Breast Cancer Screening

Background

A test for a medical condition is often classified according to sensitivity and specificity.

Sensitivity (also called the true positive rate) measures the percentage of sick people who are correcly identified as having the condition. Specificity (also called the true negative rate) measures the percentage of healthy people who are correcly identified as not having the condition.

Suppose that a mammogram test for breast cancer using is 80% sensitive and 90% specific. That is, the test will produce 80% true positive results for patients with breast cancer and 90% true negative results for patients without breast cancer. Suppose that 1% of people have breast cancer. What is the probability that a randomly selected patient with a positive mammogram has breast cancer?

Bayes' Theorem
  • A and B are events and P(B) ≠ 0
  • P(A | B) is a conditional probability: the likelihood of event A occurring given that B is true
  • P(B | A) is also a conditional probability: the likelihood of event B occurring given that A is true
  • P(A) and P(B) are the probabilities of observing A and B independently of each other; this is known as the marginal probability

The likelihood that a patient with a positive mammogram has cancer is approximately 8%.

This chart represents a sample of 2,000 women, aged 40, who participated in routine screening for breast cancer. Each block represents 1 woman.

Scroll down to find out more.

Diagnosis

The color indicates whether a woman actually had breast cancer.

1% (20 in 2,000) had breast cancer, and 99% (1,980 in 2,000) did not.

Diagnosis Grouped

The data is now grouped by whether each woman actually had breast cancer.

Diagnosis by Test Result

The color now indicates whether a mammogram showed a positive or negative result.

80% of women with breast cancer had positive mammograms. 90% of women without breast cancer had negative mammograms. In other words, a mammogram is 80% sensitive (true positives), and 90% specific (true negatives).

Test Result Grouped

The data is now grouped by mammogram result.

Diagnosis Highlighted

Positive mammograms are highlighted. For a mammogram which is 80% specific (true positives) and 90% selective (true negatives), with a 1% occurrence of breast cancer, in 2,000 women there will be 16 true positives and 190 false positives. There will also be 4 false negatives and 1,790 true negatives.

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