Download Modeling developmental changes in strength and aerobic power in

Modeling developmental changes in strength
and aerobic power in children
ALAN M. NEVILL,1 ROGER L. HOLDER,2 ADAM BAXTER-JONES,3
JOAN M. ROUND,4 AND DAVID A. JONES4
1School of Human Sciences, Liverpool John Moores University, Liverpool L3 3AK;
2School of Mathematics and Statistics and 4School of Sport and Exercise Sciences,
University of Birmingham, Birmingham B15 2TT; and 3Department of Child Health,
University of Aberdeen, Aberdeen AB9 2ZD, United Kingdom
mial model is unlikely to satisfactorily explain developmental changes over time. The authors propose an
alternative multiplicative (proportional) model with
allometric body size components to describe these developmental changes that should successfully accommodate
the nonlinear but proportional changes with body mass
and naturally help overcome the heteroscedastic (multiplicative) errors observed with such variables.
Hence, in the present study we shall examine the two
multilevel model structures (i.e., the additive polynomial structure and the alternative multiplicative ‘‘allometric’’ structure) using two longitudinal studies: study
1, that of Baxter-Jones et al. (2), where the additive
polynomial model structure was originally applied, and
study 2, that of Round et al. (19), where the alternative
multiplicative model structure was chosen. With the
assumption that the additive polynomial model adopted
by Baxter-Jones et al. was optimal, a comparison will
be made with the equivalent multiplicative model to
examine the hypothesis that a multiplicative allometric
model structure is preferable to an additive polynomial
structure when fitted to both data sets from studies 1
and 2.
multilevel regression; longitudinal growth; multiplicative
models; allometric body size components
METHODS
IN PREVIOUS CROSS-SECTIONAL studies (14, 17), the effect
of life-style factors, e.g., physical activity, training, and
diet, on developmental changes in performance variables, e.g., aerobic power and strength, has been confounded by the simultaneous changes due to growth
and development. To identify and separate the relative
contribution of these life-style factors from the changes
due to growth and maturation, there is a need to collect
such ‘‘growth’’ data longitudinally. An appropriate
method to analyze such longitudinal data is some form
of multilevel modeling process (9). When modeling
growth data, Goldstein (7, 8) incorporates ‘‘age’’ as
additive polynomial terms, where any systematic
change in the residual error, e.g., heteroscedastic (multiplicative) errors, can also be modeled simultaneously
within the multilevel analysis. However, recently, Nevill
and Holder (14, 15) criticized the use of additive models
to explain differences in variables such as aerobic power.
They argue that because variables such as strength
and aerobic power are known to be proportional to but
nonlinear with body mass (i.e., m¯ ), an additive polyno-
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Study 1
Subjects. A detailed description of the study design and
selection criteria of the athletes has been published previously (3). Briefly, a random sample of 453 coach-nominated
athletes (231 boys and 222 girls) from four sports (gymnastics, soccer, swimming, and tennis) was studied for 3 yr
consecutively. The study used a linked longitudinal design
(11), following five age cohorts, to include prepubertal, pubertal, and postpubertal children (Table 1).
The age at which the youngest child entered the study
differed according to the requirements of each sport. On entry
into the study, the youngest athletes were 8 yr of age and the
oldest were 16 yr of age. During the course of the study the
composition of these clusters remained the same. Inasmuch
as there were overlaps in ages between the clusters, it was
possible to estimate a consecutive 11-yr development pattern
over the much shorter period of 3 yr.
One of the disadvantages of longitudinal studies is the
number of dropouts during the course of the studies. In total
187 subjects left the study. Of these, 63 were excluded
because they retired from their sport, another 34 were
excluded because they were not training intensively enough
[as defined by thresholds of intensity related to hours trained
per week (20)], and 90 (19.9% of the total sample) withdrew
themselves. Those subjects who retired from their sport or
who were not considered to be intensively training were not
0161-7567/98 $5.00 Copyright r 1998 the American Physiological Society
963
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Nevill, Alan M., Roger L. Holder, Adam Baxter-Jones,
Joan M. Round, and David A. Jones. Modeling developmental changes in strength and aerobic power in children. J.
