From my stats homework:
- Approximately 1% of women aged 40-50 have breast cancer. A woman with breast cancer has a 90% chance of a positive test from a mammogram, while a woman without has a 10% chance of a false positive result. What is the probability a woman has breast cancer given that she just had a positive test?
So.. you are 45 and receive a “positive” mammogram. Not sure what that means, but maybe it means they’ve detected a dark patch, a lump you cannot feel. In real life they don’t tell you that you have cancer. They most likely schedule a repeat mammogram, and then maybe a biopsy. But anyway, given that “positive” mammogram, make a stab at how likely it is that you have breast cancer.
My professor (whom I really like, even though he can give the most amazingly obtuse examples) had rushed through a sample problem of the same sort:
- The incidence of schizophrenia in adults is about 2%. A proposed screening test is estimated to have at least 95% accuracy in making the positive diagnosis (sensitivity) and about 97% accuracy in declaring normality (specificity). Formally stated, P(normal | Ho) = .97, P(schizophrenia|Hi) > .95. So, let
- Ho = The case is normal
- Hi = The case is schizophrenic
- D = The test result (the data) is positive for schizophrenia
- What is the probability of the case being normal (ie the patient not having schizophrenia) if s/he tests positive? ie was is p(Ho|D)?
He did not give credit where credit was due; the example is from a famous (to statisticians) piece by Cohen (1994) called “The earth is round, p < .05.” That evening I sent him an email saying I had found the article and liked it… Two minutes later he emailed the entire class, saying that he had got the example from the piece, and asking us to read it 😉 He also thanked me in class for bringing it to his attention that he had not already assigned it. I may have to dig out my bullet proof vest.
The probability of someone who tests positive for schizophrenia being, in fact, normal, is .607. 60.7% of people testing positive for schizophrenia are not schizophrenic.
The probability of me finding a parking space on campus lots that allow anyone with a permit to park (and are closer to the psych department than parking in the street would be), and during normal hours (that is, between 9:00 and 16:00, with a brief window of more availability around noon) is much smaller. I don’t have formal data here, but I’d place it at about .10; 10% of the time I will find a parking space if I arrive after nine. And if I do find one, it will be at the TOP of the parking garage, which means a long slow line to get out of the lot when I leave, so I might as well park by the duck pond (~0.75 miles from Dale Hall). Before I spent $195 on a parking permit I was consistently parking 0.5-0.6 miles from Dale Hall (psych dept), on the street. What a waste of money. As soon as it gets cooler, I’m riding my bike.
The probability of having breast cancer, given a positive mammogram is .0833; only about eight out of 100 women who have a positive mammogram actually have cancer.
Arrive before 9:00 to increase your odds. Probably worth the effort. Although, I don't think it matters, as far as when you get your mammogram or your test for schizophrenia
Ha. Today I arrived at 8:30 and still had to park on the top of the parking garage. (less than half a mile to walk though)