A test has been developed to detect a certain type of cancer and suppose that th

Question

A test has been developed to detect a certain type of cancer and suppose that the test correctly identifies 95% of the cancerous people as cancerous and correctly identifies 70% of the noncancerous people as noncancerous. Suppose that 0. 2% of population has this type of cancer. i.e. The sensitivity of the test is 95%, the specificity of the test is 70%, and the prevalence of this type of cancer in the population is 0.2%. (a) Given that the test indicates that a person has cancer what is the conditional probability that the person actually has cancer? i.e. What is the predictive value positive? (b) Given the test indicates that a person does not have cancer what is the conditional probability that the person actually does not have cancer? i.e. What is the predictive value negative?

Explanation / Answer

Given, sensitivity = 0.95, specificity = 0.70, prevalence = 0.002
a) positive predictive value, PPV = [sensitivity*prevalence]/[(sensitivity*prevalence)+((1-specificity)*(1-prevalence))]
PPV = [0.95*0.002]/[(0.95*0.002)+((1-0.70)*(1-0.002))] = 0.0019/0.3013 = 0.006306 = 0.631%

b) Negative predictive value,
NPV = [specificity*(1-prevalence)]/[((1-sensitivity)*prevalence)+(specificity*(1-prevalence))]

NPV = [0.7*(1-0.002)]/[((1-0.95)*0.002)+(0.7*(1-0.002))] = 0.6986/0.6987 = 0.9999 = 99.99%

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