Abstract
Tracking refers to the idea of maintaining a relative position within a given group of individuals as they change in time. This paper presents several approaches to study and analyze tracking (i.e., stability and predictability) and its application in physical education and sport. We will use data from a mixedlongitudinal study conducted in the city of Porto, Portugal, comprising 486 girls that were divided into two age cohorts: 1214 years and 1416 years. Body mass index (BMI) was the chosen variable in all statistical analyses of tracking. Statistical techniques to describe tracking included: autocorrelations, Foulkes & Davis gamma and Goldstein constancy index. Regardless of statistical procedure used, tracking BMI was moderate to high in each cohort, which could be due to the short followup period. However, each tracking statistics showed different aspects of interindividual differences in intraindividual changes of girls' BMI. The use of any of the suggested procedures to study aspects of stability and predictability (i.e., tracking) in longitudinal studies requires a careful scrutiny of main goals and hypotheses to be tested.
Key words:
Body mass index; Growth and development; Longitudinal studies; Monitoring; Tracking
Resumo
O termo Tracking referese à noção de manutenção de posição relativa de valores de um dado grupo de sujeitos em função do tempo. O presente artigo apresenta diversas técnicas de estudo e análise do tracking (i.e., estabilidade e previsibilidade). Os dados utilizados provêm de um estudo longitudinalmisto da Região do Grande Porto, Portugal, compreendendo 486 meninas, divididas em duas coortes que abrangem as faixas etárias dos 12 aos 14 e dos 14 aos 16 anos. A variável eleita para as análises foi o índice de massa corporal (IMC). Os procedimentos estatísticos utilizados para descrever o tracking foram: autocorrelações, gama de Foulkes & Davis e índice de constância de Goldstein. Independentemente da estatística utilizada e face à curta duração do estudo, o tracking do IMC foi moderado a elevado em cada coorte. Contudo, cada procedimento de análise mostrou aspetos distintos das diferenças interindividuais nas mudanças intraindividuais do IMC das meninas. O uso parcimonioso de qualquer um dos procedimentos sugeridos para estudar aspetos da estabilidade e previsibilidade (i.e., do tracking) em estudos longitudinais exige o estabelecimento muito criterioso dos objetivos e hipóteses a serem testados.
Palavraschave:
Crescimento e desenvolvimento; Estudos longitudinais; Índice de massa corporal; Monitoramento
INTRODUCTION
Obesity is one of the major modifiable risk factors for chronic diseases and its adverse effects on health are wellknown^{1}1 Gonzalez A, Boyle MH, Georgiades K, Duncan L, Atkinson LR, MacMillan HL. Childhood and family influences on body mass index in early adulthood: findings from the Ontario Child Health Study. BMC Public Health 2012;12:755., in addition to its elevated economic impact on health systems. For example, the annual cost of obesity between 2008 and 2010 was estimated as $ 2.1 billion, representing ~14% of the total cost of Brazilian Health Systems^{2}2 Bahia L, Coutinho ES, Barufaldi LA, Abreu Gde A, Malhao TA, de Souza CP, et al. The costs of overweight and obesityrelated diseases in the Brazilian public health system: crosssectional study. BMC Public Health 2012;12:440..
The adverse consequence of excess weight on individual's health has prompted the establishment of prevention strategies early in pediatric settings^{1}1 Gonzalez A, Boyle MH, Georgiades K, Duncan L, Atkinson LR, MacMillan HL. Childhood and family influences on body mass index in early adulthood: findings from the Ontario Child Health Study. BMC Public Health 2012;12:755.^{,}^{3}3 Bayer O, Kruger H, von Kries R, Toschke AM. Factors associated with tracking of BMI: a metaregression analysis on BMI tracking. Obesity 2011;19(5):106976.. Furthermore, childhood and adolescence are viewed as critical windows in terms of obesity development, and there is a high likelihood that behaviors consolidated in this period of life remain in adulthood^{1}1 Gonzalez A, Boyle MH, Georgiades K, Duncan L, Atkinson LR, MacMillan HL. Childhood and family influences on body mass index in early adulthood: findings from the Ontario Child Health Study. BMC Public Health 2012;12:755.^{,}^{3}3 Bayer O, Kruger H, von Kries R, Toschke AM. Factors associated with tracking of BMI: a metaregression analysis on BMI tracking. Obesity 2011;19(5):106976..There is evidence suggesting that obese children are 50 to 70% more likely to become obese adults due to family history, sedentarism and unhealthy lifestyles^{4}4 LloydJones D, Adams RJ, Brown TM, Carnethon M, Dai S, De Simone G, et al. Executive summary: heart disease and stroke statistics2010 update: a report from the American Heart Association. Circulation 2010;121(7):94854..
