A woman is worried that she might have breast cancer. The standard test for breast cancer is mammography. From research she found online, she discovers the sensitivity for the mammography screening is 86 % 86% and the specificity is 97 % 97% . Her doctor has told her that one in 52 52 women in her forties will have breast cancer. Based on the sensitivity, specificity, and prevalence, determine the positive predictive value (PPV) for the mammography screening. Express your answer to three decimal places.

PPV= 0.432

Step-by-step explanation:

Hello!

Remember:

Any medical test used to detect certain sicknesses have several probabilities associated with their results.

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) = P(+ ∩ S)

                    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) = P(- ∩ H)

                     P(H)

Positive predictive value (PPV)

The prevalence of the sickness can be expresed as the probability of being sick in the population of interest.

It's defined as the probability of being sick when the test is positive:

P(S/+)= P(S ∩ +)  

               P(+)

In this case, the population of interest is "Women in their forties"

The probability of being sick is P(S)= 1 /52= 0.019

The sensibility of this test is P(+/S)= 0.86

The specificity of the test is P(-/H)= 0.97

To calculate the positive predictive value you have to reach the probability of being sick and the test is positive. You can clear this probability using the information of the sensibility of the test and the prevalence of the sickness in the population:

P(+/S) = P(+ ∩ S)

                 P(S)

P(+ ∩ S) =P(+/S) * P(S) = 0.86*0.019= 0.016

Now you need to calculate the probability of the test being positive P(+)

You can calculate it as: P(+)= P(+ ∩ S) + P(+ ∩ H)

P(+ ∩ H)= P(H) - P(- ∩ H)

The probability of the person being healthy P(H) is complementary to the prevalence of the sickness, symbolically: P(H)= 1 - P(S)= 1 - 0.019= 0.981

Now using the information of the test specificity and the probability of being healthy you can clear P(- ∩ H)

P(- ∩ H)= P(H)*P(-/H) = 0.981*0.97= 0.95157≅0.96

P(+ ∩ H)= P(H) - P(- ∩ H)= 0.981-0.96= 0.021

P(+)= P(+ ∩ S) + P(+ ∩ H)= 0.016+0.021= 0.037

The PPV of the test is:

P(S/+)= P(S ∩ +)   = 0.016 = 0.432

               P(+)         0.037

I hope it helps!

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