# [R-meta] Transformation of meta-regression coefficients to ORs

Viechtbauer, Wolfgang (SP) wo||g@ng@v|echtb@uer @end|ng |rom m@@@tr|chtun|ver@|ty@n|
Tue Jul 2 11:49:49 CEST 2019

```Dear Dirk,

What effect size / outcome measure are you using for your analysis? If you are meta-analyzing logit transformed proportions, then exponentiating a beta coefficient will give you the odds ratio for the corresponding predictor. For example:

library(metafor)

dat <- escalc(measure="PLO", xi=xi, ni=ni, data=dat.debruin2009)
dat

res <- rma(yi, vi, mods = ~ scq + ethnicity + patients + select + sens, data=dat)
res

round(exp(coef(summary(res))[-1,c("estimate", "ci.lb", "ci.ub")]), 2)

estimate ci.lb ci.ub
scq                  1.06  1.02  1.10
ethnicityother       0.39  0.19  0.80
patientsstarting     1.41  0.81  2.46
selectyes            0.72  0.38  1.38
sens<400             0.66  0.33  1.31

So, per one-unit increase in SCQ, the odds (in this case, of having an undetectable viral load) goes up by 6% (95% CI: 2% to 10%), or in other words, the odds ratio is 1.06 (95% CI: 1.02 to 1.10). Since we are comparing studies with different levels of SCQ, it might be more appropriate to say that in studies where SCQ is one point higher, the odds of an undetectable viral load are 6% higher (and if we want to be even more precise, we really should that "are on average 6% higher"). Similarly, in studies where the majority of participants are non-Caucasian, the odds are on average 61% lower (95% CI: 20% to 81%), or in other words, the odds ratio is 0.39 (95% CI: 0.19 to 0.80).

But this only works for logit transformed proportions as the outcome.

For other ratio-type outcome measures that are log transformed (e.g., log risk ratios and log odds ratios), exponentiating a coefficient gives you a ratio as well, but it is actually the ratio of two ratios. For example:

dat <- escalc(measure="OR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg)
dat

res <- rma(yi, vi, mods = ~ ablat + year, data=dat)
res

round(exp(coef(summary(res))[-1,c("estimate", "ci.lb", "ci.ub")]), 2)

estimate ci.lb ci.ub
ablat     0.97  0.95  0.99
year      1.01  0.98  1.03

So, in studies that are one more degree away from the equator (i.e., where absolute latitude increases by one unit), the odds ratio (i.e., the ratio of a TB infection in those vaccinated compared to those not vaccinated) goes down on average by 3% (95% CI: 1% to 5%), or the ratio of odds ratios is 0.97 (95% CI: 0.95 to 0.99).

Those are essentially the two cases where you can go from the coefficients to either an odds ratio or a ratio of odds ratios (the latter can also be done, for example, with risk ratios -- then you get a ratio of risk ratios).

Best,
Wolfgang

-----Original Message-----
From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org] On Behalf Of dirk.richter using upd.unibe.ch
Sent: Saturday, 29 June, 2019 16:52
To: r-sig-meta-analysis using r-project.org
Subject: [R-meta] Transformation of meta-regression coefficients to ORs

Dear mailing list members

A reviewer of a paper I have authored has requested to transform the beta
estimates of a meta-regression result into odds ratios. The analysis was
conducted with the metafor rma.mv function. Although I understand the point
of this suggestion, I have looked into several textbooks and review articles
Before I start searching for a code to do this, I would like to kindly ask
for opinions on this issue. And if it is reasonable to do so I would be
grateful to see a coding example.

Dirk Richter

UNIVERSITÄRE PSYCHIATRISCHE DIENSTE BERN (UPD) AG
ZENTRUM PSYCHIATRISCHE REHABILITATION

Dirk Richter, Dr. phil. habil.
Leiter Forschung und Entwicklung
Murtenstrasse 46
CH-3008 Bern
Tel. +41 31 632 4707
Mobil + 41 76 717 5220
E-Mail: dirk.richter using upd.unibe.ch

Bern University Hospital for Mental Health
Center for Psychiatric Rehabilitation
Dirk Richter, Dr. phil., PhD