# Problems Diagnostic Tests

## Exercise 1

A test was applied to a sample of people in order to evaluate its effectiveness; the results are as follows:

$$ \begin{array}{l|cc} & \mbox{Test }+ & \mbox{Test }- \newline \hline \mbox{Sick} & 2020 & 140 \newline

\mbox{Healthy} & 80 & 7760 \newline \end{array} $$

Calculate for this test:

- The sensitivity and the specificity.
- The positive and negative predictive value.
- The probability of a correct diagnosis.

Naming $S$ and $H$ to the events of being sick and healthy respectively,

- Sensitivity $P(+\vert S)=0.9352$ and specificity $P(-\vert H)=0.9898$.
- PPV $P(S\vert +)=0.9619$ and NPV $P(H\vert -)=0.9823$.
- $P((S\cap +)\cup (H\cap -)) = P(S\cap +) + P(H\cap -) = 0.978$.

## Exercise 2

We know, from a research study, that 10% of people over 50 suffer a particular type or arthritis. We have developed a new method to detect the disease and after clinical trials we observe that if we apply the method to people with arthritis we get a positive result in 85% of cases, while if we apply the method to people without arthritis, we get a positive result in 4% of cases. Answer the following questions:

- What is the probability of getting a positive result after applying the method to a random person?
- If the result of applying the method to one person has been positive, what is the probability of having arthritis?

Naming $A$ to the event of having arthritis,

- $P(+)=0.121$.
- $P(A\vert +) = 0.7025$.

## Exercise 3

We have two different test $A$ and $B$ to diagnose a disease. Test $A$ have a sensitivity of 98% and a specificity of 80%, while test $B$ have a sensitivity of 75% and a specificity of 99%.

- Which test is better to confirm the disease?
- Which test is better to rule out the disease?
- Often a test is used to discard the presence of the disease in a large amount of people apparently healthy. This type of test is known as
*screening test*. Which test will work better as a screening test? What are the predictive values of this test if the prevalence of the disease is 0.01? And if the prevalence of de disease is 0.2? - The positive predictive value of a screening test used to be not too high. How can we combine the tests $A$ and $B$ to have a higher confidence in the diagnosis of the disease? Calculate the post-test probability of having the disease with the combination of both thest, if the outcome of both test is positive for a prevalence of 0.01.

- Test $B$ cause it has a greater specificity.
- Test $A$ cause it has a greater sensitivity.
- Test $A$ will perform better as a screening test.

For a prevalence of $0.01$ the PPV is $P(D\vert +)=0.0472$ and the NPV is $P(\bar D\vert -)=0.9997$.

For a prevalence of $0.2$ the PPV is $P(D\vert +)=0.5506$ and the NPV is $P(\bar D\vert -)=0.9938$. - First applying test $A$ to everybody and then applying test $B$ to positive cases of test $A$.

$P(D\vert +_A\cap +_B)=0.7878$.