The Cox proportional hazards model, commonly used in clinical trials, assumes proportional hazards. However, it does not hold when, for example, there is a delayed onset of the treatment effect. In such a situation, an acute change in the hazard ratio function is expected to exist. This paper considers the Cox model with change-points and derives Akaike information criterion (AIC)-type information criteria for detecting those change-points. The change-point model does not allow for conventional statistical asymptotics due to its irregularity, thus a formal AIC that penalizes twice the number of parameters would not be analytically derived, and using it would clearly give overfitting analysis results. Therefore, we will construct specific asymptotics using the partial likelihood estimation method in the Cox model with change-points, and propose information criteria based on the original derivation method for AIC. If the partial likelihood is used in the estimation, information criteria with penalties much larger than twice the number of parameters could be obtained in an explicit form. Numerical experiments confirm that the proposed criteria are clearly superior in terms of the original purpose of AIC, which are to provide an estimate that is close to the true structure. We also apply the proposed criterion to actual clinical trial data to indicate that it will easily lead to different results from the formal AIC.
Keywords: Brownian motion; model misspecification; model selection; statistical asymptotic theory; structural change; survival time analysis.
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