In epidemiology, the analysis that deals with the tendency of maintaining a state and/or behavior in a series of longitudinal data is generally called tracking^{3}3 Bayer O, Kruger H, von Kries R, Toschke AM. Factors associated with tracking of BMI: a metaregression analysis on BMI tracking. Obesity 2011;19(5):106976.^{,}^{5}5 Kowalski CJ, Schneiderman ED. Tracking: Concepts, Methods and Tools. Int J Anthropol 1992;7(4):3350.. Although there is no universally accepted definition of the term, tracking refers to the notion of maintaining a relative position of values of a given group of individuals as a function of time; it is also linked to the idea of prediction^{3}3 Bayer O, Kruger H, von Kries R, Toschke AM. Factors associated with tracking of BMI: a metaregression analysis on BMI tracking. Obesity 2011;19(5):106976.^{,}^{5}5 Kowalski CJ, Schneiderman ED. Tracking: Concepts, Methods and Tools. Int J Anthropol 1992;7(4):3350.. Stability, change and predictability are tracking facets, requiring longitudinal information. The statistical analysis of tracking and its application have already been researched in Portugal and Brazil, mainly in Physical Education and Sports Sciences^{6}6 Freitas D, Beunen G, Maia J, Claessens A, Thomis M, Marques A, et al. Tracking of fatness during childhood, adolescence and young adulthood: a 7year followup study in Madeira Island, Portugal. Ann Hum Biol 2012;39(1):5967.^{,}^{7}7 Da Silva SP, Beunen G, Prista A, Maia J. Shortterm tracking of performance and healthrelated physical fitness in girls: the Healthy Growth in Cariri Study. J Sports Sci 2013;31(1):10413..
This study aims to present a set of statistical techniques of tracking in order to allow researchers a better understanding of their use and interpretation. Firstly, we will deal with autocorrelations^{8}8 Malina RM. Tracking of physical activity and physical fitness across the lifespan. Res Q Exerc Sport 1996;67(3 Suppl):S4857.; secondly, we will use Foulkes & Davis gamma (γ)^{9}9 Foulkes MA, Davis LE. An index of tracking for longitudinal data. Biometrics 1981;37:43946.; thirdly, we will refer to the γ statistics again, but according to suggestions made by Rogosa^{10}10 Rogosa D, Ghandour G. TIMEPATH: Statistical analysis of individual trajectories. CA S, editor: Stanford University; 1989.; finally, we will present the Goldstein growth constancy index^{11}11 Furey A, Kowalski C, Schneiderman E, Willis S. GTRACK: A PC program for computing Goldstein's growth constancy index and an alternative measure of tracking. Int J BioMed Comput 1994;36(1):3118..
METHODOLOGICAL PROCEDURES
The data used in this study are from a mixedlongitudinal study conducted in the city of Porto, Portugal, designed to investigate the interaction among individual characteristics, environmental factors, and lifestyle that affect growth, development, and health of adolescents aged 1018 years. The project was approved by the Ethics Committee of the University of Porto (process number 15/CEUP/2012). This research, almost in its final stages, intends to analyze a total of 1000 randomly selected subjects, stratified and divided into four age cohorts and evaluated for three consecutive years. The first cohort was followed from 10 to 12 years; the second from 12 to 14 years; the third from 14 to 16 years; and the fourth from 16 to 18 years. The present study considered information from 486 girls from the second (n_{c2}=215) and third (n_{c3}=169) cohorts. Body mass index (BMI): [weight (kg)/height (m^{2})] was the chosen variable for all analyses. This marker is used in epidemiological studies to define nutritional status and/or weight categories^{12}12 Guo SS, Chumlea WC. Tracking of body mass index in children in relation to overweight in adulthood. Am J Clin Nutr 1999;70(1):145S8S.^{,}^{13}13 Malina RM, Katzmarzyk PT. Validity of the body mass index as an indicator of the risk and presence of overweight in adolescents. Am J Clin Nutr 1999;70(1):131S6S..
