# How to convert continuous relative risks per exposure unit ?

If a study for example concludes that the relative risk of a cardiovascular event is 1.04 per 10 µg/m³ increase in NO2, how can I calculate the relative risk for an air pollution concentration of for 25 µg/m³. Until now, I did this as (0.04 / 10 * 25 = 0.1) - A relative risk of 0.1 per 25 µg/m³. Is this correct ? If not, how should I calculate this?

If you calculate it like this NO2 seems actually protective now (RR < 1). Or do you do 1 + 0.1 to end up at 1.1 relative risk per 25 µg/m3?

Assuming you do this last step, you end up at roughly the correct answer in this case, but it is not precisely the correct calculation. As you calculate it now, you basically say: per 10 units my risk increases by factor 1.04 (multiply baseline risk with 1.04 for each 10 unit increase) and so per 25 units it increases 2.5 times more so 1.1 (so multiply baseline risk with 1.25 for each 25 unit increase).

As you estimated the RR to be 1.04 per 10 µg/m3 an individual with an NO2 of 20 is estimated to have a risk of: baseline risk x 1.04 x 1.04. Or more in general, for an individual with X NO2 the estimated risk is:

• baseline risk x 1.04 (X/10)

As you can see, you need to use the calculation rules for powers to calculate the correct RR per 25 µg/m3 NO2:

• 1.04(X/10) = 1.04((2.5X)/25)) = 1.04(2.5(X/25)) = 1.04(2.5(X/25)) = 1.103(X/25)

So your estimate of 1.1 is almost correct in this case. But as you can see from the examples in the table below the estimates from your and the above mentioned method diverge the larger the RR per 10 µg/m3 NO2

RR per 10 µg/m3 Per 25 µg/m3 Per 50 µg/m3 Per 100 µg/m3
Incorrect Correct Incorrect Correct Incorrect Correct
1.01 1.025 1.025 1.050 1.051 1.100 1.105
1.02 1.050 1.051 1.100 1.104 1.200 1.219
1.03 1.075 1.077 1.150 1.159 1.300 1.344
1.04 1.100 1.103 1.200 1.217 1.400 1.480
1.05 1.125 1.130 1.250 1.276 1.500 1.629
1.06 1.150 1.157 1.300 1.338 1.600 1.791
1.07 1.175 1.184 1.350 1.403 1.700 1.967
1.08 1.200 1.212 1.400 1.469 1.800 2.159
1.09 1.225 1.240 1.450 1.539 1.900 2.367
1.10 1.250 1.269 1.500 1.611 2.000 2.594