Extension of NPDE for evaluation of nonlinear mixed effect models in presence of data below the quantification limit with applications to HIV dynamic model

Data below the quantification limit (BQL data) are a common challenge in data analyses using nonlinear mixed effect models (NLMEM). In the estimation step, these data can be adequately handled by several reliable methods. However, they are usually omitted or imputed at an arbitrary value in most evaluation graphs and/or methods. This can cause trends to appear in diagnostic graphs, therefore, confuse model selection and evaluation. We extended in this paper two metrics for evaluating NLMEM, prediction discrepancies (pd) and normalised prediction distribution errors (npde), to handle BQL data. For a BQL observation, the pd is randomly sampled in a uniform distribution over the interval from 0 to the probability of being BQL predicted by the model, estimated using Monte Carlo (MC) simulation. To compute npde in presence of BQL observations, we proposed to impute BQL values in both validation dataset and MC samples using their computed pd and the inverse of the distribution function. The imputed dataset and MC samples contain original data and imputed values for BQL data. These data are then decorrelated using the mean and variance-covariance matrix to compute npde. We applied these metrics on a model built to describe viral load obtained from 35 patients in the COPHAR 3-ANRS 134 clinical trial testing a continued antiretroviral therapy. We also conducted a simulation study inspired from the real model. The proposed metrics show better behaviours than naive approaches that discard BQL data in evaluation, especially when large amounts of BQL data are present.