The decision to use only girls' information is based on the fact that their body composition undergoes marked changes during the pubertal period usually resulting in larger fat deposits^{14}14 Malina R, Bouchard C, BarOr O. Growth, maturation and physical activity. Champaign, editor: Human Kinetics; 2004.. The increased body fat, combined with differences in maturational timing and tempo may result in a lower engagement in physical activity and practice of sports^{7}7 Da Silva SP, Beunen G, Prista A, Maia J. Shortterm tracking of performance and healthrelated physical fitness in girls: the Healthy Growth in Cariri Study. J Sports Sci 2013;31(1):10413.^{,}^{14}14 Malina R, Bouchard C, BarOr O. Growth, maturation and physical activity. Champaign, editor: Human Kinetics; 2004., as well as in increased weight^{14}14 Malina R, Bouchard C, BarOr O. Growth, maturation and physical activity. Champaign, editor: Human Kinetics; 2004..
The concept of Tracking
As there is no universal definition for tracking, different approaches have been proposed to define tracking from a statistical point of view^{5}5 Kowalski CJ, Schneiderman ED. Tracking: Concepts, Methods and Tools. Int J Anthropol 1992;7(4):3350.. In 1991, Foulkes & Davis^{9}9 Foulkes MA, Davis LE. An index of tracking for longitudinal data. Biometrics 1981;37:43946. were the first to systematize the two main methodological views about tracking.The first approach focuses on the study of correlations between successive measures (autocorrelations), and linear or nonlinear regression that allows future predictions^{15}15 Maia J, Silva R, Seabra A, Lopes VP. A importância do estudo do tracking (estabilidade e previsão) em delineamentos longitudinais: um estudo aplicado à epidemiologia da actividade física e à performance desportivomotora. Rev Port Cien Desp 2002;2(4):4156.. A substantial number of studies within Physical Education and Sports Science have adopted several ideas from this view ^{8}8 Malina RM. Tracking of physical activity and physical fitness across the lifespan. Res Q Exerc Sport 1996;67(3 Suppl):S4857.^{,}^{16}16 Malina RM. Adherence to physical activity from childhood to adulthood : a perspective from tracking studies. Quest 2001;53(1):34655.. The second approach is based on the recognition that the distribution of values changes naturally at each point of time and it is expected that individuals maintain the same relative position in each of these distributions. Several analytical procedures are based on this suggestion, but the problem lies in the precision of how "relative position" is defined^{5}5 Kowalski CJ, Schneiderman ED. Tracking: Concepts, Methods and Tools. Int J Anthropol 1992;7(4):3350..
Tracking: statistics, results and meaning
Descriptive statistics for BMI values from each cohort are shown in Table 1. Mean BMI values increased with time. It is important to highlight that kurtosis and skewness values suggest violations to normality.
Autocorrelations
An important part of tracking studies in Physical Activity Epidemiology and Physical Fitness resorts to the calculation of correlations (r) among the same variables sequentially measured in time, calculating what is known as autocorrelations^{16}16 Malina RM. Adherence to physical activity from childhood to adulthood : a perspective from tracking studies. Quest 2001;53(1):34655.. Regardless of interpretation of r values based on formal tests of the null hypothesis(H_{0}:r=0), Malina^{16}16 Malina RM. Adherence to physical activity from childhood to adulthood : a perspective from tracking studies. Quest 2001;53(1):34655. subjectively suggested cutoff points for autocorrelation interpretation (r<0.3=low; 0.30≤r≤0.60=moderate; r>0.60=moderate to high).However, as well known, the use of any statistical procedure in inferential terms is based on a series of assumptions for results to be valid. When computing simple correlations it is assumed that: (i) there is a linear relationship among variables; (ii) variables are randomly distributed; (iii) have homoscedasticity; and (iv) have bivariate or multivariate normal distributions^{17}17 Pestana M, Gageiro J. Análise de dados para ciências sociais. A complementaridade do SPSS: Edições Sílabo; 2003.. In Table 1, skewness and kurtosis suggest potential violation to normality of the BMI distributions at each age group in each cohort.
