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2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Age, years figure 3-6 Cross-sectional means of distance and velocity curves. An example of two boys. (Data from Chrzastek-Spruch's unpublished report on the Lubin Longitudinal Growth Study.53)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Age, years figure 3-6 Cross-sectional means of distance and velocity curves. An example of two boys. (Data from Chrzastek-Spruch's unpublished report on the Lubin Longitudinal Growth Study.53)

data. The two subjects differ in timing of their adolescent growth spurt, the age at maximum increment in height being at, respectively, 12.5 and 14.5 years of age. By taking the averages of the heights at each age, without taking account of this difference in timing of the adolescent growth spurt, one comes up with an average curve that does not show the steep slope at adolescence, seen in each individual curve. The effect of taking the cross-sectional mean becomes even more striking by a comparison of the yearly increments in height of the two curves with the average of these yearly increments. While the two individuals show a clear adolescent spurt with a maximum yearly increment in height of respectively 9.9 and 7.8 cm/yr, the cross-sectional mean of these curves is much flatter. The peak of the mean velocity curve is lower than in the individual curves and the "spurt" is spread out over a longer period than in the two individuals, a phenomenon denoted as the "phase-difference" effect.7

Although the example shown in Figure 3-6 is based on averages of longitudinal growth data, it is exactly what happens with the pattern of the means in cross-sectional growth data and the pattern of the increments of the means. By the way, the mean velocity curve in Figure 3-6 corresponds exactly to a plot of the increments of the means. This illustrates that the pattern of individual growth differs a lot from the pattern of cross-sectional mean growth, especially during adolescence. It is also the reason why the growth records of an individual over time do not match any of the centile lines shown by cross-sectional growth charts and why such charts are not useful to evaluate the normality of the pattern of growth over time. Pure cross-sectional growth standards are also called unconditioned for tempo. The differences between individual and average growth have long been recognized (Boas, 1892, and Shuttleworth, 1937, are cited by Tanner28), but it was not until the mid-1960s that Tanner, Whitehouse, and Takaishi7,54 introduced tempo-conditioned growth standards for height, weight, height velocity, and weight velocity based on longitudinal data of the British population. In addition to the classical centile distribution for attained size and velocity at each age, these references also show the "normal" variation in the shape of the growth curves. Their technique to produce the latter reference curves (the so-called tempo-conditioned standards) was based essentially on an analysis of the longitudinal growth data after centering each individual's growth data around the age at peak velocity.

Figure 3-7A illustrates this technique and its effect on average height velocity of the two boys shown in Figure 3-6, after their height measurements were peak height velocity centered. The so-obtained mean velocity curve indeed has a pattern that can be considered representative for both individuals; that is, with an age at peak velocity and a peak velocity that is the average of the two subjects. Later on, Tanner and Davies4 produced clinical longitudinal standards for height and height velocity in North American children using the same graphical principle as in their 1966 British standards. Wachholder and Hauspie,2,3 on the contrary, used a technique, derived from curve fitting, to achieve similar goals when producing clinical standards for growth and growth velocity in the Belgian population. They used "mean-constant" curves to estimate the typical average pattern of growth in the population. The mean-constant curves were obtained by fitting the Preece-Baines model 1 to each individual in the sample and feeding the mean values of the function parameters into the model. The resulting mean-constant curve represents the growth pattern of the typical average child in the population; that is, with a peak velocity and an age at peak velocity characteristic or typical for the group.12 Hence, a mean-constant curve is very much like the average of peak velocity centered curves. Figure 3-7B shows the PB1 velocity curves of the same two boys together with their mean-constant curve. Note that the PB1 curves slightly underestimate peak velocity, a minor weakness of the PB1 model that has been acknowledged elsewhere.16

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