Sequential Effect eXistence and sIgnificance Testing (SEXIT)

The SEXIT is a new framework to describe Bayesian effects, guiding which indices to use. Accordingly, the sexit() function returns the minimal (and optimal) required information to describe models' parameters under a Bayesian framework. It includes the following indices:

Centrality: the median of the posterior distribution. In probabilistic terms, there is 50% of probability that the effect is higher and lower. See point_estimate().
Uncertainty: the 95% Highest Density Interval (HDI). In probabilistic terms, there is 95% of probability that the effect is within this confidence interval. See ci().
Existence: The probability of direction allows to quantify the certainty by which an effect is positive or negative. It is a critical index to show that an effect of some manipulation is not harmful (for instance in clinical studies) or to assess the direction of a link. See p_direction().
Significance: Once existence is demonstrated with high certainty, we can assess whether the effect is of sufficient size to be considered as significant (i.e., not negligible). This is a useful index to determine which effects are actually important and worthy of discussion in a given process. See p_significance().
Size: Finally, this index gives an idea about the strength of an effect. However, beware, as studies have shown that a big effect size can be also suggestive of low statistical power (see details section).

Usage

sexit(x, significant = "default", large = "default", ci = 0.95, ...)

Arguments

x: A vector representing a posterior distribution, a data frame of posterior draws (samples be parameter). Can also be a Bayesian model.
significant, large: The threshold values to use for significant and large probabilities. If left to 'default', will be selected through sexit_thresholds(). See the details section below.
ci: Value or vector of probability of the (credible) interval - CI (between 0 and 1) to be estimated. Default to .95 (95%).
...: Currently not used.

Value

A dataframe and text as attribute.

Details

Rationale

The assessment of "significance" (in its broadest meaning) is a pervasive issue in science, and its historical index, the p-value, has been strongly criticized and deemed to have played an important role in the replicability crisis. In reaction, more and more scientists have tuned to Bayesian methods, offering an alternative set of tools to answer their questions. However, the Bayesian framework offers a wide variety of possible indices related to "significance", and the debate has been raging about which index is the best, and which one to report.

This situation can lead to the mindless reporting of all possible indices (with the hopes that with that the reader will be satisfied), but often without having the writer understanding and interpreting them. It is indeed complicated to juggle between many indices with complicated definitions and subtle differences.

SEXIT aims at offering a practical framework for Bayesian effects reporting, in which the focus is put on intuitiveness, explicitness and usefulness of the indices' interpretation. To that end, we suggest a system of description of parameters that would be intuitive, easy to learn and apply, mathematically accurate and useful for taking decision.

Once the thresholds for significance (i.e., the ROPE) and the one for a "large" effect are explicitly defined, the SEXIT framework does not make any interpretation, i.e., it does not label the effects, but just sequentially gives 3 probabilities (of direction, of significance and of being large, respectively) as-is on top of the characteristics of the posterior (using the median and HDI for centrality and uncertainty description). Thus, it provides a lot of information about the posterior distribution (through the mass of different 'sections' of the posterior) in a clear and meaningful way.

Threshold selection

One of the most important thing about the SEXIT framework is that it relies on two "arbitrary" thresholds (i.e., that have no absolute meaning). They are the ones related to effect size (an inherently subjective notion), namely the thresholds for significant and large effects. They are set, by default, to 0.05 and 0.3 of the standard deviation of the outcome variable (tiny and large effect sizes for correlations according to Funder and Ozer, 2019). However, these defaults were chosen by lack of a better option, and might not be adapted to your case. Thus, they are to be handled with care, and the chosen thresholds should always be explicitly reported and justified.

For linear models (lm), this can be generalised to 0.05 * SD_y and 0.3 * SD_y for significant and large effects, respectively.
For logistic models, the parameters expressed in log odds ratio can be converted to standardized difference through the formula π/√(3), resulting a threshold of 0.09 and 0.54.
For other models with binary outcome, it is strongly recommended to manually specify the rope argument. Currently, the same default is applied that for logistic models.
For models from count data, the residual variance is used. This is a rather experimental threshold and is probably often similar to 0.05 and 0.3, but should be used with care!
For t-tests, the standard deviation of the response is used, similarly to linear models (see above).
For correlations,0.05 and 0.3 are used.
For all other models, 0.05 and 0.3 are used, but it is strongly advised to specify it manually.

Examples

The three values for existence, significance and size provide a useful description of the posterior distribution of the effects. Some possible scenarios include:

The probability of existence is low, but the probability of being large is high: it suggests that the posterior is very wide (covering large territories on both side of 0). The statistical power might be too low, which should warrant any confident conclusion.
The probability of existence and significance is high, but the probability of being large is very small: it suggests that the effect is, with high confidence, not large (the posterior is mostly contained between the significance and the large thresholds).
The 3 indices are very low: this suggests that the effect is null with high confidence (the posterior is closely centred around 0).

