# What is the sensitivity / specificity of a symptom?

I know the two terms in the context of a classification problem:

A classification algorithms sensitivity/specificity is defined as

sensitivity = TP / (TP + FN)
specificity = TN / (TN + FP)

I like the comparison with a fire alarm:

• If the sensitivity is high, that means it actually starts ringing when there is a fire. It could still be that FP is high, meaning it just rings all the time.
• If the specificity is high, it means it does not ring when there is no fire. It could still be that FN is high which means it does not ring when there is a fire.

Now I found a table for Influenza symptoms and I'm a bit confused what it means in this context. Does it essentially define each isolated symptom as a classification algorithm, e.g. for fever:

• I get a patient who has Influenza. I only check for fever. If the patient has fever, I conclude that he has Influenza. I'm right in 68–86% of the cases. (For simplicity, assume 77%)
• I get a patient and and apply my "fever => Influenza" logic. In 25–73% of the cases I correctly say that they don't have fever (for simplicity, assume 49%).

So essentially the number come from filling this table:

|                         | Correct Pred. (True) | Wrong Prediction (False) |
| ----------------------- | -------------------- | ------------------------ |
| Has Fever (Positive)    |                      |                          |
| Has no Fever (Negative) |                      |                          |

Specificity: Hence, if I divide the number of people without fever and who also don't have influenza by the number of all people without influenza, it should be 49%? Or is it all people who visit the doctor? All people who feel sick? All people who think they have Influenza?

Sensitivity: I divide the number of people who have fever and influenza by the number of people with Influenza. Same question here: If people are asymptomatic, they probably don't go to the doctor. If there is no check if they had influenza, how can one have FN?

Did I get something completely wrong?

• Did you mean to ask (in the title Q) about S&S of symptom or did you mean test, as in a diagnostic test?
– BobE
Commented Apr 30, 2020 at 2:41
• In your algorithm, does "FP" represent false-positive results, while "FN" represents false-negatives results (of a test).
– BobE
Commented Apr 30, 2020 at 2:47

## 1 Answer

In medical diagnosis, test sensitivity is the ability of a test to correctly identify those with the disease (true positive rate), whereas test specificity is the ability of the test to correctly identify those without the disease (true negative rate). If 100 patients known to have a disease were tested, and 43 test positive, then the test has 43% sensitivity. If 100 with no disease are tested and 96 return a negative result, then the test has 96% specificity. Sensitivity and specificity are prevalence-independent test characteristics, as their values are intrinsic to the test and do not depend on the disease prevalence in the population of interest.[10] Positive and negative predictive values, but not sensitivity or specificity, are values influenced by the prevalence of disease in the population that is being tested. These concepts are illustrated graphically in this applet Bayesian clinical diagnostic model which show the positive and negative predictive values as a function of the prevalence, the sensitivity and specificity.

Sensitivity (also called the true positive rate, the recall, or probability of detection[1] 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).

I. e. sensitivity refers to the proportion of people with disease who have a positive test result.

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).

I. e. specificity refers to the proportion of people without disease who have a negative test result.

In summary:

• Sensitivity is the proportion of patients with the disease who have the symptom.
• Specificity is the proportion of patients without the disease who do not have the symptom.
• Not diagnosis, disease. Diagnosis is a social construction. Some people with a diagnosis don’t have the diagnosed disease; having a disease without it being diagnosed is common. Commented Mar 30, 2020 at 23:15
• True, thanks for the correction. Commented Mar 30, 2020 at 23:20