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.