Worm plot: a simple diagnostic device for modelling growth reference curves

The worm plot visualizes differences between two distributions, conditional on the values of a covariate. Though the worm plot is a general diagnostic tool for the analysis of residuals, this paper focuses on an application in constructing growth reference curves, where the covariate of interest is age. The LMS model of Cole and Green is used to construct reference curves in the Fourth Dutch Growth Study 1997. If the model fits, the measurements in the reference sample follow a standard normal distribution on all ages after a suitably chosen Box-Cox transformation. The coefficients of this transformation are modelled as smooth age-dependent parameter curves for the median, variation and skewness, respectively. The major modelling task is to choose the appropriate amount of smoothness of each parameter curve. The worm plot assesses the age-conditional normality of the transformed data under a variety of LMS models. The fit of each parameter curve is closely related to particular features in the worm plot, namely its offset, slope and curvature. Application of the worm plot to the Dutch growth data resulted in satisfactory reference curves for a variety of anthropometric measures. It was found that the LMS method generally models the age-conditional mean and skewness better than the age-related deviation and kurtosis.

[1]  S. Claus,et al.  Zwet, w. r. van: convex transformations of random variables. mathematica centre tracts 7. mathematisch centrum amsterdam, 1964, 116 seiten , 1968 .

[2]  Maximum likelihood estimation of reference centiles. , 1990, Statistics in medicine.

[3]  M. Kendall,et al.  Kendall's advanced theory of statistics , 1995 .

[4]  F. Mosteller,et al.  Exploring Data Tables, Trends and Shapes. , 1988 .

[5]  Patrick Royston,et al.  A method for estimating age‐specific reference intervals (‘normal ranges’) based on fractional polynomials and exponential transformation , 1998 .

[6]  T. Cole.  Fitting Smoothed Centile Curves to Reference Data , 1988 .

[7]  R. Fisher,et al.  148: Moments and Cumulants in the Specification of Distributions. , 1938 .

[8]  Patrick J. Heagerty,et al.  Semiparametric estimation of regression quantiles with application to standardizing weight for height and age in US children , 1999 .

[9]  J. C. van Wieringen,et al.  Growth diagrams 1965 Netherlands; second national survey on 0-24-year-olds , 1971 .

[10]  Beat Kleiner,et al.  Graphical Methods for Data Analysis , 1983 .

[11]  J. Royston.  An Extension of Shapiro and Wilk's W Test for Normality to Large Samples , 1982 .

[12]  A. Benard,et al.  Het uitzetten van waarnemingen op waarschijnlijkheids‐papier1 , 1953 .

[13]  T J Cole,et al.  Smoothing reference centile curves: the LMS method and penalized likelihood. , 1992, Statistics in medicine.

[14]  M A van 't Hof,et al.  A method to construct age references for skewed skinfold data, using Box-Cox transformations to normality. , 1985, Human biology.

[15]  Eric R. Ziegel,et al.  SAS System for Statistical Graphics , 1993 .

[16]  Patrick Royston,et al.  A Comparison of Statistical Methods for Age‐related Reference Intervals , 1997 .

[17]  J. H. D. Haas,et al.  Groeidiagrammen van 1-25 jarigen in Nederland , 1960 .

[18]  W. R. vanZwet.  Convex transformations of random variables , 1964 .

[19]  M. Wade.  Sense and Nonsense of Statistical Inference: Controversy, Misuse, and Subtlety , 1994 .

[20]  M. Healy,et al.  Distribution-free estimation of age-related centiles. , 1988, Annals of human biology.

[21]  Michael Friendly,et al.  SAS System for Statistical Graphics,First Edition , 1991 .

[22]  J. V. Wieringen,et al.  Secular changes of growth in the Netherlands , 1998 .

[23]  P Royston,et al.  A strategy for modelling the effect of a continuous covariate in medicine and epidemiology. , 2000, Statistics in medicine.

[24]  T J Cole,et al.  British 1990 growth reference centiles for weight, height, body mass index and head circumference fitted by maximum penalized likelihood. , 1998, Statistics in medicine.

[25]  R. Tibshirani,et al.  Generalized Additive Models , 1991 .

[26]  P. Royston.  Estimation, reference ranges and goodness of fit for the three-parameter log-normal distribution. , 1992, Statistics in medicine.

[27]  Stef van Buuren,et al.  Continuing Positive Secular Growth Change in the Netherlands 1955–1997 , 2000, Pediatric Research.

[28]  J M Wit,et al.  Body index measurements in 1996–7 compared with 1980 , 2000, Archives of disease in childhood.