The analysis of univariate, bivariate and multivariate normality of BMI distributions at different time points in each cohort was performed in STATA 12. The results (not included in the text) showed violation of these assumptions. When BMI values were transformed (1/BMI), univariate normality was achieved, but violations to bivariate and multivariate normality still could be found. To solve this problem, some authors suggest using Spearman correlation coefficient (less efficient than Pearson, but not sensitive to kurtoses problems in the distributions or presence of outliers^{17}17 Pestana M, Gageiro J. Análise de dados para ciências sociais. A complementaridade do SPSS: Edições Sílabo; 2003.). However, our choice was different as we decided to make a robust analysis suggested by Hadi^{18}18 Hadi A. A modification of a method for the detection of outliers in multivariate samples. J R Statist Soc B 1994;Series (B), 56. and implemented in SYSTAT 13 in which a resampling method (bootstrap) with 500 samples equal to the size of samples for each cohort was added in order to obtain standard errors to helps us in the construction of confidence intervals for all r values, providing a more precise view of tracking coefficients (see Table 2).
Autocorrelations and their respective confidence intervals (CI 95%) between BMI measurements over 3 years in each cohort.
Based on the correlational strength rubric of Malina^{16}16 Malina RM. Adherence to physical activity from childhood to adulthood : a perspective from tracking studies. Quest 2001;53(1):34655., there was a strong stability of BMI in girls aged 1214 years and 1416 years. In addition, the width of confidence intervals is extremely low, which confirms the accuracy of estimates. It was also possible to verify that the autocorrelation is lower between BMI1 and BMI3 in both cohorts, which reflects the higher temporal spacing between measurements. Despite the simplicity in the interpretation of rvalues, some authors^{16}16 Malina RM. Adherence to physical activity from childhood to adulthood : a perspective from tracking studies. Quest 2001;53(1):34655.^{,}^{19}19 Glenmark B, Hedberg G, Jansson E. Prediction of physical activity level in adulthood by physical characteristics, physical performance and physical activity in adolescence: an 11year followup study. Eur J Appl Physiol Occup Physiol 1994;69(6):5308.highlighted the need to consider (i) the age of the first observation (the lower the child's age, the lower the correlation coefficients), and (ii) individual characteristics given the obvious biological variation among subjects (maturational timing and tempo).
In short, although autocorrelations are widely used in tracking studies, and the BMI values obtained in both cohorts were very high, Rogosa et al.^{20}20 Rogosa D, Floden R, Willet J. Assessing the stability of teacher behavior. J Educ Psychol 1984;76(1):100027. and Twisk et al.^{21}21 Twisk JW, Kemper HC, Mellenbergh GJ. Mathematical and analytical aspects of tracking. Epidemiol Rev 1994;16(2):16583. pointed out some of the problems with this approach: (i) the assumptions of bivariate and multivariate normality are rarely tested and/or reported, and no alternatives are presented to solve this problem; (ii) the Malina^{16}16 Malina RM. Adherence to physical activity from childhood to adulthood : a perspective from tracking studies. Quest 2001;53(1):34655. cutoff points are arbitrary; and (iii) there is no single autocorrelation value to describe stability. In our example, three autocorrelations were reported, because we have three time measurements; if we had 6 time points, we would have an autocorrelation matrix with 15 values (the general formula to compute the number of possible autocorrelations is k(k1)/2, where k=number of points in time).
Foulkes & Davis γ^{9}9 Foulkes MA, Davis LE. An index of tracking for longitudinal data. Biometrics 1981;37:43946.
Foulkes & Davis γ examined the probability that two growth curves do not intersect (cross) over time; in addition, it is based on the notion that the greater the number of pairs of individuals that maintain their relative position within a distribution over the study time frame, the greater the tracking^{9}9 Foulkes MA, Davis LE. An index of tracking for longitudinal data. Biometrics 1981;37:43946.. The γ only takes positive values, ranging from 0 to 1. The higher the g, the lower the number of crossings among growth curves. Foulkes & Davies presented reference values for γ: γ≤0.50 no tracking; 0.50<γ<1.00 tracking is present; γ= 1:00, perfect tracking^{9}9 Foulkes MA, Davis LE. An index of tracking for longitudinal data. Biometrics 1981;37:43946..
Foulkes & Davisg have two formulations: a simple (FD1) and a more complex version (FD2)^{22}22 Schneiderman E, Kowalski C, Ten Have T. A GAUSS prgram for computing an index of tracking from longitudinal observations. Am J Hum Biol 1990;2(1):47590.. The simple version does not require any a priori definition of the change trajectory shape (linear or nonlinear), since it is a nonparametric statistic that assumes the following: 1) the simpler the trajectory, the higher the γ value ; and 2) the tracking of the extremes is much higher when compared to those who are situated close to the mean trajectory^{21}21 Twisk JW, Kemper HC, Mellenbergh GJ. Mathematical and analytical aspects of tracking. Epidemiol Rev 1994;16(2):16583.
