Compute the (conditional) equivalence test for frequentist models.

## Usage

```
# S3 method for class 'lm'
equivalence_test(
x,
range = "default",
ci = 0.95,
rule = "classic",
vcov = NULL,
vcov_args = NULL,
verbose = TRUE,
...
)
# S3 method for class 'merMod'
equivalence_test(
x,
range = "default",
ci = 0.95,
rule = "classic",
effects = c("fixed", "random"),
vcov = NULL,
vcov_args = NULL,
verbose = TRUE,
...
)
# S3 method for class 'ggeffects'
equivalence_test(
x,
range = "default",
rule = "classic",
test = "pairwise",
verbose = TRUE,
...
)
```

## Arguments

- x
A statistical model.

- range
The range of practical equivalence of an effect. May be

`"default"`

, to automatically define this range based on properties of the model's data.- ci
Confidence Interval (CI) level. Default to

`0.95`

(`95%`

).- rule
Character, indicating the rules when testing for practical equivalence. Can be

`"bayes"`

,`"classic"`

or`"cet"`

. See 'Details'.- vcov
Variance-covariance matrix used to compute uncertainty estimates (e.g., for robust standard errors). This argument accepts a covariance matrix, a function which returns a covariance matrix, or a string which identifies the function to be used to compute the covariance matrix.

A covariance matrix

A function which returns a covariance matrix (e.g.,

`stats::vcov()`

)A string which indicates the kind of uncertainty estimates to return.

Heteroskedasticity-consistent:

`"HC"`

,`"HC0"`

,`"HC1"`

,`"HC2"`

,`"HC3"`

,`"HC4"`

,`"HC4m"`

,`"HC5"`

. See`?sandwich::vcovHC`

Cluster-robust:

`"CR"`

,`"CR0"`

,`"CR1"`

,`"CR1p"`

,`"CR1S"`

,`"CR2"`

,`"CR3"`

. See`?clubSandwich::vcovCR`

Bootstrap:

`"BS"`

,`"xy"`

,`"residual"`

,`"wild"`

,`"mammen"`

,`"fractional"`

,`"jackknife"`

,`"norm"`

,`"webb"`

. See`?sandwich::vcovBS`

Other

`sandwich`

package functions:`"HAC"`

,`"PC"`

,`"CL"`

,`"OPG"`

,`"PL"`

.

- vcov_args
List of arguments to be passed to the function identified by the

`vcov`

argument. This function is typically supplied by the**sandwich**or**clubSandwich**packages. Please refer to their documentation (e.g.,`?sandwich::vcovHAC`

) to see the list of available arguments. If no estimation type (argument`type`

) is given, the default type for`"HC"`

equals the default from the**sandwich**package; for type`"CR"`

, the default is set to`"CR3"`

.- verbose
Toggle warnings and messages.

- ...
Arguments passed to or from other methods.

- effects
Should parameters for fixed effects (

`"fixed"`

), random effects (`"random"`

), or both (`"all"`

) be returned? Only applies to mixed models. May be abbreviated. If the calculation of random effects parameters takes too long, you may use`effects = "fixed"`

.- test
Hypothesis test for computing contrasts or pairwise comparisons. See

`?ggeffects::test_predictions`

for details.

## Details

In classical null hypothesis significance testing (NHST) within a
frequentist framework, it is not possible to accept the null hypothesis, H0 -
unlike in Bayesian statistics, where such probability statements are
possible. "... one can only reject the null hypothesis if the test
statistics falls into the critical region(s), or fail to reject this
hypothesis. In the latter case, all we can say is that no significant effect
was observed, but one cannot conclude that the null hypothesis is true."
(*Pernet 2017*). One way to address this issues without Bayesian methods is
*Equivalence Testing*, as implemented in `equivalence_test()`

. While you
either can reject the null hypothesis or claim an inconclusive result in
NHST, the equivalence test - according to *Pernet* - adds a third category,
*"accept"*. Roughly speaking, the idea behind equivalence testing in a
frequentist framework is to check whether an estimate and its uncertainty
(i.e. confidence interval) falls within a region of "practical equivalence".
Depending on the rule for this test (see below), statistical significance
does not necessarily indicate whether the null hypothesis can be rejected or
not, i.e. the classical interpretation of the p-value may differ from the
results returned from the equivalence test.

