Testing in an audit sampling context

In an audit sampling test the auditor generally assigns performance materiality, \(\theta_{max}\), to the population which expresses the maximum tolerable misstatement (as a fraction or a monetary amount). The auditor then inspects a sample of the population to compare the following two hypotheses:

\[H_-:\theta<\theta_{max}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, H_+:\theta\geq\theta_{max}\].

The evaluation() function allows you to make a statement about the credibility of these two hypotheses after inspecting a sample. The output for testing as discussed in this vignette is only displayed when you enter a value for materiality argument.

Frequentist hypothesis testing using the p-value

This will be added in a future version of jfa.

Bayesian hypothesis testing using the Bayes factor

Bayesian hypothesis testing uses the Bayes factor, \(BF_{-+}\) or \(BF_{+-}\), to make a statement about the evidence provided by the sample in support for one of the two hypotheses \(H_-\) or \(H_+\). The subscript The Bayes factor denotes which hypothesis it favors. By default, the evaluation() function returns the value for \(BF_{-+}\).

As an example of how to interpret the Bayes factor, the value of \(BF_{-+} = 10\) (provided by the evluation() function) can be interpreted as: the data are 10 times more likely to have occurred under the hypothesis \(H_-:\theta<\theta_{max}\) than under the hypothesis \(H_+:\theta\geq\theta_{max}\). \(BF_{-+} > 1\) indicates evidence for \(H_-\), while \(BF_{-+} < 1\) indicates evidence for \(H_+\).

\(BF_{-+}\) Strength of evidence
\(< 0.01\) Extreme evidence for \(H_+\)
\(0.01 - 0.033\) Very strong evidence for \(H_+\)
\(0.033 - 0.10\) Strong evidence for \(H_+\)
\(0.10 - 0.33\) Moderate evidence for \(H_+\)
\(0.33 - 1\) Anecdotal evidence for \(H_+\)
\(1\) No evidence for \(H_-\) or \(H_+\)
\(1 - 3\) Anecdotal evidence for \(H_-\)
\(3 - 10\) Moderate evidence for \(H_-\)
\(10 - 30\) Strong evidence for \(H_-\)
\(30 - 100\) Very strong evidence for \(H_-\)
\(> 100\) Extreme evidence for \(H_-\)

Example

As an example, consider that an auditor wants to verify whether the population contains less than 5 percent misstatement, implying the hypotheses \(H_-:\theta<0.05\) and \(H_+:\theta\geq0.05\). They have taken a sample of 40 transactions, of which 1 contained an error. The prior distribution is assumed to be a non-informative \(beta(1,1)\) prior.

The output below shows that \(BF_{-+}=30.28\), implying that there is very strong evidence for \(H_-\), the hypothesis that the population contains misstatements lower than 5 percent of the population.

prior <- auditPrior(confidence = 0.95, likelihood = "binomial", method = "none", materiality = 0.05)
evaluation(confidence = 0.95, materiality = 0.05, nSumstats = 40, kSumstats = 1, prior = prior)
## # ------------------------------------------------------------ 
## #             jfa Evaluation Summary (Bayesian)
## # ------------------------------------------------------------
## # Input: 
## #
## # Confidence:               95%   
## # Materiality:              5% 
## # Minium precision:         Not specified 
## # Sample size:              40 
## # Sample errors:            1 
## # Sum of taints:            1  
## # Method:                   binomial 
## # Prior distribution:       beta(α = 1, β = 1) 
## # ------------------------------------------------------------
## # Output:
## #
## # Posterior distribution:   beta(α = 2, β = 40) 
## # Most likely error:        2.38% 
## # Upper bound:              11.06% 
## # Precision:                8.67% 
## # Bayes factor-+:           30.28 
## # Conclusion:               Do not approve population 
## # ------------------------------------------------------------

Sensitivity to the prior distribution

In audit sampling, the Bayes factor is dependent on the prior distribution for \(\theta\). As a rule of thumb, when the prior distribution is very uninformative with respect to the misstatement parameter \(\theta\), the Bayes factor overestimates the evidence in favor of \(H_-\). You can mitigate this dependency using method = "median" in the auditPrior() function, which constructs a prior distribution that is impartial with respect to the hypotheses \(H_-\) and \(H_+\).

The output below shows that \(BF_{-+}=3.08\), implying that there is anecdotal evidence for \(H_-\), the hypothesis that the population contains misstatements lower than 5 percent of the population.

prior <- auditPrior(confidence = 0.95, likelihood = "binomial", method = "median", materiality = 0.05)
evaluation(confidence = 0.95, materiality = 0.05, nSumstats = 40, kSumstats = 1, prior = prior)
## # ------------------------------------------------------------ 
## #             jfa Evaluation Summary (Bayesian)
## # ------------------------------------------------------------
## # Input: 
## #
## # Confidence:               95%   
## # Materiality:              5% 
## # Minium precision:         Not specified 
## # Sample size:              40 
## # Sample errors:            1 
## # Sum of taints:            1  
## # Method:                   binomial 
## # Prior distribution:       beta(α = 1, β = 13.513) 
## # ------------------------------------------------------------
## # Output:
## #
## # Posterior distribution:   beta(α = 2, β = 52.513) 
## # Most likely error:        1.83% 
## # Upper bound:              8.56% 
## # Precision:                6.73% 
## # Bayes factor-+:           3.08 
## # Conclusion:               Do not approve population 
## # ------------------------------------------------------------