22 Schneiderman E, Kowalski C, Ten Have T. A GAUSS prgram for computing an index of tracking from longitudinal observations. Am J Hum Biol 1990;2(1):47590.^{}^{23}23 Maia J, Garganta R, Seabra A, Lopes VP, Silva S, Meira Júniro C. Explorando a noção e significado do tracking. Um percurso didático para investigadores. Psicologiapt 2007.. In our example, we chose the complex version (FD2), which requires, in addition, a formal and sequential test of the better function (limited to a 4^{th}degree polynomial) that best describes individual trajectories^{24}24 Schneiderman E, Willis S, Kowalski C, Ten Have T. A GAUUS program for computing the FoulkesDavis Tracking Index for polynomial growth curves. Int J BioMed Comput 1993;32(1):3543.^{,}^{25}25 Schneiderman ED, Kowalski CJ. Analysis of longitudinal data in craniofacial research: some strategies. Crit Rev Oral Biol Med 1994;5(34):187202.. This analysis was performed on the Longitudinal Data Analysis program (LDA) developed by Schneiderman and Kowalski^{26}26 Schneiderman E, Kowalski C. LDA. Software system for longitudinal data analysis. Version 3.2. . Texas: Baylor College of Dentistry; 1993..
Results of the best function that describes data and the Foulkes & Davis γ statistics for cohort 2.
LDA software output has a wealth of numerical and graphical information. Table 3 highlights only the main results: (1) the test for the best fitting model (F statistics and the corresponding pvalue); and (2) γ value, its standard error and the 95% CI. Thus, Foulkes & Davis γ of girls from cohort 2 is 0827 ± 0.006 (95% CI = 08150840) and 0.828 ± 0.008 (95% CI = 08120845) of those from cohort 3. It could be concluded that girls aged 1214 years, as well as those aged 1416 years, show a high BMI tracking over 3 years.
Although Foulkes & Davis γ have a more complex mathematicalstatistical structure when compared to autocorrelations and requires specialized software for its computation, its formulation has several advantages^{24}24 Schneiderman E, Willis S, Kowalski C, Ten Have T. A GAUUS program for computing the FoulkesDavis Tracking Index for polynomial growth curves. Int J BioMed Comput 1993;32(1):3543.^{,}^{25}25 Schneiderman ED, Kowalski CJ. Analysis of longitudinal data in craniofacial research: some strategies. Crit Rev Oral Biol Med 1994;5(34):187202.including: 1) collected data do not need to be equally spaced in time; 2) there is no special need for Gaussian distributions; 3) a test is available to identify the best fitting model describing change; 4) it has a unique tracking statistics and is associated with 95% CI; and 5) it allows the identification of individuals whose growth curves are more or less stable and therefore more or less predictable.
γ Statistics according to David Rogosa^{10}10 Rogosa D, Ghandour G. TIMEPATH: Statistical analysis of individual trajectories. CA S, editor: Stanford University; 1989.^{,}^{20}20 Rogosa D, Floden R, Willet J. Assessing the stability of teacher behavior. J Educ Psychol 1984;76(1):100027.^{,}^{27}27 Rogosa D. Myths and methods: Myths about longitudinal research plus supplemen. tal questions. In: Gottman J, editor. The analysis of change. New Jersey: Lawrence Erlbaum Associates; 1995. p. 365.
The method proposed by Rogosa^{20}20 Rogosa D, Floden R, Willet J. Assessing the stability of teacher behavior. J Educ Psychol 1984;76(1):100027., detailed in the software developed with Ghandour (TIMEPATH)^{10}10 Rogosa D, Ghandour G. TIMEPATH: Statistical analysis of individual trajectories. CA S, editor: Stanford University; 1989., is based on a seminal paper published with Sanner^{28}28 Rogosa D, Saner H. Longitudinal data analysis examples with random coefficient models. J Educ Behav Statist 1995;20., as well as in his "classic" work reprinted in 1995^{27}27 Rogosa D. Myths and methods: Myths about longitudinal research plus supplemen. tal questions. In: Gottman J, editor. The analysis of change. New Jersey: Lawrence Erlbaum Associates; 1995. p. 365.. The software output offers the following: 1) best fitting models for individual growth curves as well as group statistics; 2) values of a single individual g, and 3) population estimates with standarderrors, which allows for computations of 95% CI.
