Sensitivity specificity and precision of a diagnostic test

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$1000$ patients undergo a test to detect uranium in their blood. The results are in the table.

Test result	U present	U absent	Total
Positive	48	92	140
Negative	12	848	860
Total	60	940	1000

(a) Draw a Venn diagram with the probabilities.

(b) Build the tree diagram considering the test result first, then the presence of uranium.

(c) Build the tree diagram in the opposite order: first the presence of uranium, then the test result.

(d) Compute the sensitivity $P(\text{pos}\mid U)$, the specificity $P(\text{neg}\mid\overline U)$, and the precision $P(U\mid\text{pos})$.

Solution

From the relative frequencies in the table we get the joint probabilities, e.g. $P(U\cap\text{pos})=\dfrac{48}{1000}=0.048$, $P(U\cap\text{neg})=\dfrac{12}{1000}=0.012$, $P(\overline U\cap\text{pos})=\dfrac{92}{1000}=0.092$, $P(\overline U\cap\text{neg})=\dfrac{848}{1000}=0.848$: these give both the Venn diagram (part a) and the two tree diagrams, which condition respectively on the test result (b) and on the presence of uranium (c), using the marginal probabilities $P(U)=0.06$, $P(\overline U)=0.94$, $P(\text{pos})=0.14$, $P(\text{neg})=0.86$ and the corresponding conditional probabilities.

(d) Sensitivity is the probability of testing positive among the diseased: $P(\text{pos}\mid U) = \frac{48}{60} = 0.80$ Specificity is the probability of testing negative among the healthy: $P(\text{neg}\mid\overline U) = \frac{848}{940} \approx 0.902$ Precision is the probability of actually being diseased among those who test positive: $P(U\mid\text{pos}) = \frac{48}{140} \approx 0.343$

$\boxed{\text{sensitivity}=0.80;\ \text{specificity}\approx0.902;\ \text{precision}\approx0.343}$