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 '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 class '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
orbrmsfit
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 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 toc(-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.
- rvar_col
A single character - the name of an
rvar
column in the data frame to be processed. See example inp_direction()
.- 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__
orprior_
) are filtered by default, so only parameters that typically appear in thesummary()
are returned. Useparameters
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.
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{
library(rstanarm)
model <- suppressWarnings(
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 %
#>
library(emmeans)
rope(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 %
#>
#>
library(brms)
model <- 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 8e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.08 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1:
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#> Chain 1:
#> Chain 1: Elapsed Time: 0.025 seconds (Warm-up)
#> Chain 1: 0.023 seconds (Sampling)
#> Chain 1: 0.048 seconds (Total)
#> Chain 1:
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
#> Chain 2:
#> Chain 2: Gradient evaluation took 4e-06 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
#> Chain 2: Adjust your expectations accordingly!
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#> Chain 2:
#> Chain 2: Elapsed Time: 0.022 seconds (Warm-up)
#> Chain 2: 0.019 seconds (Sampling)
#> Chain 2: 0.041 seconds (Total)
#> Chain 2:
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3:
#> Chain 3: Gradient evaluation took 4e-06 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
#> Chain 3: Adjust your expectations accordingly!
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#> Chain 3:
#> Chain 3: Elapsed Time: 0.021 seconds (Warm-up)
#> Chain 3: 0.023 seconds (Sampling)
#> Chain 3: 0.044 seconds (Total)
#> Chain 3:
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
#> Chain 4:
#> Chain 4: Gradient evaluation took 4e-06 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
#> Chain 4: Adjust your expectations accordingly!
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#> Chain 4: 0.02 seconds (Sampling)
#> Chain 4: 0.042 seconds (Total)
#> Chain 4:
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 %
#>
#>
library(brms)
model <- brm(
bf(mvbind(mpg, disp) ~ wt + cyl) + set_rescor(rescor = TRUE),
data = mtcars
)
#> Compiling Stan program...
#> Start sampling
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 2.8e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.28 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1:
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#> Chain 1:
#> Chain 1: Elapsed Time: 0.402 seconds (Warm-up)
#> Chain 1: 0.159 seconds (Sampling)
#> Chain 1: 0.561 seconds (Total)
#> Chain 1:
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
#> Chain 2:
#> Chain 2: Gradient evaluation took 2e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.2 seconds.
#> Chain 2: Adjust your expectations accordingly!
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#> Chain 2:
#> Chain 2: Elapsed Time: 0.392 seconds (Warm-up)
#> Chain 2: 0.189 seconds (Sampling)
#> Chain 2: 0.581 seconds (Total)
#> Chain 2:
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3:
#> Chain 3: Gradient evaluation took 2e-05 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.2 seconds.
#> Chain 3: Adjust your expectations accordingly!
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#> Chain 3:
#> Chain 3: Elapsed Time: 0.312 seconds (Warm-up)
#> Chain 3: 0.179 seconds (Sampling)
#> Chain 3: 0.491 seconds (Total)
#> Chain 3:
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
#> Chain 4:
#> Chain 4: Gradient evaluation took 2e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.2 seconds.
#> Chain 4: Adjust your expectations accordingly!
#> Chain 4:
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#> Chain 4:
#> Chain 4: Elapsed Time: 0.365 seconds (Warm-up)
#> Chain 4: 0.189 seconds (Sampling)
#> Chain 4: 0.554 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 = 0.79). 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 = 0.78),
#> 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]
#>
#>
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(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 %
#>
#>
# }