# odds ratio to relative risk equation, how is it derived?

How is this equation derived? thanks

RR=OR/((1−Pref)+(Pref∗OR))

RR = risk ratio; OR = odds ratio; Pref = Prevalence of the outcome in the reference group

Relative Risk

RR = p1/p0


Odds Ratio

OR = (p1/(1-p1)) * ((1-p0)/p0)

OR * (p0/(1-p0)) = p1/(1-p1)

(1-p0)/(OR*p0) = (1-p1)/p1

(1-p0)/(OR*p0) = 1/p1 - 1

1 + (1-p0)/(OR*p0) = 1/p1

(OR*p0+(1-p0))/(OR*p0) = 1/p1

(OR*p0)/(OR*p0+(1-p0)) = p1

p1 = (OR*p0)/(OR*p0+(1-p0))


Since RR = p1/p0

p1 = (OR*p0)/(OR*p0+(1-p0))

p1/p0 = (OR*p0)/((OR*p0+(1-p0))*p0)

p1/p0 = OR / OR*p0+(1-p0)


which means:

RR = OR / ((1-p0)+(OR*p0))


RR = risk ratio; OR = odds ratio; p1 = the outcome in the experiment/exposed group; p0 = the outcome in the reference group