Fitting Age Related Reference Intervals to Ocular Axial Length in Children Using a Four-Parameter Curve Family

Abstract

There exist a number of methods to determine age dependent reference intervals. Some of those are based on standard parametric classes of distributions like normal or lognormal and standard parametric classes of age functions like linear or polynomial of some order. Others are based on more flexible distribution classes like Box-Cox transformation of the normal distribution, which allows for skewness. There exist also purely nonparametric methods, where the bounds of the reference intervals are only assumed to be nondecreasing and they are directly estimated by a suitable error function without any distributional assumption. In this paper we propose a flexible four-parameter age function class for the reference interval bounds and a method to estimate those. The four parameters in the class have concrete meanings; starting value at age 0, asymptotic value at increasing age, time scale and shape. The function class satisfies some desirable properties, which are discussed. The estimation of the parameters in the model uses the same type of error function as in the purely nonparametric methods. With our method we also get an estimate of the distributional position of an observation for a new individual given its age. The method is illustrated by an application example, where a 90% reference interval for ocular axis length of children up to age 18 years are determined.