because the treatments (aspirin versus no aspirin) were applied and then future outcome
was determined. This will become important shortly. Prospective studies are often called
cohort studies(because we identify two or more cohorts of participants) or, especially in
medicine, a randomized clinical trialbecause participants are randomized to conditions.
On the other hand, a retrospective study,frequently called a case-control design,would
select people who had, or had not, experienced a heart attack and then look backward in
time to see whether they had been in the habit of taking aspirin in the past.
For these data on one degree of freedom, which is statistically significant
at a 5 .05, indicating that there is a relationship between whether or not one takes aspirin
daily, and whether one later has a heart attack.^7d-Family: Risks and Odds
Two important concepts with categorical data, especially for 2 3 2 tables, are the concepts
of risks and odds. These concepts are closely related, and often confused, but they are basi-
cally very simple.
For the aspirin data, 0.94% (104/11,037) of people in the aspirin group and 1.71%
(189/11,034) of those in the control group suffered a heart attack during the course of the
study. (Unless you are a middle-aged male worrying about your health, the numbers look
rather small. But they are important.) These two statistics are commonly referred to as risk
estimates because they describe the risk that someone with, or without, aspirin will suffer
a heart attack. For example, I would expect 1.71% of men who do not take aspirin to suffer
a heart attack over the same period of time as that used in this study. Risk measures offer a
useful way of looking at the size of an effect.
The risk differenceis simply the difference between the two proportions. In our exam-
ple, the difference is 1.71% 2 0.94% 5 .77%. Thus there is about three-quarters of a per-
centage point difference between the two conditions. Put another way, the difference in risk
between a male taking aspirin and one not taking aspirin is about three-quarters of one per-
cent. This may not appear to be very large, but keep in mind that we are talking about heart
attacks, which are serious events.
One problem with a risk difference is that its magnitude depends on the overall level of
risk. Heart attacks are quite low-risk events, so we would not expect a huge difference be-
tween the two conditions. (When we looked at the death sentence data, the probability of
being sentenced to death was 11.6% and 6.1% for a risk difference of 5% points, which
appearsto be a much greater effect than the 0.75% difference in the aspirin study. Doesx^2 =25.014160 Chapter 6 Categorical Data and Chi-Square
(^7) It is important to note that, while taking aspirin daily is associated with a lower rate of heart attack, more recent
data have shown that there are important negative side effects. Current literature suggests other treatments are at
least as effective with fewer side effects.
cohort studies
randomized
clinical trial
retrospective
study
case-control
design
risk
risk difference
Table 6.10 The effect of aspirin on the incidence
of heart attacks
Outcome
Heart No Heart
Attack Attack
Aspirin 104 10,933 11,037
Placebo 189 10,845 11,034
293 21,778 22,071