Solving non-standard optimization problems using SAS/IML software with application to growth modeling and optimal design theory

It is often necessary in applied statistical research to perform direct numerical optimization because the problem can not be transformed to be solved by standard procedures implemented in the statistical packages. One example of such a problem comes from the field of growth modeling: The Polynomial Gompertz model is known to provide a good fit to individual growth curves and growth percentiles. However, this non-linear regression model cannot be fitted by PROC NLIN because of the presence of an integral in expression of the Polynomial Gompertz function. Therefore, direct optimization is necessary to obtain the least squares estimates of the model parameters. Another example is provided by optimal design theory: in some applications, such as Phase I cancer studies, it is extremely important to implement a design that provides maximal information from a small sample, so that a minimal number of patients is exposed to the potentially toxic experimental drug. The formal optimality criteria based on Fisher's information matrix can be introduced to derive a design providing the most precise estimates for a given sample size. In many practical cases this leads to a highly non-linear optimization problem with constraints. Both problems can be successfully solved using the advanced numerical optimization package in SAS/IML. This paper gives an overview of the package's main features and provides several examples of how optimization methods can be used to solve two very different in nature optimization problems. Some computational aspects of the problem are also discussed. INTRODUCTION In this section we introduce the two problems that are used to illustrate an application of SAS/IML numerical optimization routines. These problems arise from different areas of statistics: one is used for fitting non-linear curves to the growth data and the other is very helpful in planning an efficient experiment in clinical trials. We will refer to the former as the Polynomial Gompertz growth model and to the latter as the Optimal Design for the Proportional Odds model. Although these two problems are very different in nature they have a common property element: they can be solved by applying very similar non-linear optimization techniques. We describe the Proportional Odds model in detail and give only a brief overview of the Polynomial Gompertz model since it was already discussed in (Perevozskaya I., Kuznetsova O.M. (2000)). POLYNOMIAL GOMPERTZ GROWTH MODEL The patterns of human growth have been studied quite extensively. Besides the obvious fact that height increases with age until the final adult height is reached, it has been noticed that the growth velocity is high at birth, rapidly decelerates in infancy, slightly declines during the long period of juvenile growth, shoots up in adolescence and declines to zero while the growth curve approaches the final adult height. The parametric modeling of growth allows to summarize this complicated process with a reasonable number of parameters and facilitate many tasks in applications such as clinical trials, epidemiology and pediatrics. For instance, in some clinical studies the height, length, weight and head circumference individual curves are often plotted against the population percentile curves to detect the growth faltering in treatment groups. If the analytic expressions for percentile curves are available, this task is greatly simplified. (Perevozskaya I., Kuznetsova O.M. (2000)) The Polynomial Gompertz model was one of the many models developed for this purpose. Although it has a very simple and natural foundation, it has not been studied extensively until recently due to software limitations. The model is based on the fundamental concept of relative growth rate (RGR), which is analogous to the concept of a hazard function in survival analysis. It is defined as the ratio of the growth rate (velocity) to the achieved growth: