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## Likelihood Ratio

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

- See Also
- Screening Test
- Contingency Grid or Cross Tab (includes Statistics Example)
- Bayes Theorem (Bayesian Statistics)
- Fagan Nomogram
- Experimental Error (Experimental Bias)
- Lead-Time Bias
- Length Bias
- Selection Bias (Screening Bias)
- Number Needed to Screen (Number Needed to Treat, Absolute Risk Reduction, Relative Risk Reduction)
- Negative Predictive Value
- Positive Predictive Value
- Pre-Test Odds or Post-Test Odds
- Receiver Operating Characteristic
- Test Sensitivity (False Negative Rate)
- Test Specificity (False Positive Rate)
- U.S. Preventive Services Task Force Recommendations

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

- Definition
- General
- Numerator
- Test Sensitivity (or its reciprocal when calculating negative likelihood)

- Denominator
- Test Specificity (or its reciprocal when calculating positive likelihood)

- Numerator
- Positive Likelihood Ratio (LR+): Rule-In Condition
- Extent to which a positive test increases the likelihood that a patient has that disease
- Calculation 1: LR+ = (true positive probability) / (False Positive probability)
- Calculation 2: LR+ = P (test positive | disease) / P (test positive | no disease)
- P (test positive | disease)
- Probability that a person with the condition has a positive test (true positive, Test Sensitivity)

- P (test positive | No disease)
- Probablity that a person without the condition has a positive test (False Positive, 1-Test Specificity)

- P (test positive | disease)
- Calculation 3: LR+ = (Test Sensitivity) / (1 - Test Specificity)

- Negative Likelihood Ratio (LR-): Rule-Out Condition
- Extent to which a negative test decreases the likelihood that a patient has that disease
- Calculation 1: LR- = (False Negative probability) / (true negative probability)
- Calculation 1: LR- = (pFalseNeg / pTrueNeg)= P (test negative | disease) / P (test negative | no disease)
- P (test negative | disease)
- Probability that a person with the condition has a negative test (False Negative, 1-Test Sensitivity)

- P (test negative | no disease)
- Probablity that a person without the condition has a negative test (true negative, Test Specificity)

- P (test negative | disease)
- Calculation 2: LR- = (1 - Test Sensitivity) / (Test Specificity)

- General
- Interpretation: Positive Likelihood Ratio (LR+)
- LR+ over 5 - 10: Significantly increases likelihood of the disease
- LR+ between 0.2 to 5 (esp if close to 1): Does not modify the likelihood of the disease
- LR+ below 0.1 - 0.2: Significantly decreases the likelihood of the disease

- Interpretation: Application
- Once Likelihood Ratio is known, this can be applied to an individual patient
- Start with a patient's pretest probability of a given condition
- Method 1: Using a Likelihood Ratio nomogram, calculate the Post-Test Probability
- Method 2: Rough estimation of Post-Test Probability
- Indication: Pretest probability between 10 and 90%
- Do not use this estimate when the pretest probability <10% or >90%

- Positive Likelihood Ratio (LR+)
- LR+ 2: Post-test Prob. = Pretest Prob + 15%
- LR+ 5: Post-test Prob. = Pretest Prob + 30%
- LR+ 10: Post-test Prob. = Pretest Prob + 45%

- Negative Likelihood Ratio (LR-, significant values are the inverse of 2, 5 and 10)
- LR+ 0.5: Post-test Prob. = Pretest Prob - 15%
- LR+ 0.2: Post-test Prob. = Pretest Prob - 30%
- LR+ 0.1: Post-test Prob. = Pretest Prob - 45%

- References
- Krise in Herbert (2017) EM:Rap 17(2): 7-8
- McGee (2002) J Gen Intern Med 17(8): 646-9 [PubMed]

- Indication: Pretest probability between 10 and 90%

- Example: Mammogram Likelihood Ratios
- Given
- Mammogram Test Sensitivity: 77-95%
- Mammogram Test Specificity: 94-97%
- (2009) Ann Intern Med 151: 716-26 [PubMed]

- Best case analysis (using 95% sensitivity and 97% Specificity)
- LR Positive (LR+) = (0.95)/(1-0.97) = 31
- A positive Mammogram is highly suggestive of Breast Cancer

- LR Negative (LR-) = (1-0.95)/(0.97) = 0.05
- A negative Mammogram is very reassuring

- LR Positive (LR+) = (0.95)/(1-0.97) = 31
- Worst case analysis (using 77% sensitivity and 94% Specificity)
- LR Positive (LR+) = (0.77)/(1-0.94) =12
- A positive Mammogram is still suggestive of Breast Cancer

- LR Negative (LR-) = (1-0.77)/(0.94) = 0.24
- A negative Mammogram does not exclude Breast Cancer with adequate Likelihood Ratio

- LR Positive (LR+) = (0.77)/(1-0.94) =12

- Given
- Example: Prostate Specific Antigen likelihood
- Given
- PSA Test Sensitivity: 80%
- PSA Test Specificity: 30%

- Analysis
- LR Positive (LR+) = (0.8)/(1-0.3) = 1.1
- A positive PSA does not increase the likelihood of Prostate Cancer

- LR Negative (LR-) = (1-0.8)/(0.3) = 0.7
- A negative PSA does not decrease the likelihood of Prostate Cancer

- LR Positive (LR+) = (0.8)/(1-0.3) = 1.1

- Given
- Resources
- References
- Desai (2014) Clinical Decision Making, AMIA’s CIBRC Online Course