MEASURES OF DISEASE
ASSOCIATION
Nigel Paneth
MEASURES OF DISEASE
ASSOCIATION
The chances of something happening can be
expressed as a risk or as an odds:
RISK = the chances of something happening
the chances of all things happening
ODDS= the chances of something happening
the chances of it not happening
Thus a risk is a proportion,
But an odds is a ratio.
An odds is a special type of ratio,
one in which the numerator and
denominator sum to one.
Example 1. Bookies are taking bets on the
World Series. They are giving 3:1 odds on
the Yankees. What does this mean?
It means that they think that there it is three
times as likely that the Yankees will not win
the world series as that they will win.
Expressed as a risk, the Yankees are
expected to win one in four opportunities
Example 2. Among 100 people at baseline,
20 develop influenza over a year.
• The risk is 1 in 5 (i.e. 20 among 100)
• The odds is 1 to 4 (i.e. 20 compared to
80)
THE RELATIVE RISK
(RISK OR RATE RATIO)
The relative risk is a ratio of two risks.
Assume that among the 100 people at risk, 50
are men and 50 women. If 15 men and 5 women
develop influenza, then the relative risk of
developing influenza in men, as compared with
women, is:
Risk in men = 15/50
divided by
Risk in women = 5/50
15/50 : 5/50 = 3.0
(Note that from the way the question was put, the
two risks are cumulative incidence rates.)
ODDS RATIO
The odds ratio is a ratio of two odds
The odds in men = 15/35
divided by
The odds in women = 5/35
15/35 : 5/45 = 3.9
We conclude that the odds of men getting
influenza over the year are 3.9 times as high
as the odds of women getting influenza.
Thought question: note that the odds ratio
in this example (3.9) is larger than the
relative risk (3.0). Is this always the case?
Is this important?
MEASURES OF
PUBLIC HEALTH IMPACT
Four closely related measures are used:
1.Attributable risk
2.Attributable (risk) fraction
3.Population attributable risk
4.Population attributable (risk) fraction
Note: all of these measures assume
that the association between exposure
and disease has already been shown to
be causal.
1. ATTRIBUTABLE RISK (AR)
The incidence of disease in the
exposed population whose
disease can be attributed to the
exposure.
AR = Ie - Iu
2. ATTRIBUTABLE RISK
FRACTION (ARF)
The proportion of disease in
the exposed population whose
disease can be attributed to the
exposure.
ARF = (Ie - Iu)/Ie
3. POPULATION ATTRIBUTABLE
RISK (PAR)
The incidence of disease in the
total population whose disease
can be attributed to the exposure.
PAR = Ip - Iu
4. POPULATION ATTRIBUTABLE
RISK FRACTION (PARF)
The proportion of disease in the
total population whose disease
can be attributed to the exposure.
PARF = (Ip - Iu)/Ip
Note:
Ip can be linked to Ie and Iu if
one knows the proportions of the
population who are exposed (P) and
unexposed (Q), (P and Q add to 1).
Ip = P (Ie) + Q (Iu)
EXAMPLE OF THESE MEASURES
(data are invented)
• Red-meat eaters have a relative risk of 2.0
for colon cancer.
• If Iu = 50/100,000/year, then
Ie = 100/100,00/year.
• If 25% of the population are red-meat
eaters, what is Ip?
• Ip = P (Ie) + Q (Iu) , so
• Ip = .25(100/100,000) + .75 (50/100,000)
• Population incidence of colon cancer is
thus 62.5 /100,000/year
INFERRING AN ATTRIBUTABLE
RISK FRACTION FROM A
RELATIVE RISK
Note that Ie = Iu times the relative
risk (RR)
So substituting Iu x RR for Ie in the
equation for attributable risk
fraction:
(Ie - Iu)/Ie
We get:
ARF = RR (Iu) - Iu
RR (Iu)
Dividing through by Iu gives
ARF = RR - 1
RR
In other words, if we find a truly causal
relative risk of 2.0 for a disease in relation
to an exposure, we can assume that 50%
of the disease in the exposed population is
due to the exposure.
Since the courts use a probability of 50%
or greater as a threshold in liability cases,
RR of 2.0 has recently taken on great
significance in lawsuits. It has been
argued that when RR > 2.0, it is more likely
than not that the disease was due to the
exposure in an exposed individual. What
do you think of this legal reasoning?
