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Hello! My name is Rosemarie, and I am the new writer for this Mama, Ph.D. column. Before I begin, I would like to thank Anjalee for her work in this column in the last few months. She will certainly be a tough act to follow!
As you can read in Mama, Ph.D., I am a math professor, although my Ph.D. is in Economics. You can expect, therefore, that my entries, many about economics, will have a mathematical flavor to them, and I am hinting at this by calling myself the “Math Geek Mom.” In this column you can expect to read about how math and economics inform current topics, with a large smattering of entries dealing with labor markets, especially labor markets as seen by women and mothers in general, and women professors who are mothers in particular.
Although I will generally be writing about topics that involve math as it relates to economics, I want my first submission to be a little different. Since October is Breast Cancer Awareness Month, I want to draw from a topic that I discuss in my Quantitative Reasoning course that explores the statistics behind the detection of breast cancer. In my class we discuss what it means to have a mammogram come back positive, in need of a biopsy to determine if one has cancer. We come to the surprising conclusion that a positive mammogram, while something to follow up on, does not necessarily mean that one has cancer. If you are interested in learning more about this, you can see a detailed analysis in the text we use, Quantitative Reasoning by Bennett and Briggs published by Addison-Wesley.
Mammograms, although a wonderful tool, are not perfect in the detection of breast cancer. They have become more accurate in recent years, but even so, there are still opportunities for mistakes. There are some times that a mammogram will fail to detect cancer when it is there, and other times that it will indicate cancer when, in fact, one is cancer-free. This allows for false positive and false negative results.
To understand a result from a mammogram, one should understand conditional probability. While probability in general asks what the chance is of some event occurring, conditional probability asks what the chance is of an event occurring, given that we know that something else has occurred. My favorite example of conditional probability has to do with being invited to a dinner party that Jessica Fletcher is planning to attend. As a character in an old “Murder She Wrote” movie asks, “Jessica Fletcher shows up and someone gets killed. What is the chance of that?” Going to a party that Jessica Fletcher is expected to attend should give one reason to pause, but would not necessarily keep one from going to the party. Similarly, a positive mammogram gives us information that should be followed up on, but does not necessarily mean that one has cancer.
Some of the positive results that mammograms are detecting are from cancer, but some are false positives. Since mammograms are still not perfect, the percent from false positives is greater than zero. And since so few people actually have cancer, the percent from false positives can be a large portion of all positives, which includes true and false positives. Follow-up biopsies are therefore very important when a mammogram comes back positive, so it may be determined whether a patient truly has cancer. Negative mammograms, since so few people actually have cancer, are much more able to be trusted. A fear of a positive mammogram should not keep you from taking advantage of this marvel of modern medicine.
What statistics do not tell us about breast cancer is just as important as what they do say. Statistics are based on averages, and do not reflect the variation among individual people. I know a woman who was diagnosed with an aggressive form of breast cancer in 1997. At the time, the statistics suggested that she had about two years left to live. Well, whoever decided that forgot to tell her. She is now 68, still works full time and is the primary caregiver to her mother, a 97 year old Alzheimer’s patient. The statistics in this case were irrelevant to this woman’s recovery. I propose that any woman who can survive graduate school would also have a comparative advantage in beating such a disease.
How does this relate to us, who have families and students who depend on us? It is vitally important that we take care of ourselves so that we can be there for those who need us. Our campus, a woman’s college, has been recently struck hard by breast cancer, probably because of the large number of women faculty working here. One of those faculty members sent us an e-mail from home as she was recovering, reminding us all to remember the mammograms and doctor’s visits. I hope that the information I have shared today will help take some of the fear out of measures that might lead to early detection and, in the long run, the cure we all desire.
