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?
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Wikipedia (https://en.wikipedia.org/wiki/Sensitivity_and_specificity#Medical_examples) says:
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.
On en.wikipedia.org/wiki/Sensitivity_and_specificity it says:
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.
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