Estimating marginal hazard function from the correlated failure time data arising from case-control family studies is complicated by noncohort study design and risk heterogeneity due to unmeasured, shared risk factors among the family members. Accounting for both factors in this article, we propose a two-stage estimation procedure. At the first stage, we estimate the dependence parameter in the distribution for the risk heterogeneity without obtaining the marginal distribution first or simultaneously. Assuming that the dependence parameter is known, at the second stage we estimate the marginal hazard function by iterating between estimation of the risk heterogeneity (frailty) for each family and maximization of the partial likelihood function with an offset to account for the risk heterogeneity. We also propose an iterative procedure to improve the efficiency of the dependence parameter estimate. The simulation study shows that both methods perform well under finite sample sizes. We illustrate the method with a case-control family study of early onset breast cancer.

Drought is one of the most widespread and destructive hazards over the Loess Plateau (LP) of China. Due to climate change, extremely high temperature accompanied with drought (expressed as hot drought) may lead to intensive losses of both properties and human deaths in future. A hot drought probabilistic recognition system is developed to investigate how potential future climate changes will impact the simultaneous occurrence of drought and hot extremes (hot days exceeding certain values) on the LP. Two regional climate models, coupled with multiple bias‐correction techniques and multivariate probabilistic inference, are innovative integrated into the hot drought probabilistic recognition system to reveal the concurrence risk of droughts and hot extremes under different Representative Concentration Pathway (RCP) scenarios. The hot‐day index, TX90p, indicating the number of days with daily maximum temperature (Tmax) exceeding the 90th percentile threshold, and the Standardized Precipitation Index are applied to identify the joint risks on the LP using copula‐based methods. The results show that precipitation will increase throughout most of the LP under both RCP4.5 and RCP8.5 scenarios of 2036–2095, while Tmax may increase significantly all over the LP (1.8–2.7 °C for RCP4.5 and 2.7–3.6 °C for RCP8.5). The joint return periods of Standardized Precipitation Index and TX90p show that fewer stations will experience severe drought with long‐term hot extremes in two future scenarios. However, some stations may experience hot droughts that are more frequent and extreme, particularly certain stations in the southwest and south‐central regions of the LP with recurrence period less than 10 years.

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