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.

This will be added in a future version of `jfa`

.

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 `evaluation()`

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 \(H_-\) |

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 items, 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.

```
<- auditPrior(materiality = 0.05, method = "none", likelihood = "binomial")
prior <- evaluation(materiality = 0.05, nSumstats = 40, kSumstats = 1, prior = prior)
stage4 summary(stage4)
```

```
## # ------------------------------------------------------------
## # jfa Evaluation Summary (Bayesian)
## # ------------------------------------------------------------
## # Input:
## #
## # Confidence: 95%
## # Materiality: 5%
## # Minimum precision: Not specified
## # Sample size: 40
## # Sample errors: 1
## # Sum of taints: 1
## # Method: binomial
## # Prior distribution: beta(a = 1, ß = 1)
## # ------------------------------------------------------------
## # Output:
## #
## # Posterior distribution: beta(a = 2, ß = 40)
## # Most likely error: 2.5%
## # Upper bound: 11.055%
## # Precision: 8.555%
## # Bayes factor-+: 30.282
## # Conclusion: Do not approve population
## # ------------------------------------------------------------
```

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.

```
<- auditPrior(materiality = 0.05, method = "median", likelihood = "binomial")
prior <- evaluation(materiality = 0.05, nSumstats = 40, kSumstats = 1, prior = prior)
stage4 summary(stage4)
```

```
## # ------------------------------------------------------------
## # jfa Evaluation Summary (Bayesian)
## # ------------------------------------------------------------
## # Input:
## #
## # Confidence: 95%
## # Materiality: 5%
## # Minimum precision: Not specified
## # Sample size: 40
## # Sample errors: 1
## # Sum of taints: 1
## # Method: binomial
## # Prior distribution: beta(a = 1, ß = 13.513)
## # ------------------------------------------------------------
## # Output:
## #
## # Posterior distribution: beta(a = 2, ß = 52.513)
## # Most likely error: 1.904%
## # Upper bound: 8.561%
## # Precision: 6.656%
## # Bayes factor-+: 3.078
## # Conclusion: Do not approve population
## # ------------------------------------------------------------
```