Test for Practical Equivalence — equivalence

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

Usage

equivalence_test(x, ...)

# Default S3 method
equivalence_test(x, ...)

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

# S3 method for class 'brmsfit'
equivalence_test(
  x,
  range = "default",
  ci = 0.95,
  effects = "fixed",
  component = "conditional",
  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. 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 another list (one for each response variable) of numeric vectors . 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. See 'Examples'.

ci

The Credible Interval (CI) probability, corresponding to the proportion of HDI, to use for the percentage in ROPE.

rvar_col

A single character - the name of an rvar column in the data frame to be processed. See example in p_direction().

verbose

Toggle off warnings.

effects

Should results for fixed effects ("fixed", the default), random effects ("random") or both ("all") be returned? Only applies to mixed models. May be abbreviated.

component

Which type of parameters to return, such as parameters for the conditional model, the zero-inflated part of the model, the dispersion term, etc. See details in section Model Components. May be abbreviated. Note that the conditional component also refers to the count or mean component - names may differ, depending on the modeling package. There are three convenient shortcuts (not applicable to all model classes):

component = "all" returns all possible parameters.
If component = "location", location parameters such as conditional, zero_inflated, smooth_terms, or instruments are returned (everything that are fixed or random effects - depending on the effects argument - but no auxiliary parameters).
For component = "distributional" (or "auxiliary"), components like sigma, dispersion, beta or precision (and other auxiliary parameters) are returned.

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 is 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).

Model components

Possible values for the component argument depend on the model class. Following are valid options:

"all": returns all model components, applies to all models, but will only have an effect for models with more than just the conditional model component.
"conditional": only returns the conditional component, i.e. "fixed effects" terms from the model. Will only have an effect for models with more than just the conditional model component.
"smooth_terms": returns smooth terms, only applies to GAMs (or similar models that may contain smooth terms).
"zero_inflated" (or "zi"): returns the zero-inflation component.
"location": returns location parameters such as conditional, zero_inflated, or smooth_terms (everything that are fixed or random effects - depending on the effects argument - but no auxiliary parameters).
"distributional" (or "auxiliary"): components like sigma, dispersion, beta or precision (and other auxiliary parameters) are returned.

For models of class brmsfit (package brms), even more options are possible for the component argument, which are not all documented in detail here. See also ?insight::find_parameters.

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 |      8.21 % | [-2.02, 1.95]
#> 
#> 
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 % | [0.28, 1.65]
#> 
#> 
#> H0        | inside ROPE |       99% HDI
#> ---------------------------------------
#> Undecided |      5.45 % | [-1.58, 3.53]
#> 
#> 

# 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 % | [0.3113, 1.6862]
#> 
#> 
#> 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 2.1e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds.
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#> 
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 2).
#> Chain 2: 
#> Chain 2: Gradient evaluation took 1e-05 seconds
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#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 3).
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#> 
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 4).
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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 % | [36.06, 43.20]
#> wt          | Rejected |      0.00 % | [-4.77, -1.57]
#> cyl         | Rejected |      0.00 % | [-2.36, -0.66]
#> 
#> 
# multiple ROPE ranges - asymmetric, symmetric, default
equivalence_test(model, range = list(c(10, 40), c(-5, -4), "default"))
#> Possible multicollinearity between cyl and wt (r = 0.78). This might
#>   lead to inappropriate results. See 'Details' in '?equivalence_test'.
#> # Test for Practical Equivalence
#> 
#> Parameter   |        H0 | inside ROPE |        95% HDI |           ROPE
#> -----------------------------------------------------------------------
#> (Intercept) | Undecided |     57.37 % | [36.06, 43.20] | [10.00, 40.00]
#> wt          | Undecided |     12.97 % | [-4.77, -1.57] | [-5.00, -4.00]
#> cyl         |  Rejected |      0.00 % | [-2.36, -0.66] |  [-0.10, 0.10]
#> 
#> 
# named ROPE ranges
equivalence_test(model, range = list(wt = c(-5, -4), `(Intercept)` = c(10, 40)))
#> Possible multicollinearity between cyl and wt (r = 0.78). This might
#>   lead to inappropriate results. See 'Details' in '?equivalence_test'.
#> # Test for Practical Equivalence
#> 
#> Parameter   |        H0 | inside ROPE |        95% HDI |           ROPE
#> -----------------------------------------------------------------------
#> (Intercept) | Undecided |     57.37 % | [36.06, 43.20] | [10.00, 40.00]
#> wt          | Undecided |     12.97 % | [-4.77, -1.57] | [-5.00, -4.00]
#> cyl         |  Rejected |      0.00 % | [-2.36, -0.66] |  [-0.10, 0.10]
#> 
#> 

# 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 0.0924


equivalence_test(emmeans::emtrends(model, ~1, "wt", data = mtcars))
#> # Test for Practical Equivalence
#> 
#>   ROPE: [-0.10 0.10]
#> 
#> X1      |       H0 | inside ROPE |        95% HDI
#> -------------------------------------------------
#> overall | Rejected |      0.00 % | [-4.77, -1.57]
#> 
#> 

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 6e-06 seconds
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#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
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equivalence_test(model)
#> Possible multicollinearity between b_cyl and b_wt (r = 0.79). 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 % | [36.14, 43.39]
#> wt        | Rejected |      0.00 % | [-4.77, -1.57]
#> cyl       | Rejected |      0.00 % | [-2.38, -0.63]
#> 
#> 

bf <- BayesFactor::ttestBF(x = rnorm(100, 1, 1))
# equivalence_test(bf)
# }