A method for estimating age-specific reference intervals (‘normal ranges’) based on fractional polynomials and exponential transformation


Eileen M. Wright Department of Medical Statistics and Evaluation, Division of Primary Care and Population Health Sciences, Imperial College School of Medicine, Hammersmith Hospital, Du Cane Road, London, W12 0NN, UK. ewright@rpms.ac.uk


The age-specific reference interval is an important screening tool in medicine. Put crudely, an individual whose value of a variable of interest lies outside certain extreme centiles may be suspected of abnormality. We propose a parametric method for constructing such intervals. It provides smooth centile curves and explicit formulae for the centile estimates and for standard deviation (SD) scores (age-standardized values). Each parameter of an exponential–normal or modulus–exponential–normal density is modelled as a fractional polynomial function of age. Estimation is by maximum likelihood. These three- and four-parameter models involve transformations of the data towards normality which remove non-normal skewness and/or kurtosis. Fractional polynomials provide more flexible curve shapes than do conventional polynomials. The method easily accommodates binary covariates facilitating, for example, parsimonious modelling of age- and sex-specific centile curves. A method of calculating precision profiles for centile estimates is proposed. Goodness of fit is assessed by using Q–Q-plots and Shapiro–Wilk W-tests of the SD scores, and likelihood ratio tests of the parameters of an enlarged model. Four substantial real data sets are used to illustrate the method. Comparisons are made with the semiparametric LMS method of Cole and Green.