### Calculation of equivalence testing

"bayes" - Bayesian rule (Kruschke 2018)

This rule follows the "HDI+ROPE decision rule" (

*Kruschke, 2014, 2018*) used for the`Bayesian counterpart()`

. This means, if the confidence intervals are completely outside the ROPE, the "null hypothesis" for this parameter is "rejected". If the ROPE completely covers the CI, the null hypothesis is accepted. Else, it's undecided whether to accept or reject the null hypothesis. Desirable results are low proportions inside the ROPE (the closer to zero the better)."classic" - The TOST rule (Lakens 2017)

This rule follows the "TOST rule", i.e. a two one-sided test procedure (

*Lakens 2017*). Following this rule...practical equivalence is assumed (i.e. H0

*"accepted"*) when the narrow confidence intervals are completely inside the ROPE, no matter if the effect is statistically significant or not;practical equivalence (i.e. H0) is

*rejected*, when the coefficient is statistically significant, both when the narrow confidence intervals (i.e.`1-2*alpha`

) include or exclude the the ROPE boundaries, but the narrow confidence intervals are*not fully covered*by the ROPE;else the decision whether to accept or reject practical equivalence is undecided (i.e. when effects are

*not*statistically significant*and*the narrow confidence intervals overlaps the ROPE).

"cet" - Conditional Equivalence Testing (Campbell/Gustafson 2018)

The Conditional Equivalence Testing as described by

*Campbell and Gustafson 2018*. According to this rule, practical equivalence is rejected when the coefficient is statistically significant. When the effect is*not*significant and the narrow confidence intervals are completely inside the ROPE, we accept (i.e. assume) practical equivalence, else it is undecided.

### Levels of Confidence Intervals used for Equivalence Testing

For `rule = "classic"`

, "narrow" confidence intervals are used for
equivalence testing. "Narrow" means, the the intervals is not 1 - alpha,
but 1 - 2 * alpha. Thus, if `ci = .95`

, alpha is assumed to be 0.05
and internally a ci-level of 0.90 is used. `rule = "cet"`

uses
both regular and narrow confidence intervals, while `rule = "bayes"`

only uses the regular intervals.

### p-Values

The equivalence p-value is the area of the (cumulative) confidence
distribution that is outside of the region of equivalence. It can be
interpreted as p-value for *rejecting* the alternative hypothesis and
*accepting* the "null hypothesis" (i.e. assuming practical equivalence). That
is, a high p-value means we reject the assumption of practical equivalence
and accept the alternative hypothesis.

### Second Generation p-Value (SGPV)

Second generation p-values (SGPV) were proposed as a statistic that
represents *the proportion of data-supported hypotheses that are also null
hypotheses* *(Blume et al. 2018, Lakens and Delacre 2020)*. It represents the
proportion of the *full* confidence interval range (assuming a normally or
t-distributed, equal-tailed interval, based on the model) that is inside the
ROPE. The SGPV ranges from zero to one. Higher values indicate that the
effect is more likely to be practically equivalent ("not of interest").

Note that the assumed interval, which is used to calculate the SGPV, is an
estimation of the *full interval* based on the chosen confidence level. For
example, if the 95% confidence interval of a coefficient ranges from -1 to 1,
the underlying *full (normally or t-distributed) interval* approximately
ranges from -1.9 to 1.9, see also following code:

```
# simulate full normal distribution
out <- bayestestR::distribution_normal(10000, 0, 0.5)
# range of "full" distribution
range(out)
# range of 95% CI
round(quantile(out, probs = c(0.025, 0.975)), 2)
```

This ensures that the SGPV always refers to the general compatible parameter
space of coefficients, independent from the confidence interval chosen for
testing practical equivalence. Therefore, the SGPV of the *full interval* is
similar to the ROPE coverage of Bayesian equivalence tests, see following
code:

```
library(bayestestR)
library(brms)
m <- lm(mpg ~ gear + wt + cyl + hp, data = mtcars)
m2 <- brm(mpg ~ gear + wt + cyl + hp, data = mtcars)
# SGPV for frequentist models
equivalence_test(m)
# similar to ROPE coverage of Bayesian models
equivalence_test(m2)
# similar to ROPE coverage of simulated draws / bootstrap samples
equivalence_test(simulate_model(m))
```

### ROPE range

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

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

for further information.

