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In this guide they try to teach how to calculate relative and absolute risk reduction.

Problem: My understanding of the formulas leads to different result than they say.

Data and results given

  1. control: disease A occurs in 1 in 100,000 people

  2. experiment: taking drug x reduces occurence to 1 in 10,000,000

The are saying that

  • absolute risk of disease is 0.001%.
  • relative risk is 0.00001/0.001 = 0.1
  • relative risk reduction (RRR) is 1-0.1 = 0.9
  • absolute risk reduction is 0.00001-0.001=-0.00099

How to calculate

outcome of treated and control

Experimental Event Rate (EER) means taking the drug.

$$ EER = \frac{a}{a+b} = \frac{1}{1 + 9,999,999} = 0.0000001 $$

Control Event Rate (CER) means not taking the drug (control, placebo)

$$ CER = \frac{c}{c+d} = \frac{1}{1 + 99,999} = 0.00001 $$

To caclulate the RRR and ARR

$$ RRR = \frac{CER-EER}{CER} = \frac{0.00001 -0.0000001}{0.00001} = 0.99 $$ $$ ARR = CER-EER = 0.00001 - 0.0000001 = 0,0000099 $$

Question - why the result mismatch?

If I undertood everything correctly, the formulas above should be fault-free and return the right results. But my results differ from theirs.

  • my RRR is 0.99 and theirs is 0.9
  • my ARR is 0,0000099 and theirs is -0.00099

Where are my mistakes?

Why do they use 1-0.1 = 0.9 for RRR. Which formula was used?

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surfmuggle
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1 Answers1

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You are correct.

Someone there goofed up their math, as is obvious from the statement "relative risk is $0.00001/0.001 = 0.1$". Quite obviously, $$\frac{0.00001}{0.001} = 0.01$$ not $0.1$ Since they then used this number in their calculation of RRR, they got it wrong too.

But the ARR is a different problem. You should notice that your EER and CER are $\frac1{100}$ times theirs. Why? Because they are giving the values as percentages, not straight ratios. For the RRR, this cancels out because it is a ratio. The multiplier of $100$ in the numerator is cancelled by the multiplier of $100$ in the denominator. But with the ARR, there is no such cancellation. I would say it was a somewhat minor error of dropping % signs where they should have them. But when I looked up exactly what they said in your link, it got worse:

while the absolute risk reduction is 0.00001-0.001=-0.00099 or 0.099%.

Ouch! They failed to notice that what they had was already a percentage and converted it a second time!

What they should have said was:

The absolute risk of disease is $0.001\%$. The relative risk is $\frac{0.00001\%}{0.001\%} = 0.01$ and the relative risk reduction is $1- 0.01 = .99$ or $99\%$ while the absolute risk reduction is $0.00001\%-0.001\%=-0.00099\%$ or $0.0000099$.

(The dropping of the negative sign is a matter of wording: ARR is a reduction, so the number should be the amount it is reduced. While the change in values is negative, the amount of the reduction is the positive value by which they differ.)

So they made two basic arithmetic errors in just two sentences. The author(s) of this paper are obviously sloppy in their mathematics. I would not put a lot of trust in them.

Paul Sinclair
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