Since there is an individual γ and therefore 169 g's in cohort 3, the software presents relevant descriptive statistics (mean, standard deviation, minimum and maximum), and the fivenumber summary: minimum, P25 (quartile 1), median (P50), P75 (quartile 3), as well as Rates R^{2} and γ statistics. It is important to highlight that individual γ refers to the probability of an individual trajectory to cross other trajectories. The information provided by the software is very rich in order to have a detailed description of modal BMI trajectory as well as an individualized view of its stability (given by γ) and change (given by Rates). Finally, tracking population estimates are presented.
In Table 4, the standard deviation of Rate (i.e., the slope) is much higher than the mean (0.917 to 0.336), indicating great variability in BMI individual changes over the three years. The lowest γ value was 0.411, indicating no individual tracking. From P5, γ values rises to 0.613, and the median is 0.839, which is already high. The global γ is 0.852 ± 0.008 showing high BMI tracking in this cohort. In cohort 2, results (not shown) were similar.
Rogosa's suggestions^{10}10 Rogosa D, Ghandour G. TIMEPATH: Statistical analysis of individual trajectories. CA S, editor: Stanford University; 1989.^{,}^{20}20 Rogosa D, Floden R, Willet J. Assessing the stability of teacher behavior. J Educ Psychol 1984;76(1):100027.^{,}^{27}27 Rogosa D. Myths and methods: Myths about longitudinal research plus supplemen. tal questions. In: Gottman J, editor. The analysis of change. New Jersey: Lawrence Erlbaum Associates; 1995. p. 365. as well as the versatility and richness of the TIMEPATH output are very important in order to have a detailed view of tracking, allowing researchers a more detailed examination, in modal and individual ways, of BMI trajectories over the three years.
Goldstein's growth constancy index^{11}11 Furey A, Kowalski C, Schneiderman E, Willis S. GTRACK: A PC program for computing Goldstein's growth constancy index and an alternative measure of tracking. Int J BioMed Comput 1994;36(1):3118.^{,}^{29}29 Goldstein H. Measuring the stability of individual growth patterns. Ann Hum Biol 1981;8(6):54957.
The Goldstein's growth constancy index, represented as ξ by Furey et al.^{11}11 Furey A, Kowalski C, Schneiderman E, Willis S. GTRACK: A PC program for computing Goldstein's growth constancy index and an alternative measure of tracking. Int J BioMed Comput 1994;36(1):3118., is a tracking measure aimed at determining the stability and variability of individual growth (i.e., height) trajectories. According to Goldstein^{29}29 Goldstein H. Measuring the stability of individual growth patterns. Ann Hum Biol 1981;8(6):54957., the analysis of change patterns occurring in children and adolescents' growth would provide insight into the detection of stable (maintenance of a relative position) or unstable growth curves (relatively high proportion of intersecting curves). Its importance in Auxology and paediatrics is evident to timely identify children or adolescents with instability in their physical growth. Goldstein^{29}29 Goldstein H. Measuring the stability of individual growth patterns. Ann Hum Biol 1981;8(6):54957. proposed that in a random sample of individuals, an individual whose growth curve crosses a relatively high proportion of other subjects´ curves is characterized as having a low tracking.
