A paper describes a study of the use of MRI (Magnetic Resonance Imaging) exams i

Question

A paper 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 980 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 131 of those exams Indicated possible cancer. After undergoing biopsies. It was determined that 40 of the 131 did in fact have cancer in the other breast, whereas 91 did not. The women were all followed for one year, and four 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 cancer-free. Is this a Type 1 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. A Type II error would be coming to the conclusion that the woman cancer in the other breast when in fact she cancer in the other breast. Use the information in the given table to approximate the probability of this type of error.

Explanation / Answer

(a)  a type I error is the incorrect rejection of a true null hypothesis (a "false positive")

P("False Positive") = 91 / 980 = 0.0928

b.) A type II error would be coming to a conclusion that the woman does not have cancer in the other breast when in fact she has cancer in the other breast. This is also called "False Negative"

P("False Negative") = 4/980 = 0.00408  

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