References

Makowski, D., Ben-Shachar, M. S., & Lüdecke, D. (2019). bayestestR: Describing Effects and their Uncertainty, Existence and Significance within the Bayesian Framework. Journal of Open Source Software, 4(40), 1541. doi:10.21105/joss.01541
Makowski D, Ben-Shachar MS, Chen SHA, Lüdecke D (2019) Indices of Effect Existence and Significance in the Bayesian Framework. Frontiers in Psychology 2019;10:2767. doi:10.3389/fpsyg.2019.02767

Examples

# \donttest{
library(bayestestR)

s <- sexit(rnorm(1000, -1, 1))
s
#> # Following the Sequential Effect eXistence and sIgnificance Testing (SEXIT) framework, we report the median of the posterior distribution and its 95% CI (Highest Density Interval), along the probability of direction (pd), the probability of significance and the probability of being large. The thresholds beyond which the effect is considered as significant (i.e., non-negligible) and large are |0.05| and |0.30|.
#> 
#> The effect (Median = -0.96, 95% CI [-2.80, 1.08]) has a 84.70% probability of being negative (< 0), 83.50% of being significant (< -0.05), and 76.70% of being large (< -0.30)
#> 
#> Median |        95% CI | Direction | Significance (> |0.05|) | Large (> |0.30|)
#> -------------------------------------------------------------------------------
#>  -0.96 | [-2.80, 1.08] |      0.85 |                    0.83 |             0.77
#> 
print(s, summary = TRUE)
#> # The thresholds beyond which the effect is considered as significant (i.e., non-negligible) and large are |0.05| and |0.30|.
#> 
#> The effect (Median = -0.96, 95% CI [-2.80, 1.08]) has 84.70%, 83.50% and 76.70% probability of being negative (< 0), significant (< -0.05) and large (< -0.30)

s <- sexit(iris)
s
#> # Following the Sequential Effect eXistence and sIgnificance Testing (SEXIT) framework, we report the median of the posterior distribution and its 95% CI (Highest Density Interval), along the probability of direction (pd), the probability of significance and the probability of being large. The thresholds beyond which the effect is considered as significant (i.e., non-negligible) and large are |0.05| and |0.30|.
#> 
#> - Sepal.Length (Median = 5.80, 95% CI [4.47, 7.70]) has a 100.00% probability of being positive (> 0), 100.00% of being significant (> 0.05), and 100.00% of being large (> 0.30)
#> - Sepal.Width (Median = 3.00, 95% CI [2.27, 3.93]) has a 100.00% probability of being positive (> 0), 100.00% of being significant (> 0.05), and 100.00% of being large (> 0.30)
#> - Petal.Length (Median = 4.35, 95% CI [1.27, 6.46]) has a 100.00% probability of being positive (> 0), 100.00% of being significant (> 0.05), and 100.00% of being large (> 0.30)
#> - Petal.Width (Median = 1.30, 95% CI [0.10, 2.40]) has a 100.00% probability of being positive (> 0), 100.00% of being significant (> 0.05), and 72.67% of being large (> 0.30)
#> 
#> Parameter    | Median |       95% CI | Direction | Significance (> |0.05|) | Large (> |0.30|)
#> ---------------------------------------------------------------------------------------------
#> Sepal.Length |   5.80 | [4.47, 7.70] |         1 |                       1 |             1.00
#> Sepal.Width  |   3.00 | [2.27, 3.93] |         1 |                       1 |             1.00
#> Petal.Length |   4.35 | [1.27, 6.46] |         1 |                       1 |             1.00
#> Petal.Width  |   1.30 | [0.10, 2.40] |         1 |                       1 |             0.73
#> 
print(s, summary = TRUE)
#> # The thresholds beyond which the effect is considered as significant (i.e., non-negligible) and large are |0.05| and |0.30|.
#> 
#> - Sepal.Length (Median = 5.80, 95% CI [4.47, 7.70]) has 100.00%, 100.00% and 100.00% probability of being positive (> 0), significant (> 0.05) and large (> 0.30)
#> - Sepal.Width (Median = 3.00, 95% CI [2.27, 3.93]) has 100.00%, 100.00% and 100.00% probability of being positive (> 0), significant (> 0.05) and large (> 0.30)
#> - Petal.Length (Median = 4.35, 95% CI [1.27, 6.46]) has 100.00%, 100.00% and 100.00% probability of being positive (> 0), significant (> 0.05) and large (> 0.30)
#> - Petal.Width (Median = 1.30, 95% CI [0.10, 2.40]) has 100.00%, 100.00% and 72.67% probability of being positive (> 0), significant (> 0.05) and large (> 0.30)

if (require("rstanarm")) {
  model <- suppressWarnings(rstanarm::stan_glm(mpg ~ wt * cyl,
    data = mtcars,
    iter = 400, refresh = 0
  ))
  s <- sexit(model)
  s
  print(s, summary = TRUE)
}
#> # The thresholds beyond which the effect is considered as significant (i.e., non-negligible) and large are |0.30| and |1.81| (corresponding respectively to 0.05 and 0.30 of the outcome's SD).
#> 
#> - (Intercept) (Median = 52.52, 95% CI [40.70, 64.08]) has 100.00%, 100.00% and 100.00% probability of being positive (> 0), significant (> 0.30) and large (> 1.81)
#> - wt (Median = -8.04, 95% CI [-12.59, -3.18]) has 100.00%, 99.88% and 99.50% probability of being negative (< 0), significant (< -0.30) and large (< -1.81)
#> - cyl (Median = -3.49, 95% CI [-5.59, -1.60]) has 100.00%, 99.88% and 96.00% probability of being negative (< 0), significant (< -0.30) and large (< -1.81)
#> - wt:cyl (Median = 0.71, 95% CI [0.08, 1.32]) has 98.88%, 89.12% and 0.00% probability of being positive (> 0), significant (> 0.30) and large (> 1.81)
# }