# Mixed-Effects Scaling Law Analysis — Corrected Summary
## Manuscript: "Periodic supra-compartment organisation of eukaryotes"
## Date: 2026-04-08

---

## 1. Data

- **Source:** `cross_species_scaling_m2_recomputed.csv` (Paul m=2 CWT, recomputed)
- **Total chromosomes:** 71 → **n=67 after quality filters** (period_frac<0.45, n_peaks≥1000)
- **Excluded:** MCF-7 chr18/chr20 (COI), GM12878 chr21 (n<1000), Drosophila chr2L (COI)
- **Species (3-group):** H. sapiens (43), M. musculus (20), D. melanogaster (4)
- **Datasets (4-group):** MCF-7 (21), GM12878 (22), Mouse heart (20), Drosophila S2 (4)

---

## 2. Model Results

| Model | b | 95% CI | CI width | n | Notes |
|-------|---|--------|----------|---|-------|
| **OLS (primary)** | **0.848** | **[0.529, 1.167]** | 0.638 | 67 | Cross-species estimator |
| MixedLM rand-intercept 3-group | 0.741 | [0.427, 1.055] | 0.628 | 67 | Within-species estimator; ICC=0.488 |
| MixedLM rand-intercept 4-group | 0.753 | [0.447, 1.060] | 0.613 | 67 | Within-species estimator; ICC=0.495 |
| MixedLM rand-slope 3-group | 0.632 | [0.157, 1.107] | 0.950 | 67 | **Did not converge** — unreliable |
| Tobit censored (n=70) | 0.703 | [0.397, 1.008] | 0.611 | 70 | 3 COI chromosomes right-censored |
| Bootstrap 4-group (2.5/97.5) | 0.789 | [0.460, 1.170] | 0.709 | 67 | Most conservative |
| Bootstrap 3-group (2.5/97.5) | 0.806 | [0.159, 1.695] | 1.537 | 67 | Very wide; dominated by mouse |

**Within-species (stratified OLS):**
| Species/Dataset | b | 95% CI | p | R² | n |
|-----------------|---|--------|---|----|---|
| H. sapiens MCF-7 | 0.868 | [0.481, 1.255] | <0.001 | 0.538 | 21 |
| H. sapiens GM12878 | 0.732 | [0.194, 1.271] | 0.010 | 0.287 | 22 |
| H. sapiens pooled | 0.806 | [0.472, 1.141] | <0.001 | 0.366 | 43 |
| M. musculus Heart | **0.159** | [−1.016, 1.333] | **0.780** | **0.004** | 20 |
| D. melanogaster S2 | 1.695 | [0.398, 2.992] | 0.030 | 0.941 | 4 |

---

## 3. Critical Interpretive Issues

### 3.1 OLS vs MixedLM estimate different quantities

**This is the most important finding.** The random-intercept MixedLM estimates the
**within-species** slope (after removing species-level mean differences). The OLS estimates
the **cross-species** slope (using all variation, including between-species contrasts).
These are different estimands:

- **OLS b=0.848** answers: "How does period scale with length across all 67 chromosomes
  from three species spanning a 40× range in chromosome length?"
- **MixedLM b=0.741** answers: "How does period scale with length within each species,
  after accounting for species-level baseline differences?"

The manuscript's scaling law claim is a **cross-species** claim. OLS is therefore the
**correct primary estimator**. The MixedLM is a useful sensitivity check that asks whether
the scaling holds within species — but it is not a more "honest" version of the same estimate.

### 3.2 Mouse within-species scaling is flat

Mouse heart chromosomes show no within-species scaling (b=0.159, p=0.780, R²=0.004).
This is not a COI artefact (all period_frac values are well below 0.45). The flat slope
reflects high within-species period variance in mouse heart, with several chromosomes
showing anomalously short periods (chr13: 2.04 Mbp, chr1: 4.24 Mbp on chromosomes
>120 Mbp). This may reflect tissue-specific chromatin organisation in post-mitotic
cardiomyocytes, or genuine biological noise in the mouse heart CWT signal.

**Consequence:** The MixedLM b=0.741 is pulled down from OLS b=0.848 primarily by the
flat mouse slope. The cross-species scaling law is driven by the human-Drosophila contrast
and the absolute size difference between species, not by within-species scaling in mouse.

### 3.3 Reader-writer exclusion: model-dependent

The claim "reader-writer spreading (b=0.5) is excluded by the CI" is supported by OLS
(lower bound 0.529 > 0.5) but NOT by the mixed-effects model (lower bound 0.427 < 0.5)
or the bootstrap (lower bound 0.460 < 0.5).

However, since the reader-writer exclusion is a claim about the **cross-species** scaling
law, OLS is the appropriate estimator. The MixedLM CI is not directly relevant to this
claim because it estimates a different quantity (within-species slope).

The honest statement is: "The cross-species OLS slope marginally excludes b=0.5 at 95%
(lower bound 0.529). Within-species scaling in mouse is flat, so the exclusion of b=0.5
rests on the cross-species contrast rather than within-species evidence."

### 3.4 MixedLM Wald CI is anti-conservative with n=3 groups

The statsmodels MixedLM uses Wald (z=1.96) CIs for fixed effects. With only 3 species
groups, the random intercept variance is estimated from 3 data points, and the appropriate
degrees of freedom for the fixed-effect t-test are approximately 2 (Kenward-Roger
approximation). Using t(df=2) critical value (4.303 vs 1.96) would give a much wider CI:
[0.052, 1.430] for the 3-group model. The Wald CI [0.427, 1.055] is therefore
anti-conservative and should not be reported as a tighter alternative to OLS.

---

## 4. Manuscript Recommendation

### Primary result: Keep OLS b=0.848 [0.529, 1.167]

OLS is the correct estimator for the cross-species scaling claim. The CI marginally
excludes b=0.5 at 95%. This should remain the primary reported result.

### Methods addition (robustness paragraph):

> "To assess within-species scaling, we fitted a linear mixed-effects model
> log₁₀(T) ~ log₁₀(L) + (1|species) (statsmodels MixedLM, REML, 3 biological species
> as groups). The within-species slope was b=0.741 (Wald 95% CI [0.427, 1.055]; note
> that with n=3 groups the Wald CI is anti-conservative — a conservative t(df≈2) CI
> would be substantially wider). Within-species OLS slopes were b=0.868 (MCF-7),
> b=0.732 (GM12878), b=0.159 (mouse heart, p=0.780, R²=0.004), and b=1.695
> (Drosophila, n=4). The flat within-species slope in mouse heart indicates that the
> cross-species scaling law is driven primarily by the contrast between species of
> different genome sizes rather than by within-species scaling in mouse. A species-block
> bootstrap (resampling the 4 datasets with replacement, 10,000 iterations) gave a 95%
> CI of [0.460, 1.170] for the OLS slope. A Tobit regression treating COI-limited
> chromosomes as right-censored (n=70) gave b=0.703 [0.397, 1.008]."

### Results addition (one sentence):

> "Within-species scaling is significant in human (b=0.806 [0.472, 1.141], n=43) but
> not in mouse heart (b=0.159, p=0.780), indicating that the cross-species law is
> driven primarily by the contrast between species of different genome sizes."

---

## 5. Files

- `scaling_law_mixed_effects_results.csv` — full model comparison table
- `scaling_law_mixed_effects_figure.png` — 4-panel summary figure
- `scaling_law_mixed_effects_summary.md` — this document (corrected)
