Relative risk

I’m throwing up this post on the concept of relative risk as a time saver. I’m going to be doing some posting soon in which relative risk plays a role as it has in many of the postings in the past. Instead of taking time in each post to explain relative risk, I will simply be able to link to this one and get on with it.

Let’s say we are studying the rate at which people taking a particular medication get cancer. If we randomize our study population into two groups and give one group the drug and the other group a placebo, we can wait around for 5 years (or 10 years or 20 or whatever our protocol is) to see how many develop cancer in each group. Then we can compare.

If after 5 years we find that 10 of the subjects taking the drug develop cancer and only 5 of the subjects taking the placebo get cancer, we can say (assuming there were equal numbers of people in both study groups to start) that twice as many people taking the drug got cancer. We can then say that the relative risk (usually written as RR) of getting cancer from taking the drug is 2.

The relative risk is the number of subjects who develop the problem divided by the number who didn’t. In our example above it would be 10/5 = 2.

Another way to present this data is to say that the relative risk of not getting cancer by not taking the drug is 0.5. In other words, people who didn’t take the drug had half the risk of getting cancer compared to those who did.

In our simple example we assumed that there were exactly the same number of subjects in both groups. In reality this usually isn’t the case. There will be a difference in group sizes so instead of using numbers of subjects, researchers will use percentages. So in our case above, let’s say that 4 percent of the subjects taking the drug got cancer and 2 percent of those taking the placebo got cancer. We have the same relative risk of 2. (4 divided by 2 = 2)

It all seems pretty straight forward. But it often isn’t as it seems. There can be two studies that show the same relative risk but there is a world of different meaning for the relative risk. This idea is what we want to focus on because this is the one that researchers use all the time to scare us needlessly.

I’m going to create an example loosely based on the facts to show you what I mean. Let’s compare the safety records of two airlines over the past 25 years. In 1985 Delta had a crash in Dallas (in which one of my best friends was killed) and in 2001 American had three crashes. The two from 9/11 and one going from New York to San Juan, Puerto Rico. Let say that there were 200 people killed in the Dallas Delta crash and 600 people killed in the three American crashes.

Let’s assume that both airlines fly about the same number of miles each year. American has about 4000 flights per day, so if we assume an average distance (probably too short) of 500 miles per flight, we get 730 million miles flown per year. If we multiply by 25 to get the miles flown in 25 years we come up with 18.25 billion miles flown. Let’s assume the same for Delta. If we divide the number of victims of crashes from the two airlines by the number of miles flown, we can come up with a risk of dying per mile flown on each of the airlines. If we do this for Delta we discover that the risk of dying comes out to be 200 divided by 18.25 billion or about 0.00000001096. This tiny number represents the number of people who have died per mile flown by Delta. If we do the same calculation for American we find that the number is 0.00000003288 deaths per mile flown.

Now if we want to calculate the relative risk of flying on American verses flying on Delta we divide the American risk by the Delta risk (0.00000003288 divided by 0.00000001096) and we get 3. So, the relative risk of flying on American is 3 compared to flying on Delta based on the last 25 years of crash history. This means that it is 3 times more risky to fly American than it is to fly Delta. Based on these figures you are 3 times more likely to die if you fly American than if you fly Delta. It’s true, and these figures prove it.

We all know that it’s hogwash. There is no difference in risk between flying these airlines because the numbers are so low as to be ridiculous, which is the precise reason I used this example. The total numbers make a difference.

What if there were an airline that had 90 deaths for every 100,000 miles flown and another that had 30 deaths for each 100,000 miles flown. The relative risk for flying the first as compared to the second would be 3 just as it is with American verses Delta, but the real risk would be much, much different. In fact, you wouldn’t want to fly on either of these airlines because they are both too risky. But if you had to fly on one, you would much prefer the one with only 30 deaths per 100,000 miles.

Many drug companies use the relative risk in their propaganda. They might do a study with two groups of 40,000 people with one group taking their drug and the other taking a placebo and monitor them for 5 years. If the drug is supposed to prevent heart disease, then the researchers would tabulate the number of cases of heart disease that arose over the 5 years, then see how many cases developed in the subjects taking the drugs compared to the ones taking the placebo.

