True Negative

What is a True Negative?

Sensitivity and specificity are statistical measures of the performance of a binary classification test, also known in statistics as a classification function, that are widely used in medicine AI:

  • Sensitivity (also called the true positive rate, the epidemiological/clinical sensitivity, the recall, or probability of detection in some fields) measures the proportion of actual positives that are correctly identified as such (e.g., the percentage of sick people who are correctly identified as having the condition). It is often mistakenly confused with the detection limit, while the detection limit is calculated from the analytical sensitivity, not from the epidemiological sensitivity.
  • Specificity (also called the true negative rate) measures the proportion of actual negatives that are correctly identified as such (e.g., the percentage of healthy people who are correctly identified as not having the condition).

The terms “positive” and “negative” do not refer to the value of the condition of interest, but to its presence or absence; the condition itself could be a disease, so that “positive” might mean “diseased”, while “negative” might mean “healthy”.

In many tests, including diagnostic medical tests, sensitivity is the extent to which actual positives are not overlooked (so false negatives are few), and specificity is the extent to which actual negatives are classified as such (so false positives are few). Thus, a highly sensitive test rarely overlooks an actual positive (for example, showing nothing wrong despite a problem existing); a highly specific test rarely registers a positive classification for anything that is not the target of testing (for example, finding one bacterial species and mistaking it for another closely related one that is the true target). A test that is both highly sensitive and highly specific rarely records either false positives or false negatives.

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For any test, there is usually a trade-off between the measures – for instance, in airport security, since testing of passengers is for potential threats to safety, scanners may be set to trigger alarms on low-risk items like belt buckles and keys (low specificity) in order to increase the probability of identifying dangerous objects and minimize the risk of missing objects that do pose a threat (high sensitivity). This trade-off can be represented graphically using a receiver operating characteristic curve. A perfect predictor would be described as 100% sensitive, meaning all sick individuals are correctly identified as sick, and 100% specific, meaning no healthy individuals are incorrectly identified as sick. In reality, however, any non-deterministic predictor will possess a minimum error bound known as the Bayes error rate. The values of sensitivity and specificity are agnostic to the percent of positive cases in the population of interest (as opposed to, for example, precision).

The terms “sensitivity” and “specificity” were introduced by American biostatistician Jacob Yerushalmy in 1947.

True Negative Table

Evaluating a 2 by 2 table is simple if you are methodical in your approach.

Disease
present
Disease
absent
Test
positive
TP FP Total
positive
Test
negative
FN TN Total
negative
Total with
disease
Total with-
out disease
Grand
total

 

  1. Choose an arbitrary (large) number of hypothetical patients and write it in the Grand total cell.
  2. Multiply the Grand total by the Pretest probability to get the Total with the disease.
  3. Compute the Total without disease by subtraction.
  4. Multiply the Total with the disease by the Sensitivity to get the number of True positives.
  5. Multiply the Total without disease by the Specificity to get the number of True Negatives.
  6. Compute the number of False positives and False negatives by subtraction.
  7. Compute the Total positive tests and Total negative tests by addition across the rows.
  8. The predictive value of a positive test is True positives divided by Total positive tests.
  9. The predictive value of a negative test is True negatives divided by Total negative tests.

Conclusion

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