Imagine you're the formulary manager of a community health center that
has just seen its funding slashed. A drug rep has shown up at your door
with a sample of Cardicare, which she says will clearly save the lives
of patients with heart disease by preventing heart attacks at the first
signs of trouble. Trouble is, you realize when you look carefully at the
materials she has given you, the drug costs $200 a dose and patients
need to take it for a week at a time when they have angina. Is it worth
it?
In my last column (January), I discussed absolute and relative risk,
focusing on the differences between these two concepts and what they
each represent. There's another way to think about absolute and relative
risk that helps put all of these kinds of findings into perspective.
It's known as the number needed to treat. What this tells you is
how many patients you would have to treat with a given medication to see
a benefit in one patient. That number can be used, as in the
hypothetical above, to figure out how much preventing each event or
saving each life would cost. You can also do this calculation for side
effects; this is called the number needed to harm.
The calculation is simple if you know the absolute risk reduction: you
simply divide the absolute risk reduction into 100. Take the raloxifene
(Evista, Lilly) example from my last column, where we parsed the data
from the Multiple Outcomes of Raloxifene Evaluation (MORE) trial (
Arch Intern Med. 2002;162:1140-1143). There were 2,292 subjects in
the placebo group, which had 19 fractures, and 2,259 subjects in the
treatment group, which had six fractures. That gave a rate of new
clinical vertebral fractures of 0.83% (19/2,292) in the placebo group
and 0.27% (6/2,259) in the treatment group, so the difference in risk is
0.5%. Dividing 100 by 0.5 gives you 200, which is the number of women
(whose characteristics are the same as those in the study) you would
need to treat to prevent one clinical vertebral fracture--aka the number
needed to treat.
Or take the Women's Health Initiative's findings on hormone replacement
therapy (HRT) and stroke (JAMA 2002;288:321-333). The absolute
number of strokes per 10,000 person-years attributable to HRT was 8,
according to the researchers, for an absolute risk increase of 0.08.
Dividing 100 by 0.08 gives you 1,250, which is the number of women
(whose characteristics are the same as those in the study) who would
have to be placed on HRT for every one who would suffer a stroke--also
known as the number needed to harm.
In the abstract, the number needed to treat/harm is fairly meaningless.
Who's to say whether preventing one stroke or one fracture is worth not
treating women with HRT or raloxifene? But these numbers become more
powerful when you add data such as cost, or when you're looking at
outcomes such as death or those that are likely to lead to death--stroke
certainly being among them.
Take the example of recombinant human erythropoietin (rHuEPO). Following
on a 2002 study that determined the number needed to treat with rHuEPO
to prevent adverse events (JAMA 2002; 288:2827-2835), researchers
at Johns Hopkins University, Baltimore, found that preventing a single
transfusion complication, including benign ones, via the use of rHuEPO
would require treating 5,246 patients at a cost of $4.7 million. And the
costs ballooned for preventing more serious events: 28,785 patients at a
cost of $25.6 million to prevent an acute hemolytic reaction, and $71.8
million (81,000 patients) to prevent a fatal complication such as
hepatitis C infection, the Johns Hopkins researchers reported at the
2004 annual meeting of the American College of Clinical Pharmacy in
Dallas.
Worth it? The Johns Hopkins team said no, even though the original
JAMA study had found that the medication reduced the need for
transfusions by 19%.
Such calculations change over time, and with each study that is
released. In fact, at the very same meeting where the Johns Hopkins
study was presented, another team of researchers offered
quality-adjusted life-years data suggesting that rHuEPO is
cost-effective. Coupled with favorable contract pricing and other
variables that can dramatically impact the cost of drugs, you can see
how difficult it is to get a handle on the true cost of a pharmaceutical
intervention.
And then there's the question of what is a reasonable cost for saving a
human life. The U.S. government figures that each life is worth
somewhere between $2.3 million and $6.1 million, although Medicare
obviously doesn't give every hospital that much to treat every patient.
The important thing is to do these calculations, as the Johns Hopkins
team did, so you have enough information to make a decision that will
make the best use of your resources.
Rational About Ratios
There are, of course, even more ways to express risk. There are risk
ratios, odds ratios, and hazard ratios, which are often confused with
one another. Let's start with the difference between risk and odds. Risk
is basically an absolute probability that something will happen. Odds,
on the other hand, are what horse racing enthusiasts use to handicap
races, and reflect the chance that a particular event will happen
compared to another event. It's helpful to think of what risk ratios and
odds ratios of 1 mean. A risk ratio of 1 means there is no difference
between the risks of given outcomes, while an odds ratio of 1 means that
it's just as likely a given event will happen as won't happen.
