true positive rate

How many positives are correctly identified?

Sensitivity is the accuracy of positive events, given the condition/signal is present (“detection ability”).
E.g. how likely it is that a drug test classifies a drug-user as a drug-user.
Sensitivity alone is not sufficient, e.g. a test that is always positive would have 100% sensitivity, but terrible specificity.

Sensitivity / True Positive Rate (TPR) / Recall

$$\text{TPR}=\frac{\text{TP}}{\text{TP}+\text{FN}}=P(\text{positive test }|\text{ condition present})$$

What is another name for sensitivity (recall)?

What is another name for true positive rate?

What is the formula for recall?

$$\text{Recall}=\frac{\text{TP}}{\text{TP}+\text{FN}}=P(\text{positive test }|\text{ condition present})$$

Sensitivity and specificity

Assume a drug test which has 90% sensitivity (true positive rate or “recall”), i.e. the probability of a positive result given it’s a drug user $P(+\mid \text{user})=0.9$.
And 80% specificity (true negative rate or “selectivity”), i.e. $P(-\mid \text{non-user})=0.8$.

This test is quite bad, 20% of non-users get a false positive. Let’s also say 10% of the population use the drug $P(H)=0.1$.
The probability of someone being a drug-user after testing positive is:

$$\begin{array}{rl}P(\text{user}\mid +)& =\frac{P(+\mid \text{user})P(\text{user})}{P(+\mid \text{user})P(\text{user})+P(+\mid \text{non-user})P(\text{non-user})}=\frac{0.9\cdot 0.1}{0.9\cdot 0.1+0.2\cdot 0.9}=\frac{1}{3}\end{array}$$

So the probability that you did actually use the drug is only 33%. Even if the sensitivity is 1, we only reach ~35%.
This error gets better for more common events.

This is also relevant for machine learning: We can get close to 100% accuracy just by ignoring very rare samples (high sensitivity, low specificity).

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