Determining proportional hazards modeling under frailty terms with applications to Yunnan plateau ethnic census data

Introduction

When two individuals sharing identical measured covariates experience different event times, the discrepancy points to unmeasured risk factors that the standard Cox model cannot capture [1] . This hidden variation termed frailty in the survival analysis literature induces a form of selection bias: high-risk individuals fail earlier, leaving a progressively lower-risk survivor pool whose apparent hazard declines over time [2] . The result is attenuated hazard ratio estimates that understate the true effect of every measured covariate. Frailty models address this by introducing a random effect, usually at the cluster level, that represents shared unobserved risk within groups such as families, villages, or clinics [3] . The gamma distribution is the most common frailty choice because it yields a closed-form marginal likelihood, while the log-normal distribution offers greater flexibility at the cost of requiring numerical integration [4] . In demographic research, frailty terms have been applied to infant mortality in sub-Saharan Africa and to old-age survival in Scandinavian twin registries, but their use in multi-ethnic census settings remains limited [5,6] . Yunnan Province, home to 25 of China's 55 officially recognized ethnic minorities, provides a natural laboratory for frailty modeling. Ethnic groups on the Yunnan Plateau differ in altitude of settlement, dietary customs, and access to healthcare, creating substantial village-level heterogeneity that a standard Cox model would absorb into residual variation [7] . This research fits gamma and log-normal frailty models to census-linked event-history records from six Yunnan ethnic groups and quantifies how ignoring frailty distorts hazard ratio estimates [8] .

Model Description

Let hᵢⱼ (t) = h₀ (t) wⱼ exp (β′xᵢⱼ) denote the hazard for individual i in village j, where h₀ is the baseline hazard, xᵢⱼ the covariate vector, and wⱼ the village-level frailty [2] . Under the gamma Specification, wⱼ ~ Gamma (1/θ, 1/θ) with E (w) = 1 and Var (w) = θ; under the log-normal specification, log (wⱼ) ~ N (0, σ²) [3] . The marginal log-likelihood for the gamma model has a closed form involving the Laplace transform of the gamma density, while the log-normal model requires Gauss-Hermite quadrature with 20 nodes [4] . Parameter estimation uses penalized partial likelihood maximization, and model comparison relies on AIC and the likelihood ratio test against the nested Cox model (θ = 0) [9] .

Material and Methods

Material

Methods

Three models were fitted: (1) standard Cox proportional hazards, (2) Cox with gamma frailty at the village level, and (3) Cox with log-normal frailty at the village level. All models used the Efron approximation for tied event times. The proportional hazards assumption was tested by Schoenfeld residual plots for each covariate [10] . Frailty variance was estimated by restricted maximum likelihood, and 95% confidence intervals were obtained by profile likelihood [11] . Hazard ratios were compared across models for each covariate. The analysis used R 4.2 with the survival (v3.5-5) and coxme (v2.2-18.1) packages. Convergence was verified by examining the gradient norm at the final iteration (≤ 10⁻⁸ required). Table 1 shows that the gamma-frailty model produced hazard ratios 14-31% larger than the standard Cox model across all covariates, with the altitude effect showing the greatest discrepancy.

Results

Sensitivity analysis

To check robustness, the gamma-frailty model was refit on three subsamples: (a) excluding Han individuals (n = 16,281), (b) restricting to altitudes above 2,000 m (n = 21,407), and (c) using only the 2015-2022 observation window. The frailty variance estimates ranged from 0.64 to 0.81 across subsamples, and the altitude hazard ratio remained between 1.11 and 1.18 per 100 m, suggesting that the main findings are stable. Replacing gamma with inverse-Gaussian frailty yielded an AIC within 3.1 units of the gamma model, indicating that the specific frailty distribution matters less than including a frailty term at all [12] .

Discussion

The 14-31% increase in hazard ratios when moving from Cox to gamma-frailty is consistent with the theoretical prediction that ignoring positive frailty attenuates covariate effects [2,4] . The altitude effect showed the greatest attenuation correction because altitude is strongly confounded with village-level factors access to clinics, road quality, and dietary patterns that the village frailty absorbs [7] . The gamma model's superiority over the log-normal is expected in settings where the frailty distribution is right-skewed, as high-risk villages drive the upper tail [3] . A limitation is that the village-level frailty aggregates all unmeasured sources of heterogeneity; a twolevel model separating household and village effects would give finer resolution but requires more data [13] .

Conclusion

This research demonstrated that gamma-frailty Cox models recover substantially larger hazard ratio estimates than standard Cox models when applied to multi-ethnic census data with village-level clustering. The frailty variance of 0.73 in the Yunnan cohort signals that unobserved heterogeneity is too large to ignore, and altitude-related mortality risk is underestimated by nearly a third without frailty correction.

Demographers working with clustered census data should routinely compare frailty and non-frailty specifications before interpreting covariate effects.

Acknowledgement

Acknowledgements Funding sourcesThis research received general departmental funding without specific project or grant numbers. Institutional supportThe authors thank the Guangzhou College of Applied Mathematics for computing resources and library access.

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