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. A Type I error in this context is diagnosing a patient with cancer when they don't have it, leading to unnecessary stress and treatments. c. A Type II error is failing to diagnose a patient who has cancer leading to late treatment. d. The statement from the article illustrates the trade-off between the probabilities of Type I and Type II errors i.e. decreasing false-positives (Type I errors) may increase missed diagnoses (Type II errors).

Step-by-step solution

Identification of Type of Error

A false-positive error means thinking that cancer is present when in fact it is not. This corresponds to rejecting the null hypothesis of 'no cancer is present' when it is actually true, hence a false-positive is a Type I error.

Describe Type I Error and Its Consequences

A Type I error in this context is when the screening test indicates that a patient has cancer (rejecting the null hypothesis), when in fact, they do not have cancer (i.e., the null hypothesis is true). The consequence of a Type I error in this instance would be the unnecessary stress and possibly additional invasive tests and treatments for a person who does not have cancer.

Describe Type II Error and Its Consequences

A Type II error, on the other hand, would occur if the test indicates that a patient does not have cancer (not rejecting the null hypothesis), when in reality, they do have cancer (i.e., the null hypothesis is false). The consequence of a Type II error could be far more dangerous, as it could potentially lead to a late diagnosis, resulting in a delay in necessary treatments.

Discussing Probability of Error Types Trade-Off

The statement from the article suggests that if doctors become less aggressive in following up on suspicious tests, the frequency of Type I errors (false-positives) would decrease; however, the frequency of Type II errors (missed cancer diagnoses) would increase. This statement is illustrating the trade-off between the probability of Type I and Type II errors; as steps are taken to decrease one type of error, it may lead to an increase in the other type of error.

Key concepts

Type I Error

In the context of hypothesis testing, a Type I error occurs when the null hypothesis is rejected even though it is true. For medical tests, like breast cancer screenings, a Type I error is also known as a false-positive.
This means the test result suggests a problem (cancer) that doesn't actually exist. In this scenario, the patient is told they might have cancer, but in reality, they do not.
Such an error can cause unnecessary fear and anxiety, leading to further testing, which can often be invasive or stressful.

• Patients might undergo unnecessary treatment.
• This could lead to avoidable expenses and physical discomfort.

Understanding the impacts of a Type I error is crucial in developing testing methods and communicating results with patients.

Type II Error

A Type II error happens when the test fails to reject the null hypothesis when the alternative hypothesis is true. In simpler terms, a Type II error is a false-negative.
In the breast cancer screening example, this error means the test fails to detect cancer when it is indeed present in the patient.
Missing a diagnosis could have severe consequences because the patient does not receive timely treatment, potentially worsening their health condition.

• Cancer remains untreated, which can affect prognosis.
• There may be a delay in the patient receiving necessary care or follow-up.

The importance of being aware of the potential for a Type II error is clear: it underscores the need for accuracy and reliability in medical testing.

False Positives

False positives are test results that indicate the presence of a condition, such as cancer, that is not actually present. In statistical terms, false positives are considered Type I errors.
They are particularly concerning in medical diagnostics as they can lead to emotional distress and potentially unnecessary medical procedures.
False positives speak to the balance that must be found between sensitivity (the ability to detect true positives) and specificity (the ability to avoid false positives) in a test.

• Sensitive tests maximize true positive rates but might increase false positives.
• False positives can drain resources if many healthy individuals undergo further unnecessary testing.

Healthcare providers strive to minimize false positives through refined testing strategies and communication with patients.

Trade-Off in Error Types

The relationship between Type I and Type II errors involves a delicate balance. Reducing the likelihood of one error type often increases the likelihood of another.
This is captured in the medical context by the statement: less aggressive follow-ups might reduce false positives but increase missed diagnoses.

• Investing in highly sensitive tests can increase Type I errors.
• Alternatively, highly specific tests may increase Type II errors.

Understanding this trade-off helps in designing tests that are not only accurate but also aligned with the health priorities and concerns of patients and healthcare systems.