Appl. Physiol. 84(3): 963–970, 1998.—The present study
examined two contrasting multilevel model structures to
describe the developmental (longitudinal) changes in strength
and aerobic power in children: 1) an additive polynomial
structure and 2) a multiplicative structure with allometric
body size components. On the basis of the maximum loglikelihood criterion, the multiplicative ‘‘allometric’’ model was
shown to be superior to the additive polynomial model when
fitted to the data from two published longitudinal studies and
to provide more plausible solutions within and beyond the
range of observations.The multilevel regression analysis of
study 1 confirmed that aerobic power develops approximately
in proportion to body mass, m¯. The analyses from study 2
identified a significant increase in quadriceps and biceps
strength, in proportion to body size, plus an additional
contribution from age, centered at about peak height velocity
(PHV). The positive ‘‘age’’ term for boys suggested that at
PHV the boys were becoming stronger in the quadriceps and
biceps in relation to their body size. In contrast, the girls’ age
term was either negligible (quadriceps) or negative (biceps),
indicating that at PHV the girls’ strength was developing in
proportion to or, in the case of the biceps, was becoming
weaker in relation to their body size.
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MODELING GROWTH CHANGES IN STRENGTH AND AEROBIC POWER
Table 1. Number of children recruited by birth year,
sport, and gender
Gymnastics
Soccer
Swimming
Tennis
Birth
Year
F
M
M
F
M
F
M
1971
1973
1974
1975
1976
13
18
18
15
17
7
10
10
6
5
12
25
28
14
16
15
15
10
14
15
15
14
17
19
20
11
12
15
18
19
10
Total
81
38
65
60
54
81
74
Total sample 5 453. F, female; M, male. Assessment started in 1987
when those born in 1971 were 16 yr of age.
Study 2
Subjects. Details of the study design have been published
previously (19). Briefly, 100 North London schoolchildren
were studied: 50 boys and 50 girls. Three sets of measurements were made each year at ,4-mo intervals. The duration
of follow-up was 5 yr, with children being recruited in groups
commencing at 8–12 yr of age and finishing at 13–17 yr of age
at the end of the study.
Permission to approach the children was obtained from the
Local Education Authority, the head teachers of the schools,
and the Committee on the Ethics of Human Investigation at
the Middlesex and University College Hospitals (now the
University College London Hospitals). Children volunteered
for the study after being fully informed about the tests
involved and how long they would continue in ‘‘the team.’’
Parental consent in writing was then obtained for all the
children who had volunteered. The parents were asked to
supply the name of their general practitioner, who was
informed in writing about the study and given the names of
the children on their list who were participating.
As in all long-term studies, there was some loss of subjects
during the 5-yr period. Sixty percent of the children com-
Statistical Methods
An appropriate method of analyzing longitudinal (repeatedmeasures) data is to adopt this multilevel modeling approach
(9), using the program Multilevel Models Project MLn (18).
Multilevel modeling is an extension of ordinary multiple
regression, where the data have a hierarchical or clustered
structure. A hierarchy consists of units or measurements
grouped at different levels. One example is repeatedmeasures data, where individuals are measured on more
than one occasion. Here, the subjects or individuals, assumed
to be a random sample, represent the level 2 units, with the
subjects’ repeated measurements recorded at each visit being
the level 1 units. In contrast to traditional repeated-measures
analyses, the number of visits is also assumed to be a random
variable over time. The two levels of random variation take
into account the fact that growth characteristics of individual
children, such as their average growth rate, vary around a
population mean and also that each child’s observed measurements vary around his or her own growth trajectory.
In the present work the multilevel regression analyses
were performed using the MLn software to identify those
factors (different sports and levels of maturity in study 1 and
sex differences in study 2) associated with the development of
aerobic power and strength, respectively, with adjustment for
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invited to return for reassessment because of financial constraints, a common problem with longitudinal studies (21).
Measurements. Body height, weight, pubertal development, and maximal O2 uptake (V̇O2 max) were measured annually for 3 yr consecutively. Subjects were grouped into three
pubertal stages to ensure adequate cell sizes within each
sport: prepubertal (PP), midpubertal (MP), and late pubertal
(LP). Puberty was determined by visually assessing sex
characteristics: stage of breast development in girls and
genitalia development in boys. Five discrete stages have been
clearly described (23) in this study: stage 1 (PP), stages 2 and
3 (MP), and stages 4 and 5 (LP). V̇O2 max was measured with
the subject running on a motor-driven treadmill. Subjects ran
at an individually predetermined rate, on a 3.4% grade, for 1
min, then at 0.5 km/h every minute until exhaustion. Measurements of gas exchange were obtained by standard opencircuit techniques. Subjects breathed through a facemask,
and ventilation was measured through a turbine volume
transducer attached to a control unit with digital display.