## Note

There is also a `plot()`

-method
implemented in the **see**-package.

## Statistical inference - how to quantify evidence

There is no standardized approach to drawing conclusions based on the
available data and statistical models. A frequently chosen but also much
criticized approach is to evaluate results based on their statistical
significance (*Amrhein et al. 2017*).

A more sophisticated way would be to test whether estimated effects exceed
the "smallest effect size of interest", to avoid even the smallest effects
being considered relevant simply because they are statistically significant,
but clinically or practically irrelevant (*Lakens et al. 2018, Lakens 2024*).

A rather unconventional approach, which is nevertheless advocated by various
authors, is to interpret results from classical regression models either in
terms of probabilities, similar to the usual approach in Bayesian statistics
(*Schweder 2018; Schweder and Hjort 2003; Vos 2022*) or in terms of relative
measure of "evidence" or "compatibility" with the data (*Greenland et al. 2022;
Rafi and Greenland 2020*), which nevertheless comes close to a probabilistic
interpretation.

A more detailed discussion of this topic is found in the documentation of
`p_function()`

.

The **parameters** package provides several options or functions to aid
statistical inference. These are, for example:

`equivalence_test()`

, to compute the (conditional) equivalence test for frequentist models`p_significance()`

, to compute the probability of*practical significance*, which can be conceptualized as a unidirectional equivalence test`p_function()`

, or*consonance function*, to compute p-values and compatibility (confidence) intervals for statistical modelsthe

`pd`

argument (setting`pd = TRUE`

) in`model_parameters()`

includes a column with the*probability of direction*, i.e. the probability that a parameter is strictly positive or negative. See`bayestestR::p_direction()`

for details. If plotting is desired, the`p_direction()`

function can be used, together with`plot()`

.the

`s_value`

argument (setting`s_value = TRUE`

) in`model_parameters()`

replaces the p-values with their related*S*-values (*Rafi and Greenland 2020*)finally, it is possible to generate distributions of model coefficients by generating bootstrap-samples (setting

`bootstrap = TRUE`

) or simulating draws from model coefficients using`simulate_model()`

. These samples can then be treated as "posterior samples" and used in many functions from the**bayestestR**package.

Most of the above shown options or functions derive from methods originally
implemented for Bayesian models (*Makowski et al. 2019*). However, assuming
that model assumptions are met (which means, the model fits well to the data,
the correct model is chosen that reflects the data generating process
(distributional model family) etc.), it seems appropriate to interpret
results from classical frequentist models in a "Bayesian way" (more details:
documentation in `p_function()`

).

## References

Amrhein, V., Korner-Nievergelt, F., and Roth, T. (2017). The earth is flat (p > 0.05): Significance thresholds and the crisis of unreplicable research. PeerJ, 5, e3544. doi:10.7717/peerj.3544

Blume, J. D., D'Agostino McGowan, L., Dupont, W. D., & Greevy, R. A. (2018). Second-generation p-values: Improved rigor, reproducibility, & transparency in statistical analyses. PLOS ONE, 13(3), e0188299. https://doi.org/10.1371/journal.pone.0188299

Campbell, H., & Gustafson, P. (2018). Conditional equivalence testing: An alternative remedy for publication bias. PLOS ONE, 13(4), e0195145. doi: 10.1371/journal.pone.0195145

Greenland S, Rafi Z, Matthews R, Higgs M. To Aid Scientific Inference, Emphasize Unconditional Compatibility Descriptions of Statistics. (2022) https://arxiv.org/abs/1909.08583v7 (Accessed November 10, 2022)

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

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

Lakens, D. (2017). Equivalence Tests: A Practical Primer for t Tests, Correlations, and Meta-Analyses. Social Psychological and Personality Science, 8(4), 355–362. doi: 10.1177/1948550617697177

Lakens, D. (2024). Improving Your Statistical Inferences (Version v1.5.1). Retrieved from https://lakens.github.io/statistical_inferences/. doi:10.5281/ZENODO.6409077