Goldstein^{29}29 Goldstein H. Measuring the stability of individual growth patterns. Ann Hum Biol 1981;8(6):54957. presented two ways of estimating the growth constancy index, i.e., tracking measures. In the first approach, ξ and its confidence interval are based on a Jackknife estimator. In the second approach, the use of the intraclass correlation obtained from the analysis of variance (ANOVA) was suggested. These two options can be formulated, as stressed by Furey et al.^{11}11 Furey A, Kowalski C, Schneiderman E, Willis S. GTRACK: A PC program for computing Goldstein's growth constancy index and an alternative measure of tracking. Int J BioMed Comput 1994;36(1):3118., in the context of two ANOVA models^{29}29 Goldstein H. Measuring the stability of individual growth patterns. Ann Hum Biol 1981;8(6):54957.. Thus, in model I or II, the problem lies in the way the true value of each individual is formulated, its true stability and interpretation. In model I, this value is considered as an unknown (i.e., a constant), whereas in model II, it is considered as a random variable. The interpretation in the case of ANOVA I is the following: tracking inferences are valid only for cases included in the study; in ANOVA II, the inferences are made to the population consisting of individuals from where the sample was randomly extracted. Table 5 shows examples of these analyses. In Model I, a typical ANOVA table is shown, and ξ is presented (its values vary from 0 to 1, and 1 is the perfect tracking). According to Furey et al.^{11}11 Furey A, Kowalski C, Schneiderman E, Willis S. GTRACK: A PC program for computing Goldstein's growth constancy index and an alternative measure of tracking. Int J BioMed Comput 1994;36(1):3118., this index may be overestimated and may assume positive values even when there is no evident tracking. In this case, it is necessary to consider its modified or corrected value (ξ*) with a maximum value of 1, but it becomes 0 if autocorrelations are equal to 0 in successive values. The last part of model I shows point estimates as well as confidence intervals obtained by the Jackknife resampling technique. In Model II, an ANOVA table is also presented. However, tracking is expressed as an intraclass correlation coefficient. In addition, the 95% confidence interval is also shown^{11}11 Furey A, Kowalski C, Schneiderman E, Willis S. GTRACK: A PC program for computing Goldstein's growth constancy index and an alternative measure of tracking. Int J BioMed Comput 1994;36(1):3118..
Model I and II of the Golsdstein's growth constancy index^{11}11 Furey A, Kowalski C, Schneiderman E, Willis S. GTRACK: A PC program for computing Goldstein's growth constancy index and an alternative measure of tracking. Int J BioMed Comput 1994;36(1):3118.^{,}^{29}29 Goldstein H. Measuring the stability of individual growth patterns. Ann Hum Biol 1981;8(6):54957. of the BMI of girls from cohort 2.
In our example, we used model I (without any purpose of generalization); ξ values were high in both cohorts indicating strong stability for BMI (ξ_{c2}=0.918 andξ_{c3}=0.940). As expected, the values of the modified index decreased, but remained high (ξ*_{c2}=0.878 and ξ*_{c3}=0.910). The Jackknife estimates and corresponding 95% CI were 0.916 (0.873, 0.946) and 0.874 (0.809, 0.919) for cohort 2; 0.939 (0.915, 0.957) and 0.909 (0.873, 0.936) for cohort 3.
Results of model II (generalization allowed for girls of the same population), the intraclass correlation coefficient and 95% CI was 0.878 (0.834, 0.912) for girls in cohort 2 and 0.910 (0.876, 0.935) in cohort 3. These results show, once again, BMI stability for girls aged 1214 years and 1416 years.
In short, the wealth of Goldstein's suggestion^{11}11 Furey A, Kowalski C, Schneiderman E, Willis S. GTRACK: A PC program for computing Goldstein's growth constancy index and an alternative measure of tracking. Int J BioMed Comput 1994;36(1):3118.^{,}^{29}29 Goldstein H. Measuring the stability of individual growth patterns. Ann Hum Biol 1981;8(6):54957. can be extended to tracking studies covering other variables without additional problems. In addition to obtaining a single tracking measure, its computation does not require equidistant observations, and Gaussian distribution is not required. One of the advantages of Goldstein propositions is the identification of individuals with high or low separation to the mean, allowing efficient monitoring and intervention of flagged cases.
CONCLUSION
This paper presents a set of different tracking approaches aimed at helping a novice researcher in the field. The importance of the tracking concept, its various statistical analysis techniques, and meaning may constitute important methodological lessons in the Physical Education and Sports Science fields, especially when dealing with longitudinal data arising from observational and/or intervention designs. Notwithstanding, the variety of statistical approaches, it may be important for researchers to become acquainted with their versatility, implementation in different software, utility and application.
In our example, a moderatetohigh tracking was expected given its short duration (2 years), regardless of procedure used. Differences among subjects in their BMI individual changes are relatively small autocorrelations, so Foulkes & Davies γ and Goldstein's constancy index showed similar high values. However, when LDA or TIMEPATH outputs were explored, a more individualized view of each subject becomes more evident, allowing identifying adolescents who may require individual care and more efficient interventions from Physical Education teachers or paediatricians.
Acknowledgment
To CAPES Foundation, the Ministry of Education of Brazil, BrasíliaDF, for the scholarship granted to Professor Michele Caroline de Souza.
REFERENCES

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Publication Dates

Publication in this collection
MayJun 2015
History

Received
17 Mar 2015 
Accepted
13 Apr 2015