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?

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

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

• The OP might find it simpler to work exclusively in the log scale. Do you want to add edit that into your post? Aug 22 '17 at 15:34
• @mdewey Edit done. Aug 23 '17 at 9:30