Clustered data are a special type of correlated data where units within a cluster are correlated while units between different clusters are independent. The number of units in a cluster can be associated with that cluster’s outcome. This is called the informative cluster size (ICS), which is known to impact clustered data inference. However, when comparing the outcomes from multiple groups of units in clustered data, investigating ICS may not be enough. This is because the number of units belonging to a particular group in a cluster can be associated with the outcome from that group in that cluster, leading to an informative intra-cluster group size or IICGS. This phenomenon of IICGS can exist even in the absence of ICS. Ignoring the existence of IICGS can result in a biased inference for group-based outcome comparisons in clustered data. In this article, we mathematically formulate the concept of IICGS while distinguishing it from ICS and propose a nonparametric bootstrap-based statistical hypothesis-testing mechanism for testing any claim of IICGS in a clustered data setting. Through simulations and real data applications, we demonstrate that our proposed statistical testing method can accurately identify IICGS, with substantial power, in clustered data.

Aims: To evaluate the association between diabetic corneal neuropathy and diabetic foot risk by examining corneal nerve features, clinical characteristics, and tear mediators in patients with different diabetic foot risk categories. Methods: In this cross‐sectional study, 710 participants underwent diabetic foot examination, in vivo confocal microscopy scans, ocular surface assessments, and tear mediators measurements. Ordinal regression analysis was applied to determine the associations between foot risk categories and corneal nerve metrics. ROC curve analysis assessed the ability of corneal nerve parameters to identify high‐risk diabetic foot. Results: All diabetic foot risk categories exhibited significantly impaired corneal nerve fibre length, density, fractal dimension, and more swollen width compared with controls (all p < 0.001). These patients also demonstrated significantly shorter tear break‐up time, more severe ocular symptoms, elevated tear MMP‐9, and reduced tear Substance P levels (all p < 0.05). Patients with high‐risk diabetic foot showed significantly more pronounced corneal nerve alterations than those in low and moderate‐risk groups (all p < 0.05). The more severe diabetic foot risk category was significantly associated with reduced nerve fibre length, density, fractal dimension, and increased width (all p < 0.05). A predictive model incorporating corneal nerve fibre length, width, and serum creatinine achieved an AUC of 0.80 for identifying high‐risk diabetic foot. A CNFL < 5.7 mm/mm ² was associated with 4.37‐fold higher odds of having a high‐risk diabetic foot. Conclusion: Corneal nerve impairment is evident even in low‐risk diabetic foot patients and is associated with foot risk severity. Corneal nerve metrics may serve as early indicators for identifying high‐risk diabetic foot.

Brain state modulates sensory processing across visual cortex, yet how it relates to the organization of functional circuitry at the level of individual neurons and cell types remains largely unknown. To address this, we constructed one of the largest microscale directed functional circuit maps in mouse visual cortex from calcium imaging of more than 57,000 neurons across four visual areas and five cortical layers. Using a time-aware causal inference framework, we found that intra-areal connections dominate across arousal states, consistent with experimental findings on the local bias of cortical anatomy. Among intra-areal connections, anterolateral area (AL) had the highest density, and among inter-areal connections, the AL$\leftrightarrow$rostrolateral area (RL) axis formed the strongest pathway. Laminar circuit organization was dominated by layer 6 recurrence within-layer, while the most prominent between-layer pathway was layer 5-to-layer 6 in low arousal and layer 4-to-layer 5 in high arousal. Spatial extent was selectively greater for excitatory-to-inhibitory connections in high arousal, but not for excitatory-to-excitatory connections. Across 6,597 electron-microscopy reconstructions of neuron pairs, synapse count predicted functional connection strength in both arousal states, but structure-function coupling was weaker in high arousal. In stimulus-driven response prediction, neuron pairs with stronger functional connections exhibited more similar predictive performance in both states, with performance varying by layer and cell type. Overall, our findings map, at single-neuron resolution, the multi-scale organization of directed functional circuitry in mouse visual cortex across brain states.