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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!

Giacomo Margelli
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1 Answers1

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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

$$ log(OR_{switch}) = log(\frac{1}{OR}) = -log(OR)$$

Łukasz Deryło
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