Question

Researchers at the University of Washington and Harvard University analyzed records of breast cancer screening and diagnostic evaluations ("Mammogram Cancer Scares More Frequent Than Thought," USA Today, April 16,1998\()\). Discussing the benefits and downsides of the screening process, the article states that although the rate of falsepositives is higher than previously thought, if radiologists were less aggressive in following up on suspicious tests, the rate of false-positives would fall, but the rate of missed cancers would rise. Suppose that such a screening test is used to decide between a null hypothesis of \(H_{0}:\) no cancer is present and an alternative hypothesis of \(H_{a}:\) cancer is present. (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) a. Would a false-positive (thinking that cancer is present when in fact it is not) be a Type I error or a Type II error? b. Describe a Type I error in the context of this problem, and discuss the consequences of making a Type I error. c. Describe a Type II error in the context of this problem, and discuss the consequences of making a Type II error. d. Recall the statement in the article that if radiologists were less aggressive in following up on suspicious tests, the rate of false-positives would fall but the rate of missed cancers would rise. What aspect of the relationship between the probability of a Type I error and the probability of a Type II error is being described here?

Short answer

a. A false positive would be a Type I error. b. In this context, a Type I error means diagnosing a patient with cancer when they don't have it. The consequences include unnecessary treatment and stress for the patient. c. A Type II error means failing to diagnose a patient with cancer when they do have it, leading to delayed treatment and worse patient outcomes. d. The statement in the article describes the trade-off between Type I and Type II errors: reducing one type of error usually increases the other.

Step-by-step solution

Understand Types of Errors

Type I and Type II errors are common concepts in statistical testing. In the context of this exercise, a Type I error refers to a false positive where the test wrongly identifies a person as having cancer (rejecting the null hypothesis that there is no cancer when it's actually true). A Type II error refers to a false negative where the test wrongly identifies a person as not having cancer (accepting the null hypothesis when it's actually false). Understanding these definitions is the first step towards solving this problem.

Identify Type of Error for False Positive

A false positive (thinking that cancer is present when in fact it is not) would be a Type I error. This is because the null hypothesis (\(H_{0}:\) no cancer is present) is rejected when it shouldn’t be.

Describe Type I Error and Its Consequences

A Type I error in this context would be diagnosing a patient with cancer when they do not actually have it. The consequences could include unnecessary stress for the patient and potential harm from unnecessary treatment.

Describe Type II Error and Its Consequences

A Type II error in this context would be failing to diagnose a patient with cancer when they do actually have it. This could lead to delayed treatment and worse health outcomes for the patient.

Link Article Statement to Type I and II Errors

The statement emphasizes the inherent trade-off between Type I and Type II errors. If radiologists choose to be less aggressive in following up on suspicious tests (thereby reducing the Type I error rate), the rate of missed cancers (i.e., the Type II error rate) would increase, and vice versa. The balance between these two types of errors is key in considering optimal testing strategies.