Question

The paper "MRI Evaluation of the Contralateral Breast in Women with Recently Diagnosed Breast Cancer" (New England Journal of Medicine \([2007]: 1295-1303)\) describes a study of the use of MRI (Magnetic Resonance Imaging) exams in the diagnosis of breast cancer. The purpose of the study was to determine if MRI exams do a better job than mammograms of determining if women who have recently been diagnosed with cancer in one breast have cancer in the other breast. The study participants were 969 women who had been diagnosed with cancer in one breast and for whom a mammogram did not detect cancer in the other breast. These women had an MRI exam of the other breast, and 121 of those exams indicated possible cancer. After undergoing biopsies, it was determined that 30 of the 121 did in fact have cancer in the other breast, whereas 91 did not. The women were all followed for one year, and three of the women for whom the MRI exam did not indicate cancer in the other breast were subsequently diagnosed with cancer that the MRI did not detect. The accompanying table summarizes this information. Suppose that for women recently diagnosed with cancer in only one breast, the MRI is used to decide between the two "hypotheses" \(H_{0}\) : woman has cancer in the other breast \(H_{a}:\) woman does not have cancer in the other breast (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) a. One possible error would be deciding that a woman who does have cancer in the other breast is cancerfree. Is this a Type I or a Type II error? Use the information in the table to approximate the probability of this type of error. b. There is a second type of error that is possible in this setting. Describe this error and use the information in the given table to approximate the probability of this type of error.

Short answer

a. This represents a Type II error. The approximate probability of this error is 0.752. \n b. The second possible error, which is a Type I error, could occur if a woman is diagnosed as having cancer in the other breast when she actually does not. The approximate probability of this error is 0.0035.

Step-by-step solution

Identify Type I Error and Approximate its Probability

A Type I error would occur if we decide that a woman does not have cancer in the other breast when in fact she does. In the given scenario, we have 121 MRI exams indicating possible cancer but only 30 were confirmed to have cancer after biopsy. Therefore, the number of Type I errors is 91 (121 - 30). The probability of a Type I error can be approximated as the number of Type I errors divided by the total number of MRI exams indicating possible cancer, i.e. \(\frac{91}{121} = 0.752\).

Identify Type II Error and Approximate its Probability

A Type II error would occur if we decide that a woman has cancer in the other breast when in fact she does not. In the given scenario, three women for whom the MRI exam did not indicate cancer were subsequently diagnosed with cancer. The total number of women for whom the MRI did not indicate cancer can be calculated as 969 (total participants) - 121 (MRI indicated possible cancer) = 848. Therefore, the probability of a Type II error can be approximated as the number of Type II errors divided by the total number of women for whom the MRI did not indicate cancer, i.e. \(\frac{3}{848} = 0.0035\).

Interpretation of the Result

Based on the computed probabilities, we can say that the MRI exam has a higher chance of making a Type I error (falsely diagnosing a woman as cancer-free when she actually has cancer) as compared to a Type II error (falsely diagnosing a woman as having cancer when she is actually cancer-free). This has important implications in the medical field as both types of errors can have serious consequences.

Key concepts

Hypothesis Testing

Hypothesis testing is a fundamental process in statistics used to determine the likelihood that a given hypothesis about a dataset is true. Think of it like a courtroom trial, where the null hypothesis, denoted as \(H_{0}\), represents the assumption of innocence until proven guilty. In the context of medical diagnostics, such as MRI exams for cancer, \(H_{0}\) might state that there is no cancer present in the patient's other breast. Conversely, the alternative hypothesis, \(H_{a}\), suggests that cancer is present.

In practice, hypothesis testing involves calculating a test statistic, which is then compared against a threshold value to decide whether to reject the null hypothesis in favor of the alternative. If we reject \(H_{0}\) when it’s actually true, we've made a Type I error. If we fail to reject \(H_{0}\) when \(H_{a}\) is true, it's a Type II error. The study in question utilizes MRI exams to execute this testing process, with the goal of correctly identifying the presence or absence of cancer in the opposite breast of women already diagnosed with breast cancer in one breast.

MRI exams in cancer diagnosis

In the world of oncology, MRI exams are hailed for their non-invasive nature and detailed imaging capabilities. Unlike mammograms, which use low-dose X-rays, MRIs employ strong magnetic fields and radio waves to produce detailed images of the inside of the body, in this case, the breast tissue. This high-resolution imaging is particularly useful in detecting abnormalities that might indicate cancer.

In the mentioned study, MRIs were used to check for cancer in the contralateral breast of women who had already been diagnosed with cancer in one breast. The ability of MRI to detect cancer often not seen on mammograms can lead to earlier and potentially life-saving interventions. However, the precision of MRI can also raise challenges related to false positives (Type I errors) and false negatives (Type II errors), which influence the treatment paths and emotional well-being of patients.

Probability of Diagnostic Errors

Diagnostic errors in medical exams, such as those highlighted by the MRI exam study, can be classified into two main types: Type I and Type II errors. The probability of each error type is a critical factor in evaluating the reliability of a diagnostic test.

A Type I error in this context refers to a false positive diagnosis – when an MRI indicates possible breast cancer that is not present after a biopsy. Its probability was found to be roughly \(0.752\), meaning around 75% of the positive indications by MRI were false. Meanwhile, a Type II error, or a false negative, occurs when the test fails to identify cancer that is actually present. With a computed probability of approximately \(0.0035\) or 0.35%, Type II errors were significantly less common in this study. These probabilities serve as a basis for assessing the diagnostic accuracy of MRI exams and help inform healthcare professionals about the risks involved in the decision-making process.

Statistical Error Interpretation

Interpreting statistical errors, particularly in a medical diagnosis setting, involves understanding the implications of each error type on patient care. A Type I error could potentially lead to unnecessary procedures, anxiety, and increased healthcare costs. Although less prominent in the given MRI study, Type I errors must be minimized to avoid overtreatment. On the other hand, a Type II error might mean missed opportunities for early cancer treatment, which could be life-threatening.

Understanding these errors' statistical interpretation helps clinicians weigh the benefits and risks of diagnostic methods. In the example provided, a higher probability of Type I errors suggests that caution should be exercised regarding positive MRI findings, potentially involving additional tests or screenings to confirm the presence of cancer. The low probability of Type II errors indicates MRI's effectiveness in identifying most actual cancer cases, but even a small risk requires careful patient monitoring. Accurately interpreting these statistics is essential for optimizing patient outcomes and ensuring efficient use of medical resources.