Dear Colleagues,
Further to my recent email, I've had 4 replies so far, including 2 suggestions
that this might be Simpson's paradox. One respondent quoted "Pagano & Gauvreau
Principles of biostatistics $16.1 'Simpson's paradox occurs when either the
magnitude or the direction of the relationship between two variables is
influenced by the presence of a third factor.' " I'm aware of Simpson's
paradox, but I don't believe this is the explanation here. The issue relates to
how different effect size measures behave - the RR behaves as one would expect,
but the OR does not.
Let OR1 and OR2 denote the ORs for the separate stages, both assumed >1, and
let OR3 denote the final odds ratio (in our example, relating to those having
unprotected sex as a proportion of total respondents). Let
Q1=OR3/min(OR1,OR2)
Q2=OR3/max(OR1,OR2)
Q3=OR3/(OR1*OR2)
Then empirically:
Q1 ranges on (1,infinity) - so, reassuringly, Q3 can't be lower than *both* Q1
and Q2 - but as our example showed, it can be intermediate between them, and
it's easy to produce an example with Q1 arbitrarily close to 1.
Q2 ranges on (0, infinity) - if both OR1 and OR2 are large, OR3 *may* be much
larger still - but doesn't have to be.
Q3 ranges on (0,1) - so OR3 will always "look" less extreme than one might
"expect" given OR1 and OR2, and with the corresponding relationship for relative
risks RR3=RR1*RR2 in the back of ones mind.
It's straightforward to prove algebraically that min(OR1,OR2) < OR3 < OR1*OR2
in non-boundary cases.
Any thoughts on this?
Robert G. Newcombe PhD CStat FFPH
Reader in Medical Statistics
Wales College of Medicine
Cardiff University
Heath Park
Cardiff CF14 4XN
Phone 029 2074 2329
Fax 029 2074 2898
http://www.cardiff.ac.uk/medicine/epidemiology_statistics/research/statistics/newcombe.htm
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