Likelihood Ratio

Aka: Likelihood Ratio, Positive Likelihood Ratio, LR+, Negative Likelihood Ratio, LR-

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II. Indications

  1. Determine whether a test offers value in a patient's evaluation for a particular condition
  2. Likelihood combines Test Sensitivity and Test Specificity to apply a test's value to an individual patient
    1. Test Sensitivity and Test Specificity each in isolation apply only to a patient with a known diagnosis
    2. Tests are only used in patients with an unknown diagnosis
    3. Likelihood Ratio puts Test Sensitivity in the context of Test Specificity (in a single value)

III. Definition

  1. General
    1. Numerator
      1. Test Sensitivity (or its reciprocal when calculating negative likelihood)
    2. Denominator
      1. Test Specificity (or its reciprocal when calculating positive likelihood)
  2. Positive Likelihood Ratio (LR+): Rule-In Condition
    1. Extent to which a positive test increases the likelihood that a patient has that disease
    2. Calculation 1: LR+ = (true positive probability) / (False Positive probability)
    3. Calculation 2: LR+ = P (test positive | disease) / P (test positive | no disease)
      1. P (test positive | disease)
        1. Probability that a person with the condition has a positive test (true positive, Test Sensitivity)
      2. P (test positive | No disease)
        1. Probablity that a person without the condition has a positive test (False Positive, 1-Test Specificity)
    4. Calculation 3: LR+ = (Test Sensitivity) / (1 - Test Specificity)
  3. Negative Likelihood Ratio (LR-): Rule-Out Condition
    1. Extent to which a negative test decreases the likelihood that a patient has that disease
    2. Calculation 1: LR- = (False Negative probability) / (true negative probability)
    3. Calculation 1: LR- = (pFalseNeg / pTrueNeg)= P (test negative | disease) / P (test negative | no disease)
      1. P (test negative | disease)
        1. Probability that a person with the condition has a negative test (False Negative, 1-Test Sensitivity)
      2. P (test negative | no disease)
        1. Probablity that a person without the condition has a negative test (true negative, Test Specificity)
    4. Calculation 2: LR- = (1 - Test Sensitivity) / (Test Specificity)

IV. Interpretation: Positive Likelihood Ratio (LR+)

  1. LR+ over 5 - 10: Significantly increases likelihood of the disease
  2. LR+ between 0.2 to 5 (esp if close to 1): Does not modify the likelihood of the disease
  3. LR+ below 0.1 - 0.2: Significantly decreases the likelihood of the disease

V. Interpretation: Application

  1. Once Likelihood Ratio is known, this can be applied to an individual patient
  2. Start with a patient's pretest probability of a given condition
  3. Method 1: Using a Likelihood Ratio nomogram, calculate the Post-Test Probability
    1. http://www.cebm.net/index.aspx?o=1043
  4. Method 2: Rough estimation of Post-Test Probability
    1. Indication: Pretest probability between 10 and 90%
      1. Do not use this estimate when the pretest probability <10% or >90%
    2. Positive Likelihood Ratio (LR+)
      1. LR+ 2: Post-test Prob. = Pretest Prob + 15%
      2. LR+ 5: Post-test Prob. = Pretest Prob + 30%
      3. LR+ 10: Post-test Prob. = Pretest Prob + 45%
    3. Negative Likelihood Ratio (LR-, significant values are the inverse of 2, 5 and 10)
      1. LR+ 0.5: Post-test Prob. = Pretest Prob - 15%
      2. LR+ 0.2: Post-test Prob. = Pretest Prob - 30%
      3. LR+ 0.1: Post-test Prob. = Pretest Prob - 45%
    4. References
      1. Krise in Herbert (2017) EM:Rap 17(2): 7-8
      2. McGee (2002) J Gen Intern Med 17(8): 646-9 [PubMed]

VI. Example: Mammogram Likelihood Ratios

  1. Given
    1. Mammogram Test Sensitivity: 77-95%
    2. Mammogram Test Specificity: 94-97%
    3. (2009) Ann Intern Med 151: 716-26 [PubMed]
  2. Best case analysis (using 95% sensitivity and 97% Specificity)
    1. LR Positive (LR+) = (0.95)/(1-0.97) = 31
      1. A positive Mammogram is highly suggestive of Breast Cancer
    2. LR Negative (LR-) = (1-0.95)/(0.97) = 0.05
      1. A negative Mammogram is very reassuring
  3. Worst case analysis (using 77% sensitivity and 94% Specificity)
    1. LR Positive (LR+) = (0.77)/(1-0.94) =12
      1. A positive Mammogram is still suggestive of Breast Cancer
    2. LR Negative (LR-) = (1-0.77)/(0.94) = 0.24
      1. A negative Mammogram does not exclude Breast Cancer with adequate Likelihood Ratio

VII. Example: Prostate Specific Antigen likelihood

  1. Given
    1. PSA Test Sensitivity: 80%
    2. PSA Test Specificity: 30%
  2. Analysis
    1. LR Positive (LR+) = (0.8)/(1-0.3) = 1.1
      1. A positive PSA does not increase the likelihood of Prostate Cancer
    2. LR Negative (LR-) = (1-0.8)/(0.3) = 0.7
      1. A negative PSA does not decrease the likelihood of Prostate Cancer

VIII. Resources

  1. OmeRad EBM
    1. http://omerad.msu.edu/ebm/Diagnosis/Diagnosis6.html
  2. Wikipedia
    1. https://en.wikipedia.org/wiki/Likelihood_ratios_in_diagnostic_testing

IX. References

  1. Desai (2014) Clinical Decision Making, AMIA’s CIBRC Online Course

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