Lakens, D., and Delacre, M. (2020). Equivalence Testing and the Second Generation P-Value. Meta-Psychology, 4. https://doi.org/10.15626/MP.2018.933

Lakens, D., Scheel, A. M., and Isager, P. M. (2018). Equivalence Testing for Psychological Research: A Tutorial. Advances in Methods and Practices in Psychological Science, 1(2), 259–269. doi:10.1177/2515245918770963

Makowski, D., Ben-Shachar, M. S., Chen, S. H. A., and Lüdecke, D. (2019). Indices of Effect Existence and Significance in the Bayesian Framework. Frontiers in Psychology, 10, 2767. doi:10.3389/fpsyg.2019.02767

Pernet, C. (2017). Null hypothesis significance testing: A guide to commonly misunderstood concepts and recommendations for good practice. F1000Research, 4, 621. doi: 10.12688/f1000research.6963.5

Rafi Z, Greenland S. Semantic and cognitive tools to aid statistical science: replace confidence and significance by compatibility and surprise. BMC Medical Research Methodology (2020) 20:244.

Schweder T. Confidence is epistemic probability for empirical science. Journal of Statistical Planning and Inference (2018) 195:116–125. doi:10.1016/j.jspi.2017.09.016

Schweder T, Hjort NL. Frequentist analogues of priors and posteriors. In Stigum, B. (ed.), Econometrics and the Philosophy of Economics: Theory Data Confrontation in Economics, pp. 285-217. Princeton University Press, Princeton, NJ, 2003

Vos P, Holbert D. Frequentist statistical inference without repeated sampling. Synthese 200, 89 (2022). doi:10.1007/s11229-022-03560-x

## See also

For more details, see `bayestestR::equivalence_test()`

. Further
readings can be found in the references. See also `p_significance()`

for
a unidirectional equivalence test.

## Examples

```
data(qol_cancer)
model <- lm(QoL ~ time + age + education, data = qol_cancer)
# default rule
equivalence_test(model)
#> # TOST-test for Practical Equivalence
#>
#> ROPE: [-1.99 1.99]
#>
#> Parameter | 90% CI | SGPV | Equivalence | p
#> -----------------------------------------------------------------
#> (Intercept) | [59.33, 68.41] | < .001 | Rejected | > .999
#> time | [-0.76, 2.53] | 0.905 | Undecided | 0.137
#> age | [-0.26, 0.32] | > .999 | Accepted | < .001
#> education [mid] | [ 5.13, 12.39] | < .001 | Rejected | 0.999
#> education [high] | [10.14, 18.57] | < .001 | Rejected | > .999
# using heteroscedasticity-robust standard errors
equivalence_test(model, vcov = "HC3")
#> # TOST-test for Practical Equivalence
#>
#> ROPE: [-1.99 1.99]
#>
#> Parameter | 90% CI | SGPV | Equivalence | p
#> -----------------------------------------------------------------
#> (Intercept) | [59.22, 68.52] | < .001 | Rejected | > .999
#> time | [-0.80, 2.57] | 0.899 | Undecided | 0.144
#> age | [-0.27, 0.32] | > .999 | Accepted | < .001
#> education [mid] | [ 4.95, 12.58] | < .001 | Rejected | 0.998
#> education [high] | [10.17, 18.54] | < .001 | Rejected | > .999
# conditional equivalence test
equivalence_test(model, rule = "cet")
#> # Conditional Equivalence Testing
#>
#> ROPE: [-1.99 1.99]
#>
#> Parameter | 90% CI | SGPV | Equivalence | p
#> -----------------------------------------------------------------
#> (Intercept) | [59.33, 68.41] | < .001 | Rejected | > .999
#> time | [-0.76, 2.53] | 0.905 | Undecided | 0.137
#> age | [-0.26, 0.32] | > .999 | Accepted | < .001
#> education [mid] | [ 5.13, 12.39] | < .001 | Rejected | 0.999
#> education [high] | [10.14, 18.57] | < .001 | Rejected | > .999
# plot method
if (require("see", quietly = TRUE)) {
result <- equivalence_test(model)
plot(result)
}
```