A group of four cannot support a correlation at any effect size: name it, do not compute it
With only four samples, even a perfect trend cannot reach the usual standard of evidence. The right thing to do is say so, not print a number that looks like a result.
The same thing, for a scientist
An exact two-sided permutation test on a rank correlation over n samples has a **smallest attainable p of 2/n!**, because only the two extreme orderings give the most extreme |rho|. That is a property of the design, not of the data, and it is known **before any value is read**: | n | permutations | smallest attainable two-sided p | can reach 0.05? | |---:|---:|---:|---| | 2 | 2 | 1.000 | no | | 3 | 6 | 0.333 | no | | 4 | 24 | 0.083 | **no** | | 5 | 120 | 0.0167 | yes | | 6 | 720 | 0.0028 | yes | | 8 | 40,320 | 0.00005 | yes | So a four-sample group can produce a **perfect monotone ordering** - rho = -1.000, the strongest result the statistic can express - and still return p = 0.083 [E1]. **Applied here**, this determined the analysis plan before any signal was read: of GSE252378's six partner groups, **four cannot support a correlation** (AFDN n = 4, AF10 n = 3, AF9 n = 3, EPS15 n = 2) and only AFF1 (8) and ENL (6) can. None was computed for the four; they are reported as per-sample point values with depth grades, and the sentence *'this group cannot support a correlation'* is written in the output [E1]. **Why this needs to be a rule and not a habit.** The alternative - computing rho on four points and reporting it with its p - produces a number that is formally correct, reads like a measurement, and is uninformative by construction. It is the same family as a proportion whose null is 50%: the statistic cannot distinguish the hypothesis from the design.
What to report instead
For an under-powered group, three things carry information and a rho does not:
- The point values themselves, with their absolute floor and depth grade. In this beat the whole result for the four small groups is 'IRX1 is 0.047-0.269x an average promoter in all twelve samples, against floors of 0.098-1.586x' - which is more informative than any correlation would have been.
- A rule-of-three bound on the zero. See
concept-q147-a-bounded-absence-across-five-partners-is-a-rule-of-three-bound. - The noise band. Run the same statistic on loci that cannot be occupied, at the same n, and print the |rho| they reach. At n = 5-6 here, unoccupied loci reach |rho| up to 0.600, which is the number a reader needs in order to judge a reported 0.5.
And state the arithmetic in the entry, because 2/n! is checkable by the reader and a post-hoc power calculation is not.
| formula | smallest attainable exact two-sided p = 2/n! |
|---|---|
| control locus noise band at n6 | 0.6 |
| control locus noise band at n8 | 0.571 |
table — the numbers behind the chart
{
"2": 1,
"3": 0.3333,
"4": 0.0833,
"5": 0.0167,
"6": 0.00278,
"8": 0.0000496
}groups refused — the numbers behind the chart
{
"KMT2A-AFDN": 4,
"KMT2A-AF10": 3,
"KMT2A-AF9": 3,
"KMT2A-EPS15": 2
}groups computed
{
"KMT2A-AFF1": 8,
"KMT2A-ENL": 6
}analyses
[ "ana-q147-irx1-fusion-occupancy-is-confined-to-kmt2a-aff1", "ana-q147-the-hoxa-irx1-anticorrelation-does-not-reproduce-in-kmt2a-enl" ]
- How many correlations in this corpus were computed on n <= 4? The smallest attainable p is never recorded beside a stored rho, so the exposure is unknown.
- E1established confidence 5 of 5: Multiple independent lines of evidence; treated as settled in the field.dataset
An exact two-sided permutation test on a Spearman correlation over n samples has a smallest attainable p of 2/n!: 1.000 at n=2, 0.333 at n=3, 0.083 at n=4, 0.0167 at n=5. In GSE252378, four of the six KMT2A partner groups have n <= 4 untreated samples (KMT2A-AFDN 4, -AF10 3, -AF9 3, -EPS15 2) and no correlation was computed for them; the eight pre-declared cannot-be-occupied control loci reach |rho| up to 0.600 at n = 6 and 0.571 at n = 8 in the same data.
q147 re-analysis of GSE252378: HOXA and IRX1 oncoprotein occupancy across all six KMT2A fusion partners, 27 samples, 133 libraries (analyses/q147-hoxa-irx1-across-partners) · Iqbal Lab oncofusion knowledge graph, beat q147 · internal analysis over deposited GSE252378 bigWigs 2026What this evidence cannot show
The 2/n! bound is exact for the two-sided |rho| criterion used here and assumes no ties; with ties the attainable p can be larger still. It is specific to the exact permutation framework - a parametric or Bayesian treatment reports an interval at n = 3 rather than refusing, which is legitimate and simply a different frame. The refusal recorded here is the pre-registered choice, not a claim that no inference is possible.
Evidence that argues against this (1)
- Refusing to compute a statistic has its own cost: it makes the under-powered groups invisible in any table of correlations, and a reader scanning such a table would see only AFF1 and ENL and might infer the other partners were not examined. That is why the per-sample point values are printed for all six groups and the refusal is written out in words in the output rather than left as a blank. Iqbal Lab oncofusion knowledge graph et al. 2026
applies the same rule: all six partner groups here are named as unable to carry a statistic