INFERRING A POPULATION
ATTRIBUTABLE RISK FRACTION
FROM A RELATIVE RISK (this is
a little heavier going)
Remember that:
• PARF = (Ip - Iu)/Ip
• and that Ip = P(Ie)+ Q(Iu)
• and that Ie = Iu x RR
Therefore, the equation for PARF can
be rewritten in terms of RR:
PARF =
P(Ie) + Q(Iu) – Iu
P(Ie)+ Q(Iu)
Replacing Ie with Iu x RR, we get:
PARF =
P(Iu)RR + Q(Iu) – Iu
P(Iu)RR + Q(Iu)
Going From A Relative Risk To An
Attributable Risk Fraction Cont’d
Iu can be factored out and cancelled:
XIu (P x RR + Q - 1)
PARF =
Iu (P x RR + Q)
X
If we now replace Q with 1-P (since P + Q = 1):
PARF =
P x RR + 1 - P - 1
P x RR + 1 - P
or:
P (RR - 1)
P (RR - 1) + 1
In other words, if we find a truly
causal relative risk of 2.0 for a disease
in relation to an exposure, and if 50%
of the population has the exposure,
then 33% of the disease in the
population is due to the exposure.
(Again, always assuming that we are
discussing a exposure whose causal
role has been established).
EXAMPLE OF HOW FAILURE TO
UNDERSTAND WHAT AN ODDS
RATIO MEANS CAN LEAD TO
TROUBLE
Schulman et al: The Effect of Race and Sex on
Physicians' Recommendations for Cardiac
Catheterization. N Eng J Med 1999; 340: 619-625
To study doctors’ recommendations for
managing chest pain, the study used actors
to portray patients with particular
characteristics in scripted interviews about
their symptoms.
720 primary care physicians viewed a
recorded interview and were given other
data about a hypothetical patient. He or she
then made recommendations about that
patient's care.
The study used multivariate logisticregression analysis to assess the effects of
the race and sex of the patients on
treatment recommendations
The number of White and Black
patients who doctors thought should
be referred for cardiac catheterization
based on their symptoms
White
Black
Referred
Not
referred
326
(90.6%)
305
(84.7%)
331
34
360
55
360
89
720
Risk ratio and odds ratio in this table
• Relative risk or risk ratio for Blacks is:
305
360
divided by 326
360
or 0.93
• Odds ratio for Blacks is:
305 x 35
326 x 55
or 0.58
THIS IS HOW THE AUTHORS
DESCRIBED THEIR FINDINGS
Logistic-regression analysis
indicated that blacks (odds ratio,
0.60; 95 percent confidence interval,
0.4 to 0.9; P=0.02) were less likely to
be referred for cardiac
catheterization than whites.
HEART BIAS STUDY WAS
MISINTERPRETED (AP – 8/15/99)
The editors of the NEJM say they “take
responsibility” for media reports which greatly
exaggerated conclusions in a study about
possible gender and sex bias in heart care. The
study, published in the journal on Feb 25,
reported what happened when doctors viewed
taped interviews of actors describing their
identical symptoms and asked what treatment
they would recommend. It found that in cases of
equally sick patients, doctors were less likely to
refer blacks and women than they were white and
men to have cardiac catheterization, a test used
to diagnose heart disease. Several news
organizations, including the AP, interpreted the
study to show that doctors were 40% less likely
to order the tests for women and blacks than for
men and whites
However, a follow up published in the Journal recently
concluded that the likelihood of women and blacks
being referred for the tests was actually 7 percent less
than for men and whites.
The follow up, written by Dr. Lisa M. Schwartz and
others from the VA Outcomes Group in White River
Junction, Vt., said the misunderstanding resulted from
the original study's use of an "odds ratio" to report the
differences rather than a more commonly used "risk
ratio."
The researchers calculated the odds in favor of blacks
being offered the test and of whites being offered the
test. Then they calculated the ratio of these two
figures. The ratio of blacks' odds to whites' odds
worked out to 0.6, as did the ratio of women's odds to
men's. The media interpreted this to mean that
women and blacks were 40 percent less likely to be
offered catheterization. But the true difference is much
A table published with the study shows
that actually 85 percent of women and
blacks were referred for catheterization as
were 91 percent of men and whites. This
means that the risk ratio was .93. In other
words, the probability of referral was 7
percent lower for blacks and women than
for whites and men.
The journal editors said they "take
responsibility for the media's
overinterpretation" of the study's findings
and said they should not have allowed the
use of odds ratios in the study's summary.