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\begin{flushright}
SCI199Y: November 14, 1995
\end{flushright}

\bigskip
\noindent{\bf Required for next week}
\begin{description}
\item[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, {\it Chance} {\bf 6} (4), 11--18.
\item[Your task] Come prepared to discuss these articles, and to 
ask questions about the parts you didn't understand.
\end{description}
\bigskip

\noindent{\bf 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 one or more books, 
summarizing a series of articles on a topic, 
following up on an article of interest, 

\bigskip
\noindent{\bf Reading List on DNA fingerprinting}
\begin{itemize}
\item Berry, D.A. (1990) DNA Fingerprinting:  What does it prove?
{\it Chance} {\bf 3} (3), 15--25.
\item Cohen, J.E. (1990) DNA Fingerprinting: What (really) are the odds?
{\it Chance} {\bf 3} (3), 26--32.
\item National Research Council (NRC) (1992).  DNA typing: Statistical
basis for interpretation.  In {\it DNA Technology
in Forensic Science} 74--96.  National Academy Press, Washington DC.
(This is available on the WWW: the URL is
{\tt http://www.geom.umn.edu/docs/snell/chance/teaching\_aids/DNAtyping/}).
\item Roeder, K. (1994) DNA fingerprinting: a review of the controversy.
(with discussion) {\it Statistical Science} {\bf 9}, 222--278.
\end{itemize}
(See also the reference last week to an article by Balding and Donnelly.)

\bigskip
\noindent{\bf Some more applications of Bayes' theorem}

\noindent{\it To spousal abuse and murder}

This is taken from the article in {\it Chance} that was handed out on
October 31, attached to the Paulos article.\footnote
{Propensity to Abuse-- Propensity to Murder?, J.F. Merz and J.P. Caulkins,
{\it Chance} {\bf 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.)
$$
{\rm prob}(M|A)=\frac{{\rm prob}(A|M){\rm prob}(M)}
{{\rm prob}(A|M){\rm prob}(M) + {\rm prob}(A|{\rm not}~M){\rm prob}({\rm not}~M)}
$$
The article uses the following estimates:
\begin{eqnarray*}
{\rm prob(A|M)} & = 0.5 \\
{\rm prob(M)} & = 0.29 \\
{\rm prob(not M)} &= 0.71 \\
{\rm prob(A|not M)} &= 0.05 \\
\end{eqnarray*}
My Hewlett-Packard gives the result on the left hand side as 0.8032; very
close to 80\%.  If ${\rm prob(A|M)}$ 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?

\bigskip
\noindent{\it Random screening for HIV-prevalence}

In any kind of screening for disease, the probability that we're
interested in is
$ {\rm prob(patient~has~antibodies~to~HIV | test~is~positive)} $.
Using Bayes' theorem again, we'll need the following pieces:
$ {\rm prob(test~is~positive | patient~has~antibodies~to~HIV)} $
and
$ {\rm prob(patient~has~antibodies~to~HIV)}  $.\hfill
We'll also need
$ {\rm 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\hfill
{\tt (URL: http://www.geom.umn.edu/docs/snell/chance/course/topics/aids.html)}
which includes the text of a {\it New York Times} editorial piece.
The editorial took the figures from an article in the {\it 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
$$
{\rm prob(patient~has~antibodies~to~HIV|test~is~positive)} = 0.667
$$
or about 2/3.  That leaves a 1 in 3 chance that the patient does not
have antibodies 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 antibodies 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
$$
{\rm prob(patient~has~antibodies~to~HIV|test~is~positive)} = 0.0909,
$$
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.

\bigskip
\noindent{\bf 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.

\bigskip
\noindent{\bf In the Globe and Mail this week}
\begin{itemize}
\item ``Switch prevents rampaging cells'' November 10, A7
Toronto research found that a deficiency in CTLA-4 leads to symptoms of
autoimmune diseases in laboratory mice.  The discovery is expected
to lead to more information about the treatment
of immune diseases.  The full report appears in this week's {\it Science}.
\item ``Hope rises for vaccine to prevent AIDS infection'' November 10, A7
Also taken from an article in this week's {\it Science}.  At the last 
world AIDS conference, though, there seemed to be general agreement that
a vaccine for AIDS was nowhere in sight.
\item ``From monster home to white elephant'' November 10, A7
A report released by Statscan on November 9 projects increasing demand
for small homes and decreasing demand for large homes, over the next 20 years.
\item ``U.S. swimmer's lax penalty raises ire'' November 13, D5
A young (age 14) U.S. swimmer who tested positive for steroids was given
2 years probation; a penalty much milder than usually taken by national
sports organizations.  This reminded me about a recent article in
{\it Applied Statistics}.
\item ``Eateries live hand-to-mouth'' November 13, B6.
The graphic caught my eye, but the story that might be of more interest
is that this is the first of a series of articles
examining industry sectors using data provided by Dun \& Bradstreet, Canada.
\item ``U of T again dominates in ranking'' November 13, A6.
The dreaded MacLeans report is out again.  How do they come up with these
numbers anyway?
\item ``Entrepreneurs now outnumber civil servants'' November 13, A6.
This reports data from the Labor Force Survey of Statscan.  So what is
the labour force survey, and how does one get hold of a copy?
\end{itemize}
\vfill\eject
\noindent{\bf Some titles of {\it Chance} articles}
\begin{itemize}
\item Wringing {\it The Bell Curve}: a cautionary tale about
the relationships among race, genes and IQ.
\item A paradox in the ranking of figure skaters.
\item Small cars, big cars: what is the safety difference?
\item Racetrack betting: do bettors understand the odds?
\item Baseball: pitching no-hitters.
\item Baseball's earned run average: what kind of an average is it?
\item The baby boom generation and how they grew.
\item Resizing triathlons for fairness
\item Can TQM improve athletic performance?
\item The best NFL field goal kickers: are they lucky or good?
\item What happened to HIV transmission among drug injectors in New Haven?
\item Answering questions about baseball using statistics.
\item The state of state election polls.
\item How much more efficiently can humans run than swim?
\item Statistical evidence of cheating on multiple choice tests.
\item Should pregnant women move?  Linking risks for birth defects with
proximity to toxic waste sites.
\item Scientific inferences and environmental health problems.
\end{itemize}
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