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A Bayesian analysis returns a posterior distribution for each parameter (or effect). To minimally describe these distributions, we recommend reporting a point-estimate of centrality as well as information characterizing the estimation uncertainty (the dispersion). Additionally, one can also report indices of effect existence and/or significance.
Based on the previous comparison of point-estimates and indices of effect existence, we can draw the following recommendations.
We suggest reporting the median as an index of centrality, as it is more robust compared to the mean or the MAP estimate. However, in case of a severely skewed posterior distribution, the MAP estimate could be a good alternative.
The 95% or 89% Credible Intervals (CI) are two reasonable ranges to characterize the uncertainty related to the estimation (see here for a discussion about the differences between these two values). We also recommend computing the CIs based on the HDI rather than quantiles, favouring probable over central values.
Note that a CI based on the quantile (equal-tailed interval) might be more appropriate in case of transformations (for instance when transforming log-odds to probabilities). Otherwise, intervals that originally do not cover the null might cover it after transformation (see here).
The Bayesian framework can neatly delineate and quantify different aspects of hypothesis testing, such as effect existence and significance. The most straightforward index to describe existence of an effect is the Probability of Direction (pd), representing the certainty associated with the most probable direction (positive or negative) of the effect. This index is easy to understand, simple to interpret, straightforward to compute, robust to model characteristics, and independent from the scale of the data.
Moreover, it is strongly correlated with the frequentist p-value, and can thus be used to draw parallels and give some reference to readers non-familiar with Bayesian statistics. A two-sided p-value of respectively .1
, .05
, .01
and .001
correspond approximately to a pd* of 95%, 97.5%, 99.5% and 99.95%.
Thus, for convenience, we suggest the following reference values as an interpretation helpers:
The percentage in ROPE is a index of significance (in its primary meaning), informing us whether a parameter is related or not to a non-negligible change (in terms of magnitude) in the outcome. We suggest reporting the percentage of the full posterior distribution (the full ROPE) instead of a given proportion of CI in the ROPE, which appears to be more sensitive (especially to delineate highly significant effects). Rather than using it as a binary, all-or-nothing decision criterion, such as suggested by the original equivalence test, we recommend using the percentage as a continuous index of significance. However, based on simulation data, we suggest the following reference values as an interpretation helpers:
Note that extra caution is required as its interpretation highly depends on other parameters such as sample size and ROPE range (see here).
Based on these suggestions, a template sentence for minimal reporting of a parameter based on its posterior distribution could be:
“the effect of X has a probability of pd of being negative (Median = median, 89% CI [ HDI_{low} , HDI_{high} ] and can be considered as significant (ROPE% in ROPE).”
Although it can also be used to assess effect existence and significance, the Bayes factor (BF) is a versatile index that can be used to directly compare different models (or data generation processes). The Bayes factor is a ratio that informs us by how much more (or less) likely the observed data are under two compared models - usually a model with versus a model without the effect. Depending on the specifications of the null model (whether it is a point-estimate (e.g., 0) or an interval), the Bayes factor could be used both in the context of effect existence and significance.
In general, a Bayes factor greater than 1 is taken as evidence in favour of one of the model (in the nominator), and a Bayes factor smaller than 1 is taken as evidence in favour of the other model (in the denominator). Several rules of thumb exist to help the interpretation (see here), with > 3 being one common threshold to categorize non-anecdotal evidence.
If you have any advice, opinion or such, we encourage you to let us know by opening an discussion thread or making a pull request.