Expired air was analyzed. The system was calibrated before
each session with standard gases of known O2 and CO2
concentration. Heart rate was continuously recorded during
exercise. The highest O2 uptake was accepted as V̇O2 max if a
plateau occurred (an increase of ,2 ml/kg with an increase in
workload) or if one of the following criteria was met: heart
rate .95% of the predicted maximum corrected for age
(predicted maximum heart rate 5 220 2 age) or respiratory
exchange ratio .1.1.
pleted the full 5 yr, with similar proportions of boys and girls.
The main loss occurred when children left school at 15 and 16
yr of age, but most of these children had been in the study for
at least 4 yr by this time. The other major loss occurred when
the 11-yr-old children changed from junior to senior school,
with some of the children continuing their education at
schools outside the area. The duration of the follow-up was 5
yr, with 84% of the girls and 82% of the boys being studied for
at least 4 yr.
Measurements. Maximum voluntary contraction strengths
of the knee extensors and forearm flexors (hereafter referred
to as quadriceps and biceps muscles, respectively) were
measured using a chair described by Parker et al. (17). The
investigators were experienced in the use of the various
procedures, and the main members of the team responsible
for overseeing the measurements remained the same throughout the study.
Each child was allowed to become familiar with the apparatus and procedures and then performed three maximum
voluntary contractions, the highest of which was recorded. As
in a previous study (17), there was no consistent trend among
the three strength measurements. On the rare occasions
when there was appreciable variation among the three efforts, the contractions were repeated until three values were
obtained that differed by #4% from the highest force.
Standing height was measured using a portable stadiometer (Holtain). Children removed their shoes and stood with
their heels and back against the upright with the neck gently
extended by upward manual pressure. Height was recorded
to the nearest millimeter. Weight was measured with portable scales to the nearest 0.1 kg with the children in light
indoor clothing without shoes.
The multilevel analysis reported was conducted on a subset
of the children (33 boys and 25 girls) who had their growth
spurt during the time of the study. For these children, the
number of visits was 9.8 6 1.9 and 9.6 6 2.3 (SD) for the boys
and girls, respectively. In these cases, the time of peak height
velocity (PHV) was identified and used in the multilevel
analysis to assess the contribution of height, weight, age, and
sex to the differences in strength observed between boys and
girls.
MODELING GROWTH CHANGES IN STRENGTH AND AEROBIC POWER
differences in body size (height and weight) and age. The two
levels of hierarchical or nested observational units used in
both studies were the number of visits at level 1 (within
individuals) and the sample of children (between individuals)
at level 2.
965
As with the additive polynomial models described above,
categorical factors can be incorporated into subsequent analyses by introducing them as fixed indicator variables, e.g.,
‘‘sex’’ (boys 5 0, girls 5 1).
Model Comparison
Additive Model Structure
Additive polynomial models were proposed by Goldstein (7,
8) to describe longitudinal changes in growth-related data
over time. On the basis of these models, Baxter-Jones et al. (2)
adopted the following additive polynomial model to describe
the developmental changes in aerobic power (y 5 V̇O2 max, in
l/min)
y 5 k1 · weight 1 k2 · height 1 k3 · height2
1 ai 1 bi · age 1 c · age2 1 eij
(1)
Multiplicative Model Structure
However, in contrast to the additive polynomial model
adopted by Baxter-Jones et al. (2), an alternative multiplicative allometric model was proposed for strength (y 5 strength,
in N) by Round et al. (19) on the basis of the work of Nevill (12)
and Nevill and Holder (15) as follows
y 5 weightk1 · heightk2 · exp (ai 1 bi · age 1 c · age2) e ij
(2)
where, as with the additive model structure proposed by
Baxter-Jones et al., all the parameters were fixed with the
exception of the constant and age parameters ai and bi, which
were allowed to vary randomly from child to child (level 2),
and the multiplicative error ratio eij, which that was used to
describe the error variance between visit occasions (level 1).
Following Baxter-Jones et al., the coefficient c has been
incorporated as a fixed effect, but whether this term should be
incorporated as a random effect will be explored in the
subsequent multilevel analyses.
