What are the advantages and disadvantages of your choice?

In this mini-project, we’ll consider this (fictional) medical test.  In a population, 2% of all people have a certain medical condition.  This is called the prevalence of the condition.  

A new test for this condition is being developed.  Here is some data that was gathered during the development of the test.  The test was administered to people without this condition, and also people with the condition.  The result of the test is a real number, and it is rounded to the nearest one-tenth.  

(pic shown in the file).

For example, 50% of people who do not have the condition had a test value of 0.0.  30% of people without the condition had a test value of 0.1, and so on.  

For people with the condition, 5% of them had a test value of 0.0, another 5% had a test value of 0.1, and so on.  

It is our job to choose a cut-off value for the test, LaTeX: cc.  When someone takes the test, if they have a test value which is greater than or equal to LaTeX: cc, then the test result will be indicated as positive, meaning it is suspected the individual has the condition.  If they have a test value strictly less than LaTeX: cc, the test result will be indicated as negative, meaning it is suspected the individual does not have the condition.  

What value of LaTeX: cc should we pick?  

1.  Let’s say we pick LaTeX: c=0.1c = 0.1.  This means anyone who tests at 0.1 or higher will have a positive test result, and anyone who tests at 0.0 will have a negative test result.  Using the table above, and the prevalence of 2%, find each of these conditional probabilities for this choice of LaTeX: cc:

(a)  LaTeX: \text{P}(\text{positive} \mid \text{has condition})P ( positive ∣ has condition ) (this is called the sensitivity, power, or true positive rate)

(b)  LaTeX: \text{P}(\text{negative} \mid \text{has condition})P ( negative ∣ has condition ) (this is called the Type II error, or false negative rate)

(c)  LaTeX: \text{P}(\text{negative} \mid \text{does not have condition})P ( negative  does not have condition ) (this is called the specificity, or true negative rate)

(d)  LaTeX: \text{P}(\text{positive} \mid \text{does not have condition})P ( positive  does not have condition ) (this is called the Type I error, or the false positive rate)

(e)  LaTeX: \text{P}(\text{has condition} \mid \text{positive})P ( has condition  positive ) (this is called the Positive Predictive Value)

(f)  LaTeX: \text{P}(\text{does not have condition} \mid \text{negative})P ( does not have condition ∣ negative ) (this is called the Negative Predictive Value)

2.  Instead, say we pick LaTeX: c = 0.3c = 0.3.  Find each of the six conditional probabilities from #1 for this new choice of LaTeX: cc. 

3.  Now it’s up to you to pick a value for LaTeX: cc.  Which value of LaTeX: cc would you recommend?  Why?  What are the advantages and disadvantages of your choice?  

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