Oncofusion

start / graph / concept-q130-a-spearman-over-four-quartiles-has-24-orderings-so-a-perfect-monotone-is-one-draw-in-twenty-four

Explainerestablished confidence 5 of 5: Multiple independent lines of evidence; treated as settled in the field.Q130-does-the-eviction-gradient-exist-in-h3k79me2

A Spearman over four quartiles has 24 orderings and 11 attainable values: rho = +1.000 is 1/24 = 0.042, |rho| = 1.000 is 2/24 = 0.083, and a +-20% band around 0.80 contains exactly one attainable value - so a perfect monotone over four strata is a weak statistic and must never be scored alone

also called the n=4 Spearman problem · one perfect monotone in four cells is p = 0.29 · why the span and not the rho is the scored statistic

When you rank only four things, there are just twenty-four possible orders, so getting a perfect one by luck happens about one time in twenty-four. A perfect-looking trend across four groups is much weaker evidence than it appears and needs a second, size-based measurement beside it.

The same thing, for a scientist

**Enumerated, not asserted.** All 4! = 24 orderings of four values were generated and their Spearman against `[4, 3, 2, 1]` computed [E1]: | rho | -1.0 | -0.8 | -0.6 | -0.4 | -0.2 | 0.0 | +0.2 | +0.4 | +0.6 | +0.8 | +1.0 | |---|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:| | orderings | 1 | 3 | 1 | 4 | 2 | 2 | 2 | 4 | 1 | 3 | 1 | - `P(rho = +1.000) = 1/24 = 0.0417` - `P(|rho| = 1.000) = 2/24 = 0.0833` - With **four** contrasts scored, the expected number of perfect monotones is `4 x 2/24 = 0.33` and `P(at least one) = 1 - (22/24)^4 = 0.29`. **One perfect monotone among four cells is not evidence.** **Three consequences this corpus has already needed.** 1. **A +-20% band around 0.80 contains exactly one attainable value, 0.80 itself.** Q124's criterion H3 required `rho >= 0.80 at n = 4`, which is a criterion about a permutation class (4 of 24 orderings, 1/6) dressed as a correlation threshold [E2]. 2. **A null can produce perfect monotones routinely.** In this beat's own DMSO-decile-matched null, **3 of 20 draws gave |rho| = 1.000** against a chance expectation of 1.67. If the Spearman had been the scored statistic, the null would have 'replicated' the finding in 15% of draws [E3]. 3. **So the scored statistic must be a magnitude, judged against a null computed on the same bins, the same strata sizes and the same normalisation.** In this beat that is the Q1-minus-Q4 span, and the Spearman is reported only as the shape it takes [E3]. **What the Spearman IS good for.** Sign. A rho of exactly -1.000 in one mark and exactly +1.000 in another, measured in the same bins, is a statement no bin-level artefact can make - and that sign-reversal argument does not depend on either rho being unlikely.

Go deeper
The general rule, which is not about Spearman

When a statistic is computed over a small number of strata, enumerate its exact null before quoting it. The enumeration takes three lines and it tells you two things a p-value will not: which values are attainable (so you know whether your threshold falls between two of them, as Q124's 0.80 did) and how coarse the statistic is.

This is the same family as the failure recorded in the brief - '4 of 93 exceed the 95th percentile' with no comparison against the 4.65 the percentile implies. In both cases the criterion is sound in form and the number it should be compared against was never written down. Write the null's number in the same sentence as the observed one.

Where this corpus should apply it next

Every eviction-quartile Spearman in Q125 and in this beat - nine of them across four marks and two genotypes - is an n = 4 statistic. None is wrong, and none should be quoted without its span and its matched null. The most exposed single figure is Q125's H3K4me3 -1.000, which is the free artefact control the whole gradient argument rests on: its span is 0.2534, the smallest of the five contrasts in the panel, and its matched null's own Spearman is +0.000 and +0.200 in the two arms. The control holds, and it holds on the span, not on the rho.