The model can be linearized with a logarithmic transformation, and a multilevel regression analysis on loge(y) can be
used to estimate the unknown parameters. The transformed
log-linear multilevel regression model becomes
loge ( y) 5 k1 · loge (weight) 1 k2 · loge (height)
1 ai 1 bi · age 1 c · age2 1 loge (e ij )
(3)
RESULTS
Study 1
Although Baxter-Jones et al. (2) did not report the
maximum log-likelihood criteria when fitting the model
(Eq. 1) to the 231 young male and 222 young female
athletes separately (Tables 4 and 5 in Ref. 2), the
criteria were found to be 295.07 and 29.47 for the boys
and girls, respectively. Simple replacement of V̇O2 max
with loge(V̇O2 max) increased the maximum log-likelihood criteria for boys and girls to 291.58 and 28.25,
respectively. However, replacement of the three terms
weight, height, and height2 with just two terms, loge(weight) and loge(height), further increased the maxi-
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where all parameters were fixed, with the exception of the
constant and age parameters ai and bi, which were allowed to
vary randomly from child to child (level 2), and eij, which was
assumed to have a constant error variance between visit
occasions (level 1). The subscripts i and j are used to indicate
random variation at levels 2 and 1, respectively.
Other explanatory variables were incorporated into the
analysis by introducing them as indicator variables. For example, in study 2, sex was introduced as an indicator variable
(boys 5 0, girls 5 1), since, in this way, the boys’ constant term
would be incorporated within a baseline parameter ai, from
which the girls’ constant term would deviate. Similarly, in
study 1 the performance of PP swimmers was used as the
baseline parameter (the constant ai ) with which the performance of other sports (tennis, soccer, and gymnastics) and
levels of maturity (MP and LP subjects) were compared, i.e.,
allowed to deviate from the constant baseline ai. To allow
different growth rates to be associated with various groups,
the product of age and a group indicator variable was
introduced as an additional predictor variable into the multilevel analysis, i.e., by introducing a sport * age interaction
term.
Clearly, because the models are not nested or hierarchical,
a direct comparison between two competing model forms,
given by Eqs. 1 and 3, is not possible using traditional criteria
such as the residual sum of squares, the standard error, and
the coefficient of determination (R2 ). However, when proposing tests to compare separate model forms, Cox (6) chose the
same maximum-likelihood criterion that was subsequently
developed by Box and Cox (5) to help select the most suitable
transformation to provide normally distributed errors with
constant variance.
Because the multilevel regression analysis software MLn
also adopts the maximum log likelihood as its standard
criterion of model assessment (quality of fit), we use this
criterion to compare the two competing models (Eqs. 1 and 3).
This will be done in two stages. First, by use of the transformation methods of Box and Cox (5), a comparison will be made by
allowing the dependent variable ( y) to be untransformed or
logarithmically transformed, e.g., y 5 V̇ O2 max or y 5
loge(V̇O2 max), and the independent predictor variables to be
incorporated as the additive polynomial model (Eq. 1). Briefly,
the method of Box and Cox examines a wide range of
transformations and, on the basis of a maximum loglikelihood criterion, selects the optimum transformation for
the chosen multiple linear regression model.
The second stage will examine the use of the three polynomial terms, weight, height, and height2, and consider replacing them with the two logarithmically transformed terms
loge(weight) and loge(height). Once again, the competing
models will be assessed using the maximum log-likelihood
criterion. A final check would be to reassess the first stage, if
the second stage indicates that the independent variables of
weight and height should be included as logarithmically
transformed terms.
A simple modification of the maximum log-likelihood criterion would produce the Akaike information criterion (1)
(AIC 5 22 3 maximum log likelihood 1 2 3 number of
parameters fitted) that would take into account the different
number of fitted parameters in the two model structures to be
compared. When the performance of competing models is
assessed, the model that fits the data best is the one with the
largest maximum log likelihood or with the minimum AIC
value. However, because the maximum log-likelihood criterion is always negative, we are seeking to identify the model
that minimizes both criteria in absolute terms.
