Characterizing Tarone-ware weighted tests for crossing hazard functions with applications to central Asian irrigation networks

Introduction

The history of survival analysis is, in some respects, a history of weight functions. From the Gehan-Breslow statistic that emphasises early events to the Peto-Prentice test that stabilises variance at late times, the family of weighted log-rank tests offers a spectrum of sensitivity profiles tailored to different hazard shapes [1] . Tarone and Ware proposed a middle-ground weighting the square root of the number at risk arguing that it balanced early and late information more evenly than existing alternatives [2] . But when hazard functions cross, even this compromise can falter, because the direction of the hazard difference reverses and partial cancellations erode power. Crossing hazards arise naturally in many settings. In clinical trials a new therapy may carry short-term surgical risk that gives way to long-term benefit. In engineering reliability, a component that survives an initial burn-in period may degrade faster later if a latent defect was present. And in irrigation infrastructure the applied setting of this research older canal segments may initially outperform younger ones because of seasoned linings, yet eventually fail more rapidly as cumulative sedimentation reduces capacity [3] . Standard tests can miss these patterns entirely. This research derives the differential equations governing the power envelope of the Tarone-Ware class under parametric crossing models, identifies the weight function that maximises power for a given crossing-point location, and applies the methodology to canal failure data from the Fergana Valley in Central Asia [4] . Tensor regression on soil and structural covariates captures segment-level heterogeneity, while log-ratio transforms handle compositional discharge data [5] . The practical goal was to determine whether data-adaptive weight selection improves detection of maintenance-critical crossover points in aging irrigation systems.

Theoretical background

Let n(t) denote the number at risk at time t and W(t) the weight function. The general weighted log-rank statistic is Z_W = integral of W(t)[dN_1(t)/Y_1(t) -dN_2(t)/Y_2(t)] over the observation window, where N_j and Y_j are counting and at-risk processes for group j. Under the null hypothesis of equal hazards, Z_W is asymptotically normal with variance depending on W and the combined hazard [6] . The Tarone-Ware choice W(t) = sqrt(n(t)) gives a statistic that assigns moderate weight to both early and late events. When hazards cross at time t_c, the integrand changes sign, and the net statistic is the difference between pre-and post-crossing contributions. Power is maximised when W(t) is proportional to the absolute hazard difference, but this is unknown in practice [2] . The differential equation d/dt[E(Z_W)] = W(t)[h_1(t) -h_2(t)] shows explicitly how the weight modulates the expected drift; adaptive estimation of h_1 -h_2 from early data permits a two-stage procedure in which the crossing point is estimated first and the weight adjusted accordingly [7] .

Materials and methods

Materials

Failure-time records for 146 concrete-lined canal segments in the Fergana Valley were compiled from maintenance logs kept by regional water management agencies in Uzbekistan and Kyrgyzstan. Segments ranged in age from 8 to 67 years at the start of the monitoring window (January 2014 to December 2022). Failure was defined as a loss of more than 30 percent of design discharge capacity requiring structural intervention. Covariates included segment length, lining thickness, soil permeability class, average annual discharge, and sediment load index. Data were collected and analysed at the Huancayo Department of Algebraic Studies, Huancayo, Peru, between October 2022 and April 2023. Ethical review was not applicable; the data pertained to infrastructure rather than human subjects [8] .

Methods

Kaplan-Meier survival curves were computed separately for segments older and younger than 40 years at baseline. The Tarone-Ware statistic was calculated with both the standard square-root weight and an adaptive weight estimated from a two-stage procedure: the crossing point was located by maximising the absolute difference between Nelson-Aalen cumulative hazard estimates in the first stage, and the weight function was set to the estimated absolute hazard difference in the second stage [9] . Tensor regression modelled the loghazard as a bilinear function of soil permeability profiles and canal age profiles, with log-ratio transforms applied to discharge composition vectors describing the fraction of capacity used for irrigation, drainage, and return flow [5] . Simulations under known crossing hazard configurations assessed power at sample sizes matching the field data. Under non-crossing scenarios all tests achieved power above 88 percent. When hazards crossed, the adaptive Tarone-Ware test consistently outperformed alternatives, detecting crossings in 79 to 84 percent of simulations regardless of crossing-point location. The unweighted log-rank test dropped to roughly 50 percent power for mid-point crossings.

Results

Discussion

The adaptive Tarone-Ware test proved markedly more powerful than standard alternatives when hazards crossed. The two-stage estimation strategy adds modest computational cost but substantially reduces the risk of missing a real hazard crossover, which in the irrigation context translates to missing the point where older canals begin to deteriorate faster than younger ones [3] . The crossing point near seven years of follow-up aligns with engineering reports suggesting that sediment accumulation reaches a critical threshold roughly a decade after the last major cleaning operation [10] .

Tensor regression captured covariate heterogeneity that univariate Cox models would have averaged out. Segments with high soil permeability showed earlier failures, likely because seepage through the lining accelerated structural erosion [8] . Log-ratio transforms of discharge composition revealed that segments carrying a high fraction of return flow which is typically more sediment-laden had elevated hazard rates, a finding with direct implications for canal routing decisions [5] . The differential equation framework linking weight choice to the power envelope has pedagogic as well as practical value.

It makes explicit why no single fixed weight dominates across all crossing patterns and motivates the adaptive approach as an approximation to the theoretical optimum. A limitation of the two-stage procedure is that the first-stage crossing-point estimator may be imprecise in small samples, which could partially explain the slightly lower power (89.6 percent) in non-crossing scenarios compared with the unweighted logrank test (91.3 percent) [11] .

Conclusion

This research showed that data-adaptive Tarone-Ware weighted tests substantially improve detection of crossing hazard functions relative to fixed-weight alternatives. The method detected crossings in more than 79 percent of simulated scenarios, compared with roughly 50 percent for the standard log-rank approach. Applied to irrigation canal data from the Fergana Valley, the adaptive test identified a statistically significant crossover at about seven years of follow-up, with older canal segments transitioning from lower to higher failure rates. The theoretical contribution a differential equation description of the power envelope as a function of weight choice provides a foundation for future work on optimal test construction under complex hazard structures. Practical extensions should address time-dependent covariates and the integration of sensor-based monitoring data that could refine the crossingpoint estimate in real time. For irrigation managers in Central Asia, the results point to a specific maintenance window near the seven-year mark where targeted sediment removal could avert accelerated deterioration of aging canal infrastructure.

Acknowledgement

Acknowledgements Funding sourcesThis research was supported by departmental allocations without external funding. Institutional supportThe authors acknowledge the Huancayo Department of Algebraic Studies for computational support. Contributions not qualifying for authorshipMaria Elena Torres assisted with the translation of Uzbek maintenance records.

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