Estimating Transformations for Population Models of Continuous, Closed Interval Data
Matthew M. Hutmacher(1) and Jonathan L. French(2)
(1)Ann Arbor Pharmacometrics Group, Ann Arbor, MI, USA; (2)Pfizer, Inc., New London, CT, USA
Objectives: The Stanford Health Assessment Questionnaire Disability Index (HAQ-DI) is a self reported tool used in rheumatoid diseases. These clinical data have a closed interval range from 0 to 3, inclusive of the endpoints. Presumptively, the closed range vitiates the normality assumption for random effects. The standard Box-Cox transformation family mitigates these issues for positive open interval data, but is not appropriate for closed or semi-closed (closed on one end) interval data. Maintaining probability support for a model within the data range is difficult, and is noticeable when simulating data, since unrealistic (negative) data can be generated. We propose a general nonlinear mixed-effects approach to estimating transformations for longitudinal closed interval data (semi-closed interval data are a special case), for which a transformation family other than Box-Cox is required. Data with values on the range limits (endpoint data) are considered censored observations in the likelihood. This assumption is predicated on the transient nature (non-permanent) and incomplete effect of most drugs. The likelihood can be extended to account for ‘inflation' of the endpoint data.
Methods: A case study is presented to introduce the transformation methodology. The model results are compared to an approach which uses shift parameters to translate the closed interval into an open one to facilitate transformation, and also to the model fitted to the original data. Simulation studies were performed to understand the properties of these approaches.
Results: Posterior predictive distributions of selected statistics indicated that only the proposed methodology maintains simulated data within the feasibility range. Simulations with the model fitted to the original scale indicates that setting data outside the closed interval to the range endpoints induces estimation bias, and this bias increases with the quantity of data set to the endpoints. Further, using arbitrary shift parameters can induce unpredictable biases. These biases also increase with greater amounts of data near the range endpoints. Such biases yield biased predictions of the population mean on the original scale.
Conclusions: A general likelihood-based approach is proposed for closed interval data. The approach is principled in that it restricts the probability support to the feasible data range, and does not suffer from biases induced by selection of arbitrary shift parameters.
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