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 default
rope(x, ...)
# S3 method for numeric
rope(x, range = "default", ci = 0.95, ci_method = "HDI", verbose = TRUE, ...)
# S3 method for data.frame
rope(x, range = "default", ci = 0.95, ci_method = "HDI", verbose = TRUE, ...)
# S3 method for emmGrid
rope(x, range = "default", ci = 0.95, ci_method = "HDI", verbose = TRUE, ...)
# S3 method for BFBayesFactor
rope(x, range = "default", ci = 0.95, ci_method = "HDI", verbose = TRUE, ...)
# S3 method for MCMCglmm
rope(x, range = "default", ci = 0.95, ci_method = "HDI", verbose = TRUE, ...)
# S3 method for stanreg
rope(
x,
range = "default",
ci = 0.95,
ci_method = "HDI",
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 = "HDI",
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).

## 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
#> -----------
#> 10.52 %
#>
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
#> -----------
#> 5.11 %
#>
#>
#> ROPE for the 95% HDI:
#>
#> inside ROPE
#> -----------
#> 4.84 %
#>
#>
if (FALSE) {
library(rstanarm)
model <- stan_glm(mpg ~ wt + gear, data = mtcars, chains = 2, iter = 200, refresh = 0)
rope(model)
rope(model, ci = c(.90, .95))
library(emmeans)
rope(emtrends(model, ~1, "wt"), ci = c(.90, .95))
library(brms)
model <- brms::brm(mpg ~ wt + cyl, data = mtcars)
rope(model)
rope(model, ci = c(.90, .95))
library(brms)
model <- brms::brm(brms::mvbind(mpg, disp) ~ wt + cyl, data = mtcars)
rope(model)
rope(model, ci = c(.90, .95))
library(BayesFactor)
bf <- ttestBF(x = rnorm(100, 1, 1))
rope(bf)
rope(bf, ci = c(.90, .95))
}
```