Stating that an effect is absent

Many of the most interesting results in standstill research are nulls: no effect of the environment, no seasonal rhythm, no coupling between people standing together, no association between how loud a room is and how much a body moves.

The problem is that a non-significant test does not support any of those claims. It says only that the data are compatible with no effect, and they are equally compatible with an effect too small for this sample to resolve. Writing that up as "no effect was found" is an overclaim, and reviewers rightly say so.

Equivalence testing states the claim that was actually meant. Two one-sided tests invert the usual logic, so that the null becomes "the effect is at least this large", and rejecting it supports the statement that the effect is smaller than that. The bound is a smallest effect size of interest, and choosing it is a scientific judgement rather than a statistical one. That is the point: "no effect" is not a testable claim, while "smaller than half a millimetre per second" is.

Four outcomes, not two

import micromotion as mm

r = mm.tost_paired(silence, music, bound=0.5)      # bound in mm/s, the data's own units
r["verdict"]      # 'effect' | 'equivalent' | 'trivial' | 'inconclusive'
mm.equivalence.interpret(r)                        # one sentence a report can quote

verdict	meaning	what to write
effect	significant, and the interval reaches past the bound	there is an effect worth caring about
equivalent	not significant, and the interval fits inside the bound	the effect is smaller than the bound
trivial	significant and inside the bound	real, and too small to matter
inconclusive	neither	this sample cannot decide

inconclusive is the most useful of the four, because it covers the case usually reported as "no effect". A small study that finds nothing has not shown that nothing is there.

The functions

mm.tost_paired(a, b, bound)                  # paired difference, bound in the data's units
mm.tost_independent(a, b, bound)             # two groups, Welch standard error
mm.equivalence_correlation(r, n, bound)      # a correlation, from r and n alone

equivalence_correlation takes r and n rather than the raw series, so that it can be applied to a correlation that is already published without recomputing it.

Choosing the bound

Give the bound in the units the reader thinks in. A bound of 0.5 mm/s says that differences smaller than half a millimetre per second are not interesting here, which is a claim someone can argue with. A bound of d = 0.2 is a claim about a standardised quantity that nobody has an intuition for.

For correlations, 0.2 is a common convention for "small". Better still is to state what magnitude would change the conclusion and use that.

The package offers no default bound, deliberately. The whole content of the claim is in the bound, and a package-supplied default would become a convention by accident—a convention about what matters in a research question, which is not something a measurement library knows.

Worked example

Five nulls from one standstill corpus, at a bound of r = 0.20:

claim	r	n	90 % CI	verdict
body tracks the environment, within-day	+0.010	365	[−0.076, +0.096]	equivalent
year-long habituation, one device	−0.050	331	[−0.140, +0.041]	equivalent
movement settling vs heart-rate settling	+0.100	365	[+0.014, +0.185]	equivalent
coupling between three performers	+0.015	69	[−0.185, +0.214]	inconclusive
coupling between sixteen performers	+0.070	16	[−0.368, +0.483]	inconclusive

The first three can be stated as findings. The last two cannot, and that is the useful part. With sixteen people the interval runs from −0.37 to +0.48, so the same data are compatible with nothing and with a substantial coupling, and no further analysis will separate them. Here the sample size is the binding constraint rather than the choice of measure, which is worth reporting in place of a p-value of 0.8.