Compute Support Intervals

A support interval contains only the values of the parameter that predict the observed data better than average, by some degree k; these are values of the parameter that are associated with an updating factor greater or equal than k. From the perspective of the Savage-Dickey Bayes factor, testing against a point null hypothesis for any value within the support interval will yield a Bayes factor smaller than 1/k.

Usage

si(posterior, prior = NULL, BF = 1, verbose = TRUE, ...)

# S3 method for numeric
si(posterior, prior = NULL, BF = 1, verbose = TRUE, ...)

# S3 method for stanreg
si(
  posterior,
  prior = NULL,
  BF = 1,
  verbose = TRUE,
  effects = c("fixed", "random", "all"),
  component = c("location", "conditional", "all", "smooth_terms", "sigma", "auxiliary",
    "distributional"),
  parameters = NULL,
  ...
)

# S3 method for brmsfit
si(
  posterior,
  prior = NULL,
  BF = 1,
  verbose = TRUE,
  effects = c("fixed", "random", "all"),
  component = c("location", "conditional", "all", "smooth_terms", "sigma", "auxiliary",
    "distributional"),
  parameters = NULL,
  ...
)

# S3 method for blavaan
si(
  posterior,
  prior = NULL,
  BF = 1,
  verbose = TRUE,
  effects = c("fixed", "random", "all"),
  component = c("location", "conditional", "all", "smooth_terms", "sigma", "auxiliary",
    "distributional"),
  parameters = NULL,
  ...
)

# S3 method for emmGrid
si(posterior, prior = NULL, BF = 1, verbose = TRUE, ...)

# S3 method for data.frame
si(posterior, prior = NULL, BF = 1, verbose = TRUE, ...)

Arguments

posterior: A numerical vector, stanreg / brmsfit object, emmGrid or a data frame - representing a posterior distribution(s) from (see 'Details').
prior: An object representing a prior distribution (see 'Details').
BF: The amount of support required to be included in the support interval.
verbose: Toggle off warnings.
...: Arguments passed to and from other methods. (Can be used to pass arguments to internal logspline::logspline().)
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 containing the lower and upper bounds of the SI.

Note that if the level of requested support is higher than observed in the data, the interval will be [NA,NA].

Details

For more info, in particular on specifying correct priors for factors with more than 2 levels, see the Bayes factors vignette.

This method is used to compute support intervals based on prior and posterior distributions. For the computation of support intervals, the model priors must be proper priors (at the very least they should be not flat, and it is preferable that they be informative - note that by default, brms::brm() uses flat priors for fixed-effects; see example below).

Note

There is also a plot()-method implemented in the see-package.

Choosing a value of `BF`

The choice of BF (the level of support) depends on what we want our interval to represent:

A BF = 1 contains values whose credibility is not decreased by observing the data.
A BF > 1 contains values who received more impressive support from the data.
A BF < 1 contains values whose credibility has not been impressively decreased by observing the data. Testing against values outside this interval will produce a Bayes factor larger than 1/BF in support of the alternative. E.g., if an SI (BF = 1/3) excludes 0, the Bayes factor against the point-null will be larger than 3.

Setting the correct `prior`

For the computation of Bayes factors, the model priors must be proper priors (at the very least they should be not flat, and it is preferable that they be informative); As the priors for the alternative get wider, the likelihood of the null value(s) increases, to the extreme that for completely flat priors the null is infinitely more favorable than the alternative (this is called the Jeffreys-Lindley-Bartlett paradox). Thus, you should only ever try (or want) to compute a Bayes factor when you have an informed prior.

(Note that by default, brms::brm() uses flat priors for fixed-effects; See example below.)

It is important to provide the correct prior for meaningful results.

When posterior is a numerical vector, prior should also be a numerical vector.
When posterior is a data.frame, prior should also be a data.frame, with matching column order.
When posterior is a stanreg, brmsfit or other supported Bayesian model:
- prior can be set to NULL, in which case prior samples are drawn internally.
- prior can also be a model equivalent to posterior but with samples from the priors only. See unupdate().
- Note: When posterior is a brmsfit_multiple model, prior must be provided.
When posterior is an emmGrid / emm_list object:
- prior should also be an emmGrid / emm_list object equivalent to posterior but created with a model of priors samples only. See unupdate().
- prior can also be the original (posterior) model. If so, the function will try to update the emmGrid / emm_list to use the unupdate()d prior-model. (This cannot be done for brmsfit models.)
- Note: When the emmGrid has undergone any transformations ("log", "response", etc.), or regriding, then prior must be an emmGrid object, as stated above.

