23.3 Audit Sampling: Attribute, Variable, and Monetary Unit Sampling

Audit Sampling: Attribute, Variable, and Monetary Unit Sampling

Auditors rarely examine every item in a population. This section covers audit sampling under PSA 530: sampling and non-sampling risk, statistical and non-statistical approaches, attribute sampling for tests of controls, classical variable sampling (mean-per-unit, ratio, and difference estimation), and monetary unit sampling for tests of details.

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1. PSA 530: Audit Sampling Fundamentals

Audit Sampling is the application of audit procedures to less than 100% of items within a population of audit relevance such that all sampling units have an equal or known chance of selection in order to provide the auditor with a reasonable basis on which to draw conclusions about the entire population.

Statistical vs Non-Statistical Sampling

• Statistical Sampling: Any approach to sampling that has the following characteristics:
  1. Random selection of sample items (e.g., random number tables or computer-generated random selection).
  2. The use of probability theory to evaluate sample results, including measurement of sampling risk.
• Non-Statistical Sampling (Judgmental Sampling): An approach that does not exhibit characteristics (1) and (2). The auditor uses professional judgment to determine sample size, select items, and evaluate results. CPALE Principle: Both statistical and non-statistical sampling are acceptable under PSA 530. Both require the exercise of professional judgment in design and execution. A properly designed non-statistical sample can provide results that are just as valid as a statistical sample, but only statistical sampling mathematically measures sampling risk.

Sampling Risk vs Non-Sampling Risk

Whenever an auditor tests less than 100% of a population, the auditor is exposed to two categories of risk:

                                        Audit Testing Risk
                                                │
                       ┌────────────────────────┴────────────────────────┐
                       ▼                                                 ▼
                 Sampling Risk                                   Non-Sampling Risk
        (Sample does not represent population)              (Auditor human / technical error)
                       │                                                 │
         ┌─────────────┴─────────────┐                     ┌─────────────┴─────────────┐
         ▼                           ▼                     ▼                           ▼
    Type I Risk                 Type II Risk         Applying Wrong Audit     Misinterpreting Evidence
   (Alpha Risk)                 (Beta Risk)               Procedure             or Failing to Recognize
 (Affects Efficiency)     (Affects Effectiveness)    (e.g., counting goods       Misstatement in Sample
                           **AUDIT FAILURE RISK**     on consignment as owned)  (Human fallibility)

1. Non-Sampling Risk: The risk that the auditor reaches an erroneous conclusion for any reason not related to sampling risk. Examples include using inappropriate audit procedures (e.g., inspecting physical inventory to verify legal ownership), misinterpreting audit evidence, or failing to recognize a misstatement in an examined document. Non-sampling risk is controlled through adequate planning, direction, supervision, and firm-wide quality management under PSQM 1.

2. Sampling Risk: The risk that the auditor's conclusion based on a sample may be different from the conclusion if the entire population were subjected to the same audit procedure. Sampling risk manifests in two distinct forms across tests of controls and substantive tests:

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2. Attribute Sampling for Tests of Controls

Attribute sampling is a statistical sampling method used to estimate the rate of occurrence of a specific characteristic or attribute (e.g., presence of an approval signature, internal verification of prices, matching of receiving reports to vendor invoices). It is used almost exclusively in Tests of Controls to evaluate whether internal controls operated effectively throughout the period.

Core Parameters in Attribute Sampling

1. Tolerable Deviation Rate (TDR): The maximum rate of deviations from a prescribed internal control procedure that the auditor is willing to accept without altering the planned assessed level of control risk. (Inverse relationship to sample size: Higher TDR = Smaller Sample Size).
2. Expected Population Deviation Rate (EPDR): The deviation rate that the auditor expects to find in the population based on prior audits or a pilot sample. (Direct relationship to sample size: Higher EPDR = Larger Sample Size).
3. Acceptable Risk of Over-Reliance (ARO / Beta Risk): The risk that the auditor will conclude that internal controls are effective when they are not. Typically set at 5% or 10%. (Inverse relationship to sample size: Lower ARO = Larger Sample Size).

