Probability When Screening for Pre-clinical Disease

A basic understanding of probability is essential for interpreting screening tests for early disease. The validity of screening tests is evaluated by screening a large number of people and then evaluating the probability that the screen test correctly identified people with and without the disease by comparing the results to a more definitive determination of whether the disease was truly present or not. Sometimes the definitive answer is based on a more extensive battery of diagnostic tests, and sometimes it is determined by simply waiting to see if the disease occurs within a certain follow up period of time.

Results from an evaluation of a screening test are summarized in a different type of contingency table as shown in the table below showing results for a screening test for breast cancer in 64,810 women. 

	True Disease Status
	Diseased	Not Diseased	Total
Test Positive	132	983	1,115
Test Negative	45	63,650	63,695
Column Totals	177	64,633	64,810

There were 177 women who were ultimately found to have had breast cancer, and 64,633 women remained free of breast cancer during the observation period. Among the 177 women with breast cancer, 132 had a positive screening test (true positives), but 45 of the women with breast cancer had negative tests (false negatives). Among the 64,633 women without breast cancer, 63,650 appropriately had negative screening tests (true negatives), but 983 incorrectly had positive screening tests (false positives).

If we focus on the rows, we find that 1,115 subjects had a positive screening test, i.e., the screening results were abnormal and suggested disease. However, only 132 of these were found to actually have disease. Also note that 63,695 people had a negative screening test, suggesting that they did not have the disease, but, in fact 45 of these people were actually diseased.

Sensitivity of a Screening Test

One measure of test validity is sensitivity, i.e., the probability of a positive screening test in people who truly have the disease. When thinking about sensitivity, focus on the individuals who really were diseased - in this case, the left-hand column.

Table - Sensitivity of a Screening Test

 

	Diseased	Not Diseased	Total
Test Positive	132	983	1,115
Test Negative	45	63,650	63,695
Column Totals	177	64,633	64,810

The probability of a positive screening test in women who truly have breast cancer is:

P(Screen+ | breast cancer) = 132/177 = 0.746 = 74.6%.

Interpretation: "The probability of the screening test correctly identifying women with breast cancer was 74.6%."

Specificity of a Screening Test

Specificity focuses on the probability that non-diseased subjects will be classified as non-diseased by the screening test. Now we focus on the column for women who were truly not diseased.

Table - Specificity of a Screening Test

	Diseased	Not Diseased	Total
Test Positive	132	983	1,115
Test Negative	45	63,650	63,695
Column Totals	177	64,633	64,810

 The probability of a positive screening test in women who truly have breast cancer is:

P(Screen- | no breast cancer) = 63,650/64,633 = 0.985 = 98.5%.

Interpretation: "The probability of the screening test correctly identifying women without breast cancer was 98.5%."

Test Yourself

In the above example, what was the prevalence of disease among the 64,810 women in the study population? Compute the answer on your own before looking at the answer.

Answer

Positive and Negative Predictive Value of a Screening Test

Another way of thinking about the same results is from the patient's point of view after receiving the results of their screening test. Consider men who are getting the results of a blood test that is used as a screening test for prostate cancer (the PSA test [prostate specific antigen]). If the screening test was positive, what is the probability that they really have cancer; how worried should they be? Conversely, if a patient is told that his prostate cancer screening test was negative, how reassured should she be?

Positive Predictive Value

If a test subject has an abnormal screening test (i.e., it's positive), what is the probability that the subject really has the disease?  Now, we shift our focus to the first row in the contingency table we have been using.

Table - Positive Predicative Value Focuses on the First Row

	Diseased	Not Diseased	Total
Test Positive	132	983	1,115
Test Negative	45	63,650	63,695
Column Totals	177	64,633	64,810

There were 1,115 women with positive screening tests, but only 132 of these actually had the disease. Therefore, even if a subject's screening test was positive, the probability of actually having breast cancer was:

P(breast cancer | Screen+) = 132/1,115 = 11.8%

Interpretation: Among those who had a positive screening test, the probability of disease was 11.8%.

Negative Predictive Value

If a test subject has a negative screening test, what is the probability that she doesn't have breast cancer? In our example, there were 63,695 subjects whose screening test was negative, and 63,650 of these were, in fact, free of disease. Now we focus on the "test negative" row in the contingency table.

Table - Negative Predicative Value

 

	Diseased	Not Diseased	Total
Test Positive	132	983	1,115
Test Negative	45	63,650	63,695
Column Totals	177	64,633	64,810

 The negative predictive value is:

P(no breast cancer | Screen-) = 63,650/63,950=0.999, or 99.9%.

Interpretation: Among those who had a negative screening test, the probability of being disease-free was 99.9%.

Test Yourself

Problem #1

A group of women aged 40 are screened for breast cancer, and 1% of these women will have breast cancer. 80% of women with breast cancer will have a positive mammogram, while 9.6% of women without breast cancer will have a positive mammogram.

What is the probability that a woman with a positive mammogram has breast cancer? [Hint: Using the information provided, create a contingency table based on a total of 10,000 screened women, and use the percentages to derive the number of subjects in each category.]

Answer in Word file

Problem #2

Computed tomography (CT) scans are used as a screening test for lung cancer. The test involves X-ray at low doses of radiation to take detailed images of the lungs. The images are read by a radiologist and classified as needing further clinical examination or not. Diagnostic tests for lung cancer are based on sputum cytology and evaluated by a certified cytopathologist. 

A study was done in which 1200 persons aged 55 to 80 years who currently smoke or quit within the past 15 years and have at least a 30-pack-year history of cigarette smoking underwent both CT scans (as a possible screening test) and sputum cytology (as a diagnostic test). CT scans were read and classified as needing further clinical examination (screen positive) or not (screen negative). Analysis of sputum cytology indicated whether the patient actually had a lung cancer or not. The results are summarized in the table below.

	Lung Cancer	No Lung Cancer	Total
Screen positive	29	93	122
Screen Negative	4	1074	1078
Total	33	1167	1200

Compute the sensitivity, specificity, false negative fraction, false positive fraction, the positive predictive value, and the negative predictive value of the CT as a screening test for lung cancer.

Link to Answers in a Word file

Is this a good screening test for lung cancer?

US Preventative Services Task Force Recommendations

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