Penalized Least Squares Estimation And Variable Selection For Growth Curve Model

Growth curve model is a general multivariable linear model. As a branch of linear model, growth curve model plays an important role in modern statistics. And it is used extensively in in economics, biology, medical research and epidemiological study. In this paper, we consider two kinds of inference problems of growth curve model i.e. estimation of the parameter matrix and variable selection. Penalized least squares approaches are proposed to simultaneously select variables and estimate coefficients, and it has much better properties than other methods. Thus this proposed idea is widely applicable to parameter estimation and variable selection. In this paper, firstly, we define the penalized least squares for growth curve model, after transforming it by matrix stacking transfor-mation or the Potthoff-Roy transformation. By using different penalty function (such as Hard Thresholding function, LASSO, ENET, adaptive LASSO, SACD), we can get different corresponding estimations, as well as achieve the variable selection. In addi-tion, we discuss the properties of the penalized least squares estimations of the growth curve model, which is transformed by Potthoff-Roy transformation, and the properties, which are Oracle properties, are proved in this paper. Furthermore, we define penalized least squares for the unchanged original growth curve model, and we can get numerical solution for the algorithm based on Nelder-Mead method. In addition to theoretical re-search, we carry out a simulation study to validate the proposed algorithm. Simulation is mainly done by programming in MATLAB. By using the criteria to measure estimation and variable selection, we compare several penalized least squares estimations and the effect of variable selection of different penalty functions, the result shows that the adap-tive LASSO performs better in parameter estimation and variable selection. Besides, we compare different transformations. Results indicate that Potthoff-Roy transformation performs better than matrix stacking transformation when considering variable selection and parameter estimation comprehensively.