# [R-meta] Back transformation of double arscine transformed estimates in metafor

Wolfgang Viechtbauer wo||g@ng@v|echtb@uer @end|ng |rom m@@@tr|chtun|ver@|ty@n|
Fri Oct 4 13:30:39 CEST 2019

1) The logistic model also doesn't require adjustments to the counts, so
yes, it has the same advantage as the PFT (and PAS) transformation.

2) You are doing a meta-analysis of proportions, but the analysis is
carried out on a transformed scale (like with PFT). When you use a
logistic model, you are (implicitly) doing the analysis on a logit scale.
For easier interpretation, we then typically transform the results back to
odds or directly to proportions.

Best,
Wolfgang

> A few questions remain:
> 1. The reason for using FT transformation was due to many outcomes being
> equal to 0 in proportions and thereby to avoid adding numbers that distort
> the estimates. Would the PAS or PLO have the same advantages as the FT
> transformation in this regard?
> 2. The meta-regression is performed on proportions and not on relative
> risks or odds. Can a meta-regression on proportions be performed in PAS or
> PLO?
>
> Regards,
> Daniel
>
> fre. 4. okt. 2019 11.35 skrev Viechtbauer, Wolfgang (SP) <
> wolfgang.viechtbauer using maastrichtuniversity.nl>:
>
>> Dear Daniel,
>>
>> predict(metareg, transf=transf.ipft.hm, targ=list(ni=a\$total)) gives you
>> the fitted values (proportions) for the 11 studies. So these are the
>> predicted values (based on the model) for whatever values these studies
>> take on for the moderator variables. If you want to compute predicted
>> values for other combinations of moderator values, you need to use the
>> 'newmods' argument. For example:
>>
>> predict(metareg, newmods = c(0,1,1,0),
>>         transf=transf.ipft.hm, targ=list(ni=a\$total))
>>
>> will give the predicted value (proportion) for continent = North America,
>> age = infant, and pm = LA (based on your post on Stack Overflow --
>> https://stackoverflow.com/questions/58203464/reverse-transformation-of-double-arscine-transformed-estimates-when-doing-meta-r
>> -- I can see that pm has two levels, LA (reference level) and TA).
>>
>> By varying one moderator and holding the other moderators constant, you
>> can illustrate how a moderator affects the results (you cannot just take
>> the model coefficients and transform them).
>>
>> But: The back-transformation for the FT transformation is problematic.
>> Please take a look at:
>>
>> https://onlinelibrary.wiley.com/doi/full/10.1002/jrsm.1348
>>
>> Even though the FT transformation has some nice properties, I would
>> therefore avoid it (because we typically do want to back-transform in the
>> end). You could either just use the 'standard' arcsine square root
>> transformation (measure="PAS") or maybe even better switch to a logistic
>> mixed-effects model, which you can fit with rma.glmm():
>>
>> metareg <- rma.glmm(measure="PLO", ai=compl, nu=total, data=a,
>>                     mods = ~ continent + age + pm)
>>
>> The results are then analyzed on the logit (log odds) scale. So, the
>> back-transformation to odds would be:
>>
>> predict(metareg, newmods = c(0,1,1,0), transf=exp)
>>
>> In fact, here, you can exponentiate the coefficients themselves, which
>> then reflect odds ratios:
>>
>> round(exp(coef(summary(metareg))[,c("estimtate", "ci.lb", "ci.ub")]), 3)
>>
>> The back-transformation to proportions would be:
>>
>> predict(metareg, newmods = c(0,1,1,0), transf=transf.ilogit)
>>
>> Best,
>> Wolfgang
>>
>> -----Original Message-----
>> From: R-sig-meta-analysis [mailto:
>> r-sig-meta-analysis-bounces using r-project.org] On Behalf Of Daniel Mønsted
>> Sent: Friday, 04 October, 2019 10:47
>> To: r-sig-meta-analysis using r-project.org
>> Subject: [R-meta] Back transformation of double arscine transformed
>> estimates in metafor
>>
>> Hey
>>
>> I am performing a meta-regression of multiple single arm
>> studies. The outcome is proportions of complications following a specific
>> surgical treatment which is the same for all included studies. I want to
>> explore if variables such as age, continent or medications have an impact
>> on the outcome. Since some of the identified studies have 0 complications
>> events I have performed Freeman-Tuckey double arscine transformation of
>> data.
>>
>> Data transformation
>> b<-escalc(xi=compl, ni=total, data=a, measure="PFT", add=0)
>>
>> Meta-regression of multiple identified studies
>> metareg<-rma(yi, vi, data=b, mods=~continent+age+pm)
>> print(metareg)
>>
>> Mixed-Effects Model (k = 11; tau^2 estimator: REML)
>>
>> tau^2 (estimated amount of residual heterogeneity):     0.0091 (SE =
>> 0.0060)
>> tau (square root of estimated tau^2 value):             0.0952
>> I^2 (residual heterogeneity / unaccounted variability): 91.15%
>> H^2 (unaccounted variability / sampling variability):   11.30
>> R^2 (amount of heterogeneity accounted for):            28.85%
>>
>> Test for Residual Heterogeneity:
>> QE(df = 6) = 78.3204, p-val < .0001
>>
>> Test of Moderators (coefficient(s) 2:5):
>> QM(df = 4) = 7.6936, p-val = 0.1035
>>
>> Model Results:
>>
>>                         estimate      se     zval    pval    ci.lb   ci.ub
>> intrcpt                   0.3197  0.1079   2.9640  0.0030   0.1083
>> 0.5311  **
>> continentAsia            -0.1666  0.1062  -1.5685  0.1168  -0.3747  0.0416
>> continentNorth America   -0.1755  0.1067  -1.6452  0.0999  -0.3845
>> 0.0336   .
>> ageinfant                 0.1824  0.0741   2.4616  0.0138   0.0372
>> 0.3277   *
>> pmTA                     -0.1484  0.0973  -1.5252  0.1272  -0.3392  0.0423
>>
>> ---
>> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>>
>> These estimates and CI are transformed. Usually I would transform them back
>> to proportions with predict in presence of simple models. But I am not sure
>> hos to do it in multiple models.
>>
>> predict(metareg, transf=transf.ipft.hm, targ=list(ni=a\$total)). This gives
>> us multiple lines of estimates which I cannot interpretate:
>>
>>      pred  ci.lb  ci.ub  cr.lb  cr.ub
>> 1  0.0259 0.0017 0.0715 0.0000 0.1388
>> 2  0.0202 0.0005 0.0594 0.0000 0.1245
>> 3  0.0000 0.0000 0.0348 0.0000 0.0692
>> 4  0.0202 0.0005 0.0594 0.0000 0.1245
>> 5  0.1058 0.0290 0.2206 0.0056 0.2976
>> 6  0.0175 0.0000 0.0940 0.0000 0.1478
>> 7  0.1174 0.0380 0.2310 0.0100 0.3110
>> 8  0.0202 0.0005 0.0594 0.0000 0.1245
>> 9  0.0283 0.0000 0.1236 0.0000 0.1799
>> 10 0.0259 0.0017 0.0715 0.0000 0.1388
>> 11 0.0259 0.0017 0.0715 0.0000 0.1388
>>
>> How do I obtain estimates of proportions for the impact of each variable
>> explored in the model?
>>
>> Regards,
>> Daniel