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 false-positives 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. What aspect of the relationship between the probability of Type I and Type II errors is being described by 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?

Short answer

a. A false positive is a Type I error. b. In this context, a Type I error is falsely diagnosing a patient with cancer, leading to possible stress, unnecessary testing, or harmful treatment. c. A Type II error would be falsely assuming a patient is cancer-free, causing delayed treatment and worse health outcomes. d. The statement in the article describes a trade-off relationship between Type I and Type II errors.

Step-by-step solution

Identify Type I and Type II Errors

A false positive, or thinking that cancer is present when in fact it is not, is a Type I error. A Type I error occurs when the null hypothesis (in this case \(H_{0}:\) no cancer is present) is rejected when it is true. A Type II error is when the null hypothesis is accepted when it is false, or believing there is no cancer when there actually is.

Describe Type I Error and its Consequences

In this situation, a Type I error would be diagnosing a patient with cancer when they do not have it. The consequences of making a Type I error could be unnecessary stress, anxiety, further invasive testing, and potential physical harm from unnecessary treatment for the patients.

Describe Type II Error and its Consequences

A Type II error, in this context, would be telling a patient that they do not have cancer when, in fact, they do. The consequences of making a Type II error would be that the patient's health would continue to deteriorate as the cancer was left untreated, reducing their chance of survival and possibility of a full recovery.

Analyze The Relationship Between Type I and Type II Errors

The statement in the article implies a trade-off between Type I and Type II errors in radiological diagnosis. If radiologists were less aggressive in following up on suspicious tests, they would make fewer Type I errors (false positives), but the rate of Type II errors (missed diagnoses of real cancers) would increase.