Prostate-specific antigen (PSA) test measures the level of PSA in blood and is widely used to
screen for prostate cancer. A newer, related parameter, PSA density, is computed as the serum PSA level (in ng/mL) divided by the volume of the prostate gland (in cubic centimeters (cc)). Unlike the PSA test itself, PSA density (PSAD) accounts for the fact larger prostates gland generally have higher levels of PSA under normal baseline physiologic conditions (i.e., in the absence of prostate cancer). Accordingly, consideration of PSA density is increasingly favored over PSA alone for assessing prostate cancer risk.
The results of PSAD test conducted in USA and reported in a Journal was read by a clinician in India (wherein 1.5% are affected with prostate cancer),
489 1400  10 263 
Now the clinician in India want to know what is the probability of a person detected PSAD to have the disease and probability that a person detected PSAD negative to not have the disease?

To solve this problem, we need to apply Bayes' Theorem to calculate the probability of a person having prostate cancer given a positive PSA density (PSAD) test result, and the probability of not having prostate cancer given a negative PSAD test result. 

Here are the data points provided and relevant notations:

- Number of positive PSAD tests: 489
- Number of total PSAD tests: 1400
- Number of true positive cases (detected PSAD and having cancer): 10
- Number of true negative cases (detected negative PSAD and not having cancer): 263
- Prevalence of prostate cancer in India: 1.5% (or 0.015)

### 1. Probability of having prostate cancer given a positive PSAD test (Positive Predictive Value - PPV):
This is calculated using Bayes' Theorem as follows:

\[
P(\text{Cancer} | \text{Positive PSAD}) = \frac{P(\text{Positive PSAD} | \text{Cancer}) \cdot P(\text{Cancer})}{P(\text{Positive PSAD})}
\]

Where:
- \( P(\text{Positive PSAD} | \text{Cancer}) = \frac{\text{Number of true positives}}{\text{Total number of positives}} = \frac{10}{489} \)
- \( P(\text{Cancer}) = 0.015 \) (prevalence in India)
- \( P(\text{Positive PSAD}) = \frac{489}{1400} \)

Substitute the values:

\[
P(\text{Cancer} | \text{Positive PSAD}) = \frac{\left(\frac{10}{489}\right) \times 0.015}{\frac{489}{1400}}
\]

### 2. Probability of not having prostate cancer given a negative PSAD test (Negative Predictive Value - NPV):
This is calculated as follows:

\[
P(\text{No Cancer} | \text{Negative PSAD}) = \frac{P(\text{Negative PSAD} | \text{No Cancer}) \cdot P(\text{No Cancer})}{P(\text{Negative PSAD})}
\]

Where:
- \( P(\text{Negative PSAD} | \text{No Cancer}) = \frac{263}{\text{Total negatives}} \) (We need to compute the total negatives)
- \( P(\text{No Cancer}) = 1 - P(\text{Cancer}) = 1 - 0.015 = 0.985 \)
- \( P(\text{Negative PSAD}) = 1 - P(\text{Positive PSAD}) = 1 - \frac{489}{1400} \)

Let's calculate these probabilities.

I'll start by calculating the Positive Predictive Value (PPV) and Negative Predictive Value (NPV).The calculations yield the following results:

1. **Probability of having prostate cancer given a positive PSAD test (Positive Predictive Value - PPV):** Approximately 0.088% (0.00088). This means that if a person tests positive on the PSAD test, there is about a 0.088% chance that they actually have prostate cancer.

2. **Probability of not having prostate cancer given a negative PSAD test (Negative Predictive Value - NPV):** Approximately 43.7% (0.437). This means that if a person tests negative on the PSAD test, there is about a 43.7% chance that they truly do not have prostate cancer.

These values suggest that the PSAD test might have limited effectiveness in accurately identifying prostate cancer in this population, especially given the low PPV.

Would you like further details or have any questions? Here are some related questions that might interest you:

1. How does the prevalence of a disease affect the predictive value of diagnostic tests?
2. What are the sensitivity and specificity of the PSAD test, and how do they impact the results?
3. How does PSAD compare with other diagnostic methods for prostate cancer?
4. What are the implications of low PPV and NPV for clinical decision-making?
5. Can PSA and PSAD tests be combined with other markers to improve accuracy?

**Tip:** When interpreting diagnostic test results, always consider the prevalence of the disease in the population, as it significantly impacts the test's predictive values.

Learn how to calculate the probability of having prostate cancer given a positive PSA density (PSAD) test and the probability of not having prostate cancer given a negative PSAD test. Understand the impact of prevalence rates and test results on diagnostic accuracy.

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