On 10/27/2012 6:07 PM, Stephanie Chan wrote:

> Thanks, everyone, for their responses to my question about
> person-years.  I mostly wanted to get a sense of whether I could
> convert them into annualized event rates, for the purposes of
> calculating absolute risk reductions and increases,  For example, in
> the BMJ article I'm reading, the thromboembolism rate in afib
> patients with a CHADS2 score of 1 is 4.6 per 100 person-years.  Since
> we know from RCTs that warfarin leads to a 60% RRR in
> thromboembolism, if I could convert the 4.6 per 100 person-years to
> 4.6%, I can figure out the ARR and the NNT for warfarin in this
> population.  Just wasn't sure if that was OK to do.

This is an area of confusion, so it helps to review some basic 
mathematical definitions.

A proportion is a ratio of two counts, where the numerator is the count 
in a subset of the set being counted in the denominator. Using an 
example I worked on a while back: there were 70,412 infants born in the 
state of Missouri in 1995, and 29,637 of these infants had one or more 
visits to the ER during the first year of their life. The proportion, 
0.42, is the probability that a randomly selected infant will visit the 
ER during their first year of life.

A proportion, by definition, must be between 0 and 1.

A percentage, of course, is just a proportion multiplied by 100.

Contrast a proportion/percentage with a rate. A rate is a count divided 
by a measure of time or area. In Epidemiology it is usually a measure of 
patient days of exposure.

Here's an example of a rate. There were 22 central line infections in 
2006 in a critical care unit of a hospital. There were 7,560 patient 
days of exposure, meaning that on an average day, there were 20.7 
(7560/365) beds filled in that unit. The central line infection rate was 
0.0029 (22/7560) infections per patient day.

Often the denominator will be rescaled to make the numbers more 
manageable. In this example, you can multiply the rate per patient day 
by 365 to get a rate per patient year. It works out to be 1.06 central 
line infections per patient year.

Notice that the rate does not have to be between 0 and 1. In fact, you 
can always create a rate that is bigger than 1 if you change the unit of 
time by a large enough factor. In the infection example, you have the 
added possibility of more than one infection per patient, but even 
without this, the rate can exceed one.

There is a bit of ambiguity here, though. Often you will see a rate like 
"The final infant mortality rate in the United States for 2008 was 6.61 
infant deaths per 1,000 live births." 
(http://www.sidscenter.org/Statistics.html).

While that is described as a "rate" and it is indeed larger than 1, it 
is no different than a percentage. The set "infant deaths" is a subset 
of "live births" so what that website calls a "rate" is more accurately 
thought of as a proportion. The mortality rate per 1,000 live births is 
bounded above by 1,000 just like the percentage is bounded above by 100.

Now, can you calculate a NNT or NNH from a rate (actually from two 
rates)? Yes, you can but the interpretation is a bit tricky. Subtract 
rate 1 from rate 2 and invert it. It represents the amount of time 
(rather than the number of patients) that you have to treat with the new 
therapy until you see one additional cure (or one additional harm).

For example, aspirin as a primary prevention of heart attacks has a 
positive benefit. "There was a 44 percent reduction in the risk of 
myocardial infarction (relative risk, 0.56; 95 percent confidence 
interval, 0.45 to 0.70; P less than 0.00001) in the aspirin group (254.8 
per 100,000 per year as compared with 439.7 in the placebo group)." 
(Source: Final Report on the Aspirin Component of the Ongoing 
Physicians’ Health Study. New England Journal of Medicine. 
1989;321(3):129–135.)

Take the difference in these two rates and invert it. You get 
1/(439.7-254.8)=0.0054. Multiply by 100,000 to get 540. That means that 
you need to wait 540 patient years on average to see one fewer 
myocardial infarction. The numbers of ulcers in the two groups were 169 
and 138 per 100,000 patient years respectively. With about 54,500 
patient days of observation in each group, you can an NNH of about 1,800.

So aspirin prevents about 3 myocardial infarctions on average for every 
ulcer that it causes. I'm not a doctor, but that sounds like a good 
trade-off to me.

Here's another way of looking at it. There are about 80 million people 
in the United States between the ages of 45 to 64. If all of them took 
aspirin, we would see 80,000,000/540 = 150,000 fewer myocardial 
infarctions per year, but we'd be stuck with 80,000,000/1,800 = 44,000 
more ulcers.

Now the NNT for a rate is no longer bounded below by 1.0. An NNT of 0.5, 
for example, means that you'd see one fewer event on average for every 
half of a patient year of treatment.

There are several big cautions here. First, as someone else already 
noted, you have to assume uniformity across time because 1,800 patient 
days could mean 60 patients each seen for a month or 5 patients each 
seen for a year. The NNT calculation presumes that both groups are 
equivalent.

Second, rates are often calculated in observational studies. The aspirin 
example I cited above is an exception, but rates are very frequently 
used in observational studies. Can you calculate the NNT or NNH for an 
observational study? Maybe, but at times it requires a large leap of faith.

Finally, as I noted, I am not a doctor, so I may have mangled the 
aspirin example.

Steve Simon, [log in to unmask], Standard Disclaimer.
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