Region of Practical Equivalence (ROPE)

Compute the proportion of the HDI (default to the 89% HDI) of a posterior distribution that lies within a region of practical equivalence.

Usage

rope(x, ...)

# S3 method for class 'numeric'
rope(x, range = "default", ci = 0.95, ci_method = "ETI", verbose = TRUE, ...)

# S3 method for class 'data.frame'
rope(
  x,
  range = "default",
  ci = 0.95,
  ci_method = "ETI",
  rvar_col = NULL,
  verbose = TRUE,
  ...
)

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

Arguments

x

Vector representing a posterior distribution. Can also be a stanreg or brmsfit model.

...

Currently not used.

range

ROPE's lower and higher bounds. Should be "default" or depending on the number of outcome variables a vector or a list. For models with one response, range can be:

a vector of length two (e.g., c(-0.1, 0.1)),
a list of numeric vector of the same length as numbers of parameters (see 'Examples').
a list of named numeric vectors, where names correspond to parameter names. In this case, all parameters that have no matching name in range will be set to "default".

In multivariate models, range should be a list with another list (one for each response variable) of numeric vectors . Vector names should correspond to the name of the response variables. If "default" and input is a vector, the range is set to c(-0.1, 0.1). If "default" and input is a Bayesian model, rope_range() is used. See 'Examples'.

ci

The Credible Interval (CI) probability, corresponding to the proportion of HDI, to use for the percentage in ROPE.

ci_method

The type of interval to use to quantify the percentage in ROPE. Can be 'HDI' (default) or 'ETI'. See ci().

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.

Note

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

ROPE

Statistically, the probability of a posterior distribution of being different from 0 does not make much sense (the probability of a single value null hypothesis in a continuous distribution is 0). Therefore, the idea underlining ROPE is to let the user define an area around the null value enclosing values that are equivalent to the null value for practical purposes (Kruschke 2010, 2011, 2014).

Kruschke (2018) suggests that such null value could be set, by default, to the -0.1 to 0.1 range of a standardized parameter (negligible effect size according to Cohen, 1988). This could be generalized: For instance, for linear models, the ROPE could be set as 0 +/- .1 * sd(y). This ROPE range can be automatically computed for models using the rope_range() function.

Kruschke (2010, 2011, 2014) suggests using the proportion of the 95% (or 89%, considered more stable) HDI that falls within the ROPE as an index for "null-hypothesis" testing (as understood under the Bayesian framework, see equivalence_test()).

Sensitivity to parameter's scale

It is important to consider the unit (i.e., the scale) of the predictors when using an index based on the ROPE, as the correct interpretation of the ROPE as representing a region of practical equivalence to zero is dependent on the scale of the predictors. Indeed, the percentage in ROPE depend on the unit of its parameter. In other words, as the ROPE represents a fixed portion of the response's scale, its proximity with a coefficient depends on the scale of the coefficient itself.

Multicollinearity - Non-independent covariates

When parameters show strong correlations, i.e. when covariates are not independent, the joint parameter distributions may shift towards or away from the ROPE. Collinearity invalidates ROPE and hypothesis testing based on univariate marginals, as the probabilities are conditional on independence. Most problematic are parameters that only have partial overlap with the ROPE region. In case of collinearity, the (joint) distributions of these parameters may either get an increased or decreased ROPE, which means that inferences based on rope() are inappropriate (Kruschke 2014, 340f).

rope() performs a simple check for pairwise correlations between parameters, but as there can be collinearity between more than two variables, a first step to check the assumptions of this hypothesis testing is to look at different pair plots. An even more sophisticated check is the projection predictive variable selection (Piironen and Vehtari 2017).

Strengths and Limitations

Strengths: Provides information related to the practical relevance of the effects.

Limitations: A ROPE range needs to be arbitrarily defined. Sensitive to the scale (the unit) of the predictors. Not sensitive to highly significant effects.

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.

References

Cohen, J. (1988). Statistical power analysis for the behavioural sciences.
Kruschke, J. K. (2010). What to believe: Bayesian methods for data analysis. Trends in cognitive sciences, 14(7), 293-300. doi:10.1016/j.tics.2010.05.001 .
Kruschke, J. K. (2011). Bayesian assessment of null values via parameter estimation and model comparison. Perspectives on Psychological Science, 6(3), 299-312. doi:10.1177/1745691611406925 .
Kruschke, J. K. (2014). Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan. Academic Press. doi:10.1177/2515245918771304 .
Kruschke, J. K. (2018). Rejecting or accepting parameter values in Bayesian estimation. Advances in Methods and Practices in Psychological Science, 1(2), 270-280. doi:10.1177/2515245918771304 .
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
Piironen, J., & Vehtari, A. (2017). Comparison of Bayesian predictive methods for model selection. Statistics and Computing, 27(3), 711–735. doi:10.1007/s11222-016-9649-y

