ODDS RATIO- META ANALYSIS-Reversing the reference category Reversing the reference category
Hello!
I am doing a meta-analysis.
I explain the problem with an example: 
All the study included in the meta-analysis have reference B (OR) except one that has reference A (OR). For this study, unfortunately, I do not have the data available to calculate the Odds Ratio inverted directly from Table 2X2.
Is there any possibility of getting OR B vs A directly from OR A vs B?
Thanks in advance! 
 A: Using standard notation of:


*

*$a$ being number of events in non-reference group

*$b$ being number of non-events in non-reference group

*$c$ being number of events in reference group

*$d$ being number of non-events in reference group


we have 
$$ OR = \frac{ad}{bc}  $$.
Switching groups means switching roles of $a$ and $c$ and of $b$ and $d$, so
$$ OR_{switch} = \frac{bc}{ad}  = \frac{1}{OR}  $$.
For meta-analysis you probably need to re-calculate SE as well. Recall that
$$ SE(log(OR)) = \sqrt{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}} $$
so 
$$ SE(log(OR)) =SE(log(OR_{switch})) $$
For SE without logarithm, see this post and Maarten Buis answer. By delta method 
$$ SE(OR) = SE(log(OR)) \cdot OR $$ so
$$ SE(OR_{switch}) = SE(log(OR_{switch})) \cdot OR_{switch} = \frac{SE(log(OR))}{OR} = \frac{SE(OR)}{OR^2}$$
If you wish to work exclusively on log-scale (as @mdewey suggested) it's even more straightforward:
$$ SE(log(OR_{switch})) =SE(log(OR)) $$
and since 
$$ OR_{switch} = \frac{1}{OR}  $$
we have that 
$$ log(OR_{switch}) = log(\frac{1}{OR}) = -log(OR)$$
