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The R package metafor offers various ways of back-transforming the results of a meta-analysis with a transformed effect size/outcome variable. In this case, I will refer to a reliability generalization meta-analysis using a random-effects model and coefficient alpha as the outcome variable, which has been transformed using with a Bonett transfromation 'ABT' using escalc.

The pooled estimate can be back-transformed using the predict() and transf functions. For example,

predict(model_name, transf=transf.iabt)

will provide a back-tranformed point estimate and confidence intervals.

My question is: How can one back-transform other aspects of an rma output that are not metric-free, such as tau and its confidence intervals, using the metafor package/R? I have tried summary(model_name, transf=transf.iabt) but that gives me the same results as summary(model_name). Perhaps I have misunderstood the calculation of tau-square, hence tau, which may be something that is not so easily 'back-transformable'.

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You don't need to back-transform the estimate of $\tau^2$. predict() gives you the prediction interval and its bounds can be back-transformed, so that tells you how much heterogeneity there is in the back-transformed scale.

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  • $\begingroup$ Much appreciated, Prof V. So you are suggesting to use the prediction interval as an index of heterogeneity (instead of tau, for instance)? Isn't tau quite informative too though, like an SD of sorts? One could in fact solve for the back-transformed tau value using the back-transformed prediction interval (e.g., 1.96*tau = upper/lower bound - point estimate). $\endgroup$
    – PyjamaNinja
    Commented Aug 27, 2021 at 14:05
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    $\begingroup$ 1) Yes, you can use tau as well, but you were asking about back-transformations. You can easily back-transform the bounds of the PI, but back-transforming tau is tricky and not something that is typically done. 2) The back-transformation is non-linear, so the back-transformed CI/PI is not symmetric and hence back-calculating tau this way is not the right way of doing this. In any case, don't worry about back-transforming tau - just look at the back-transformed bounds of the PI. $\endgroup$
    – Wolfgang
    Commented Aug 27, 2021 at 14:34
  • $\begingroup$ Will do, Prof. Thanks again for your time. $\endgroup$
    – PyjamaNinja
    Commented Aug 27, 2021 at 14:47

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