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.