Stats Sensitivity and specificity
Last update: 20210719
Sensitivity and specificity
The third paper on the list of BMJ statistics notes by Altman and Bland, ( lit-altman_bland.md ), altman1994diagnostic1 3. Altman DG, Bland JM. (1994) Diagnostic tests 1: sensitivity and specificity. 308, 1552. 1
The simple diagnostic test such as an x-ray is used to classify patients into two groups:
- Presence of a symptom or sign
- Yes
- No
Altman and Bland use the following cited example; The results of a scan (test) compared to the correct diagnosis (true positive) based on either necropsy, biopsy, or surgical inspection. i.e. How good is the scan for correct diagnosis?
Table 1. Relation between results of liver scan and correct diagnosis.
| Liver scan | : Pathology (diagnosis) : | ||
|---|---|---|---|
| Abnormal (+) | Normal (-) | Total | |
| Abnormal(+) | 231 | 32 | 263 |
| Normal(-) | 27 | 54 | 81 |
| Total | 258 | 86 | 344 |
Patients who are correctly labelled are:
- Disease signs and abnormal liver
- 258 true positives
- No signs and healthy liver
- 86 true negatives
The proportions of these two groups that were also correctly diagnosed by the scan were \(231/258=0.90\) and \(54/86=0.63\), respectively.
- Sensitivity
- Proportion of true positives that are correctly identified by the test.
- Specificity
- Proportion of true negatives that are correctly identified by the test.
Based on Altman and Bland’s example sample, we expect 90% true positives (patients with abnormal pathology to have abnormal (positive) liver scans), and 63% true negatives (those with normal pathology would have normal (negative) liver scans).
Confidence intervals
The sensitivity and specificity are proportions, so confidence intervals can be calculated. This uses standard methods for proportions gardner1989calculating.
Quantifying the diagnostic ability
Sensitivity and specificity are one approach to quantifying the diagnostic ability of the test. In this case, we already have the final results of tests and diagnosis from the sample set. For an individual patient we only have the test result. We want to quantify how well the test can predict true positives.
This is answered in the next statistical note; predictive values. It defines positive and negative predictive values and requires the use of sensitivity, specificity, and prevalence.
References
Footnote 1 This article is almost identical to the original version in acknowledgment to Altman and Bland. It is adapted here as part of a set of curated, consistent, and minimal examples of statistics required for human genomic analysis. ↩