Likelihood Ratios in Diagnostic Testing - Example

Example

A medical example is the likelihood that a given test result would be expected in a patient with a certain disorder compared to the likelihood that same result would occur in a patient without the target disorder.

Some sources distinguish between LR+ and LR−. A worked example is shown below.

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Relationships among terms
Condition
(as determined by "Gold standard")
Condition Positive Condition Negative
Test
Outcome		 Test
Outcome
Positive		 True Positive False Positive
(Type I error)		 Positive predictive value =
Σ True Positive Σ Test Outcome Positive
Test
Outcome
Negative		 False Negative
(Type II error)		 True Negative Negative predictive value =
Σ True Negative Σ Test Outcome Negative
Sensitivity =
Σ True Positive Σ Condition Positive 		Specificity =
Σ True Negative Σ Condition Negative
A worked example
The fecal occult blood (FOB) screen test was used in 2030 people to look for bowel cancer:
Patients with bowel cancer
(as confirmed on endoscopy)
Condition Positive Condition Negative
Fecal
Occult
Blood
Screen
Test
Outcome		 Test
Outcome
Positive		 True Positive
(TP) = 20		 False Positive
(FP) = 180		 Positive predictive value = TP / (TP + FP)
= 20 / (20 + 180)
= 10%
Test
Outcome
Negative		 False Negative
(FN) = 10		 True Negative
(TN) = 1820		 Negative predictive value = TN / (FN + TN)
= 1820 / (10 + 1820)
≈ 99.5%
Sensitivity = TP / (TP + FN)
= 20 / (20 + 10)
≈ 67% 		Specificity = TN / (FP + TN)
= 1820 / (180 + 1820)
= 91%

Related calculations

  • False positive rate (α) = type I error = 1 − specificity = FP / (FP + TN) = 180 / (180 + 1820) = 9%
  • False negative rate (β) = type II error = 1 − sensitivity = FN / (TP + FN) = 10 / (20 + 10) = 33%
  • Power = sensitivity = 1 − β
  • Likelihood ratio positive = sensitivity / (1 − specificity) = 66.67% / (1 − 91%) = 7.4
  • Likelihood ratio negative = (1 − sensitivity) / specificity = (1 − 66.67%) / 91% = 0.37

Hence with large numbers of false positives and few false negatives, a positive FOB screen test is in itself poor at confirming cancer (PPV = 10%) and further investigations must be undertaken; it did, however, correctly identify 66.7% of all cancers (the sensitivity). However as a screening test, a negative result is very good at reassuring that a patient does not have cancer (NPV = 99.5%) and at this initial screen correctly identifies 91% of those who do not have cancer (the specificity).

Read more about this topic:  Likelihood Ratios In Diagnostic Testing

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