Arm-specific versus total-sample fragility quotients in large vessel vasculitis trials

The fragility quotient normalizes the fragility index by total sample size but can obscure arm-specific fragility when trial arms are unequal, since the fragility index systematically alters outcomes in only one arm. We defined the modified-arm fragility quotient as the fragility index divided by the size of the arm in which outcomes are toggled. We compared it to the fragility quotient using three approaches: (1) re-analysis of 29 categorical outcomes from 18 large-vessel vasculitis trials (allocation ratios 1.0-2.0), (2) R grid simulation across 1:1 to 4:1 allocation ratios with sample sizes 60-240 (90,000 simulations), and (3) Python Monte Carlo simulation with 3,000 replications across 1:1, 1:2, and 1:3 allocations. In balanced 1:1 trials, the modified-arm fragility quotient equaled twice the fragility quotient exactly, as predicted. The proportion of trials in which the modified-arm fragility quotient exceeded twice the fragility quotient increased systematically with allocation imbalance: 0% (1:1), 81% (2:1), 93% (3:1), and 95% (4:1). In the R simulation, mean divergence between the modified-arm fragility quotient and twice the value of the fragility quotient increased from 0.00 (1:1) to 0.10 (4:1). Empirical data showed a strong correlation (r = 0.98) between the two quotients, but divergence increased in the 25 unbalanced allocation trials. The ratio of the modified-arm fragility quotient to twice the value of the fragility quotient scaled linearly with allocation ratio, with mean values of 1.0 (1:1), 1.36 (2:1), 1.91 (3:1), and 2.41 (4:1). These findings show that the modified-arm fragility quotient and total-sample fragility quotient convey equivalent information under equal allocation. In contrast, under unequal randomization, the total-sample fragility quotient can mischaracterize trial fragility due to dilution from the larger, unmodified arm, whereas the modified-arm fragility quotient avoids this bias by restricting normalization to the modified arm.