966
MODELING GROWTH CHANGES IN STRENGTH AND AEROBIC POWER
Table 3. Parsimonious multilevel regression analysis
of log-transformed aerobic power of young athletes
adjusted for boys’ size (weight and height) and age
Fixed Explanatory Variables
Value
Boys
Constant (a)
Loge (weight) (k1 )
Loge (height) (k2 )
Age (b)
Age2 (c)
Gymnastics-swimming (Da)
Soccer-swimming (Da)
Tennis-swimming (Da)
LP-PP (Da)
Interaction
Soccer p age (Db)
25.393 6 1.007
0.6962 6 0.0690
0.7309 6 0.2348
0.0366 6 0.0078
20.0033 6 0.0009
20.1285 6 0.0233
20.1042 6 0.0272
20.0770 6 0.0163
0.0361 6 0.0171
0.0134 6 0.0067
Variance-Covariance Matrix of Random Variables
Constant (a)
Age (b)
Level 1 (within individuals)
0.0063 6 0.0005
Constant (aij)
Level 2 (between individuals)
0.0071 6 0.0014
20.0008 6 0.0004
Constant (ai )
Age (bi )
0.00016 6 0.00013
Girls
24.154 6 1.072
0.6652 6 0.0727
0.4789 6 0.2468
0.0610 6 0.0103
20.069 6 0.0012
20.0883 6 0.0263
20.0467 6 0.0238
Constant (a)
Loge (weight) (k1 )
Loge (height) (k2 )
Age (b)
Age2(c)
Gymnastics-swimming (Da)
Tennis-swimming (Da)
Interaction
Gymnastics p age (Db)
Tennis p age (Db)
20.0251 6 0.0082
20.0177 6 0.0074
Variance-Covariance Matrix of Random Variables
Constant (a)
Study 2
Separate multilevel regression analyses were performed on the strength of quadriceps and biceps, centered about age at PHV, of 58 normal British schoolTable 2. MLL and AIC together with number of fitted
parameters for competing models in study 1
V̇O2 max
Boys
Competing
Models
Additive polynomial
(Eq. 1)
Additive polynomial
(Eq. 1)*
Log-transformed
multiplicative
(Eq. 3)
Total no. of visit
occasions
Age (b)
Level 1 (within individuals)
0.00844 6 0.00081
Level 2 (between individuals)
Constant (ai )
Age (bi )
0.00453 6 0.00124
20.00080 6 0.00038
0.00036 6 0.00015
Values are means 6 SEE. Aerobic power is expressed as V̇O2 max in
l/min [log (l/min)]. Age was adjusted about origin using mean age 6
12 yr. Swimming was used as baseline measure (a), and other sport
groups were compared with it, indicated by (Da). LP-PP, difference
between late puberty and prepuberty, also indicated by Da.
Girls
MLL
AIC
n
MLL
AIC
n
295.07
226.1
18
29.47
50.9
16
291.58
219.2
18
28.25
48.5
16
286.49
202.0
14
25.07
36.1
13
503
Constant (aij )
408
MLL, maximum log likelihood; AIC, Akaike information criterion;
V̇O2 max , maximal O2 uptake; n, number of fitted parameters.
AIC 5 (22 3 MLL) 1 (2 3 no. of parameters fitted). * With logtransferred dependent variable [loge (V̇O2 max )].
children (19). PHV was at 12.2 6 0.94 yr among the
girls (n 5 25) and 13.4 6 0.85 yr among the boys (n 5
33). When the additive polynomial model (Eq. 1) was
fitted to the untransformed quadriceps and biceps
strength measurements, the maximum log-likelihood
criteria were 22,649.4 and 22,350.5, respectively.
Simple replacement of y 5 strength with y 5
loge(strength) increased the maximum log-likelihood
criteria for quadriceps and biceps strength to 22,612.5
and 22,302.0, respectively. Indeed, for the quadriceps
strength measurements, replacement of the three terms
weight, height, and height2 with just the two terms
loge(weight) and loge(height) further increased the maxi-
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mum log-likelihood criteria to 286.49 and 25.07 for the
boys and girls, respectively, confirming the better fit of
the logarithmically transformed multiplicative allometric model (Eq. 3) than the additive polynomial model
(Eq. 1) originally adopted by Baxter-Jones et al. These
results, together with the number of fitted parameters
for each model, are summarized in Table 2.
Indeed, on the basis of the logarithmically transformed multiplicative allometric model (Eq. 3), the
resulting parsimonious solution for the aerobic power
of young male athletes was simpler (Table 3), no longer
requiring the maturity variable MP-PP (thus allowing
the same constant for MP and PP male athletes) or the
tennis * age and gymnastics * age interaction terms.
The predicted V̇O2 max values of the young male athletes,
based on the results in Table 3, are plotted in Fig. 1.