If there were 43 people who developed heart disease while taking the placebo and 31 who developed heart disease while taking the drug, then the relative risk of not taking the drug would be 43 divided by 31 which is 1.39. The drug company then proclaims that people not taking their drug had a 30 percent greater chance of developing heart disease, which sounds pretty significant. But it’s a lot like the Delta/American example: the numbers aren’t all that large.

What it really means is that out of the 40,000 people who took the placebo for 5 years only 12 more developed heart disease than in the group of 40,000 people who took the drug. Thats 2.4 people out of 40,000 per year. (12 people divided by 5 years = 2.4 people per year) If the drug costs a lot of money and has unpleasant side effects associated with it (think statins), would you be willing to pay the bucks and put up with the side effect hassle on the chance that you might be one of the 2.4 out of 40,000 who might be saved from developing heart disease? Or would you just take your chances? I would definitely take my chances.

2.4 people out of 40,000 per year sounds a lot less risky than a 39 percent increased risk that the relative risk implies.

One other way the relative risk can be used to flim flam people is when it is too small to be meaningful yet is touted as having great merit.

Without going into all the statistical detail (which if you’re interested in you can read about here and here from previous posts) suffice it to say that relative risk ratios below 2 are pretty much meaningless. So when you see a relative risk of 1.28 and hear someone say that represents a 28 percent increased risk, you can blow it off. Once the relative risk gets to the 2 or greater level, then it starts to have some meaning. But, once again, only if the numbers aren’t so small as to be meaningless. Remember, the relative risk of flying American is three times that of flying Delta if you simply look at the relative risk. In reality, however, the numbers are so small that the differences are meaningless.

Please note: I reserve the right to delete comments that are offensive or off-topic.

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11 thoughts on “Relative risk

  1. Very interesting. Perhaps if people were more educated on the topic of statistics, there would be less people wasting their money on slot machines, statins and other unuseful things.

    Hi Max–

    I don’t know.  I’m pretty good with statistics, yet I continue to bet football.



  2. Isn’t this the same numbers game that people were using to convince the public that ‘Second Hand Smoke’ is worse for you than if you smoked cigarettes yourself?

    I think most people STILL believe it’s true. When you’ve got a catchy phrase with an emotional hook and something that SOUNDS true, it’s pretty much impossible to stop the mass public from believing it…

    These are the same numbers.  It goes to show how easy statistical humbug can override plain ol’ common sense.



  3. What, going not by CDC/NIH projections and suppositions, but the most conservative and hard-science-based guess, is the relative risk of smoking to cause cancer, then?

    Anyway, failure to include actual risk in a fearful calculation is one of the biggest problems our authoritarian society has.

    The odds of an asteroid impact is almost zero. But all the NASA bureaucrats seeking bigger budgets focus upon is the consequences if one actually did occur.

    Every activity or item that gets effectively banned by government regulations or lawsuits gains that status without any sane examination of the actual odds. Ten million people could do/use something, and ten die, and it might get banned, directly or through lawsuits making it too expensive to provide.

    Likewise, the odds of Hussein having had WMD was almost nil, going by the intelligence available at the time, but the consequences were so scary that people accepted ridiculous lies, exaggerations, and suppositions.

    The only models that claim humans are causing greenhouse-based global warming fail 100% of the time, when historic data is plugged into them. The only global increase in temperatures is of surface temperatures, which cannot be caused by the greenhouse effect, but must have some other source. So the odds of human greenhouse emissions causing global warming, based on what we know, are actually a negative number. But it’s scary…even scarier because they talk only about worst-case consequences that are guesswork…so people blindly accept, without examining the science for themselves.

    Our society is hagridden by fear of consequences without the actual risk being weighed.

    Or, for that matter, the cost. All of society may suffer a thousand-dollar per person burden in order to reduce the danger to 0.001% of the society. Again, only the scariness of the consequences are considered.

