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

equivalence_test(x, ...)

# S3 method for default
equivalence_test(x, ...)

# S3 method for numeric
equivalence_test(x, range = "default", ci = 0.95, verbose = TRUE, ...)

# S3 method for data.frame
equivalence_test(x, range = "default", ci = 0.95, verbose = TRUE, ...)

# S3 method for emmGrid
equivalence_test(x, range = "default", ci = 0.95, verbose = TRUE, ...)

# S3 method for BFBayesFactor
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. 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. The Credible Interval (CI) probability, corresponding to the proportion of HDI, to use for the percentage in ROPE. Toggle off warnings. Should results for fixed effects, random effects or both be returned? Only applies to mixed models. May be abbreviated. 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. 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 |      6.20 % | [-1.91 1.77]
#>
#> 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 % | [ 0.23 1.65]
#>
#>
#> H0        | inside ROPE |      99% HDI
#> --------------------------------------
#> Undecided |      6.36 % | [-1.67 3.66]
#>
#>
# 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 % | [ 0.6394 1.9564]
#>
#>
#> H0        | inside ROPE |          99% HDI
#> ------------------------------------------
#> Undecided |    4.8436 % | [-1.5668 3.4701]
#>
#> if (FALSE) {
library(rstanarm)
model <- rstanarm::stan_glm(mpg ~ wt + cyl, data = mtcars)
equivalence_test(model)
equivalence_test(model, ci = c(.50, 1))

# plot result
test <- equivalence_test(model)
plot(test)

library(emmeans)
equivalence_test(emtrends(model, ~1, "wt"))

library(brms)
model <- brms::brm(mpg ~ wt + cyl, data = mtcars)
equivalence_test(model)
equivalence_test(model, ci = c(.50, .99))

library(BayesFactor)
bf <- ttestBF(x = rnorm(100, 1, 1))
equivalence_test(bf)
equivalence_test(bf, ci = c(.50, .99))
}