Replying to S. van der Linden & M. H. Goldberg Nature Communications (2020)

The point made by van der Linder and Goldberg1 about the impact of variance estimators is valid. Random-effects meta-analysis requires an estimation of heterogeneity, and which estimator is selected matters because it influences the calculation of the effect size2,3. However, there is no universally superior estimator. Whether an estimator is considered more or less biased in simulations studies depends on a variety of parameters such as, for instance, the number of studies included, the number of participants per study (n), and how much this n varies from study to study3,4,5.

The DerSimonian–Laird (DL) estimator is the most popular estimator in the literature2,3,4,5, implemented by default in multiple meta-analysis software4. Therefore, its use facilitates replication by a wide range of audiences with varying levels of technical expertise. Nonetheless, the DL estimator has been challenged when heterogeneity is moderate to high, when meta-analysis includes only a few studies, and/or when it examines binary outcomes3,5,6. Yet, DL has been shown to perform adequately when the number of studies in the meta-analysis is moderate to large (>~30), and estimating continuous outcomes4,6,7—consistent with our data.

Van der Linder and Goldberg note that the high heterogeneity associated with the overall estimate and some subgroups analyses requires additional attention. We follow van der Linder and Goldberg’s valid suggestion and present overall effect sizes comparing multiple estimators (Table 1). These authors also present some alternative estimates, but failed to discuss important nuances in their interpretation.

Table 1 Overall effect size per estimator.

Results show that the overall estimate ranges from d = −0.058 to d = −0.219, depending on the estimator used. The Hunter–Schmidt (HS) and the Sidik–Jonkman (SJ) estimators tend to report, respectively, negative and positive biased estimates compared to other estimators4. Maximum likelihood (ML) and Restricted Maximum Likelihood (REML) have good properties when heterogeneity is high and the size of studies varies considerably5, which applies well to our case. EB (equivalent to Paule–Mandel) has been recommended as an alternative to DL when the number of studies is small, and the heterogeneity is moderate to high3,4,5. Nevertheless, when meta-analysis combines small and large studies (as it is our case), EB tends to produce a positive bias compared with DL and REML7.

Table 1 shows that several estimators produce an overall effect size higher than what we reported. Does this mean our conclusions are wrong? No—our conclusions hold and here is why.

Firstly, our primary main conclusion is that behavioural interventions have a very small average effect size. This conclusion is grounded on Cohen’s d guidelines8—arbitrary to some extent but—followed in the absence of more objective standards. All effect sizes below d = 0.2 are interpreted as very small:9 if two groups’ means differ by <0.2 standard deviations, the difference between them is trivial, even if statistically significant9,10. The highest estimates within a variety of estimators reach, at best, the threshold of the small effect classification (which ranges from d = 0.20 to d = 0.499). Moreover, the estimators that reach this threshold are associated with positive bias when there are large differences in study sizes (which is our case). We call to mind that Cohen’s d does not vary from 0 to 1, but from 0 to infinity, which puts in perspective the small differences between estimators. Moreover, these results can be compared to more intuitive measures of effect size. A Cohen’s d = 0.2 is equivalent to Cohen’s r = 0.110, and it is difficult to argue that correlations r ≤ 0.1 could be interpreted as meaningful. Under several estimators, the probability of benefit11 doubles from 7% to 14%, equivalent to a probability of superiority10 changing from 53% to 56%, not much better than the flip of a coin.

Secondly, the idea that such small effects scaled at the population level can become meaningful is misleading. Interventions scaled up to the general population imply these will target a more heterogeneous set of individuals regarding their motivation to behave pro-environmentally, compared to the small-scale interventions where people self-select to participate12, page 4, 10)—which represent more than half our sample. This suggests that effect sizes would likely approach our estimates for naïve subjects (i.e., no self-selection) (d = −0.040 95% CI−0.103, −0.016).

Thirdly, van der Linder and Goldberg failed to note that random-effects meta-analysis pays more attention to small studies when pooling the overall effect9. Yet, small studies are prone to bias because small studies tend to be published only when reaching significance, and significance in small studies occurs when there are large effect sizes. This is a noteworthy concern because small studies (n ≤ 100) correspond to 57% of our total estimates (k = 82). We make this point in the original paper12, reporting a substantial small-studies bias13 and showing that an analysis restricted to the more robust samples (moderate and large studies) produces overall effect sizes well below d = 0.2. Table 2 shows that this conclusion holds across estimators.

Table 2 Sensitivity analyses per moderator with different estimators (Cohen’s d).

Lastly, our secondary main conclusions also hold. We discouraged the isolated use of information-based interventions and its very small effect size (below d = 0.1) holds across estimators. We encouraged the use of nudges (firstly) and social comparison (secondly). The estimates for nudges hold across estimators ((I2 = 0). These results hold across estimators due to low between-studies heterogeneity. In the case of social norms, there is high between-studies heterogeneity (I2 = 72.2%), generating variation between estimators. But high heterogeneity often originates from substantial differences in sample sizes between the pooled studies14. And social norms are another good example of a strong statistically significant difference between small studies (d = −0.387, n ≤ 100, k = 13), and medium/large studies (d = −0.036, n > 100, k = 19) (meta-regression t-test = 5.21 p < 0.001).

Fundamentally, in case of high heterogeneity, DL provided more conservative estimates compared with other good alternatives, but DL estimates leaned closer to the estimates that would result from restricting the analysis to more robust samples, larger and naïve, less likely to be biased.

In summary, our conclusions hold because most overall effect sizes produced by different estimators are still interpretable as very small or, at best, borderline small. Furthermore, there is a substantial small-studies bias that pushes many estimates up. Notwithstanding, discussions about interventions and their (accurate) effect sizes are imperative and timely. The U.N. declared the next 10 years as the Decade of Action for climate change to deliver on the 2030 promises15, motivated by the awareness that action is not advancing at the speed or scale required, and calling for interventions to step-up their impact.

Stating that effect-sizes in psychology are known to be small should not be used as a justification to inflate the meaningfulness of (very) small effects. A thoughtful debate beyond statistical significance is long overdue to make psychological and behavioural science more relevant to intervention and policy-making.