Facts and figures
n draws with abs rho 13
absence classmeasured-not-analysed
absence class noteNine n = 4 Spearman coefficients have been committed to this corpus across Q124, Q125 and this beat, and none carried its exact permutation null until now. The enumeration is three lines of code.
absence class assigned2026-09-23 (Q130)
exact null
{
  "n_orderings": 24,
  "attainable_rho": [
    -1,
    -0.8,
    -0.6,
    -0.4,
    -0.2,
    0,
    0.2,
    0.4,
    0.6,
    0.8,
    1
  ],
  "counts": {
    "-1.0": 1,
    "-0.8": 3,
    "-0.6": 1,
    "-0.4": 4,
    "-0.2": 2,
    "0.0": 2,
    "0.2": 2,
    "0.4": 4,
    "0.6": 1,
    "0.8": 3,
    "1.0": 1
  },
  "P_rho_eq_plus_1": 0.0417,
  "P_abs_rho_eq_1": 0.0833,
  "P_at_least_one_abs_1_in_4_contrasts": 0.2939,
  "expected_n_perfect_monotones_in_4_contrasts": 0.3333,
  "note": "stated in PREREGISTRATION section 4 BEFORE any H3K79me2 quartile number was computed. A +-20%% band around 0.80 contains exactly one attainable value, 0.80 itself."
}
null spearmans observed in 20 matched draws
[
  0.4,
  -0.8,
  1,
  -0.6,
  -0.6,
  0.8,
  -0.4,
  -0.8,
  0.6,
  -0.6,
  -0.6,
  0.4,
  0,
  -1,
  0,
  0.8,
  0.4,
  -1,
  -0.8,
  -0.4
]
Still open
  1. Should this corpus's eviction-quartile analyses move to a per-bin rank correlation against the continuous FLAG VTP/DMSO ratio instead of four strata? That statistic has n = 1,228 rather than n = 4 and the quartiles become a presentation device rather than the test. It was not done here because Q117's quartiles are the committed comparator and changing the statistic mid-line would have made this beat incomparable to Q125.
  2. How many other small-n thresholds in this corpus's pre-registrations fall between attainable values? Nobody has swept them.
Every source behind this entry (3)
  1. E1established confidence 5 of 5: Multiple independent lines of evidence; treated as settled in the field.dataset

    Enumerating all 24 orderings of four values gives 11 attainable Spearman coefficients with multiplicities 1/3/1/4/2/2/2/4/1/3/1 from -1.0 to +1.0, so P(rho = +1.000) = 1/24 = 0.0417 and P(|rho| = 1.000) = 2/24 = 0.0833 under a uniform random ordering.

    What this evidence cannot show

    This is arithmetic about a permutation null, not about chromatin. The four quartile ratios in this beat are NOT exchangeable - they share libraries, bins and a monotone baseline - so 1/24 is the probability under a random ordering and not under 'no drug effect'. The empirical matched null is the answer to the latter and it is a different and larger number for the Spearman: 3 of 20.

    Evidence that argues against this (1)
  2. E2strong confidence 4 of 5: Replicated in peer-reviewed work by more than one group.dataset

    Q124's criterion H3 required Spearman >= 0.80 at n = 4, which selects 4 of 24 orderings (1/6) and admits only two attainable values, 0.80 and 1.00; a +-20% tolerance band around 0.80 contains exactly one attainable value.

    What this evidence cannot show

    Q124's criterion was not wrong in the sense of producing a false verdict; it was coarse in a way the threshold notation conceals. Every band-based threshold in this corpus's pre-registrations should be checked against the attainable value set of its statistic.

    Evidence that argues against this (1)
    • A threshold that admits only two attainable values is not automatically a bad threshold - a criterion demanding near-perfect monotonicity is a legitimate thing to demand. Iqbal Lab oncofusion knowledge graph 2026
  3. E3moderate confidence 3 of 5: One strong study, or several smaller consistent ones.dataset

    In this beat's DMSO-decile-matched null for the control VTP contrast, 3 of 20 draws produced |Spearman| = 1.000 against a chance expectation of 1.67, while none of the 20 reached the observed Q1-minus-Q4 span - so on these data the Spearman would have been 'replicated' by the null in 15% of draws and the span never was.

    What this evidence cannot show

    20 draws is a small null for estimating a 1-in-12 event; 3 of 20 is entirely consistent with the chance expectation of 1.67 and is quoted to make the point that perfect monotones are common, not to claim the null is inflated.

    Evidence that argues against this (1)
    • A published methods study establishes that STANDARD ChIP-seq normalisation FAILS TO DETECT a genome-wide decrease in a histone mark under an inhibitor of its writer: standard methods see no H3K27me3 reduction under an EZH2 inhibitor, and a substantial reduction appears only with a Drosophila chromatin spike-in. Every number in this entry is exactly the kind of normalisation that paper shows to be blind. Egan B et al. 2016
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