Similarly, the parsimonious solution for the aerobic
power of young female athletes (Table 3) did not require
maturity indicator variables MP-PP or LP-PP, which
were reported originally by Baxter-Jones et al. (2). The
predicted V̇O2 max values of the young female athletes,
based on the results in Table 3, are plotted in Fig. 2.
Because the multiplicative model requires fewer
parameters than the additive model, the AIC criterion
(1) would provide stronger support for the multiplicative model than the maximum log-likelihood criterion
reported above.
Baxter-Jones et al. (2) found that by allowing (fitting)
age2 to be a random effect term in their additive model
(Eq. 1) the increase in maximum log likelihood did not
justify the three additional fitted parameters (i.e., the
covariances constant * age2 and age * age2 and the
variance age2 ) at level 1 or 2. The same conclusion was
reached when age2 was declared as a random effect in
the multilevel analysis on the basis of the logarithmically transformed multiplicative model (Eq. 3).
MODELING GROWTH CHANGES IN STRENGTH AND AEROBIC POWER
967
The predicted strength of the quadriceps and biceps is
plotted in Figs. 3 and 4, respectively.
The parsimonious solutions identified a significant
increase in quadriceps and biceps strength explained
by the developmental component in body size (weight
and height), a sex difference, and an additional contribution identified by age and age2 components that were
different for boys and girls (identified by the significant
age * sex interaction terms).
When the multiplicative model is compared with the
equivalent additive model to predict the strength of
quadriceps and biceps, the AIC criterion (1) provides
evidence similar to the maximum log-likelihood criterion in favor of the multiplicative model.
DISCUSSION
mum log-likelihood criterion to 22,612.4, explaining
more variation in quadriceps strength with one less
predictor. However, for the strength in the biceps,
replacement of the three terms weight, height, and
height2 with the two terms loge(weight) and loge(height)
decreased the maximum log-likelihood criterion to
22,304.8. This reduction is not unreasonable considering that the model (Eq. 2) requires one less predictor
variable than the additive polynomial model (Eq. 1).
Once again, these results support the use of the multiplicative allometric model (Eq. 2) when explaining the
strength measurements of Round et al. (19) compared
with the additive polynomial model (Eq. 1). These
results, together with the number of fitted parameters
for each model, are summarized in Table 4.
The multilevel regression analyses of the quadriceps
and biceps strength measurements, based on the multiplicative allometric model (Eq. 2), are given in Table 5.
Fig. 2. Predicted V̇O2 max for female athletes [swimming (s), tennis
(3), and gymnastics (k)] at different ages. Weight and height were
controlled for by regressing each variable on age and age2.
When choosing multilevel modeling to explain the
developmental changes in aerobic power in young
athletes, Baxter-Jones et al. (2) argued that the use of
ratio standards, such as V̇O2 max per kilogram body
mass, to ‘‘normalize’’ or control for growth may lead to
spurious correlations, incorrect conclusions, and misinterpretation of the data [arguments originally proposed
by Tanner (22)]. To avoid these problems, the authors
argued that by choosing to analyze their data using the
additive polynomial model (Eq. 1) the contribution of
developmental growth and maturation would be separated appropriately from other factors (the effects of
different training methods in this instance).
A limitation of the additive polynomial approach is
that the fitted model is valid only within the range of
observations collected and may give absurd predictions
outside that range. However, when considering the
experimental design effects and other problems associated with scaling for growth and maturation, Nevill et
al. (16) and Nevill and Holder (15) demonstrated the
value of an important class of multiplicative models,
often referred to as allometric or power function models, that provide more plausible solutions within and
beyond the range of observations. For these models, the
concept of a ratio is an integral part of the model form,
and the variables and errors are assumed to be proportional and multiplicative, respectively. In addition,
there is also evidence to suggest that, as children grow
into adults, their leg volume increases in a greater
proportion to their body mass, m1.1 (13). To accommodate the effect of this possible disproportionate increase
in leg muscle on performance variables, such as aerobic
power and quadriceps strength, Nevill (12) suggested
introducing ‘‘height’’ as well as ‘‘mass’’ as a continuous
covariate to explain the developmental changes in such
variables. For these reasons, the multiplicative allometric model (Eq. 2) was chosen, within a multilevel
structure, to explain the developmental changes in
aerobic power and strength from studies 1 and 2,
respectively.
