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The region of practical equivalence (ROPE) is a Bayesian decision rule for testing whether a parameter’s effect is negligible rather than exactly zero. Instead of asking whether a point null hypothesis can be rejected, a ROPE analysis asks whether the posterior distribution for a parameter falls inside, outside, or across a pre-specified band of values that are, for practical purposes, indistinguishable from “no effect.” The method comes from psychologist John K. Kruschke, formalized in Kruschke (2018), Rejecting or Accepting Parameter Values in Bayesian Estimation, Advances in Methods and Practices in Psychological Science 1(2), 270-280, and extended with a formal decision rule in Kruschke & Liddell (2018).
The operational definition: what makes a value “practically equivalent”
A ROPE is a small interval around a reference value – almost always zero, for a difference or slope parameter – chosen before the decision is made, whose bounds represent the largest effect a researcher is willing to treat as negligible. An effect that falls entirely inside the ROPE is not “no effect” in a literal sense; it is an effect too small to matter for the question at hand. That distinction is the whole point of the method: a point null hypothesis (the effect is exactly 0.000…) is a measure-zero event no posterior distribution can ever confirm, but a bounded region of negligible effects is a real, testable claim a posterior can actually fall inside or outside of.
The Kruschke-Liddell decision rule
Given a ROPE and a highest density interval (HDI) – the narrowest interval containing a chosen proportion of the posterior, conventionally 95%, though Kruschke has also used 89% to avoid the “cliff-like” instability a 95% HDI can show in small samples – the decision rule from Kruschke & Liddell (2018) has three outcomes:
- Reject the null value: the HDI falls entirely outside the ROPE.
- Accept the null value: the HDI falls entirely inside the ROPE.
- Undecided: the HDI partially overlaps the ROPE. More precision (typically more data) is needed before a decision can be made either way.
The “undecided” outcome is deliberate, not a failure of the method. It gives a researcher an honest way to say “we don’t yet have enough precision to rule out either a negligible or a meaningful effect” – a state frequentist null-hypothesis significance testing has no real vocabulary for, since it forces a reject/fail-to-reject call regardless of how uninformative the data actually are.
Choosing the width of the ROPE
Kruschke (2018) proposes a default ROPE of ±0.1 on a standardized effect-size scale – an effect smaller than Cohen’s conventional “negligible” threshold of d = 0.1 in absolute value. For an ordinary linear-model parameter this generalizes to ±0.1 × SDy; for a logistic-regression coefficient expressed as a log odds ratio, converting through the standard logistic distribution (multiplying by π/√3) widens the default to roughly ±0.18. These are starting points, not fixed rules – the ROPE should ultimately reflect a domain judgment about the smallest effect that would change a real decision (a treatment worth adopting, a policy worth changing), the same judgment a minimal clinically important difference (MCID) encodes in trial design. A ROPE chosen after seeing the results, or chosen specifically to produce a favorable outcome, defeats the method’s purpose and should be pre-specified and justified in the same section of a manuscript that reports the result.
Worked example
Suppose a study estimates a standardized pre/post difference using a Bayesian hierarchical model, and the resulting posterior has a 95% HDI of [-0.04, 0.09] on a Cohen’s-d scale. Using the default ±0.1 ROPE, that entire HDI falls inside [-0.1, 0.1] – the null value is accepted: the data support treating the effect as practically negligible, not merely “not statistically significant.” Had the HDI instead been [0.03, 0.24], it would straddle the ROPE’s upper bound of 0.1, landing in the undecided zone: some credible parameter values represent a negligible effect, others a meaningful one, and the honest conclusion is that more precision is needed before deciding either way. Only an HDI such as [0.15, 0.31], sitting entirely above 0.1, would justify rejecting the null value outright.
Why ROPE is not the same as “the credible interval excludes zero”
Checking whether a 95% credible interval excludes zero is a common Bayesian shortcut, but it inherits a version of the same problem as frequentist significance testing: given enough data, almost any interval eventually excludes an exact point value, including trivially small effects nobody would call meaningful. ROPE analysis separates two questions that “does the interval touch zero” conflates: whether an effect exists at all, and whether it is large enough to matter. A large study can produce a narrow HDI of [0.02, 0.04] that excludes zero cleanly and yet sits entirely inside a sensible ROPE – a real, precisely estimated, and practically negligible effect at the same time.
ROPE and frequentist equivalence testing
The frequentist analogue is equivalence testing, most commonly implemented as the two one-sided tests (TOST) procedure: define an equivalence margin, then test whether the effect is significantly smaller than the upper margin and significantly larger than the lower margin. Both approaches require the same substantive judgment call – a pre-specified, justified boundary of practical indifference – and both exist to answer the same question journals increasingly expect answered directly: not just “was there a statistically detectable difference,” but “was there a difference large enough to matter.” ROPE folds that judgment into a single posterior-based decision rule instead of two hypothesis tests; the choice between the two generally follows whichever inferential framework, Bayesian or frequentist, the rest of the analysis already uses.
Reporting a ROPE analysis
A manuscript reporting a ROPE-based conclusion should state, alongside the model and posterior summary: the ROPE’s bounds and how they were chosen (default convention vs. a domain-specific justification, and whether that choice was pre-registered); the HDI width used (95%, 89%, or other) and its bounds; and the resulting decision – reject, accept, or undecided – stated explicitly in those terms rather than left implied by the numbers alone. Where an interpreting Bayes factors analysis is reported alongside a ROPE analysis, note that the two can disagree: a Bayes factor tests a sharp point null against a diffuse alternative, while ROPE tests an interval null against the same data, and the two are answering related but not identical questions.
Frequently asked questions
Does ROPE require a specific prior or sampler?
No. ROPE is a decision rule applied to whatever posterior distribution a Bayesian model produces, regardless of whether that posterior came from conjugate analysis, grid approximation, or MCMC sampling. The choice of prior and sampler affects the posterior itself, and therefore the resulting ROPE decision, but ROPE itself adds no additional modeling requirement on top of standard Bayesian estimation.
Can ROPE be used for a single parameter and for a whole model at once?
The rule as defined by Kruschke and Liddell applies per parameter. Some software extends the idea by reporting the percentage of the full posterior distribution – not just its HDI – that falls inside the ROPE, which behaves more like a continuous index of overlap than the three-way accept/reject/undecided call.
Is a ROPE the same thing as a minimal clinically important difference (MCID)?
They serve a similar purpose – both encode “how large an effect has to be before it matters” – but an MCID is typically derived from clinical or patient-reported judgment about a specific outcome measure, while a ROPE is a general statistical convention (often the default ±0.1 standardized-effect band) that can be replaced with an MCID-derived value whenever one exists for the outcome being modeled.
