We used nested models as a framework for characterising the consequences of a selectively measured predictor in clinical prediction models. In this framework, a model containing predictors available in the target population is referred to as the restricted model, while the extended model additionally includes a selectively measured predictor. We show that non-invariance in the restricted model between selected patients and the target population decomposes into components due to omission of the additional predictor, potential residual non-invariance, and their interaction. This framework is extended to a predictor measured via multiple routes of selection resulting in collider structures. We further show that imputation based on the conditional distribution of the additional predictor in the selected population transfers the restricted-model non-invariance to the imputed extended model as imputation bias. In a forthcoming version, this framework will be illustrated using real world and simulated data. The framework provides a formal basis for understanding how selective predictor measurement affects generalisability of clinical prediction models using routinely collected health data.

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