# For questions 1-3, refer to the following information:

Screening tests are often used in clinical practice to assess the likelihood that a person has a medical condition. The rationale is that, if disease is identified before symptom onset, then earlier treatment may lead to cure or improved survival or quality of life. The prostate-specific antigen (PSA) test for prostate cancer measures blood concentrations of PSA, a protein produced by the prostate gland. Elevated levels of PSA may help identify men with prostate cancer. A definitive diagnosis, however, requires biopsies of the prostate gland. A population of men over 50 years old who are considered at high risk for prostate cancer had both the PSA screening test and a biopsy. Among these men, 21% had both elevated PSA levels and a positive biopsy, whereas 50% had low PSA levels and a negative biopsy. 24% had a positive biopsy. **Hint:** Start by drawing a Venn diagram

1. Is a man having an elevated PSA level independent of a man having a positive biopsy? Why or why not? **(2 points)**

No, a man having an elevated PSA level and a man having a positive biopsy are not independent events.

The theoretical proportion of menwith elevated PSA level and positive biopsy is 0.1128

The actual proportion of men with elevated PSA level and positive biopsy is 0.21

Theoretical proportion is not equal to the actual proportion, these are not independent events

Let´s put the given information in a cross table

We were told that 24% of men had a positive biopsy

we were told that 21% of mean had elevated PSA levels and a positive biopsy

50% of men had a low PSA level and a negative biopsy

so

Positive biopsy AND elevated level + Positive biopsy AND low levels = 0.24

0.21 + Positive biopsy AND low levels = 0.24

Positive biopsy AND low levels = 0.24 - 0.21 = 0.03

Finally we miss a value to get the 100%

0.21 + 0.03 + 0.5 = 0.74

so

1 - 0.74 = 0.26

Those will be the proportion of people who got a negative biopsy and elevated PSA levels

This is the table:

...................................................................................PSA Levels

.....................................................elevated levels...................................Low level.........................Total

positive biopsy.........................**..0.21**..........................................................0.03.............................0.24

negative biopsy.........................0.26..........................................................0.5.................................0.76

Total................................................0.47..........................................................0.53.................................1

Is a man having an elevated PSA level independent of a man having a positive biopsy? Why or why not?

According to the definition of independent events:

2 events are independent if and only if:

P (A and B) = P (A) * P(B)

so

Let A be Elevated PSA level

Let B be Positive biopsy

si

Probability of elevated PSA level = P (elevated PSA level and negative biopsy) + P (elevated PSA level and positive biopsy)

= 0.21 + 0.26 = 0.47

Probability of positive biopsy = P (positive biopsy and elevated levels) + P (positive biopsy and low levels)

= 0.21 + 0.03 = 0.24 (this values was provided in the statement, i am putting the way to find it for reference)

so

P (A) = 0.47

P (B) = 0.24

If A and B are independentes then:

P(A and B) = 0.47 * 0.24 = 0.1128

so if positive biopsy is independent to elevated PSA levels then the proportion of positive biopsy and elevated levels should be 0.1128

But in the table we see that 21% had both elevated PSA levels and a positive biopsy, a proportion of 0.21

so

Theoretical value for independent events = 0.1128

Actual value of the intersection = 0.21

Therefore, **an elevated PSA level and positive biopsy are not independent events.**