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 numeric
rope(x, range = "default", ci = 0.95, ci_method = "ETI", verbose = TRUE, ...)

# S3 method for stanreg
rope(
x,
range = "default",
ci = 0.95,
ci_method = "ETI",
effects = c("fixed", "random", "all"),
component = c("location", "all", "conditional", "smooth_terms", "sigma",
"distributional", "auxiliary"),
parameters = NULL,
verbose = TRUE,
...
)

# S3 method for brmsfit
rope(
x,
range = "default",
ci = 0.95,
ci_method = "ETI",
effects = c("fixed", "random", "all"),
component = c("conditional", "zi", "zero_inflated", "all"),
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. In models with one response, range should be a vector of length two (e.g., c(-0.1, 0.1)). In multivariate models, range should be a list with a numeric vectors for each response variable. 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.

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.

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.

## Details

### 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.

## Note

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

## 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(.90, .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 %
#>
#>
# \dontrun{
library(rstanarm)
model <- stan_glm(mpg ~ wt + gear, data = mtcars, chains = 2, iter = 200, refresh = 0)
#> 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
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(.90, .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 %
#>
#>

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

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 1.2e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1:
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#> Chain 1:
#>
#> SAMPLING FOR MODEL '2d19b3a372313df641edf05db5e9f303' NOW (CHAIN 2).
#> Chain 2:
#> Chain 2: Gradient evaluation took 9e-06 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.09 seconds.
#> Chain 2: Adjust your expectations accordingly!
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#> Chain 2:
#> Chain 2:  Elapsed Time: 0.025216 seconds (Warm-up)
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#> Chain 2:                0.051843 seconds (Total)
#> Chain 2:
#>
#> SAMPLING FOR MODEL '2d19b3a372313df641edf05db5e9f303' NOW (CHAIN 3).
#> Chain 3:
#> Chain 3: Gradient evaluation took 9e-06 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.09 seconds.
#> Chain 3: Adjust your expectations accordingly!
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#> Chain 3:
#> Chain 3:  Elapsed Time: 0.028672 seconds (Warm-up)
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#> Chain 3:                0.055096 seconds (Total)
#> Chain 3:
#>
#> SAMPLING FOR MODEL '2d19b3a372313df641edf05db5e9f303' NOW (CHAIN 4).
#> Chain 4:
#> Chain 4: Gradient evaluation took 8e-06 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.08 seconds.
#> Chain 4: Adjust your expectations accordingly!
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#> Chain 4:
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#> Chain 4:                0.046352 seconds (Total)
#> Chain 4:
rope(model)
#> Possible multicollinearity between b_cyl and b_wt (r = 0.77). 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(.90, .95))
#> Possible multicollinearity between b_cyl and b_wt (r = 0.77). 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 %
#>
#>

library(brms)
model <- brms::brm(brms::mvbind(mpg, disp) ~ wt + cyl, data = mtcars)
#> Setting 'rescor' to TRUE by default for this model
#> Warning: In the future, 'rescor' will be set to FALSE by default for all models. It is thus recommended to explicitely set 'rescor' via 'set_rescor' instead of using the default.
#> Compiling Stan program...
#> Start sampling
#>
#> SAMPLING FOR MODEL '9d9150ff544e4b0191bf4201f9c63f72' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 7.7e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.77 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1:
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#> Chain 1:                1.33494 seconds (Total)
#> Chain 1:
#>
#> SAMPLING FOR MODEL '9d9150ff544e4b0191bf4201f9c63f72' NOW (CHAIN 2).
#> Chain 2:
#> Chain 2: Gradient evaluation took 5.9e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.59 seconds.
#> Chain 2: Adjust your expectations accordingly!
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#> Chain 2:
#>
#> SAMPLING FOR MODEL '9d9150ff544e4b0191bf4201f9c63f72' NOW (CHAIN 3).
#> Chain 3:
#> Chain 3: Gradient evaluation took 5.7e-05 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.57 seconds.
#> Chain 3: Adjust your expectations accordingly!
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#> Chain 3:
#>
#> SAMPLING FOR MODEL '9d9150ff544e4b0191bf4201f9c63f72' NOW (CHAIN 4).
#> Chain 4:
#> Chain 4: Gradient evaluation took 5.7e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.57 seconds.
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#> Chain 4:
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#> Chain 4:                1.25977 seconds (Total)
#> Chain 4:
rope(model)
#> Possible multicollinearity between b_mpg_cyl and b_mpg_wt (r = 0.78), b_disp_cyl and b_disp_wt (r = 0.78). 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(.90, .95))
#> Possible multicollinearity between b_mpg_cyl and b_mpg_wt (r = 0.78), b_disp_cyl and b_disp_wt (r = 0.78). 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]
#>
#>

library(BayesFactor)
bf <- 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(.90, .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 %
#>
#>
# }