# ODDS-LIKELIHOOD RATIOS

**ODDS-LIKELIHOOD RATIOS**is a topic covered in the

**Guide to Diagnostic Tests**.

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Another way to calculate the posttest probability of disease is to use the odds-likelihood (or odds-probability) approach. Sensitivity and specificity are combined into one entity called the **likelihood** ratio (LR):

When test results are dichotomized, every test has two likelihood ratios, one corresponding to a positive test (LR^{+}) and one corresponding to a negative test (LR^{–}):

For continuous measures, multiple likelihood ratios can be defined to correspond to ranges or intervals of test results. (See Table 1–5 for an example.)

Serum Ferritin (mcg/L) | Likelihood Ratios for Iron Deficiency Anemia |
---|---|

≥ 100 | 0.08 |

45–99 | 0.54 |

35–44 | 1.83 |

25–34 | 2.54 |

15–24 | 8.83 |

≤ 15 | 51.85 |

*Data from Guyatt G et al. Laboratory diagnosis of iron deficiency anemia*. J Gen Intern Med.

*1992 Mar–Apr;7(2):145–53*.

Likelihood ratios can be calculated using the above formulas. They can also be found in some textbooks, journal articles, and online programs (see Table 1–6 for sample values). Likelihood ratios provide an estimation of whether there will be significant change in pretest to posttest probability of a disease given the test result, and thus can be used to make quick estimates of the usefulness of contemplated diagnostic tests in particular situations. A likelihood ratio of 1 implies that there will be no difference between pretest and posttest probabilities. Likelihood ratios of > 10 or < 0.1 indicate large, often clinically significant differences. Likelihood ratios between 1 and 2 and between 0.5 and 1 indicate small differences (rarely clinically significant).

Target Disease | Test | LR^{+} | LR^{–} |
---|---|---|---|

Abscess | Abdominal CT scanning | 9.5 | 0.06 |

Coronary artery disease | Exercise electrocardiogram (1 mm depression) | 3.5 | 0.45 |

Lung cancer | Chest radiograph | 15 | 0.42 |

Left ventricular hypertrophy | Echocardiography | 18.4 | 0.08 |

Myocardial infarction | Troponin I | 24 | 0.01 |

Prostate cancer | Digital rectal examination | 21.3 | 0.37 |

The simplest method for calculating posttest probability from pretest probability and likelihood ratios is to use a nomogram (Figure 1–7). The clinician places a straightedge through the points that represent the pretest probability and the likelihood ratio and then reads the posttest probability where the straightedge crosses the posttest probability line.

**Figure 1–7.**Nomogram for determining posttest probability from pretest probability and likelihood ratios. To figure the posttest probability, place a straightedge between the pretest probability and the likelihood ratio for the particular test. The posttest probability will be where the straightedge crosses the posttest probability line. (

*Adapted and reproduced, with permission, from Fagan TJ. Nomogram for Bayes theorem*. [

*Letter*.] N Engl J Med.

*1975 Jul 31;293(5):257*.)

A more formal way of calculating posttest probabilities uses the likelihood ratio as follows:

Pretest odds × Likelihood ratio = Posttest oddsTo use this formulation, probabilities must be converted to odds, where the odds of having a disease are expressed as the chance of having the disease divided by the chance of not having the disease. For instance, a probability of 0.75 is the same as 3:1 odds (Figure 1–8).

**Figure 1–8.**Formulas for converting between probability and odds.

To estimate the potential benefit of a diagnostic test, the clinician first estimates the pretest odds of disease given all available clinical information and then multiplies the pretest odds by the positive and negative likelihood ratios. The results are the **posttest odds**, or the odds that the patient has the disease if the test is positive or negative. To obtain the posttest probability, the odds are converted to a probability (Figure 1–8).

For example, if the clinician believes that the patient has a 60% chance of having a myocardial infarction (pretest odds of 3:2) and the troponin I test is positive (LR^{+} = 24), then the posttest odds of having a myocardial infarction are

If the troponin I test is negative (LR^{–} = 0.01), then the posttest odds of having a myocardial infarction are

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Another way to calculate the posttest probability of disease is to use the odds-likelihood (or odds-probability) approach. Sensitivity and specificity are combined into one entity called the **likelihood** ratio (LR):

When test results are dichotomized, every test has two likelihood ratios, one corresponding to a positive test (LR^{+}) and one corresponding to a negative test (LR^{–}):

For continuous measures, multiple likelihood ratios can be defined to correspond to ranges or intervals of test results. (See Table 1–5 for an example.)

