points for significant differences, as a simple incremental method does. Instead, all of the data points within a stasis interval are simultaneously compared with all ...
ANNALS OF HUMAN BIOLOGY, 1998, VOL. 25, NO. 3, 203 219
An example of variation and pattern in saltation and stasis growth dynamics M. LAMPL'~, K. ASHIZAWA~, M. KAWABATA~ and M. L. JOHNSON§ t Emory University, Atlanta, GA, USA Otsuma Women's University,Tokyo, Japan §University of Virginia Health SciencesCenter, Charlottesville,VA, USA Ann Hum Biol Downloaded from informahealthcare.com by Emory University on 04/26/12 For personal use only.
Received 4 August 1996; revised 9 July 1997
Summary. The serial data from two siblings, aged 6.6 and 7.5 years of age at the initiation of the study, measured each evening for total standing height during 365 days, are analysed by two methods to investigate the nature of the underlying growth pattern. The saltation and stasis model, designed to identify the presence of statistically significant pulses in sequential data, is compared for goodness-of-fit to first to sixth degree polynomial functions, used to investigate the presence of a slowly varying smooth continuous function in the data, and high order polynomials of the same degree of flexibility as the individual's saltation and stasis results. The saltation and stasis model is found to better-fit the experimental data than the slowly varying smooth continuous functions (p < 0.01 to 0.001). The timing characteristics of the saltation and stasis patterns are investigated and the temporal patterns are suggestive of a non-random, aperiodical deterministic system.
1.
Introduction
A pattern of saltatory growth characterized by variable amplitude pulsatile daily growth events, or saltations, followed by refractory intervals of stasis during which no significant growth takes place has been previously described for a sample of infants (Lampl, Veldhuis and Johnson 1992, Lampl 1993) and an adolescent (Lampl and Johnson 1993). The saltation and stasis model was based on time-intensive empirical observations that were statistically analysed and subsequently formally modelled to provide the possibility for statistical comparison between alternative mathematical algorithms. The saltation and stasis algorithm is a statistically better descriptor of these growth data than a number of continuous mathematical models (Johnson and Lampl 1995). The saltation and stasis algorithm is designed to investigate the presence of pulse/ stasis sequences and, thus, models an underlying two-phase growth mechanism: an on/off switch that is permissive for a growth event to take place. When growth occurs and how much growth can occur are not a part of the saltation and stasis model: There are no assumptions about the frequency or amplitude of saltations or the duration of stasis intervals. The model assumes only that saltations and stasis are sequential and the process is saltatory in nature. This is an important point to clarify because the process of saltation and stasis as originally published has been misrepresented in recent literature. The presence of saltation and stasis in the growth patterns of other samples has been questioned from short daily data series of five human infants (Heinrichs, Munson, Counts, Cutler and Baron 1995), seven rabbits (Oerter Klein, Munson, Bacher, Cutler and Baron 1994) and a sample of weekly and semi-weekly height measurements from a mixed-age sample of children (Hermanussen and Geiger-Benoit 1995). These studies all employ an analytic method that is methodologically unsound because it is a poor discriminator between growth patterns (Johnson and Lampl 1995, Lampl, Cameron, Veldhuis and Johnson 0301-4460/98 $12"00 © 1998 Taylor & Francis Ltd.
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1995, Johnson, Veldhuis and Lampl 1996) and it assumes that saltatory growth patterns are simple periodic functions of uniform amplitude. Thus, at the present time, there is a controversy surrounding the underlying mechanism by which growth proceeds. It is of interest to investigate whether a pattern of saltation and stasis is evident in individuals at different developmental ages and from different populations. Presently, time-intensive longitudinal anthropometric measurements of sufficiently long duration for statistical analysis and significant modelling are rare data sets. An important data series, the daily assessments of stature on two siblings, was previously published focussing on diurnal statural changes and a general description of seasonal changes in growth rate (Ashizawa and Kawabata 1990). No curve-fitting methods were applied to these data and no statistical assessment of growth rate changes were previously published. These data are a resource for further investigating the patterns of individual growth in time-intensive data.
2.