Risk ratios are calculated by dividing the risk of an event in the
treated group by the risk of that event in the control group. Going back
to the raloxifene study, with a risk of fracture of 0.27% in the
treatment group and 0.83% in the placebo group, your risk ratio for
suffering a fracture would be 0.3% divided by 0.83%, or 0.32.
Odds ratios are calculated the same way, just using odds. For odds, the
denominator is not the total, but the number who did not have the event
of interest. In the raloxifene trial, to find the odds of having a
fracture, you would divide the six fractures in the treated group by the
2,253 nonfractures in the treated group, giving you an odds of having a
fracture of 0.0027, or 0.27%. In the placebo group, you would divide the
19 fractures by the 2,273 nonfractures, resulting in odds of 0.0084, or
0.84%. The odds ratio--0.27 divided by 0.84, or 0.32--is therefore the
same as the risk ratio.
These calculations demonstrate that when a given outcome is rare, and an
intervention or exposure has relatively little effect on the absolute
risk, the risk and odds ratios can be quite similar. But the more common
a given outcome, or the more significant the effect of the intervention
or exposure, the more different the two figures become.
For example, if half the patients in the placebo group had suffered
fractures, and 10% of those in the treatment group had suffered them,
the risk ratio of a fracture would be 0.1 divided by 0.5, or 0.2--not
tremendously different from 0.32. But the odds ratio would be quite
different. The odds of suffering a fracture in the placebo group are
even at 1:1, while the odds of suffering one in the treatment group are
10% divided by 90%, or 0.11. And 1 divided by 0.11 is about 9. Your odds
of suffering a fracture in the treatment group, in other words, are 9 to
1, which sounds much different from 0.2. And odds ratios can reach very
high numbers: to cite a recent example, a Swiss study of influenza
vaccines and Bell's palsy published last year in The New England
Journal of Medicine (2004;350:896-903) found an adjusted odds ratio
for Bell's palsy of 84.0 for those who took the intranasal flu vaccine
compared to those who took a parenteral vaccine.
The raloxifene/MORE study didn't use odds ratios, for good reason. If
you pay close attention, you'll notice that risk or hazard ratios are
what are typically used in large clinical trials like the MORE trial,
while odds ratios are what's used in case-control studies such as the
Swiss study of influenza vaccine. That's because if you don't know the
prevalence of a particular disease or event--the Swiss study
acknowledged that "there has been no systematic survey of Bell's palsy
in Switzerland"--you can't calculate relative risk and risk ratios. You
therefore have to rely on odds ratios. Still, you may find that many
meta-analyses, particularly older ones, make use of odds ratios even
when risk ratios can be calculated. That's a bit of an accident of
history, since the early rules for reporting meta-analyses relied on
odds ratios, and it's changing somewhat.
A word about hazard ratios: they're just a variation of the risk ratio
that is used when the numbers arise from a Cox proportional hazards
model. The Cox model allows you to determine the effects of several
variables on a given outcome.
In my last column, I explained why absolute risk is a better way to
express findings than relative risk. Similarly, it's important to
realize that odds ratios, while useful in certain situations, can be
more misleading than risk ratios. So while there are clear reasons to
use odds ratios instead of risk ratios, it's important to realize that
just as expressing relative risk differences rather than absolute risk
differences can dramatically change the perception of a given
intervention or hazard's influence, so too can expressing odds ratios
rather than risk ratios.
In my next column, I'll discuss the meaning and derivation of those
strange numbers that appear after odds and risk ratios--otherwise known
as confidence intervals.
Table. Calculating Number Needed To Treat, Risk Ratio, and Odds Ratio
Take, for example, a trial in which 10,000 people are treated with drug
X, and 10,000 are not treated. In the untreated group, 4,000 people get
disease Y. In the treated group, only 10 get disease Y. The absolute
risk of the disease is 40% in the untreated group, and 0.1% in the
treated group. That gives you an absolute risk reduction of 39.9%. Based
on those numbers, here's how to calculate the number needed to treat,
risk ratio and odds ratio.
Number needed to treat: Divide 100 by the absolute risk
reduction. In this case, 100/39.9 = 2.5. That means you would only need
to treat 2.5 people with drug X to prevent one case of disease Y.
Risk ratio:
Divide the absolute risk in the treated group by the absolute risk in the
control group. (10/10,000) / (4,000/10,000) = 0.0025, meaning the risk of
having the disease is almost zero
Odds ratio:
Divide the odds of an event in the treated group by the odds in the
control group. (Odds uses nonevents as the denominator.) Odds of an event
in the treated group is 10/9,990 or 0.1 Odds of an event in the untreated
group is 4,000/6,000 or 0.67 0.001/0.67 = 0.00149, or 1:670. You also
could say that the odds of not having the disease are 670 times greater if
you receive the treatment.
This example illustrates how different risk and odds ratios can be.