Compute the (conditional) equivalence test for frequentist models.

## Usage

```
# S3 method for lm
equivalence_test(
x,
range = "default",
ci = 0.95,
rule = "classic",
verbose = TRUE,
...
)
# S3 method for merMod
equivalence_test(
x,
range = "default",
ci = 0.95,
rule = "classic",
effects = c("fixed", "random"),
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'.- 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"`

.

## 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 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 of an effect (i.e. H0) is

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

)*include*or*exceed*the ROPE boundaries. 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. Else, the decision whether to accept or reject H0 is undecided.- "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 H0, 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.

### 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). This statistic
is actually computed in the same way as the percentage inside the ROPE as
returned by `equivalence_test()`

(see Lakens and Delacre 2020
for details on computation of the SGPV). Thus, the `"inside ROPE"`

column reflects the SGPV.

### 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.

## References

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

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

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

## See also

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

.
Further readings can be found in the references.

## 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 | % in ROPE | H0 | p
#> ------------------------------------------------------------------
#> (Intercept) | [59.33, 68.41] | 0% | Rejected | > .999
#> time | [-0.76, 2.53] | 83.52% | Undecided | 0.137
#> age | [-0.26, 0.32] | 100% | Accepted | < .001
#> education [mid] | [ 5.13, 12.39] | 0% | Rejected | 0.999
#> education [high] | [10.14, 18.57] | 0% | Rejected | > .999
# conditional equivalence test
equivalence_test(model, rule = "cet")
#> # Conditional Equivalence Testing
#>
#> ROPE: [-1.99 1.99]
#>
#> Parameter | 90% CI | % in ROPE | H0 | p
#> ------------------------------------------------------------------
#> (Intercept) | [59.33, 68.41] | 0% | Rejected | > .999
#> time | [-0.76, 2.53] | 83.52% | Undecided | 0.137
#> age | [-0.26, 0.32] | 100% | Accepted | < .001
#> education [mid] | [ 5.13, 12.39] | 0% | Rejected | 0.999
#> education [high] | [10.14, 18.57] | 0% | Rejected | > .999
# plot method
if (require("see", quietly = TRUE)) {
result <- equivalence_test(model)
plot(result)
}
```