Subjects and methods The present data were collected over 365 days and consist of 359 serial daily anthropometric measurements (6 days resulted in missing data) taken on each of two siblings, a female of 7.5 years of age and a male of 6.6 years of age at the initiation of the study. These individuals lived in Hachioji, a suburb of Tokyo and were measured daily during June 1984 to May 1985. Total standing height was taken each morning and evening, together with stature to the seventh cervical vertebra and the iliospinal height, as previously described (Ashizawa and Kawabata 1990). The present investigation concerns the daily total standing height measures as assessed in the evening to minimize variability in measurement due to early morning diurnal effects and measurement uncertainty of the cervical and iliospinal height measurements (Ashizawa and Kawabata 1990). A shortcoming of the data collection protocol is the absence of replicate measurements. As Himes (1989) points out, the reliability of measurements with high reliabilities, such as stature, is not enhanced by replicate measurements. However, no assessment of measurement error can be made in the present analysis directly from these data. A range of 1-3 mm of technical error of measurement was employed in the present analysis at each data point, a range that includes a number of reported technical errors for total standing height in the published literature. The question addressed in the present study is, 'Are these data better described by a saltation and stasis model or a smoothly continuous curve?' Mathematical functions that represent both possibilities are fitted to the data and the results compared by statistical tests employed to identify which of these mathematical options better represents the underlying biological process in the serial data. First, the fit of each model to the experimental data is statistically compared by least-squares procedures. The relative quality of the fit is compared by F-ratios of the weighted variance (Johnson and Lampl 1995) and F-ratios o f p < 0.01 are considered support for significantly better fit of one or the other model in the present study. Secondly, the residuals, or differences between the fitted curves and the data, are examined for autocorrelation effects. There are two possible results of this procedure. It can either be concluded that one of the models tested is a statistically significantly better fit to the serial growth data (p < 0.01) and, thus, that one of the alternatives is the better descriptor of the
Saltatory growth variation
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underlying growth process. Or, it can be concluded that the data are statistically unresolvable as better fitted by the assumptions of either a slowly varying smooth continuous model or a saltation and stasis model.
2.1. Continuous functions Traditional approaches to the identification of the biology of growth have involved the curvilinear smoothing of serial data points by equations that assume a slowly varying smooth continuous function. These mathematical descriptors have been developed by reference to data sets of infrequently collected data (annual and semi-annual intervals) and a number of such equations have been fitted to different data series. It has been found that different functions best fit serial data encompassing different developmental age ranges. For example, both the equations proposed by Karlberg (1987) and Preece and Baines (1978) have components that encompass the developmental period of the present individuals. However, these models were designed to describe human growth over a period from birth to adulthood, or about 18 years. The present daily observations span a substantially smaller time frame. Thus, it is not possible to estimate uniquely all of the parameters of these models for the present data. In lieu of these equations, a simpler smooth function is employed for comparison with the fit of the saltation and stasis model. Over the time span of the present data, any relatively smoothly varying model can be adequately described by a sixth degree polynomial and thus, first to sixth degree polynomials are used as slowly varying smooth continuous functions. It is interesting to note that for children of this age, the Karlberg equation (Karlberg 1987) is a quadratic polynomial and thus the Karlberg equation is implicitly tested here. In addition, by using the sixth degree polynomial, the smooth continuous curve is permitted to have increased flexibility in curvature and is testing for a model of slow growth between times of more rapid growth. Specifically, this permits a fit for most [. timei tim%,.~ ] {i = 2 Ltim%~sttim%~.~t31 continuous functions with good precision, yet cannot describe saltations (Johnson and Lampl 1995). The actual polynomial that is used has a scaled time axis, The first and last subscripts refer to the first and last time values. This scaling process minimizes truncation and roundoff errors within the computer algorithms. The polynomial used is 6
ZAj(y j=0 The As are the polynomial coefficients to be estimated (0 6). This analysis is aimed at identifying the goodness of fit of the traditional model of slowly varying growth during childhood that approximates accepted childhood growth rates, while permitting some flexibility in traditional childhood growth curves. In addition, a h~,, , , , ~ , polynomial is employed as a control. Specifically, high order polynomials model the data as a continuous process, exhibiting high frequency wave forms. This analysis involves fitting an orthogonal polynomial of the same degree as the number of saltations identified by the saltation and stasis model and comparing goodness of fit by F-ratio of weighted variance. The orthogonal polynomials were evaluated by the International Mathematics and Statistical
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Library (IMSL) D R F O R P procedure (double precision fit orthogonal polynomial regression model) (IMSL 1989). This high order polynomial application follows the saltation and stasis model assessment, as the polynomial order to be tested cannot be identified prior to this analysis. Thus, if the saltation and stasis model predicts m significant saltations, the fit of a high order polynomial or equivalent flexibility (order m) is also assessed. This is proposed as a rigorous test of the saltation and stasis model: The difference in the two approaches is that the smoothly continuous high order function applies continuous first derivatives between growth pulses and the saltation and the stasis model is based on discontinuous first derivatives between growth saltations. This is a test of the goodness of fit for the specific feature of the stasis intervals intervening between saltations.
2.2. The saltation and stasis algorithm Details of the saltation and stasis analysis procedure have been previously published (Johnson and Lampl 1995). The formal mathematical model was developed after statistical incremental analysis of empirical data suggested that a pattern of growth/no-growth characterized a set of time-intensive infant total body length data (p