One possible method of analyzing growth data, such as
V̇O2 max and strength, from longitudinal studies might have
been to fit ontogenetic allometric models to each subject’s data separately, allowing the mass exponent to
vary from subject to subject, i.e., between subjects (4, 10).
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Fig. 1. Predicted maximal O2 uptake (V̇O2 max) for male athletes
[swimming (s), tennis (3), gymnastics (k), and soccer (n)] at
different ages at pre- and midpuberty. Weight and height were
controlled for by regressing each variable on age and age2.
968
MODELING GROWTH CHANGES IN STRENGTH AND AEROBIC POWER
Table 4. MLL and AIC together with number of fitted parameters for competing models in study 2
Quadriceps Strength
Biceps Strength
Competing Models
MLL
AIC
n
MLL
AIC
n
Additive polynomial (Eq. 1)
Additive polynomial (Eq. 1)*
Log-transformed multiplicative (Eq. 3)
Total no. of visit occasion
22,649.4
22,612.5
22,612.4
5,322.8
5,249.0
5,246.8
502
12
12
11
22,350.5
22,303.0
22,304.8
4,725.0
4,630.0
4,631.6
521
12
12
11
See Table 2 footnote for definition of abbreviations. AIC 5 (22 3 MLL) 1 (2 3 no. of parameters fitted). * With log-transformed dependent
variable [loge (strength)].
However, this approach requires a two-stage analysis
(within and between groups), in which the overall effect
of the covariates (e.g., height, age, and maturation) and
the correlation of intermediate statistics (the slope and
intercept parameters from the individual allometric
models) can only be partially accommodated at the
second-stage (between-group) analysis. In contrast, by
Fixed Explanatory Variables
Value
Quadriceps
3.915 60.306
0.383 6 0.09434
1.233 6 0.3279
20.150 6 0.0403
0.0533 6 0.0149
20.0067 6 0.0025
Constant (a)
Loge (weight) (k1 )
Loge (height) (k2 )
Sex (girls-boys) (Da)
Age (b)
Age2 (c)
Interaction
Age p sex (Db)
20.0441 6 0.0123
Variance-Covariance Matrix of Random Variables
Constant (a)
Age (b)
Level 1 (within individuals)
Constant (aij )
0.00927 6 0.00066
Level 2 (between individuals)
Constant (ai )
Age (bi )
0.0159 6 0.0032
0.00116 6 0.0008
0.00130 6 0.00038
Biceps
Constant (a)
Loge (weight) (k1 )
Loge (height) (k2 )
Sex (girls-boys) (Da)
Age (b)
Age2 (c)
Interaction
Age p sex (Db)
3.313 6 0.298
0.361 6 0.0919
1.062 60.322
20.171 6 0.0394
.0483 6 0.0138
0.0055 6 0.0023
20.0717 6 0.0097
Level 1 (within individuals)
Constant (aij )
0.00949 6 0.0006
Level 2 (between individuals)
Constant (ai )
Age (bi )
0.0150 6 0.0031
0.00008 6 0.0006
0.00055 6 0.00023
Values are means 6 SE. Strength was expressed in N. Age was
adjusted about origin using age at peak height velocity (PHV). Boys’
result was baseline measure (a), and girls’ result was compared with
boys’ result, indicated by (Da).
Fig. 3. Predicted strength of quadriceps (Quads) for boys (s) and
girls (1) at different ages from peak height velocity (PHV). Weight
and height were controlled for by regressing each variable on age and
age2.
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Table 5. Parsimonious multilevel regression analysis
of log-transformed strength, adjusted for body size
(weight and height), age (centered about PHV), and sex
adopting the multiplicative allometric model (Eq. 2)
within a multilevel structure, we can adopt a one-stage
allometric analysis that is able to incorporate all covariates simultaneously and, in addition, is flexible enough
to allow each subject to have his or her own individual
body mass exponent, i.e., simply by fitting the variable,
log-transformed body mass, as a random component at
level 2. In fact, in the present study, when age and body
mass were allowed to vary between subjects at level 2,
the age term explained the most variation in the
multilevel analyses of V̇O2 max and strength, as described in METHODS and RESULTS.
On the basis of the maximum log-likelihood criterion,
not only did the multiplicative allometric model provide
a superior fit to the data from both studies compared
with the additive polynomial model, but the model also
provided a simpler and more plausible interpretation of
the data. In study 1 the mass exponents for boys and
girls are very close to the anticipated theoretical values, m¯. The significant height exponents for boys and
girls were 0.73 and 0.48, respectively, which might
indicate a greater relative increase in muscle mass in
relation to body mass (12).
