How do you explain the difference between relative risk and absolute risk? The other day I had a consultation with an epidemiologist.  She is an MD with a public health degree in epidemiology and has a lot of statistical savvy.  She mentors her research fellows and residents and helps them with statistical issues.  She understands hypothesis testing pretty well. She had a typical problem of comparing two groups to see if there is a difference in their risk related to getting congestive heart failure (CHF).  She tested the mean difference in the proportion of subjects getting CHF. The p-value was 0.08.  Then she also decided to look at the relative risk and got a p-value of 0.027.  So she asked why is one significant and the other not.
Looking at 95% two-sided confidence intervals for the difference and for the ratio she saw that the mean difference interval contained 0 but the upper confidence limit for the ratio was less than 1. So why do we get inconsistent results?
My answer while technically correct was not very satisfactory. I said "These are different statistics and can give different results.  The p-values are both in the area of marginally significant.  This can easily happen."
I think there must be better ways to answer this in laymen's terms to physicians to help them understand the difference between testing relative risk vs absolute risk.  In epi studies, this problem comes up a lot because they often look at rare events where the incidence rates for both groups are very small and the sample sizes are not very large.
I have been thinking about this a little and have some ideas that I will share. But first I would like to hear how some of you would handle this. I know that many of you work or consult in the medical field and have probably faced this issue. What would you do?
 A: Well, from what you've already said, I think you've got most of it covered but just need to put it in her language: One is a difference of risks, one is a ratio. So one hypothesis test asks if $p_2 - p_1 = 0$ while the other asks if $\frac{p_2}{p_1} = 1$. Sometimes these are "close" sometimes not. (Close in quotes because clearly they aren't close in the usual arithmetic sense). If the risk is rare, these are typically "far apart". e.g. $.002/.001 = 2$ (far from 1) while $.002-.001 = .001$ (close to 0); but if the risk is high, then these are "close": $.2/.1 = 2$ (far from 0) and $.2 - .1 = .1$ (also far from 0, at least compared to the rare case.
A: Mind that in both tests, you test a completely different hypothesis with different assumptions. The results are not comparable, and that is a far too common mistake.
In absolute risk you test whether the (average) difference in proportion differs significantly from zero. The underlying hypothesis in the standard test for this assumes that the differences in proportion are normally distributed. This might hold for small proportions, but not for large.  Technically you calculate the following conditional probability :
$$P( p_1 - p_2 = 0 | X )$$
with $p_1$ and $p_2$ the two proportions, and $X$ your explanatory variable. This is equivalent to testing the slope $b$ of the following model :
$$p = a + b\cdot X + \epsilon$$
where you assume that $\epsilon \sim N(0,\sigma)$.
In relative risk you do something completely different. You test the odds of having a positive outcome based on the explanatory variable $X$. So you calculate
$$P\left( \log\left(\frac{p_1}{p_2}\right) = 0 | X \right)$$
which is equivalent to testing the slope in the following logistic model:
$$\log\left(\frac{p}{1-p}\right) = a + b\cdot X + \epsilon$$
with $\log(\frac{p}{1-p})$ being the log of the odds. Note that this hypothesis is formulated in terms of the odds, and not proportions! So the assumptions of the model are also formulated in terms of the odds (or more exactly, the log of the odds). You're testing a different hypothesis.
The reason why this makes a difference is given in Peter Flom's answer: a small difference in absolute risks can lead to a big value for the odds. So in your case it means that the proportion of people getting the disease don't differ substantially, but the odds of being in one group is significantly larger than the odds of being in the other group. That is perfectly sensible.