Serum Ferritin (mcg/L) | Likelihood Ratios for Iron Deficiency Anemia |
---|---|

≥ 100 | 0.08 |

45–99 | 0.54 |

35–44 | 1.83 |

25–34 | 2.54 |

15–24 | 8.83 |

≤ 15 | 51.85 |

*Data from Guyatt G et al. Laboratory diagnosis of iron deficiency anemia*. J Gen Intern Med.

*1992 Mar–Apr;7(2):145–53*.

Likelihood ratios can be calculated using the above formulas. They can also be found in some textbooks, journal articles, and online programs (see Table 1–6 for sample values). Likelihood ratios provide an estimation of whether there will be significant change in pretest to posttest probability of a disease given the test result, and thus can be used to make quick estimates of the usefulness of contemplated diagnostic tests in particular situations. A likelihood ratio of 1 implies that there will be no difference between pretest and posttest probabilities. Likelihood ratios of > 10 or < 0.1 indicate large, often clinically significant differences. Likelihood ratios between 1 and 2 and between 0.5 and 1 indicate small differences (rarely clinically significant).

Target Disease | Test | LR^{+} | LR^{–} |
---|---|---|---|

Abscess | Abdominal CT scanning | 9.5 | 0.06 |

Coronary artery disease | Exercise electrocardiogram (1 mm depression) | 3.5 | 0.45 |

Lung cancer | Chest radiograph | 15 | 0.42 |

Left ventricular hypertrophy | Echocardiography | 18.4 | 0.08 |

Myocardial infarction | Troponin I | 24 | 0.01 |

Prostate cancer | Digital rectal examination | 21.3 | 0.37 |

The simplest method for calculating posttest probability from pretest probability and likelihood ratios is to use a nomogram (Figure 1–7). The clinician places a straightedge through the points that represent the pretest probability and the likelihood ratio and then reads the posttest probability where the straightedge crosses the posttest probability line.

**Figure 1–7.**Nomogram for determining posttest probability from pretest probability and likelihood ratios. To figure the posttest probability, place a straightedge between the pretest probability and the likelihood ratio for the particular test. The posttest probability will be where the straightedge crosses the posttest probability line. (

*Adapted and reproduced, with permission, from Fagan TJ. Nomogram for Bayes theorem*. [

*Letter*.] N Engl J Med.

*1975 Jul 31;293(5):257*.)

A more formal way of calculating posttest probabilities uses the likelihood ratio as follows:

Pretest odds × Likelihood ratio = Posttest oddsTo use this formulation, probabilities must be converted to odds, where the odds of having a disease are expressed as the chance of having the disease divided by the chance of not having the disease. For instance, a probability of 0.75 is the same as 3:1 odds (Figure 1–8).

**Figure 1–8.**Formulas for converting between probability and odds.

To estimate the potential benefit of a diagnostic test, the clinician first estimates the pretest odds of disease given all available clinical information and then multiplies the pretest odds by the positive and negative likelihood ratios. The results are the **posttest odds**, or the odds that the patient has the disease if the test is positive or negative. To obtain the posttest probability, the odds are converted to a probability (Figure 1–8).

For example, if the clinician believes that the patient has a 60% chance of having a myocardial infarction (pretest odds of 3:2) and the troponin I test is positive (LR^{+} = 24), then the posttest odds of having a myocardial infarction are

If the troponin I test is negative (LR^{–} = 0.01), then the posttest odds of having a myocardial infarction are

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### Citation

*Guide to Diagnostic Tests*, 7th ed., McGraw-Hill Education, 2017.

*Emergency Central*, emergency.unboundmedicine.com/emergency/view/GDT/619005/5/ODDS_LIKELIHOOD_RATIOS.

*Guide to Diagnostic Tests*. McGraw-Hill Education; 2017. https://emergency.unboundmedicine.com/emergency/view/GDT/619005/5/ODDS_LIKELIHOOD_RATIOS. Accessed September 26, 2022.

*Guide to Diagnostic Tests*(7th edition). McGraw-Hill Education. https://emergency.unboundmedicine.com/emergency/view/GDT/619005/5/ODDS_LIKELIHOOD_RATIOS

*Guide to Diagnostic Tests*. McGraw-Hill Education; 2017. [cited 2022 September 26]. Available from: https://emergency.unboundmedicine.com/emergency/view/GDT/619005/5/ODDS_LIKELIHOOD_RATIOS.