Answer:
Step-by-step explanation:
Remember:
Any medical test used to detect certain sicknesses have several probabilities associated with their results.
Positive (test is +) ⇒ P(+)
True positive (test is + and the patient is sick) ⇒ P(+ ∩ S)
False-positive (test is + but the patient is healthy) ⇒P(+ ∩ H)
Negative (test is -) ⇒ P(-)
True negative (test is - and the patient is healthy) ⇒ P(- ∩ H)
False-negative (test is - but the patient is sick) ⇒ P(- ∩ S)
You can arrange them in a contingency table as:
Probabilities Positive ; Negative
Sick + ∩ S ; - ∩ S S
Healthy + ∩ H ; - ∩ H H
+ - 1
The sensibility of the test is defined as the capacity of the test to detect the sickness in sick patients (true positive rate).
⇒ P(+/S) = <u>P(+ ∩ S)</u>
P(S)
The specificity of the test is the capacity of the test to have a negative result when the patients are truly healthy (true negative rate)
⇒ P(-/H) = <u>P(- ∩ H)</u>
P(H)
1) You are studying the value of the prostate-specific antigen (PSA) blood test for the detection of prostate cancer on men of 50 years of age and older.
Total 100000 men
686 men tested positive
281 of the men that tested positive had cancer
45 men that tested negative had cancer
Total - positive cases: 100000 - 686 = 99314 tested negative
; Positive ; Negative ; Total
Sick ; 281 ; 45 ; 326
Healthy ; 405 ; 99269 ; 99674
Total ; 686 ; 99314 ; 100000
2)
Sensitivity of the test is
P(+/S) = <u>P(+ ∩ S) </u>= <u>0.00281 </u>= 0.86
P(S) 0.00326
Where:
P(+ ∩ S) = 281/100000 = 0.00281
P(S) = 326/100000 = 0.00326
The test has an 86% probability of detecting PSA in sick patients.
3)
Specificity of the test is
P(-/H) = <u>P(- ∩ H) </u>= <u>0.99269 </u>= 0.995
P(H) 0.99674
Where:
P(- ∩ H)= 99269/100000= 0.99269
P(H)= 99674/100000= 0.99674
The test has a 99.5% probability of not detecting PSA in healthy patients.
4)
Positive predictive value (PPV)
It's defined as the probability of being sick when the test is positive:
P(S/+)= <u>P(S ∩ +) </u>= <u>0.00281 </u>= 0.04
P(+) 0.0686
Where
P(+)= 686/100000= 0.0686
There is a 4% probability of having cancer if the test is positive.
5)
Negative predictive value (NPV)
P(H/-)= <u>P(H ∩ -) </u>= <u>0.99269 =</u> 0.999
P(-) 0.99314
Where:
P(-)= 99314/100000= 0.99314
There is a 99.9% probability of being healthy if the test is negative.
6 to 10 are all examples of medical tests.
I hope this helps!
