DNA kinship test: precision, sensitivity, specificity

Text

A DNA test to establish whether two people are related gives a positive result $\dfrac67$ of the times when the two people are actually related, and a negative result $\dfrac56$ of the times when they are not. Moreover, if two people tested positive, they have a $\dfrac{12}{13}$ probability of being related.

• (a) Write the data as conditional probabilities for the events $A=$ ”positive” and $B=$ ”related”.
• (b) Draw a Venn diagram with the probabilities, setting up a system of equations to find them.
• (c) What are the precision, sensitivity and specificity of the test, as percentages?

Solution

(a) $\;P(A\mid B)=\dfrac67$ (sensitivity), $\;P(\bar A\mid\bar B)=\dfrac56$ (specificity), $\;P(B\mid A)=\dfrac{12}{13}$ (precision). (b) Let $P(B)=b$. Then $P(A\cap B)=\tfrac67 b,\quad P(\bar A\cap B)=\tfrac17 b,\quad P(A\cap\bar B)=\tfrac16(1-b),\quad P(\bar A\cap\bar B)=\tfrac56(1-b).$ Precision $P(B\mid A)=\dfrac{P(A\cap B)}{P(A)}=\dfrac{12}{13}$ gives $\frac{\tfrac67 b}{\tfrac67 b+\tfrac16(1-b)}=\frac{12}{13}\ \Longrightarrow\ \frac{6}{7}b=2(1-b)\ \Longrightarrow\ b=0{,}7.$ The four Venn probabilities are then $P(A\cap B)=0{,}6,\quad P(\bar A\cap B)=0{,}1,\quad P(A\cap\bar B)=0{,}05,\quad P(\bar A\cap\bar B)=0{,}25$ (sum $=1$ ✓; $P(A)=0{,}65$ and $P(B\mid A)=12/13$ ✓). (c) $\text{sensitivity}=\frac67\approx85{,}7\%,\quad \text{specificity}=\frac56\approx83{,}3\%,\quad \text{precision}=\frac{12}{13}\approx92{,}3\%.$ $\boxed{\text{precision}\approx92{,}3\%,\ \text{sensitivity}\approx85{,}7\%,\ \text{specificity}\approx83{,}3\%\ \ (P(B)=0{,}7).}$