Perform a **Test for Practical Equivalence** for Bayesian and frequentist models.

## Usage

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

- 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.

## Value

A data frame with following columns:

`Parameter`

The model parameter(s), if`x`

is a model-object. If`x`

is a vector, this column is missing.`CI`

The probability of the HDI.`ROPE_low`

,`ROPE_high`

The limits of the ROPE. These values are identical for all parameters.`ROPE_Percentage`

The proportion of the HDI that lies inside the ROPE.`ROPE_Equivalence`

The "test result", as character. Either "rejected", "accepted" or "undecided".`HDI_low`

,`HDI_high`

The lower and upper HDI limits for the parameters.

## Details

Documentation is accessible for:

For Bayesian models, the **Test for Practical Equivalence** is based on the *"HDI+ROPE decision rule"* (Kruschke, 2014, 2018) to check whether parameter values should be accepted or rejected against an explicitly formulated "null hypothesis" (i.e., a ROPE). In other words, it checks the percentage of the `89%`

HDI that is the null region (the ROPE). If this percentage is sufficiently low, the null hypothesis is rejected. If this percentage is sufficiently high, the null hypothesis is accepted.

Using the ROPE and the HDI, Kruschke (2018)
suggests using the percentage of the `95%`

(or `89%`

, considered more stable)
HDI that falls within the ROPE as a decision rule. If the HDI
is completely outside the ROPE, the "null hypothesis" for this parameter is
"rejected". If the ROPE completely covers the HDI, i.e., all most credible
values of a parameter are inside the region of practical equivalence, the
null hypothesis is accepted. Else, it’s undecided whether to accept or
reject the null hypothesis. If the full ROPE is used (i.e., `100%`

of the
HDI), then the null hypothesis is rejected or accepted if the percentage
of the posterior within the ROPE is smaller than to `2.5%`

or greater than
`97.5%`

. Desirable results are low proportions inside the ROPE (the closer
to zero the better).

Some attention is required for finding suitable values for the ROPE limits
(argument `range`

). See 'Details' in `rope_range()`

for further information.
**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. In such cases, the test for practical equivalence may
have inappropriate results. Collinearity invalidates ROPE and hypothesis
testing based on univariate marginals, as the probabilities are conditional
on independence. Most problematic are the results of the "undecided"
parameters, which may either move further towards "rejection" or away
from it (Kruschke 2014, 340f).
`equivalence_test()`

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 a `print()`

-method with a `digits`

-argument to control
the amount of digits in the output, and there is a
`plot()`

-method
to visualize the results from the equivalence-test (for models only).

## References

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

Kruschke, J. K. (2014). Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan. Academic Press

