Required for next week

Reading
"Does Exposure to Second-Hand Smoke Increase Lung Cancer Risk?", A.J. Gross and "What Evidence is Needed to Link Lung Cancer and Second-Hand Smoke?", H.E. Rockette, Chance 6 (4), 11--18.
Your task
Come prepared to discuss these articles, and to ask questions about the parts you didn't understand.

Notes on short project 4

Short project 4 hasn't been assigned yet, but it will require submission of a proposal for your final essay. My idea for this essay is that it will give an in-depth discussion of one of the topics we've come across in class. That could mean reviewing a book, such as The Bell Curve, summarizing a series of articles on a topic, such as DNA fingerprinting, following up on a newspaper article of interest, or a Chance article of interest,

Reading List on DNA fingerprinting

(See also the reference last week to an article by Balding and Donnelly.)

Some more applications of Bayes' theorem

To spousal abuse and murder

This is taken from the article in Chance that was handed out on October 31, attached to the Paulos article. (Propensity to Abuse-- Propensity to Murder?, J.F. Merz and J.P. Caulkins, Chance 8 (2), p.14.) Although they computed the odds ratio version, I find it easier to work with the ordinary version that I gave in last week's handout. In the formula at the top of p.3, replace "G" by "current or former mate murdered the victim" and "DNA match" by "a history of known abuse". To save space, I'll use "M" and "A" for these two cases. (If you hate notation, write out the formula in long-hand.) 
			prob(A|M) prob(M)
prob(M|A)=     -----------------------------------------------
		prob(A|M)prob(M) +  prob(A| not M) prob(not M)
The article uses the following estimates: 
 prob(A|M) 	 = 0.5 
 prob(M) 	 = 0.29 
 prob(not M)} 	= 0.71 
 prob(A|not M)} = 0.05
My Hewlett-Packard gives the result on the left hand side as 0.8032; very close to 80%. If prob(A|M) is bumped up to 3/4, then I get 0.8595. Recall that defense attorney Dershowitz referred to a probability of 1 in 1000 (the risk that a women in an abusive relationship has of being murdered each year). Where does this probability appear in the calculation above?

Random screening for HIV-prevalence

In any kind of screening for disease, the probability that we're interested in is prob(patient~has~antibodies~to~HIV | test~is~positive). Using Bayes' theorem again, we'll need the following pieces: prob(test~is~positive | patient~has~antibodies~to~HIV) and prob(patient~has~antibodies~to~HIV). We'll also need prob(test~is~positive | patient~doesn't~have~antibodies~to~HIV) this is also called the ``false-positive" rate of the test.

I got the following figures from the Chance News WWW page (URL: http://www.geom.umn.edu/docs/snell/chance/course/topics/aids.html) which includes the text of a New York Times editorial piece. The editorial took the figures from an article in the New England Journal of Medicine. The US Army does routine testing of recruits, and has a very careful testing procedure, with an estimated false positive rate of only 1 in 20,000. However, in the general population, the incidence of HIV is also rare: estimated to be 1 in 10,000. Assuming that the army's test misses NO true positives, we get 

prob(patient~has~antibodies~to~HIV|test~is~positive) = 0.667
or about 2/3. That leaves a 1 in 3 chance that the patient does not have antidbodies to HIV, after testing positive.

If the false positive rate is increased from 1 in 20,000 to 1 in 10,000, there is a 1 in 2 chance that the patient does not have antidbodies to HIV, after testing positive. If the false positive rate is as large as 1 in 1,000, which it may well be expected to be if widespread screening were instituted, the same formula gives 

prob(patient~has~antibodies~to~HIV|test~is~positive) = 0.0909,
i.e. there is about a 90% chance that a randomly chosen patient testing positive does NOT have antibodies to HIV. For every 10 people testing positive, 9 of them will NOT have HIV.

What's driving this calculation is the small incidence of HIV antibodies in the general population (the 1 in 10,000 above). Screening in high risk groups would give much different numbers, because this probability would be much larger.

Technical note to follow

In the ``hot hand'' articles, there are several technical terms used: null hypothesis, alternative hypothesis, significance level, power. These are all related to a particular type of statistical inference called hypothesis testing, or sometimes significance testing. More on this next week.

In the Globe and Mail this week


Some titles of Chance articles