                       Attribute Sampling Sample Size Determinants

   Factor                                     Change in Factor      Effect on Sample Size
   ──────────────────────────────────────────────────────────────────────────────────────
   Acceptable Risk of Over-Reliance (ARO)         Increases (10% vs 5%)       DECREASES
   Tolerable Deviation Rate (TDR)                 Increases (8% vs 4%)        DECREASES
   Expected Population Deviation Rate (EPDR)      Increases (3% vs 1%)        INCREASES
   Population Size (N > 5,000 items)              Increases                   VIRTUALLY NO EFFECT
   ──────────────────────────────────────────────────────────────────────────────────────

Sample Evaluation Mechanics

After testing the selected sample ($n$), the auditor counts the number of deviations ($d$) and calculates:

1. Sample Deviation Rate ($SDR$): $SDR = \frac{d}{n}$

2. Allowance for Sampling Risk ($ASR$): The statistical buffer accounting for the probability that the population deviation rate exceeds the sample deviation rate.

3. Upper Deviation Limit ($UDL$ or Computed Upper Exception Rate - CUER): $UDL = SDR + ASR$

The Attribute Sampling Decision Rule

$\begin{cases} \text{If } UDL \le TDR: & \text{Control is effective; support planned assessed Control Risk.} \\ \text{If } UDL > TDR: & \text{Control is ineffective; INCREASE assessed Control Risk and expand substantive tests.} \end{cases}$

Practical Numerical Illustration: Attribute Sampling

An auditor tests internal controls over sales invoice approvals for a wholesale client:

• Acceptable Risk of Over-Reliance ($ARO$) = 5%
• Tolerable Deviation Rate ($TDR$) = 6.0%
• Sample size ($n$) = 100 invoices
• Number of deviations found ($d$) = 2 missing approvals
• Statistical tables indicate that for $n = 100$, $d = 2$, and $ARO = 5\%$, the Upper Deviation Limit ($UDL$) is 6.2%.

   Analysis:
   - Sample Deviation Rate (SDR) = 2 / 100 = 2.0%
   - Allowance for Sampling Risk (ASR) = UDL - SDR = 6.2% - 2.0% = 4.2%
   - Computed Upper Deviation Limit (UDL) = 6.2%
   - Tolerable Deviation Rate (TDR) = 6.0%

   Comparison:
   UDL (6.2%) > TDR (6.0%)

   Conclusion:
   The auditor cannot rely on the internal control at the planned 5% risk level.
   The auditor must assess Control Risk at a higher level (or maximum) and expand
   the extent of year-end substantive tests of details of sales transactions.

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3. Variable Sampling & Monetary Unit Sampling (MUS / PPS)

Variable sampling methods are applied in substantive tests of details where the auditor's objective is to estimate a continuous monetary amount (such as the total peso balance of accounts receivable or inventory) and evaluate whether an account balance is materially misstated.

Classical Variable Sampling Models

                                  Classical Variable Sampling Models
                                                  │
         ┌────────────────────────────────────────┼────────────────────────────────────────┐
         ▼                                        ▼                                        ▼
   Mean-Per-Unit (MPU)                   Ratio Estimation                         Difference Estimation
         │                                        │                                        │
• Calculates average audited             • Multiplies book value by               • Calculates average audited
  value per sample item:                   the ratio of sample audited              difference per sample item:
  x_bar = (sum x_i) / n                    value to sample book value:              d_bar = sum(x_i - y_i) / n
• Point Estimate:                        • Point Estimate:                        • Point Estimate:
  Total = x_bar * N                        Total = Book * (sum x / sum y)           Total = Book + (d_bar * N)
• Does NOT require book value            • Highly efficient when errors           • Highly efficient when error
  for individual line items;               are proportional to recorded             size is independent of
  requires larger sample size.             book amounts.                            recorded line item value.