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)
equivalence_test(x = rnorm(1000, 0, 0.01), range = c(-0.1, 0.1))
#> # Test for Practical Equivalence
#>
#> ROPE: [-0.10 0.10]
#>
#> H0 | inside ROPE | 95% HDI
#> -------------------------------------
#> Accepted | 100.00 % | [-0.02 0.02]
#>
#>
equivalence_test(x = rnorm(1000, 0, 1), range = c(-0.1, 0.1))
#> # Test for Practical Equivalence
#>
#> ROPE: [-0.10 0.10]
#>
#> H0 | inside ROPE | 95% HDI
#> --------------------------------------
#> Undecided | 8.11 % | [-2.00 1.97]
#>
#>
equivalence_test(x = rnorm(1000, 1, 0.01), range = c(-0.1, 0.1))
#> # Test for Practical Equivalence
#>
#> ROPE: [-0.10 0.10]
#>
#> H0 | inside ROPE | 95% HDI
#> ------------------------------------
#> Rejected | 0.00 % | [0.98 1.02]
#>
#>
equivalence_test(x = rnorm(1000, 1, 1), ci = c(.50, .99))
#> # Test for Practical Equivalence
#>
#> ROPE: [-0.10 0.10]
#>
#> H0 | inside ROPE | 50% HDI
#> -------------------------------------
#> Rejected | 0.00 % | [ 0.31 1.64]
#>
#>
#> H0 | inside ROPE | 99% HDI
#> --------------------------------------
#> Undecided | 5.05 % | [-1.58 3.65]
#>
#>
# print more digits
test <- equivalence_test(x = rnorm(1000, 1, 1), ci = c(.50, .99))
print(test, digits = 4)
#> # Test for Practical Equivalence
#>
#> ROPE: [-0.1000 0.1000]
#>
#> H0 | inside ROPE | 50% HDI
#> -----------------------------------------
#> Rejected | 0.0000 % | [ 0.3115 1.7148]
#>
#>
#> H0 | inside ROPE | 99% HDI
#> ------------------------------------------
#> Undecided | 4.9495 % | [-1.7070 3.7015]
#>
#>
# \donttest{
model <- rstanarm::stan_glm(mpg ~ wt + cyl, data = mtcars)
#>
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 2e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.2 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1:
#> Chain 1:
#> Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
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#> Chain 1:
#> Chain 1: Elapsed Time: 0.057 seconds (Warm-up)
#> Chain 1: 0.057 seconds (Sampling)
#> Chain 1: 0.114 seconds (Total)
#> Chain 1:
#>
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 2).
#> Chain 2:
#> Chain 2: Gradient evaluation took 1.5e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.15 seconds.
#> Chain 2: Adjust your expectations accordingly!
#> Chain 2:
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#> Chain 2:
#> Chain 2: Elapsed Time: 0.059 seconds (Warm-up)
#> Chain 2: 0.053 seconds (Sampling)
#> Chain 2: 0.112 seconds (Total)
#> Chain 2:
#>
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 3).
#> Chain 3:
#> Chain 3: Gradient evaluation took 1.4e-05 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.14 seconds.
#> Chain 3: Adjust your expectations accordingly!
#> Chain 3:
#> Chain 3:
#> Chain 3: Iteration: 1 / 2000 [ 0%] (Warmup)
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#> Chain 3:
#> Chain 3: Elapsed Time: 0.056 seconds (Warm-up)
#> Chain 3: 0.055 seconds (Sampling)
#> Chain 3: 0.111 seconds (Total)
#> Chain 3:
#>
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 4).
#> Chain 4:
#> Chain 4: Gradient evaluation took 1.4e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.14 seconds.
#> Chain 4: Adjust your expectations accordingly!
#> Chain 4:
#> Chain 4:
#> Chain 4: Iteration: 1 / 2000 [ 0%] (Warmup)
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#> Chain 4:
#> Chain 4: Elapsed Time: 0.06 seconds (Warm-up)
#> Chain 4: 0.052 seconds (Sampling)
#> Chain 4: 0.112 seconds (Total)
#> Chain 4:
equivalence_test(model)
#> Possible multicollinearity between cyl and wt (r = 0.78). This might
#> lead to inappropriate results. See 'Details' in '?equivalence_test'.
#> # Test for Practical Equivalence
#>
#> ROPE: [-0.60 0.60]
#>
#> Parameter | H0 | inside ROPE | 95% HDI
#> ----------------------------------------------------
#> (Intercept) | Rejected | 0.00 % | [36.21 43.06]
#> wt | Rejected | 0.00 % | [-4.74 -1.62]
#> cyl | Rejected | 0.00 % | [-2.36 -0.70]
#>
#>
# plot result
test <- equivalence_test(model)
#> Possible multicollinearity between cyl and wt (r = 0.78). This might
#> lead to inappropriate results. See 'Details' in '?equivalence_test'.
plot(test)
#> Picking joint bandwidth of 0.0895
equivalence_test(emmeans::emtrends(model, ~1, "wt", data = mtcars))
#> # Test for Practical Equivalence
#>
#> ROPE: [-0.10 0.10]
#>
#> Parameter | H0 | inside ROPE | 95% HDI
#> --------------------------------------------------
#> overall | Rejected | 0.00 % | [-4.74 -1.62]
#>
#>
model <- brms::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 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!
#> Chain 1:
#> Chain 1:
#> Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
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#> Chain 1:
#> Chain 1: Elapsed Time: 0.025 seconds (Warm-up)
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#> Chain 1: 0.051 seconds (Total)
#> Chain 1:
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
#> Chain 2:
#> Chain 2: Gradient evaluation took 6e-06 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
#> Chain 2: Adjust your expectations accordingly!
#> Chain 2:
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#> Chain 2:
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#> Chain 2: 0.043 seconds (Total)
#> Chain 2:
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3:
#> Chain 3: Gradient evaluation took 6e-06 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
#> Chain 3: Adjust your expectations accordingly!
#> Chain 3:
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#> Chain 3: 0.047 seconds (Total)
#> Chain 3:
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
#> Chain 4:
#> Chain 4: Gradient evaluation took 7e-06 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.07 seconds.
#> Chain 4: Adjust your expectations accordingly!
#> Chain 4:
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#> Chain 4: Elapsed Time: 0.025 seconds (Warm-up)
#> Chain 4: 0.026 seconds (Sampling)
#> Chain 4: 0.051 seconds (Total)
#> Chain 4:
equivalence_test(model)
#> Possible multicollinearity between b_cyl and b_wt (r = 0.78). This might
#> lead to inappropriate results. See 'Details' in '?equivalence_test'.
#> # Test for Practical Equivalence
#>
#> ROPE: [-0.60 0.60]
#>
#> Parameter | H0 | inside ROPE | 95% HDI
#> --------------------------------------------------
#> Intercept | Rejected | 0.00 % | [36.09 43.30]
#> wt | Rejected | 0.00 % | [-4.78 -1.52]
#> cyl | Rejected | 0.00 % | [-2.40 -0.61]
#>
#>
bf <- BayesFactor::ttestBF(x = rnorm(100, 1, 1))
# equivalence_test(bf)
# }
```