Assessing the numerical integration of dynamic prediction formulas using the exact expressions under the joint frailty-copula model

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Joint models allow survival outcomes of a patient to be dynamically predictable based on intermediate events observed after treatment. The existing dynamic prediction methods need some numerical integration over frailty or random-effects distributions since the integral is implicit and improper. For a joint frailty-copula model, the Clayton copula and the gamma frailty model have been used to derive a dynamic prediction formula based on meta-analytic data. However, the prediction formulas under the Clayton copula involve an improper integral and need to be approximated numerically, as in all the joint models. In this paper, we consider the Gumbel copula and the FGM copula to obtain the exact (true) prediction formula without any numerical integration. The proposed formula also provides a tool for assessing approximation errors occurring to the approximation by numerical integration. Our numerical assessments show some approximation errors of the true prediction formula especially when the frailty distribution is heavy-tailed. A real data example illustrates how the proposed formulas can be used for assessing the approximation error. Our study may suggest some assessments for approximation errors in other types of joint models using more complex random-effects and frailty distributions.

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