# Inflation of Odds Ratios

Ioannidis 2005 states that, for a study with two groups (one control, one experiment), N = 2000, and true odds ratio of 1.1, the median observed odds ratio will be 1.23 among studies reaching significance ($p \leq .05$).

I'm trying to reproduce this result. I think I want to do something like:

$$\frac{1}{N}\int_x^\infty p(t)dt=.05$$

Where $p(t)$ expresses the probability of an odds ratio of $t$. Then I would solve for $x$ and find that it's 1.23.

But I worry that this is overly simplistic, and even if it's not I'm not certain what to put in for $p(t)$. I could assume $\sim N(0,1)$, but I'm not sure if that's correct.

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If you read the paper, he describes the simulation experiment in detail at the bottom of page 2 and top of page 3. If you want to validate this with math, though, you'll have problems, as you'll need to calculate the median of a ratio of two binomial (approx. by Normal, perhaps) variates given that the two variates differ by more than a certain amount (which is based on the sample standard deviations); I'd personally consider this intractable and stick with the simulation approach. –  jbowman Jun 22 '12 at 17:54
@jbowman: oh, you mean he just generated a set of data according to that distribution and those are the numbers he got? I didn't understand that from the paper. –  Xodarap Jun 22 '12 at 18:15
Yes, it didn't look like it to me either until I read that part a couple of times, then I realized his results were simulation results. Still, interesting. –  jbowman Jun 22 '12 at 18:31