by the audit authorities under Article 79 of Regulation (EU) 2021/1060. Projected/extrapolated error (SRS/equal probability selection): If an exhaustive stratum is used, the projected error in this group is the sum of the errors found in the units belonging to the strata: For the non-exhaustive strata, the projected error is: The projected error at the level of population is the sum of the two components above. Projected/extrapolated error (SRS/equal probability selection): If an exhaustive stratum is used, the projected error is the sum of the errors found in the units belonging to the strata: For the non-exhaustive strata, the projected error is: The projected error at the level of population is the sum of the two components above. Sampling precision: Precision is exclusively calculated with data pertaining to the non-exhaustive strata. Sampling precision: Precision is exclusively calculated with data pertaining to the non-exhaustive strata. 2.3. Simple random sampling – three periods ( 2 ) NON-STRATIFIED STRATIFIED Sample size calculation First period where First period where Second period where Second period Third period Third period Notes: Whenever different approximations for the standard-deviations of each period cannot be obtained/are not applicable, the same value of standard deviation may be applied to all periods. In such a case is just equal to the single standard-deviation of errors . The parameter σ refers to the standard-deviation obtained from auxiliary data (e.g. historical data) and s refers to the standard-deviation obtained from the audited sample. In the formulas, whenever s is not available, it may be substituted by σ. See also notes above for the two-period simple random sampling as regards the use of the standard approach to the recalculation of the sample size and the global approach referred to in Article 5(6). Extrapolation of errors In the framework of application of the off-the-shelf methodologies laid down in this Regulation, a single extrapolation method, ratio estimation, applies for SRS referred to in Article 5(1), point (b), and equal probability selection referred to in Article 6(1), point (b), for the purpose of simplification and legal certainty. This does not limit the application of other extrapolation methods by the audit authorities under Article 79 of Regulation (EU) 2021/1060. Projected/extrapolated error (SRS/equal probability selection): For the exhaustive strata, the projected error is the sum of the errors found in the units belonging to the strata: For the non-exhaustive strata, the projected error is: The projected error at the level of population is the sum of the two components above. Projected/extrapolated error (SRS/equal probability selection): For the exhaustive strata, the projected error is the sum of the errors found in the units belonging to the strata: For the non-exhaustive strata, the projected error is: The projected error at the level of population is the sum of the two components above. Sampling precision: Precision is exclusively calculated with data pertaining to the non-exhaustive strata. Sampling precision: Precision is exclusively calculated with data pertaining to the non-exhaustive strata. ( 1 ) MUS standard approach may be applied with more than three sampling periods by relevant adjustments of the formulas. ( 2 ) Simple Random Sampling may be applied with more than three sampling periods by relevant adjustments of the formulas.