References

Wagenmakers, E., Gronau, Q. F., Dablander, F., & Etz, A. (2018, November 22). The Support Interval. doi:10.31234/osf.io/zwnxb

Examples

library(bayestestR)

prior <- distribution_normal(1000, mean = 0, sd = 1)
posterior <- distribution_normal(1000, mean = 0.5, sd = 0.3)

si(posterior, prior, verbose = FALSE)
#> BF = 1 SI: [0.04, 1.04]
# \dontrun{
# rstanarm models
# ---------------
library(rstanarm)
contrasts(sleep$group) <- contr.equalprior_pairs # see vignette
stan_model <- stan_lmer(extra ~ group + (1 | ID), data = sleep)
#> 
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 7.2e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.72 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1: 
#> Chain 1: 
#> Chain 1: Iteration:    1 / 2000 [  0%]  (Warmup)
#> Chain 1: Iteration:  200 / 2000 [ 10%]  (Warmup)
#> Chain 1: Iteration:  400 / 2000 [ 20%]  (Warmup)
#> Chain 1: Iteration:  600 / 2000 [ 30%]  (Warmup)
#> Chain 1: Iteration:  800 / 2000 [ 40%]  (Warmup)
#> Chain 1: Iteration: 1000 / 2000 [ 50%]  (Warmup)
#> Chain 1: Iteration: 1001 / 2000 [ 50%]  (Sampling)
#> Chain 1: Iteration: 1200 / 2000 [ 60%]  (Sampling)
#> Chain 1: Iteration: 1400 / 2000 [ 70%]  (Sampling)
#> Chain 1: Iteration: 1600 / 2000 [ 80%]  (Sampling)
#> Chain 1: Iteration: 1800 / 2000 [ 90%]  (Sampling)
#> Chain 1: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 1: 
#> Chain 1:  Elapsed Time: 0.34 seconds (Warm-up)
#> Chain 1:                0.371 seconds (Sampling)
#> Chain 1:                0.711 seconds (Total)
#> Chain 1: 
#> 
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 2).
#> Chain 2: 
#> Chain 2: Gradient evaluation took 2.4e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.24 seconds.
#> Chain 2: Adjust your expectations accordingly!
#> Chain 2: 
#> Chain 2: 
#> Chain 2: Iteration:    1 / 2000 [  0%]  (Warmup)
#> Chain 2: Iteration:  200 / 2000 [ 10%]  (Warmup)
#> Chain 2: Iteration:  400 / 2000 [ 20%]  (Warmup)
#> Chain 2: Iteration:  600 / 2000 [ 30%]  (Warmup)
#> Chain 2: Iteration:  800 / 2000 [ 40%]  (Warmup)
#> Chain 2: Iteration: 1000 / 2000 [ 50%]  (Warmup)
#> Chain 2: Iteration: 1001 / 2000 [ 50%]  (Sampling)
#> Chain 2: Iteration: 1200 / 2000 [ 60%]  (Sampling)
#> Chain 2: Iteration: 1400 / 2000 [ 70%]  (Sampling)
#> Chain 2: Iteration: 1600 / 2000 [ 80%]  (Sampling)
#> Chain 2: Iteration: 1800 / 2000 [ 90%]  (Sampling)
#> Chain 2: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 2: 
#> Chain 2:  Elapsed Time: 0.302 seconds (Warm-up)
#> Chain 2:                0.268 seconds (Sampling)
#> Chain 2:                0.57 seconds (Total)
#> Chain 2: 
#> 
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 3).
#> Chain 3: 
#> Chain 3: Gradient evaluation took 2.2e-05 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.22 seconds.
#> Chain 3: Adjust your expectations accordingly!
#> Chain 3: 
#> Chain 3: 
#> Chain 3: Iteration:    1 / 2000 [  0%]  (Warmup)
#> Chain 3: Iteration:  200 / 2000 [ 10%]  (Warmup)
#> Chain 3: Iteration:  400 / 2000 [ 20%]  (Warmup)
#> Chain 3: Iteration:  600 / 2000 [ 30%]  (Warmup)
#> Chain 3: Iteration:  800 / 2000 [ 40%]  (Warmup)