    Our kids should be taught statistics, economics, and logic even in grade school…instead, even the three Rs have taken a back seat to good citizenship, meaningless “self esteem”, and other passivity training.

    It’s no wonder they’re turning into such sheep.

    Hi Kaz–

    Agreed.  It’s too bad that critical thinking isn’t taught as a subject instead of simply assuming that people know how to think critically.



  4. Hi Dr. Eades,

    The use of relative risk statistics by “big pharma” to sell their drugs is the primary reason I don’t trust physicians easily. Quite frankly, I think the use of statistics in this manner, while technically not fraud, accomplishes the same thing. There is clear intent to mislead, and influence people to make decisions they would not otherwise make. As you state 2.4 people oout of 40,000 sounds a lot less risky than 39% increase.

    The anxiety and fear used by medicine to drain our pockets of our money is dispicable enough. We don’t need the facts further confused by the use of relative risk statistics applied to “risk factors” that probably aren’t true risk factors at all.

    I believe it was Malcolm Kendrick that wrote about need for a “funeral rate.” I’m interested knowing the abosulte risk of any food, drug, vitamin, herb, or whatever that I introduce into my body. But I hope we don’t have to bankrupt our country before this other nonsense will stop.

    Thanks for writing this piece and please do some more.


    Hi Rusty–

    I definitely plan to do some more.  This one was merely teeing up the RR for use later on.  Stay tuned.



  5. Great post and comments. I would add that along with risk and cost, benefit should be part of the equation. Using smoking as an example, some adults believe the risk of disease is offset by their enjoyment of the habit and other benefits (“it keeps me slim,” “it helps me relax,” etc). Intelligent or not, adults have the right to make such trade offs as long as they do not force others to pay for the consequences. The nanny politicians and busybody public health authorities seem incapable of grasping that some of us are willing to take some measure of risk in return for pleasure.

    Hi Paul–

    That’s because in their minds they are much smarter than we are and know so much better what we should do than we do.  Therefore, since they are so much smarter, they need to set the rules for the great unwashed masses of the rest of us.



  6. I rambled on for so long that you forgot to answer my initial question, as to what the relative cancer risk of smoking actually is.

    Hi Kaz–

    I don’t know off the top of my head.  I’ll see if I can ferret it out. If I had to guess I would say around 3 for cancer.  Probably much higher if you look at overall illness including heart disease, chronic bronchitis, emphysema, etc.



  7. Here are two other medical ‘statistics’ that I’m very skeptical of, but although I’ve HEARD the counter-arguments, I don’t have the actual numbers to back them up. Perhaps you’ve heard them:

    1) The US has the worst infant mortality rate of any industrialized nation, and lags behind even Cuba.

    rebuttal: The US includes as ‘infant mortalities’ many cases that would be classified otherwise in many other countries, and ‘at risk’ births really have a FAR better survival rate in the US than any other country in the world.

    2) The Japanese eat massive quantities of rice, but have FAR lower incidence of heart attack than the US, proving that eating rice, although high carb, doesn’t cause cardio-vascular problems.

    rebuttal: While the incidence of heart attack is much lower for the Japanese, the incidence of stroke is several times greater. Combining both, the statistics are about the same in both countries where a high-carb diet is the norm. The difference between heart attack and stroke is probably due to the large amount of fish and seafood in the Japanese diet.

    On the second point, I thought I had seen the data in The Protein Power Lifeplan, but I flipped through it, albeit casually, over the weekend, and didn’t catch it. I’ll look again, but I may be remembering it incorrectly.

    Hi Bob–

    I think you’re right.  Sorry it’s taken me so long to post this comment, but somehow you got hung up in my spam filter.



  8. I have always loved claims of 50% or 100% better when they do not state the base number.

    Even though it is basically impossible to win a lottery, it actually is impossible if you don’t get a ticket. So, your chances of winning are infinitely better if you get a ticket, sounds good to me – so why do I keep losing?

    Precisely!  And if you buy two lottery tickets you’ve got twice as much of a chance to win.  You go from 1 in 75 million to 2 in 75 million.  Your odds are doubled, but even doubled are virtually non-existent.