Is the e(QE) scalar output by Robumeta the correct Q value to use for calculating I-squared?

[Sarah Kathryn]

Robumeta gives e(QE) and uses it to estimate tau-squared. Is this the q-value that should be used to calculate I-squared?

[Tiago Pereira]

Can you please provide us with an example, including a sample of the data? Maybe a fraction of anonymized estimates?

[Sarah Kathryn]

Thank you for your reply!

Let me give a sample and ask the question a little differently. When I run my meta-regression using metareg, I get the following output (below). The metreg results provide a heterogeneity value of e(Q)=115.73. I understand this to be Cochran's Q, which is used (along with the degrees of freedom, 113) to calculate the given I-squared value (25.53%) using (Q-df)/Q.

However, when I run my meta-regression using robumeta, I do not get an "e(Q)" value in my output. I only get a "e(QE) output which Stata define as "used to estimate tau-squared".

Question 1: What, exactly, is the e(QE) value and how/why does it differ from e(Q)?

Question 2: How can I get the e(Q) value for an analysis using robumeta so that I can accurately calculate I-squared values?
________________________

USING METAREG
______________________________
Meta-regression Number of obs = 115
REML estimate of between-study variance tau2 = .1348
% residual variation due to heterogeneity I-squared_res = 25.53%
Proportion of between-study variance explained Adj R-squared = 4.10%
With Knapp-Hartung modification
------------------------------------------------------------------------------
win_es_pe~me | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
explicit | .250679 .195363 1.28 0.202 -.1363704 .6377284
_cons | .5291505 .1689657 3.13 0.002 .1943989 .8639021
------------------------------------------------------------------------------

. ereturn list

scalars:
e(N) = 115
e(tau2) = .134842019909638
e(df_m) = 1
e(Q) = 151.7295974304087
e(df_Q) = 113
e(I2) = .2552540709677435
e(df_r) = 113
e(q_KH) = .2047169831261853
e(remll) = 5.754501557027961
e(remll_c) = -41.19075628042376
e(chi2_c) = 93.89051567490344
e(tau2_0) = .1406036699121426
e(F) = 1.646460068773037

macros:
e(cmd) : "metareg"
e(depvar) : "win_es_per_outcome"
e(predict) : "metareg_p"
e(method) : "REML"
e(wsse) : "se"
e(properties) : "b V"

matrices:
e(b) : 1 x 2
e(V) : 2 x 2

functions:
e(sample)


_____________
USING ROBUMETA
________________
Robust standard error estimation using random model weights
N Level 1 = 101
N Level 2 = 23
Min Level 1 n = 1
Max Level 1 n = 7
Average = 4.39
Assumed rho = 0.80
tau-squared = 0.0406
------------------------------------------------------------------------------
win_es_pe~me | Coef. Std. Err. dfs p-value [95%Conf. Interval]
-------------+----------------------------------------------------------------
_cons | 0.6953 0.0813 19.0825 0.0000 0.5251 0.8655
------------------------------------------------------------------------------

. ereturn list

scalars:
e(N) = 101
e(tau2) = .0406280814393788
e(tau2o) = .0406280814393788
e(QE) = 26.66511372616174

macros:
e(properties) : "b V"
e(depvar) : "win_es_per_outcome"

matrices:
e(b) : 1 x 1
e(V) : 1 x 1
e(dfs) : 1 x 1

[Tiago Pereira]

I realize that my answer is long overdue, but I had the same question today. In the context of meta-regression, the term 'e(QE)' refers to the weighted residual sum of squares, as specified in formula 14 of Hedges et al. (2010) (https://pubmed.ncbi.nlm.nih.gov/26056092/). Assuming independence, the value of e(QE) obtained from an intercept-only model should be similar to the Cochran's Q statistic in a traditional frequentist meta-analysis.

Similarly, assuming independence and covariates, the value of e(QE) in -robumeta- should be comparable to the residual Cochran's Q statistic obtained in -metareg-."