# Computing the pre-study probability of a finding being true

I read a fascinating free article about why most published research findings might be false. I am able to follow along almost completely, except for one tiny part.

In it, the author, Ioannidis, writes

Let R be the ratio of the number of “true relationships” to “no relationships”
among those tested in the ﬁeld.
...
The pre-study probability of a relationship being true is R⁄(R + 1).


Why is P(T)=R/(1+R)?

He writes in the same paragraph:

Let us also consider, for computational simplicity, circumscribed ﬁelds where
either there is only one true relationship (among many that can be
hypothesized) or the power is similar to ﬁnd any of the several existing true
relationships.


Perhaps this can help explain it, but I do not see how.

$R$ is the ratio of "true relationships", let's call it $S$, to the "no relationships", call it $\overline{S}$. Therefore, $p(T) = \frac{S}{S+\overline{S}}$, in other words, probabilty of being true is equal to the number of true cases divided by total number of cases. The author, instead, wrote $p(T) = \frac{R}{R+1}$. Well, lets substitute $R=\frac{S}{\overline{S}}$, and see what comes out:

$$p(S) = \frac{\frac{S}{\overline{S}}}{1+ \frac{S}{\overline{S}}} = \frac {\frac{S}{\overline{S}}} {\frac{\overline{S}+S}{\overline{S}}} = \frac {S} {S + \overline{S}}$$

so it worked out! but honestly, I have no idea why he has used this notation to represent the probability. Maybe he had a point, but I can't get it!

• It's just simple algebra - how embarrassing! – The Unfun Cat Feb 7 '14 at 23:55
• no worries dude, happens sometimes. perhaps you need a beer, 'couse it's friday night, lol – omidi Feb 7 '14 at 23:57
• I've actually had four ;) – The Unfun Cat Feb 8 '14 at 0:00
• ok glad to hear that, cheers! – omidi Feb 8 '14 at 0:01
• NB $R$ is called the odds in favour of $S$. – Scortchi - Reinstate Monica Feb 8 '14 at 0:40

Lets define the number of studies with a true relationship as T, and the number with no relationship as N.

R is T/N.

The probability you want is T/(T+N)

That value is R/(R+1).