Examples

library(bayestestR)

rope(x = rnorm(1000, 0, 0.01), range = c(-0.1, 0.1))
#> # Proportion of samples inside the ROPE [-0.10, 0.10]:
#> 
#> inside ROPE
#> -----------
#> 100.00 %   
#> 
rope(x = rnorm(1000, 0, 1), range = c(-0.1, 0.1))
#> # Proportion of samples inside the ROPE [-0.10, 0.10]:
#> 
#> inside ROPE
#> -----------
#> 8.32 %     
#> 
rope(x = rnorm(1000, 1, 0.01), range = c(-0.1, 0.1))
#> # Proportion of samples inside the ROPE [-0.10, 0.10]:
#> 
#> inside ROPE
#> -----------
#> 0.00 %     
#> 
rope(x = rnorm(1000, 1, 1), ci = c(0.90, 0.95))
#> # Proportions of samples inside the ROPE [-0.10, 0.10]:
#> 
#> ROPE for the 90% HDI:
#> 
#> inside ROPE
#> -----------
#> 4.89 %     
#> 
#> 
#> ROPE for the 95% HDI:
#> 
#> inside ROPE
#> -----------
#> 4.63 %     
#> 
#> 
# \donttest{
model <- suppressWarnings(
  rstanarm::stan_glm(mpg ~ wt + gear, data = mtcars, chains = 2, iter = 200, refresh = 0)
)
rope(model)
#> # Proportion of samples inside the ROPE [-0.60, 0.60]:
#> 
#> Parameter   | inside ROPE
#> -------------------------
#> (Intercept) |      0.00 %
#> wt          |      0.00 %
#> gear        |     43.68 %
#> 
rope(model, ci = c(0.90, 0.95))
#> # Proportions of samples inside the ROPE [-0.60, 0.60]:
#> 
#> ROPE for the 90% HDI:
#> 
#> Parameter   | inside ROPE
#> -------------------------
#> (Intercept) |      0.00 %
#> wt          |      0.00 %
#> gear        |     46.11 %
#> 
#> 
#> ROPE for the 95% HDI:
#> 
#> Parameter   | inside ROPE
#> -------------------------
#> (Intercept) |      0.00 %
#> wt          |      0.00 %
#> gear        |     43.68 %
#> 
#> 

# multiple ROPE ranges
rope(model, range = list(c(-10, 5), c(-0.2, 0.2), "default"))
#> # Proportion of samples inside the ROPE [-10.00, 5.00]:
#> 
#> Parameter   | inside ROPE
#> -------------------------
#> (Intercept) |      0.00 %
#> wt          |      0.00 %
#> gear        |     10.53 %
#> 

# named ROPE ranges
rope(model, range = list(gear = c(-3, 2), wt = c(-0.2, 0.2)))
#> # Proportion of samples inside the ROPE [-0.10, 0.10]:
#> 
#> Parameter   | inside ROPE
#> -------------------------
#> (Intercept) |      0.00 %
#> wt          |      0.00 %
#> gear        |    100.00 %
#> 

rope(emmeans::emtrends(model, ~1, "wt"), ci = c(0.90, 0.95))
#> # Proportions of samples inside the ROPE [-0.10, 0.10]:
#> 
#> ROPE for the 90% HDI:
#> 
#> X1      | inside ROPE
#> ---------------------
#> overall |      0.00 %
#> 
#> 
#> ROPE for the 95% HDI:
#> 
#> X1      | inside ROPE
#> ---------------------
#> overall |      0.00 %
#> 
#> 

model <- brms::brm(mpg ~ wt + cyl, data = mtcars, refresh = 0)
#> Compiling Stan program...
#> Start sampling
rope(model)
#> Possible multicollinearity between b_cyl and b_wt (r = 0.78). This might
#>   lead to inappropriate results. See 'Details' in '?rope'.
#> # Proportion of samples inside the ROPE [-0.60, 0.60]:
#> 
#> Parameter | inside ROPE
#> -----------------------
#> Intercept |      0.00 %
#> wt        |      0.00 %
#> cyl       |      0.00 %
#> 
rope(model, ci = c(0.90, 0.95))
#> Possible multicollinearity between b_cyl and b_wt (r = 0.78). This might
#>   lead to inappropriate results. See 'Details' in '?rope'.
#> # Proportions of samples inside the ROPE [-0.60, 0.60]:
#> 
#> ROPE for the 90% HDI:
#> 
#> Parameter | inside ROPE
#> -----------------------
#> Intercept |      0.00 %
#> wt        |      0.00 %
#> cyl       |      0.00 %
#> 
#> 
#> ROPE for the 95% HDI:
#> 
#> Parameter | inside ROPE
#> -----------------------
#> Intercept |      0.00 %
#> wt        |      0.00 %
#> cyl       |      0.00 %
#> 
#> 

