This pre-registered, cross-platform study examined whether a model-free summary of heavy-tailed gene-expression fluctuations, the Tsallis index (q), differs between human cortical samples from bipolar II disorder (BD-II) cases and controls. The authors set out two separable questions: (1) does the per-sample value of q differ between cases and controls, and (2) if q is invariant, does any disease-related signal instead appear in the collective correlation structure of the transcriptome?
The approach emphasised a leakage-free protocol and pre-registration to limit analytical flexibility. All results reported here are drawn from that pre-registered analysis as presented in the source preprint.
The analysis used a per-sample maximum-likelihood estimator to fit q-Gaussian distributions to gene-expression fluctuation data. The estimator was applied under a leakage-free, pre-registered protocol to multiple independent cohorts. Cohort identifiers reported in the source are GSE80655, GSE12649 and GSE53987. Model comparison between q-Gaussian and Gaussian forms used BIC-based metrics to assess whether the heavy-tailed q-Gaussian provided a better description of marginal fluctuations than a standard Gaussian.
The pre-registered plan specified a minimum effect of interest for equivalence testing (Δq bound = 0.03) and described procedures for pooling homogeneous cohorts and for random-matrix permutation testing to probe correlation-structure differences between groups.
Per-sample maximum-likelihood estimation produced concentrated cohort-level values of q. Reported mean q values were approximately 1.37 for GSE80655, 1.33 for GSE12649, and 1.35 for GSE53987. Across samples, model comparison strongly favoured the q-Gaussian over a Gaussian: q-Gaussian was preferred in 99–100% of samples, with median ΔBIC much smaller than zero. This pattern supports the interpretation that q is a meaningful descriptor of marginal heavy-tailed behaviour rather than an artefact of fitting.
Critically, the BD-II versus control difference in q was null in every cohort: no detectable case–control difference in the marginal Tsallis index was found in the analysed datasets.
To assess invariance more directly, the authors pooled the three homogeneous cohorts and performed an equivalence test (TOST) rather than relying on non-rejection of difference. The pooled data established equivalence at the bound |Δq| ≥ 0.037. However, the pooled analysis did not reach the originally pre-registered equivalence bound of 0.03; the authors report that achieving that pre-registered sensitivity would have required roughly twice the present sample size. They explicitly report this power shortfall and characterise the null as bounded and informative but underpowered relative to the pre-specified minimum effect of interest.
A random-matrix test was applied to probe whether between-group structural differences in correlation patterns survived label permutation. The test found no between-group structural difference that survived permutation. Taken together with the null marginal q results, the authors argue the disease question for BD-II is better addressed not at the level of the marginal non-extensivity index but in the higher-order correlation structure and in dynamical modes (which are addressed in companion work referenced by the authors).
The authors attempted a cross-tissue scale anchor using a glioma RNA-seq reference to assess whether a same-value claim for q across tissues was supportable. That cross-tissue anchoring failed: the glioma RNA-seq reference was confounded by sequencing depth in a cohort-inconsistent manner and therefore could not be used to justify cross-tissue equivalence of q. The authors report this failure transparently rather than obscuring it.
From these results the authors infer that non-extensivity, as summarised by q, behaves as a conserved organisational property of the human cortical transcriptome in the datasets analysed. For bipolar II disorder specifically, the marginal Tsallis index does not distinguish cases and controls. Consequently, the authors relocate the disease-focused question from marginal departures from Gaussianity to the architecture of correlations among expression modes and to transcriptomic dynamics. Companion analyses addressing dynamics and collective modes are cited by the authors as the next step.
The report emphasises pre-registration, leakage-free estimation, and explicit acknowledgment of the study's power limits relative to the pre-registered minimum effect size (0.03). The authors provide links to supplementary material, data and code on the preprint page. The study is a preprint and has not been peer reviewed; competing interests are declared as none. The failed glioma-based cross-tissue anchor is presented as an unresolved limitation of cross-tissue claims in this work.
Overall, the source documents a null yet bounded and informative result: the Tsallis index q, estimated per sample using a q-Gaussian maximum-likelihood estimator, is invariant between BD-II and control in the analysed cortical cohorts, shifting the locus of potential disease signal to correlation structure and dynamics rather than to marginal heavy-tailedness.