#> Chain 3: Iteration: 1000 / 2000 [ 50%]  (Warmup)
#> Chain 3: Iteration: 1001 / 2000 [ 50%]  (Sampling)
#> Chain 3: Iteration: 1200 / 2000 [ 60%]  (Sampling)
#> Chain 3: Iteration: 1400 / 2000 [ 70%]  (Sampling)
#> Chain 3: Iteration: 1600 / 2000 [ 80%]  (Sampling)
#> Chain 3: Iteration: 1800 / 2000 [ 90%]  (Sampling)
#> Chain 3: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 3: 
#> Chain 3:  Elapsed Time: 0.328 seconds (Warm-up)
#> Chain 3:                0.254 seconds (Sampling)
#> Chain 3:                0.582 seconds (Total)
#> Chain 3: 
#> 
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 4).
#> Chain 4: 
#> Chain 4: Gradient evaluation took 2.2e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.22 seconds.
#> Chain 4: Adjust your expectations accordingly!
#> Chain 4: 
#> Chain 4: 
#> Chain 4: Iteration:    1 / 2000 [  0%]  (Warmup)
#> Chain 4: Iteration:  200 / 2000 [ 10%]  (Warmup)
#> Chain 4: Iteration:  400 / 2000 [ 20%]  (Warmup)
#> Chain 4: Iteration:  600 / 2000 [ 30%]  (Warmup)
#> Chain 4: Iteration:  800 / 2000 [ 40%]  (Warmup)
#> Chain 4: Iteration: 1000 / 2000 [ 50%]  (Warmup)
#> Chain 4: Iteration: 1001 / 2000 [ 50%]  (Sampling)
#> Chain 4: Iteration: 1200 / 2000 [ 60%]  (Sampling)
#> Chain 4: Iteration: 1400 / 2000 [ 70%]  (Sampling)
#> Chain 4: Iteration: 1600 / 2000 [ 80%]  (Sampling)
#> Chain 4: Iteration: 1800 / 2000 [ 90%]  (Sampling)
#> Chain 4: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 4: 
#> Chain 4:  Elapsed Time: 0.351 seconds (Warm-up)
#> Chain 4:                0.292 seconds (Sampling)
#> Chain 4:                0.643 seconds (Total)
#> Chain 4: 
si(stan_model, verbose = FALSE)
#> Support Interval
#> 
#> Parameter   |    BF = 1 SI | Effects |   Component
#> --------------------------------------------------
#> (Intercept) | [0.41, 2.74] |   fixed | conditional
#> group1      | [0.42, 2.74] |   fixed | conditional
si(stan_model, BF = 3, verbose = FALSE)
#> Support Interval
#> 
#> Parameter   |    BF = 3 SI | Effects |   Component
#> --------------------------------------------------
#> (Intercept) | [0.80, 2.34] |   fixed | conditional
#> group1      | [0.66, 2.43] |   fixed | conditional

# emmGrid objects
# ---------------
library(emmeans)
group_diff <- pairs(emmeans(stan_model, ~group))
si(group_diff, prior = stan_model, verbose = FALSE)
#> Support Interval
#> 
#> Parameter       |      BF = 1 SI
#> --------------------------------
#> group1 - group2 | [-2.73, -0.43]

# brms models
# -----------
library(brms)
contrasts(sleep$group) <- contr.equalprior_pairs # see vingette
my_custom_priors <-
  set_prior("student_t(3, 0, 1)", class = "b") +
  set_prior("student_t(3, 0, 1)", class = "sd", group = "ID")

brms_model <- suppressWarnings(brm(extra ~ group + (1 | ID),
  data = sleep,
  prior = my_custom_priors,
  refresh = 0
))
#> Compiling Stan program...
#> Start sampling
si(brms_model, verbose = FALSE)
#> Support Interval
#> 
#> Parameter   |    BF = 1 SI | Effects |   Component
#> --------------------------------------------------
#> b_Intercept | [0.52, 2.48] |   fixed | conditional
#> b_group1    | [0.72, 2.41] |   fixed | conditional
# }