Stat 430 Handout					10/15/01

DISCUSSION  OF  THE  MEANING  OF  RELATIVE  RISKS 
VERSUS  ODDS-RATIOS.

Consider the setting of Problem 3 on HW 4, in which several students 
told me, with reference to the Cody & Smith book, that Odds-ratios 
were not appropriate to look at but Risk-ratios were. Actually, both 
of these statistics are perfectly meaningful (estimates of) ratios of cell 
probabilities for the two-way table in a particular study. The key 
issue, which neither the book nor I explained sufficiently well, was 
whether one or the other ratio made sense to generalize to the whole 
population, andthis in turn depends upon how the individuals in the 
study were sampled, i.e., were chosen for inclusion.

Here is a fuller explanation. Suppose that we are talking about a rare 
effect in the population, which can be seen either in people taking a 
particular drug or, with still smaller frequency in people not taking the 
drug. Suppose that in the general population (i.e., with people 
sampled completely at random), the typical frequencies are
            Effect        OK        Totals
   Drug       80         3920        4000
No-Drug       20         5980        6000
 Totals      100         9900       10000

Thus. 40% of people take the drug, but Drug-takers suffer 
the Effect 0.2% of the time , while others show the effect only
0.33% of the time. So in the general population, the relative 
risk for showing the Effect is  (80/4000)/(20/6000  =  6 ,
and because we are talking about very small probabilities, the 
odds-ratio of showing th Effect for Drug-takers versus Non-
Drug-takers is approximately the same. 

Now suppose that we conducted a case-control study by 
sampling 100 people at random from those known to show the 
Effect (say, at a certain hospital), and 100 people at random from 
those not showing the Effect. The data might look like this:
           Effect       OK         Totals
   Drug      80         41          121
No-Drug      20         59           79
 Totals     100        100          200

In this small study, the relative risk of getting th Effect from the Drug
is (estimated by)  (80/121)/(20/79) = 2.612 ,  while the odds-ratio
is (estimated by)  (80/41)/(20/59)  =  5.756 . 

The conclusion to draw from this little example is not that one or the 
other of these functions of cell-probabilities --- relative risk or odds
ratio --- is "right" and that one is "wrong", because both represent 
meaningful quantities for the study population. The conclusion is that, 
for populations sampled like case-control populations to fixed 
marginal totals in one direction only, the relative risk is DIFFERENT in
the special sampled population than in the general population. The odds 
ratio is in this sense the more "robust" of the two statistics: it is 
meaningful and relfective of the general population without depending 
too much on the typicality of the marginal totals.