Compute the (conditional) equivalence test for frequentist models.

# S3 method for lm
equivalence_test(
x,
range = "default",
ci = 0.95,
rule = "classic",
p_values = FALSE,
verbose = TRUE,
...
)

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

## Arguments

x A statistical model. The range of practical equivalence of an effect. May be "default", to automatically define this range based on properties of the model's data. Confidence Interval (CI) level. Default to 0.95 (95%). Character, indicating the rules when testing for practical equivalence. Can be "bayes", "classic" or "cet". See 'Details'. Logical, if TRUE, adjusted p-values for equivalence testing are calculated. Toggle warnings and messages. Arguments passed to or from other methods. Should parameters for fixed effects ("fixed"), random effects ("random"), or both ("all") be returned? Only applies to mixed models. May be abbreviated.

A data frame.

## 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.

### 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.

The calculation of p-values is somewhat "experimental". For parameters, where H0...

• ... is rejected, the p-value equals a NHST as if the upper / lower boundary of the ROPE (see range) would be the point-null to test against.

• ... is accepted, the p-value is set to 1.

• ... is undecided, the p-value equals a NHST against the point-null, however, the "uncertainty" (i.e. ROPE range) is added to the confidence intervals (so the upper confidence interval limit equals the regular upper confidence interval limit + half the ROPE range).

All p-values are then adjusted for multiple testing (using stats::p.adjust() with method = "fdr").

### 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

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        H0 inside ROPE        90% CI
#>       (Intercept)  Rejected      0.00 % [59.33 68.41]
#>              time Undecided     83.52 % [-0.76  2.53]
#>               age  Accepted    100.00 % [-0.26  0.32]
#>   education [mid]  Rejected      0.00 % [ 5.13 12.39]
#>  education [high]  Rejected      0.00 % [10.14 18.57]
#>

# conditional equivalence test
equivalence_test(model, rule = "cet")
#> # Conditional Equivalence Testing
#>
#>   ROPE: [-1.99 1.99]
#>
#>         Parameter        H0 inside ROPE        90% CI
#>       (Intercept)  Rejected      0.00 % [59.33 68.41]
#>              time Undecided     83.52 % [-0.76  2.53]
#>               age  Accepted    100.00 % [-0.26  0.32]
#>   education [mid]  Rejected      0.00 % [ 5.13 12.39]
#>  education [high]  Rejected      0.00 % [10.14 18.57]
#>

# plot method
if (require("see", quietly = TRUE)) {
result <- equivalence_test(model)
plot(result)
}
#> Error: package or namespace load failed for ‘see’:
#>  object ‘check_heterogeneity’ is not exported by 'namespace:parameters'