Beyond the Cox model: a log-logistic accelerated failure time frailty model for analysing first birth timing in...

The Cox proportional hazards (PH) model is the most common approach for analyzing time-to-event data in public health. However, its reliance on the proportional hazards' assumption is a key limitation, often violated in demographic research. This study illustrates a comprehensive model selection framework, comparing the performance of various parametric accelerated failure time (AFT) and frailty models as robust alternatives when the PH assumption fails, using data on the timing of first birth as a case study.

We conducted a secondary analysis of time to first birth among 9,955 women from the 2015 Zimbabwe Demographic and Health Survey. After confirming violations of the PH assumption using Schoenfeld residuals, we systematically compared the fit of several parametric models' exponential, Weibull, log-normal, and log-logistic with and without gamma-shared frailty terms. Model selection was based on the Akaike (AIC) and Bayesian (BIC) Information Criteria.

The log-logistic AFT model with gamma frailty showed improved relative fit for this dataset, accommodating a non-monotonic hazard and accounting for unobserved regional heterogeneity (θ = 0.671, p<0.001). The model yielded interpretable time ratio (TR) estimates, where higher education (TR=1.11) and later cohabitation (TR=1.25) were associated with delayed first birth, while marriage (TR=0.98) and contraceptive use (TR=0.97) were associated with earlier timing. In contrast, standard parametric models and the Cox model were suboptimal for this dataset.

This study provides an empirical demonstration of how AFT models with frailty terms can be applied in settings where the proportional hazards assumption is violated. The log-logistic frailty model showed improved relative fit for this dataset and offers an interpretable framework for analysing time-to-event outcomes with clustered data.