Mammograms are x-ray images of the breast use to detect breast cancer. A mammogram typically identify correctly 85% of cancer cases and 95% of cases without cancer. The Canadian Cancer Society (CCS) estimates that 3.36% of women in their 60s gets breast cancer. If a woman in her 60s get a positive mammogram result
(a) What is the probability of true-positive test?
(b) What is the probability of false-positive test?
(c) What is the probability that the woman really has breast cancer given the positive test?
a) The probability of true-positive test is the probability that a woman who has breast cancer receives a positive mammogram result. This can be calculated using Bayes' theorem:
P(True-positive test) = P(Positive result | Breast cancer) * P(Breast cancer)
P(Positive result | Breast cancer) is the sensitivity of the mammogram, which is 85% or 0.85. P(Breast cancer) is the prevalence of breast cancer in women in their 60s, which is 3.36% or 0.0336. Therefore:
P(True-positive test) = 0.85 * 0.0336 = 0.02856 or approximately 2.86%
b) The probability of false-positive test is the probability that a woman who does not have breast cancer receives a positive mammogram result. This can also be calculated using Bayes' theorem:
P(False-positive test) = P(Positive result | No breast cancer) * P(No breast cancer)
P(Positive result | No breast cancer) is the specificity of the mammogram, which is 95% or 0.95. P(No breast cancer) is the complement of the prevalence of breast cancer, which is 1 - 0.0336 = 0.9664. Therefore:
P(False-positive test) = 0.95 * 0.9664 = 0.91728 or approximately 91.73%
c) The probability that the woman really has breast cancer given the positive test is the positive predictive value (PPV) of the mammogram. This can be calculated using the formula:
PPV = P(Breast cancer | Positive result) = P(Positive result | Breast cancer) * P(Breast cancer) / P(Positive result)
P(Positive result) is the probability of a positive result, which is the sum of the probabilities of true-positive and false-positive tests:
P(Positive result) = P(True-positive test) + P(False-positive test) = 0.02856 + 0.91728 = 0.94584 or approximately 94.58%
Therefore:
PPV = 0.85 * 0.0336 / 0.94584 = 0.02999 or approximately 3.00%
This means that if a woman in her 60s gets a positive mammogram result, there is a 3.00% chance that she really has breast cancer.
