## Study Design 101

# Some helpful formulas

#### Sensitivity and Specificity

**Test result**

Positive | Negative | Total | |

Presence | a | b | a+b |

Absence | c | d | c+d |

Total | a+c | b+d | a+b+c+d |

b = false negative

c = false positive

**To estimate sensitivity**

The number of positive test results for the presence of an outcome (a) divided by the total presence of an outcome (a+b)

Sensitivity = a / (a+b)

**To estimate specificity**

Number of negative test results for the absence of an outcome (d) divided by the total absences of an outcome (c + d)

Specificity = d / (c+d)

#### False Positive and False Negative rates

**Test Result**

Positive | Negative | Total | |

Presence | a | b | a+b |

Absence | c | d | c+d |

Total | a+c | b+d | a+b+c+d |

**To calculate rate of false positives**

The number of false positive test results for an outcome (c) divided by the total number of absences of an outcome (c+d)

Rate of false positives = c / (c+d)

**To calculate the rate of false negatives**

The number of false negative test results for an outcome (b) divided by the total number of presences of an outcome (a+b)

Rate of false negatives = b / (a+b)

#### Positive Predictive Value and Negative Predictive Value

**Test Result**

Positive | Negative | Total | |

Presence | a | b | a+b |

Absence | c | d | c+d |

Total | a+c | b+d | a+b+c+d |

**To estimate positive predictive value**

The number of positive test results for the presence of an outcome (a) divided by the total number of positive test results (a+c).

Positive predictive value = a / (a+c)

**To estimate negative predictive value**

The number of negative test results for the absence of an outcome (d) divided by the total number of negative test results (b+d).

Negative predictive value = d / (b+d)

**Note:** the formulas for positive predictive value and negative predictive value are accurate if the prevalence of the outcome (presences) is known.

#### Relative Risk

**Outcome**

Yes | No | |

Variable Present (Yes) | a | b |

Variable not Present or reference (No) | c | d |

**Relative Risk** = (a / a+b) / (c / c+d)

**Smelly Shoes Example**

Yes | No | |

Variable Present (Yes) | 9 | 1 |

Reference (No) | 2 | 8 |

In this example, 9 of the 10 pairs of sneakers that were worn without socks were smelly, and 2 of the 10 pairs of sneakers worn with socks were smelly. The relative risk would be (9/10) / (2/10), or 4.5. Therefore, the data suggest it is four times more likely to have smelly shoes if shoes are worn without socks.

Things to note about this formula:

- If the relative risk < 1 the exposure/incidence is protective: it lowers the risk for expressing the outcome.
- If the relative risk = 1 there is no association between an exposure that delineates the cohorts and the outcome.
- If the relative risk > 1 there is an association between an exposure that delineates the cohorts and the outcome (as seen in the example).

#### Attributable Risk

**Outcome**

Yes | No | |

Variable Present | a | b |

Control | c | d |

**To calculate attributable risk**

Subtract the outcome incidence rate of the control group from the outcome incidence rate of the experimental group.

Attributable risk = (a-c)

Attributable risk is helpful in showing to what extent the exposure to the variable of interest relates to the outcome studied.

**Cohort**

In our smelly shoe example, attributable risk would be 7. This is interpreted as: "The risk of smelly shoes can be attributed to wearing shoes without socks in seven cases."

#### Odds Ratio

**To calculate the odds ratio**

The number of people in the "variable present" cohort that experiences an outcome (a) divided by the number of people in the reference cohort that experiences the outcome (b) to the number of people in the "variable present" cohort that experiences no outcome (c) divided by the number of people in the reference cohort that experiences no outcome (d).

Odds ratio = (a/b) / (c/d)

Helpful hint: This formula can be simplified to ad/bc.

#### Odds Ratio in an unmatched study

Subjects with disease/outcome (cases) |
exposed (a) not exposed (c) |

Subjects without disease/outcome (controls) |
exposed (b) not exposed (d) |

**Odds Ratio** = (a/c) / (b/d) = ad /bc

An Odds Ratio of unity means that cases are no more likely to be exposed to the risk factor than controls.

#### Odds ratio in a matched study

In a 1:1 matching, a case is paired with a control based on a similar characteristic (e.g. age), and the exposure is assessed in this pair.

f = a pair in which the control is not exposed and the case is exposed

g = a pair in which the control is exposed and the case is not exposed

**Odds ratio** = f / g

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Study Design 101 by Himmelfarb Health Sciences Library is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.