1. Mean-Per-Unit (MPU) Estimation:

   • Calculates the average audited value of sample items: $\bar{x} = \frac{\sum x_i}{n}$
   • Point Estimate of Population: $\text{Point Estimate} = \bar{x} \times N$ (where $N$ is total population units).
   • Advantage: Does not require knowledge of individual recorded book values for sampled items.
   • Disadvantage: Typically requires a significantly larger sample size to achieve target precision.

2. Ratio Estimation:

   • Calculates the ratio between the audited value of the sample and the book value of the sample: $R = \frac{\sum x_i}{\sum y_i}$
   • Point Estimate of Population: $\text{Point Estimate} = \text{Total Recorded Book Value } (Y) \times R$
   • Best Used: When misstatements are proportional to the recorded book amounts.

3. Difference Estimation:

   • Calculates the average difference between audited value and book value per sample item: $\bar{d} = \frac{\sum (x_i - y_i)}{n}$
   • Projected Total Misstatement: $\text{Projected Misstatement} = \bar{d} \times N$
   • Point Estimate of Population: $\text{Point Estimate} = \text{Total Recorded Book Value } (Y) + (\bar{d} \times N)$
   • Best Used: When the dollar amount of misstatements is constant and not proportional to item size.

Condition for Ratio and Difference Estimation: The auditor must expect a sufficient number of errors in the sample (typically at least 20 to 50 misstated items). If zero or very few errors exist, ratio and difference standard errors collapse, rendering statistical evaluation invalid.

Monetary Unit Sampling (MUS) / Probability-Proportional-to-Size (PPS)

Monetary Unit Sampling (also known as Probability-Proportional-to-Size sampling or dollar-unit sampling) is a statistical sampling method where the individual monetary unit (each individual Philippine Peso) within the population is defined as the sampling unit.

                                    MUS / PPS Sampling Architecture
                                                   │
     Population: Total Accounts Receivable Book Value = PHP 10,000,000
     Every single Peso (from Peso 1 to Peso 10,000,000) has an equal selection chance.
                                                   │
     A physical invoice with a face value of PHP 500,000 has 500,000 chances of being hit.
     A physical invoice with a face value of PHP 1,000 has only 1,000 chances of being hit.
                                                   │
            Automatic Stratification: Larger balances are naturally selected!

Key Characteristics of MUS/PPS

• Automatic Stratification: Items with large monetary balances have a proportionally higher probability of selection. Any item with a book value greater than the sampling interval ($Book \ge SI$) is guaranteed 100% selection.
• Highly Efficient for Low-Error Populations: Requires a smaller sample size than classical variable sampling when few or zero misstatements are anticipated.
• Limitations:
  • Inappropriate when testing for understatement or omission (a zero balance has a zero probability of selection; unrecorded liabilities cannot be detected via MUS).
  • Accounts with negative or zero balances require special handling or must be segregated into an independent testing population.

Sample Size & Interval Determination in MUS

$\text{Sampling Interval } (SI) = \frac{\text{Tolerable Misstatement } (TM)}{\text{Reliability Factor } (RF)}$ $\text{Sample Size } (n) = \frac{\text{Total Population Book Value } (BV)}{SI} = \frac{BV \times RF}{TM}$

Evaluating Sample Errors under MUS

For each selected item containing an error where Recorded Book Value ($BV$) is less than the Sampling Interval ($SI$):

1. Calculate the Tainting Percentage ($T$): $T = \frac{\text{Book Value } (BV) - \text{Audited Value } (AV)}{BV}$
2. Calculate the Projected Misstatement ($PM$): $PM = T \times SI$
3. For items where $BV \ge SI$, the projected misstatement equals the actual dollar misstatement (no tainting multiplier applied).
4. Upper Error Limit ($UEL$): Sum of Basic Precision (Sampling Interval $\times$ Reliability Factor at zero errors) $+$ Total Projected Misstatements $+$ Incremental Allowance for Sampling Risk.

If $UEL \le \text{Tolerable Misstatement}$, the auditor concludes that the account balance is not materially misstated.