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).

## Usage

eti(x, ...)

# S3 method for numeric
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,
...
)

## Arguments

x

Vector representing a posterior distribution, or a data frame of such vectors. Can also be a Bayesian model. bayestestR supports a wide range of models (see, for example, methods("hdi")) and not all of those are documented in the 'Usage' section, because methods for other classes mostly resemble the arguments of the .numeric or .data.framemethods.

...

Currently not used.

ci

Value or vector of probability of the (credible) interval - CI (between 0 and 1) to be estimated. Default to .95 (95%).

verbose

Toggle off warnings.

effects

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

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.

parameters

Regular expression pattern that describes the parameters that should be returned. Meta-parameters (like lp__ or prior_) are filtered by default, so only parameters that typically appear in the summary() are returned. Use parameters to select specific parameters for the output.

## Value

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.

## Details

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. While this can be useful to better represent the credibility mass of a distribution, the HDI also has some limitations. See spi() for details.

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.

Other ci: bci(), ci(), cwi(), hdi(), si(), spi()

## Examples

library(bayestestR)

posterior <- rnorm(1000)
eti(posterior)
#> 95% ETI: [-1.93, 1.84]
eti(posterior, ci = c(0.80, 0.89, 0.95))
#> Equal-Tailed Interval
#>
#> 80% ETI       |       89% ETI |       95% ETI
#> ---------------------------------------------
#> [-1.29, 1.25] | [-1.58, 1.53] | [-1.93, 1.84]

df <- data.frame(replicate(4, rnorm(100)))
eti(df)
#> Equal-Tailed Interval
#>
#> Parameter |       95% ETI
#> -------------------------
#> X1        | [-1.93, 2.19]
#> X2        | [-1.70, 1.96]
#> X3        | [-1.91, 1.63]
#> X4        | [-1.87, 1.87]
eti(df, ci = c(0.80, 0.89, 0.95))
#> Equal-Tailed Interval
#>
#> Parameter |       80% ETI |       89% ETI |       95% ETI
#> ---------------------------------------------------------
#> X1        | [-1.16, 1.28] | [-1.74, 1.78] | [-1.93, 2.19]
#> X2        | [-0.96, 1.62] | [-1.40, 1.70] | [-1.70, 1.96]
#> X3        | [-1.07, 0.88] | [-1.52, 1.15] | [-1.91, 1.63]
#> X4        | [-1.20, 1.18] | [-1.67, 1.47] | [-1.87, 1.87]
# \dontrun{
library(rstanarm)
model <- stan_glm(mpg ~ wt + gear, data = mtcars, chains = 2, iter = 200, refresh = 0)
#> Warning: The largest R-hat is 1.06, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#r-hat
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
eti(model)
#> Equal-Tailed Interval
#>
#> Parameter   |        95% ETI | Effects |   Component
#> ----------------------------------------------------
#> (Intercept) | [29.80, 50.38] |   fixed | conditional
#> wt          | [-6.99, -3.94] |   fixed | conditional
#> gear        | [-2.21,  1.24] |   fixed | conditional
eti(model, ci = c(0.80, 0.89, 0.95))
#> Equal-Tailed Interval
#>
#> Parameter   |        80% ETI |        89% ETI |        95% ETI | Effects |   Component
#> --------------------------------------------------------------------------------------
#> (Intercept) | [32.43, 45.66] | [31.18, 47.98] | [29.80, 50.38] |   fixed | conditional
#> wt          | [-6.44, -4.62] | [-6.66, -4.35] | [-6.99, -3.94] |   fixed | conditional
#> gear        | [-1.52,  0.79] | [-1.88,  0.99] | [-2.21,  1.24] |   fixed | conditional

library(emmeans)
eti(emtrends(model, ~1, "wt"))
#> Equal-Tailed Interval
#>
#> Parameter |        95% ETI
#> --------------------------
#> overall   | [-6.99, -3.94]

library(brms)
model <- brms::brm(mpg ~ wt + cyl, data = mtcars)
#> Compiling Stan program...
#> Start sampling
#>
#> SAMPLING FOR MODEL '2d19b3a372313df641edf05db5e9f303' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 1.3e-05 seconds
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#>
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#> Chain 2: Gradient evaluation took 8e-06 seconds
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#> SAMPLING FOR MODEL '2d19b3a372313df641edf05db5e9f303' NOW (CHAIN 3).
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eti(model)
#> Equal-Tailed Interval
#>
#> Parameter   |        95% ETI | Effects |   Component
#> ----------------------------------------------------
#> b_Intercept | [36.22, 43.28] |   fixed | conditional
#> b_wt        | [-4.81, -1.65] |   fixed | conditional
#> b_cyl       | [-2.37, -0.62] |   fixed | conditional
eti(model, ci = c(0.80, 0.89, 0.95))
#> Equal-Tailed Interval
#>
#> Parameter   |        80% ETI |        89% ETI |        95% ETI | Effects |   Component
#> --------------------------------------------------------------------------------------
#> b_Intercept | [37.48, 41.93] | [36.89, 42.57] | [36.22, 43.28] |   fixed | conditional
#> b_wt        | [-4.21, -2.19] | [-4.48, -1.93] | [-4.81, -1.65] |   fixed | conditional
#> b_cyl       | [-2.07, -0.95] | [-2.21, -0.80] | [-2.37, -0.62] |   fixed | conditional

library(BayesFactor)
bf <- ttestBF(x = rnorm(100, 1, 1))
eti(bf)
#> Equal-Tailed Interval
#>
#> Parameter  |      95% ETI
#> -------------------------
#> Difference | [0.85, 1.26]
eti(bf, ci = c(0.80, 0.89, 0.95))
#> Equal-Tailed Interval
#>
#> Parameter  |      80% ETI |      89% ETI |      95% ETI
#> -------------------------------------------------------
#> Difference | [0.92, 1.19] | [0.89, 1.22] | [0.85, 1.26]
# }