How to analyse multiple ordinal scores in a clinical trial? Multivariate vs. univariate analysis.
Laffont C. M. and Concordet D.
INRA, UMR 1331, Toxalim, F-31027 Toulouse, France. Université de Toulouse, INPT, ENVT, UPS, EIP, F-31076 Toulouse, France.
Objectives: Longitudinal measurements of ordinal responses are very frequent in clinical trials to assess drug efficacy and side effects. Usually, several scores are recorded and this multiplicity is an issue for data analysis. Most of the time, each score is analysed separately or an average/aggregated score is used. Both approaches, however, ignore the correlations between scores which very often document different aspects of a same physio-pathological process (e.g. pain, inflammation) and, in the last case, the actual metric of the scores is not taken into account. Very few methods exist to analyse several scores simultaneously [1, 2]. In 2008, Todem et al. proposed a method for the analysis of longitudinal bivariate ordinal data using probit-linear mixed effects models [2]. We propose to generalise their approach and apply it to pharmacokinetic/pharmacodynamic analyses.
Methods: We use the concept of latent variables to derive the joint distribution of K ordinal responses. Each ordinal response Yk (k = 1…K) is viewed as the categorisation of a continuous, unobserved variable denoted Zk which is the true variable of interest. We use mixed effects models for the K latent variables, assuming that the random effects for subject i at time tij (inter- and intra-individual variability) are correlated across scores. Model estimation is performed with a SAEM[3, 4]-like algorithm implemented in C++. The multivariate normal cumulative distribution function is approximated using Gauss-Legendre quadratures. Two simulation studies were carried out with different scenarios to assess the applicability of our method. Drug dose or time-varying drug concentration was used as a covariate in the model. In the end, a principal component analysis was performed to summarise the correlations between scores.
Results: Our method allowed correct estimation of all model parameters, including correlations between scores. As expected, multivariate and univariate analyses gave different results regarding the percentage of subjects within each “crossing” category. In contrast, they produced similar estimation of marginal distributions.
Conclusions: We show that a multivariate analysis can be more appropriate than separate univariate analyses for the assessment of drug efficacy and safety and offers new perspectives in terms of benefit-risk ratio evaluation. The latent variable approach provides a good framework for the modelling of drug effects through various response models.
References:
[1] Liu I and Agresti A. The analysis of ordered categorical data: an overview and a survey of recent developments. Sociedad de Estadística e Investigación Operativa Test 14 (2005):1-73.
[2] Todem D, Kim K and Lesaffre E. Latent-variable models for longitudinal data with bivariate ordinal outcomes. Statistics in Medicine 26 (2007):1034-1054.
[3] Kuhn E and Lavielle M. Maximum likelihood estimation in nonlinear mixed effects model. Computational Statistics and Data Analysis 49 (2005):1020-1038.
[4] Savic RM, Mentré F and Lavielle M. Implementation and evaluation of the SAEM algorithm for longitudinal ordered categorical data with an illustration in pharmacokinetics-pharmacodynamics. AAPS J 13 (2011):44-53.
Acknowledgments: We thank Novartis Pharma AG, Switzerland, for funding this project.