model <- brms::brm(
  brms::bf(brms::mvbind(mpg, disp) ~ wt + cyl) + brms::set_rescor(rescor = TRUE),
  data = mtcars,
  refresh = 0
)
#> Compiling Stan program...
#> Start sampling
rope(model)
#> Possible multicollinearity between b_mpg_cyl and b_mpg_wt (r = 0.79),
#>   b_disp_cyl and b_disp_wt (r = 0.79). This might lead to inappropriate
#>   results. See 'Details' in '?rope'.
#> # Proportion of samples inside the ROPE.
#> ROPE with depends on outcome variable.
#> 
#> Parameter      | inside ROPE |      ROPE width
#> ----------------------------------------------
#> mpg_Intercept  |      0.00 % |   [-0.60, 0.60]
#> mpg_wt         |      0.00 % |   [-0.60, 0.60]
#> mpg_cyl        |      0.00 % |   [-0.60, 0.60]
#> disp_Intercept |      0.00 % | [-12.39, 12.39]
#> disp_wt        |      0.00 % | [-12.39, 12.39]
#> disp_cyl       |      0.00 % | [-12.39, 12.39]
#> 
rope(model, ci = c(0.90, 0.95))
#> Possible multicollinearity between b_mpg_cyl and b_mpg_wt (r = 0.79),
#>   b_disp_cyl and b_disp_wt (r = 0.79). This might lead to inappropriate
#>   results. See 'Details' in '?rope'.
#> # Proportions of samples inside the ROPE.
#> ROPE with depends on outcome variable.
#> 
#> ROPE for the 90% HDI:
#> 
#> Parameter      | inside ROPE |      ROPE width
#> ----------------------------------------------
#> mpg_Intercept  |      0.00 % |   [-0.60, 0.60]
#> mpg_wt         |      0.00 % |   [-0.60, 0.60]
#> mpg_cyl        |      0.00 % |   [-0.60, 0.60]
#> disp_Intercept |      0.00 % | [-12.39, 12.39]
#> disp_wt        |      0.00 % | [-12.39, 12.39]
#> disp_cyl       |      0.00 % | [-12.39, 12.39]
#> 
#> 
#> ROPE for the 95% HDI:
#> 
#> Parameter      | inside ROPE |      ROPE width
#> ----------------------------------------------
#> mpg_Intercept  |      0.00 % |   [-0.60, 0.60]
#> mpg_wt         |      0.00 % |   [-0.60, 0.60]
#> mpg_cyl        |      0.00 % |   [-0.60, 0.60]
#> disp_Intercept |      0.00 % | [-12.39, 12.39]
#> disp_wt        |      0.00 % | [-12.39, 12.39]
#> disp_cyl       |      0.00 % | [-12.39, 12.39]
#> 
#> 

# different ROPE ranges for model parameters. For each response, a named
# list (with the name of the response variable) is required as list-element
# for the `range` argument.
rope(
  model,
  range = list(
    mpg = list(b_mpg_wt = c(-1, 1), b_mpg_cyl = c(-2, 2)),
    disp = list(b_disp_wt = c(-5, 5), b_disp_cyl = c(-4, 4))
  )
)
#> Possible multicollinearity between b_mpg_cyl and b_mpg_wt (r = 0.79),
#>   b_disp_cyl and b_disp_wt (r = 0.79). This might lead to inappropriate
#>   results. See 'Details' in '?rope'.
#> # Proportion of samples inside the ROPE.
#> ROPE with depends on outcome variable.
#> 
#> Parameter      | inside ROPE |    ROPE width
#> --------------------------------------------
#> mpg_Intercept  |      0.00 % | [-0.10, 0.10]
#> mpg_wt         |      0.00 % | [-1.00, 1.00]
#> mpg_cyl        |     89.82 % | [-2.00, 2.00]
#> disp_Intercept |      0.00 % | [-0.10, 0.10]
#> disp_wt        |      0.00 % | [-5.00, 5.00]
#> disp_cyl       |      0.00 % | [-4.00, 4.00]
#> 

bf <- BayesFactor::ttestBF(x = rnorm(100, 1, 1))
rope(bf)
#> # Proportion of samples inside the ROPE [-0.10, 0.10]:
#> 
#> Parameter  | inside ROPE
#> ------------------------
#> Difference |      0.00 %
#> 
rope(bf, ci = c(0.90, 0.95))
#> # Proportions of samples inside the ROPE [-0.10, 0.10]:
#> 
#> ROPE for the 90% HDI:
#> 
#> Parameter  | inside ROPE
#> ------------------------
#> Difference |      0.00 %
#> 
#> 
#> ROPE for the 95% HDI:
#> 
#> Parameter  | inside ROPE
#> ------------------------
#> Difference |      0.00 %
#> 
#> 
# }