Question

Suppose that 100,000 men were screened for prostate cancer for the first time. Of these, 4,000 men had a positive result on the screening blood test; of those who tested positive, 800 had a biopsy indicating a diagnosis of prostate cancer. Among the remaining 96,000 men who screened negative, 100 developed prostate cancer within the following year and were assumed to be false negatives to the screen. a) set-up the two-by-two table for this data. (please provide an actual table) b) what is the prevalence of prostate cancer in this population? c) calculate and interpret the sensitivity of this screening test d) calculate and interpret the specificity of this screening test. e) Calculate and interpret the positive predictive value of this screening test.

Answer 1

An actual two-by-two table is a tabular representation containing two rows and two columns.

• The columns consist of the tested True positive for prostate cancer and tested True Negative for prostate cancer
• The rows consist of the predicted positive screening and predicted negative values

a)

Mathematically, the set-up of the two-by-two table for this data can be computed as:

Tested                  True Positive for cancer   True Negative    Total

Predicted Positive         800                            3200                 4000

Predicted Negative        100                            95900              96000

Total                                900                            99100              100000  

b)

The prevalence rate of prostate cancer in this population is:

$\mathbf{ =\dfrac{900}{100000}}$

$\mathbf{ =\dfrac{9}{1000}}$

= 9 per thousand.

c)

The calculation of the sensitivity of this screening is as follows:

$\mathbf{=\dfrac{TP}{TP+PN_1}}$

where;

• TP = True positive for cancer
• PN₁ = Predicted Negative for true positive cancer

∴

$\mathbf{=\dfrac{800}{800+100}}$

= 0.889

= 88.9%

The interpretation shows that 88.9% are correctly identified to be actual positive for prostate cancer.

d)

The calculation of the specificity  of this screening is as follows:

$\mathbf{=\dfrac{PN_2}{PN_2+TN}}$

where;

• TN = True positive for cancer
• PN₂ = Predicted Negative for true negative cancer

∴

$\mathbf{=\dfrac{95900}{95900+3200}}$

= 0.9677

= 96.77%

The interpretation shows that 96.7% of an actual negative is correctly identified as such.

e)

The positive predicted value of the screening test is computed as:

$\mathbf{= \dfrac{TP}{TP + TN}}$

$\mathbf{= \dfrac{800}{800 + 3200}}$

= 0.2

= 20%

The interpretation of the positive predicted value of this screening shows that 20% that are subjected to the diagnosis of positive prostate cancer truly have the disease.

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