Equal-Tailed Interval (ETI) — eti • bayestestR

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 class 'numeric'
eti(x, ci = 0.95, verbose = TRUE, ...)

# S3 method for class 'data.frame'
eti(x, ci = 0.95, rvar_col = NULL, verbose = TRUE, ...)

# S3 method for class 'brmsfit'
eti(
  x,
  ci = 0.95,
  effects = "fixed",
  component = "conditional",
  parameters = NULL,
  verbose = TRUE,
  ...
)

# S3 method for class 'get_predicted'
eti(x, ci = 0.95, use_iterations = FALSE, 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.

rvar_col

A single character - the name of an rvar column in the data frame to be processed. See example in p_direction().

effects

Should variables for fixed effects ("fixed"), random effects ("random") or both ("all") be returned? Only applies to mixed models. May be abbreviated.

For models of from packages brms or rstanarm there are additional options:

"fixed" returns fixed effects.
"random_variance" return random effects parameters (variance and correlation components, e.g. those parameters that start with sd_ or cor_).
"grouplevel" returns random effects group level estimates, i.e. those parameters that start with r_.
"random" returns both "random_variance" and "grouplevel".
"all" returns fixed effects and random effects variances.
"full" returns all parameters.

component

Which type of parameters to return, such as parameters for the conditional model, the zero-inflated part of the model, the dispersion term, etc. See details in section Model Components. May be abbreviated. Note that the conditional component also refers to the count or mean component - names may differ, depending on the modeling package. There are three convenient shortcuts (not applicable to all model classes):

component = "all" returns all possible parameters.
If component = "location", location parameters such as conditional, zero_inflated, smooth_terms, or instruments are returned (everything that are fixed or random effects - depending on the effects argument - but no auxiliary parameters).
For component = "distributional" (or "auxiliary"), components like sigma, dispersion, beta or precision (and other auxiliary parameters) are returned.

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.

use_iterations

Logical, if TRUE and x is a get_predicted object, (returned by insight::get_predicted()), the function is applied to the iterations instead of the predictions. This only applies to models that return iterations for predicted values (e.g., brmsfit models).

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 zero 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 97.5th 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.

Model components

Possible values for the component argument depend on the model class. Following are valid options:

"all": returns all model components, applies to all models, but will only have an effect for models with more than just the conditional model component.
"conditional": only returns the conditional component, i.e. "fixed effects" terms from the model. Will only have an effect for models with more than just the conditional model component.
"smooth_terms": returns smooth terms, only applies to GAMs (or similar models that may contain smooth terms).
"zero_inflated" (or "zi"): returns the zero-inflation component.
"location": returns location parameters such as conditional, zero_inflated, or smooth_terms (everything that are fixed or random effects - depending on the effects argument - but no auxiliary parameters).
"distributional" (or "auxiliary"): components like sigma, dispersion, beta or precision (and other auxiliary parameters) are returned.

For models of class brmsfit (package brms), even more options are possible for the component argument, which are not all documented in detail here. See also ?insight::find_parameters.

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]
# \donttest{
model <- suppressWarnings(
  rstanarm::stan_glm(mpg ~ wt + gear, data = mtcars, chains = 2, iter = 200, refresh = 0)
)
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

eti(emmeans::emtrends(model, ~1, "wt", data = mtcars))
#> Equal-Tailed Interval
#> 
#> X1      |        95% ETI
#> ------------------------
#> overall | [-6.99, -3.94]

model <- brms::brm(mpg ~ wt + cyl, data = mtcars)
#> Compiling Stan program...
#> Start sampling
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 6e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1:                0.036 seconds (Total)
#> Chain 1: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
#> Chain 2: 
#> Chain 2: Gradient evaluation took 3e-06 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
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#> Chain 2: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3: 
#> Chain 3: Gradient evaluation took 3e-06 seconds
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#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
#> Chain 4: 
#> Chain 4: Gradient evaluation took 3e-06 seconds
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eti(model)
#> Equal-Tailed Interval
#> 
#> Parameter   |        95% ETI | Effects |   Component
#> ----------------------------------------------------
#> b_Intercept | [36.12, 43.25] |   fixed | conditional
#> b_wt        | [-4.78, -1.64] |   fixed | conditional
#> b_cyl       | [-2.38, -0.66] |   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 | [37.42, 41.98] | [36.82, 42.64] | [36.12, 43.25] |   fixed | conditional
#> b_wt        | [-4.19, -2.20] | [-4.45, -1.95] | [-4.78, -1.64] |   fixed | conditional
#> b_cyl       | [-2.06, -0.96] | [-2.19, -0.82] | [-2.38, -0.66] |   fixed | conditional

bf <- BayesFactor::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]
# }