Visual Predictive Checks for Censored and Categorical data
Martin Bergstrand, Andrew C. Hooker, Mats O Karlsson
Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Background: Non random censoring of data as in the presence of observations below the quantification limit (BQL) can harm not only parameter estimates but also diagnostic plots such as Visual Predictive checks (VPCs). Treating this type of data as a combination of censored continuous data and categorical data (e.g. BQL) can facilitate unbiased interpretation. VPCs can be adopted for any type of categorical data by plotting the observed and the simulated fraction of observations of each category versus an independent variable. The visual interpretation of a VPC is its strength, however it can be difficult to distinguish if lack of agreement is due to random chance or model misspecification. Calculating non-parametric confidence intervals based on simulated data for different percentiles of continuous data or for the fraction of observations in a certain category for categorical data is likely to improve the interpretability. This work aims to illustrate a new approach for VPCs in the presence of categorised data.
Methods: VPCs were created based on different models for ordered categorical data, count data and continuous data with censored low and/or high observations. Each VPC was based on 1000 datasets simulated with obtained parameter estimates. For continuous data the median and 95 % prediction interval for the observed data are plotted together with non-parametric 95 % confidence intervals for the corresponding percentiles calculated from the simulated datasets. To ensure correctly calculated percentiles all censored observations was retained in the original dataset. Percentiles for observed data can only be adequately calculated and presented for percentiles where the censored observations constitute a smaller fraction than the percentile in question. For all categorical data the fraction of observations in each category was compared to a simulation based 95 % confidence interval. In all created VPCs time was chosen as the independent variable. Individual strategies for stratification and binning across the independent variable were adopted for each data-set.
Results: The combination of VPCs for both censored data and continuous data was found to more clearly indicate the presence of important model misspecifications than VPCs focusing on only the continuous observations. The inclusion of confidence intervals for the diagnostic variable (i.e. percentiles or fraction of observations) acts as an effective support to identify actual model misspecifications.
