Compute the **Equal-Tailed Interval (ETI)** of posterior distributions using the quantiles method. The probability of being below this interval is equal to the probability of being above it. The ETI can be used in the context of uncertainty characterisation of posterior distributions as **Credible Interval (CI)**.

eti(x, ...) # S3 method for numeric eti(x, ci = 0.95, verbose = TRUE, ...) # S3 method for data.frame eti(x, ci = 0.95, verbose = TRUE, ...) # S3 method for MCMCglmm eti(x, ci = 0.95, verbose = TRUE, ...) # S3 method for sim.merMod eti( x, ci = 0.95, effects = c("fixed", "random", "all"), parameters = NULL, verbose = TRUE, ... ) # S3 method for sim eti(x, ci = 0.95, parameters = NULL, verbose = TRUE, ...) # S3 method for emmGrid eti(x, ci = 0.95, verbose = TRUE, ...) # S3 method for stanreg eti( x, ci = 0.95, effects = c("fixed", "random", "all"), component = c("location", "all", "conditional", "smooth_terms", "sigma", "distributional", "auxiliary"), parameters = NULL, verbose = TRUE, ... ) # S3 method for brmsfit eti( x, ci = 0.95, effects = c("fixed", "random", "all"), component = c("conditional", "zi", "zero_inflated", "all"), parameters = NULL, verbose = TRUE, ... ) # S3 method for BFBayesFactor eti(x, ci = 0.95, verbose = TRUE, ...)

x | Vector representing a posterior distribution, or a data frame of such
vectors. Can also be a Bayesian model ( |
---|---|

... | Currently not used. |

ci | Value or vector of probability of the (credible) interval - CI (between 0 and 1)
to be estimated. Default to |

verbose | Toggle off warnings. |

effects | Should results for fixed effects, random effects or both be returned? Only applies to mixed models. May be abbreviated. |

parameters | Regular expression pattern that describes the parameters that
should be returned. Meta-parameters (like |

component | Should results for all parameters, parameters for the conditional model or the zero-inflated part of the model be returned? May be abbreviated. Only applies to brms-models. |

A data frame with following columns:

`Parameter`

The model parameter(s), if`x`

is a model-object. If`x`

is a vector, this column is missing.`CI`

The probability of the credible interval.`CI_low`

,`CI_high`

The lower and upper credible interval limits for the parameters.

Unlike equal-tailed intervals (see `eti()`

) that typically exclude 2.5%
from each tail of the distribution and always include the median, the HDI is
*not* equal-tailed and therefore always includes the mode(s) of posterior
distributions.

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).

The 89% intervals (`ci = 0.89`

) are deemed to be more stable than, for
instance, 95% intervals (Kruschke, 2014).
An effective sample size of at least 10.000 is recommended if one wants to estimate
95% intervals with high precision (Kruschke, 2014, p. 183ff). Unfortunately, the default number of posterior samples for most Bayes packages (e.g., `rstanarm` or `brms`) is only 4.000 (thus, you might want to increase it when fitting your model). Moreover,
89 indicates the arbitrariness of interval limits - its only remarkable property is being
the highest prime number that does not exceed the already unstable 95%
threshold (McElreath, 2015).

However, 95% has some advantages too.
For instance, it shares (in the case of a normal posterior distribution) an intuitive relationship with the
standard deviation and it conveys a more accurate image of the (artificial)
bounds of the distribution. Also, because it is wider, it makes analyses more
conservative (i.e., the probability of covering 0 is larger for the 95% CI than
for lower ranges such as 89%), which is a good thing in the context of the
reproducibility crisis.

A 95% equal-tailed interval (ETI) has 2.5% of the distribution on either
side of its limits. It indicates the 2.5th percentile and the 97.5h percentile.
In symmetric distributions, the two methods of computing credible intervals,
the ETI and the HDI, return similar results.

This is not the case for skewed distributions. Indeed, it is possible that
parameter values in the ETI have lower credibility (are less probable) than
parameter values outside the ETI. This property seems undesirable as a summary
of the credible values in a distribution.

On the other hand, the ETI range does change when transformations are applied
to the distribution (for instance, for a log odds scale to probabilities):
the lower and higher bounds of the transformed distribution will correspond
to the transformed lower and higher bounds of the original distribution.
On the contrary, applying transformations to the distribution will change
the resulting HDI.

#> 95% ETI: [-2.07, 2.04]#> Equal-Tailed Interval #> #> 80% ETI | 89% ETI | 95% ETI #> --------------------------------------------- #> [-1.31, 1.31] | [-1.65, 1.70] | [-2.07, 2.04]#> Equal-Tailed Interval #> #> Parameter | 95% ETI #> ------------------------- #> X1 | [-2.10, 2.02] #> X2 | [-2.22, 1.98] #> X3 | [-2.32, 2.15] #> X4 | [-1.87, 2.00]#> Equal-Tailed Interval #> #> Parameter | 80% ETI | 89% ETI | 95% ETI #> --------------------------------------------------------- #> X1 | [-1.51, 1.40] | [-1.83, 1.62] | [-2.10, 2.02] #> X2 | [-1.44, 1.36] | [-1.59, 1.53] | [-2.22, 1.98] #> X3 | [-1.46, 1.01] | [-1.97, 1.19] | [-2.32, 2.15] #> X4 | [-1.51, 1.49] | [-1.80, 1.77] | [-1.87, 2.00]if (FALSE) { library(rstanarm) model <- stan_glm(mpg ~ wt + gear, data = mtcars, chains = 2, iter = 200, refresh = 0) eti(model) eti(model, ci = c(.80, .89, .95)) library(emmeans) eti(emtrends(model, ~1, "wt")) library(brms) model <- brms::brm(mpg ~ wt + cyl, data = mtcars) eti(model) eti(model, ci = c(.80, .89, .95)) library(BayesFactor) bf <- ttestBF(x = rnorm(100, 1, 1)) eti(bf) eti(bf, ci = c(.80, .89, .95)) }