Estimation of Medical Reference Limits by Truncated Gaussian and Truncated Power Normal Distributions

Abstract

Truncated distributions can be used to model a data set if observations below or/and above certain values should not get into the estimation procedure. In this case, the data set is truncated at below or/and above values, and the truncated part of the data is modelled. The truncated Gaussian and the truncated Gaussian mixture distributions are formulated and used to model the data. Maximum likelihood estimation of the parameters is computed using iterative methods. An algorithm is developed to optimize truncation points. A test statistic is defined to measure the goodness of fit of the estimations. The critical value of the test statistic is caculated by means of a Monte Carlo simulation method. Data are often rounded to a specific decimal position or to the nearest integer. In all the above described procedures this is considered. The same procedures are applied to model skewed data. The Power normal distribution allows modelling such a data set. The truncated power normal and the truncated power normal mixture distributions are formulated and used in this case. Maximum likelihood estimation of the parameters are obtained using the Newton-Raphson method and the EM algorithm.These models are applied to the data sets of hospitalized patients to estimate medical reference limits. The interpretation of results of medical laboratory tests are based on reference limits. Procedures for the determination of reference limits are recommended and published by the International Federation of Clinical Chemistry (IFCC). These procedures are based on obtaining a set of values from a certain reference group, comprising "normal" subjects as a rule. It is often recommended that each medical laboratory should establish its own reference limits, because they can differ amoung countries, regions and laboratories. But in practice, only a few laboratories do this, since the selection of the reference group is beyond the potential of most single laboratories. Furthermore, reference values obtained outside hospitals may not be representative for hospitalized patients. These data sets contain non-pathological as well as pathological values. The truncated distributions mentioned above are used to estimate and separate the distribution of the non-pathological values from the pathological values, with some assumption about the distribution of the whole data. The performance of the developed models is studied in relation to the overlap of the mixture components by means of Monte Carlo simulation studies. The statistical analysis was performed by usingsoftware R (version 2.7.2).