In response to a growing body of statisticians and researchers that criticize the utility of null-hypothesis testing (NHT) for science as a cumulative endeavour, the American Psychological Association Task Force on Statistical Inference avoided an outright ban on NHT, but instead suggested that researchers report effect sizes in addition to p-values derived from NHT.
However, effect sizes are not easily accumulated across studies. Meta-analytic approaches can accumulate distributions of effect sizes, but effect sizes are typically computed as a ratio between raw effect magnitude and unexplained "noise" in the data of a given experiment, meaning that the distribution of effect sizes is affected not only by the variability in the raw magnitude of the effect across studies, but also variability in the manifestation of noise across studies.
In contrast, an alternative measure of effect strength, likelihood ratios, permit both intuitive interpretation on a study-by-study basis, and can be easily aggregated across studies for meta-analysis. Within each study, the likelihood represents the weight of evidence for a model containing a given effect relative to a model that does not contain the effect, and could typically be reported as, for example, "Computation of a likelihood ratio for the effect of X revealed 8 times more evidence for the effect than for its respective null". Furthermore, the likelihood ratio also permits intuitive representation of the strength of null findings insofar as likelihood ratios below 1 represent scenarios where the null is favoured and taking the reciprocal of this value represents the weight of evidence for the null over the effect. Notably, the likelihood ratio is represented mathematically as the ratio of unexplained variances of the two models, which differ only in the variance explained by the effect and thus is not a huge conceptual departure from an effect size. On the other hand, computation of a meta-analytic likelihood ratio, representing the weight of evidence for an effect across studies, is simply a matter of taking the product of likelihood ratios across studies.
Thus, I argue that for science seeking to establish the degree of gross evidence in favour of a effect/model, likelihood ratios are the way to go.
There are more nuanced cases where models are differentiable only in the specific size of an effect, in which case some sort of representation of the interval over which we believe the data are consistent with effect parameter values might be preferred. Indeed, the APA task force also recommends reporting confidence intervals, which can be used to this end, but I suspect that this is also an ill-considered approach.
Confidence intervals are lamentably often misinterpreted (by students and researchers alike). I also fear that their ability for use in NHT (by assessment of inclusion of zero within the CI) will only serve to further delay the extinction of NHT as an inferential practice.
Instead, when theories are differentiable only by the size of effects, I suggest that Bayesian approach would be more appropriate, where the prior distribution of each effect is defined by each model separately, and the resulting posterior distributions are compared.
Does this approach, replacing p-values, effect sizes and confidence intervals with likelihood ratios and, if necessary, Bayesian model comparison, seem sufficient? Does it miss out on some necessary inferential feature that the here-maligned alternatives provide?