Statistical Modelling 16 (5) (2016), 392–408

A multivariate single-index model for longitudinal data

Jingwei Wu
Department of Epidemiology and Biostatistics,
College of Public Health, Temple University,
PA 19122,

Wanzhu Tu
Department of Biostatistics,
Indiana University School of Medicine,
IN 46202,


Index measures are commonly used in medical research and clinical practice, primarily for quantification of health risks in individual subjects or patients. The utility of an index measure is ultimately contingent on its ability to predict health outcomes. Construction of medical indices has largely been based on heuristic arguments, although the acceptance of a new index typically requires objective validation, preferably with multiple outcomes. In this article, we propose an analytical tool for index development and validation. We use a multivariate single-index model to ascertain the best functional form for risk index construction. Methodologically, the proposed model represents a multivariate extension of the traditional single-index models. Such an extension is important because it assures that the resultant index simultaneously works for multiple outcomes. The model is developed in the general framework of longitudinal data analysis. We use penalized cubic splines to characterize the index components while leaving the other subject characteristics as additive components. The splines are estimated directly by penalizing nonlinear least squares, and we show that the model can be implemented using existing software. To illustrate, we examine the formation of an adiposity index for prediction of systolic and diastolic blood pressure in children. We assess the performance of the method through a simulation study.


mixed effect model; multivariate setting; penalizing nonlinear least squares; P-spline; single-index model.


Example data and code in zipped archive.