The age parameters in Table 3, 0.0366 and 0.061 for
boys and girls, respectively, were highly significant (for
statistical accuracy, age was measured about an origin
of 12 yr). This suggests that the aerobic power of the
MODELING GROWTH CHANGES IN STRENGTH AND AEROBIC POWER
Fig. 4. Predicted strength of biceps for boys (s) and girls (1) at
different ages from PHV. Weight and height were controlled for by
regressing each variable on age and age2.
parameter (0.0533), i.e., 0.0533 2 0.0441 5 0.0092.
This suggests that at the age of PHV the girls’ quadriceps strength is developing at the same rate in proportion to their body size, whereas boys’ quadriceps
strength is developing at a greater rate or disproportionately to their body size. Although not relevant to the
present study, this finding was explained by the introduction of the hormone testosterone into the multilevel
regression analysis (19).
Similarly, in the solution for biceps strength, given in
Table 5, the gap between the boys’ and girls’ age
parameter (20.0717) is even greater, with the girls’ age
term found to be negative (0.0483 2 0.0717 5 20.0234),
i.e., the girls are becoming weaker with increasing
years in the region of PHV, having already adjusted for
the developmental changes in body size.
Hence, by adopting the multiplicative (proportional)
allometric model (Eq. 2) in the multilevel regression
analyses, further valuable insight into the developmental differences between boys’ and girls’ strength and
aerobic power was obtained. In study 1 the highly
significant age parameters from the boys’ and girls’
analyses of aerobic power indicate that, at 12 yr of age,
aerobic power was increasing at a greater proportion to
body size, a finding explained by the young athletes’
elite training status. However, the analysis of boys’
aerobic power identified an additional significant maturation indicator variable, late puberty (LP-PP), but
with no such indicator variable required for the girls’
analysis. A similar finding was observed in study 2,
with the significant age parameter indicating that boys’
strength at PHV (both quadriceps and biceps) was
increasing at a greater rate relative to their body size.
In contrast, the girls’ age parameter was negligible
(quadriceps) or negative (biceps), indicating that the
girls’ strength at PHV was increasing only ‘‘in proportion’’ to their body size and, in the case of the girls’
biceps strength, becoming marginally weaker relative
to their body size. We can only speculate that these
observed developmental differences in the boys’ and
girls’ strength and aerobic power are due to hormonal
factors such as testosterone.
Finally, although not essential, if a model provides
plausible estimates of the dependent variable beyond
the range of observation, the model is more acceptable
on theoretical grounds. For example, in study 1 the
additive polynomial model would predict a female
gymnasts’ V̇O2 max to be .1.0 l/min at birth (Table 5 and
Fig. 3 in Ref. 2). Of course, this prediction is physiologically unsound. However, by using the multiplicative
(proportional) allometric model (Eq. 2), such a prediction would be impossible, since, as weight and height
would tend toward zero, so too would the dependent
variable V̇O2 max.
The work in study 1 was supported by The (UK) Sports Council,
Research Unit and the work in study 2 by Action Research (UK).
Address for reprint requests: A. M. Nevill, Centre for Sport and
Exercise Sciences, School of Human Sciences, Liverpool John Moores
University, Byrom St., Liverpool L3 3AK, UK.
Received 3 July 1997; accepted in final form 3 November 1997.
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male and female young athletes was increasing at a
greater proportion of their body size at 12 yr of age, a
finding explained by their training status. At 12 yr of
age, postpubertal boys had greater levels of aerobic
power than prepubertal boys, indicating that biological
age as well as chronological age was an important
factor in the development of aerobic power. Furthermore, the aerobic power was increasing at a greater
rate in male soccer players than in other male athletes
at 12 yr of age (identified by the 0.0134 soccer * age
interaction term). Also, the aerobic power was increasing at a significantly greater rate at 12 yr of age in
female swimmers than in female gymnasts and tennis
players (indicated by the 20.0251 gymnast * age and
20.0177 tennis * age interaction terms).
The mass exponents in study 2 are a little more
difficult to explain (i.e., 0.38 and 0.36 for quadriceps
and biceps, respectively). However, because the two
measures of strength are specific to a localized muscle
group, using the entire body mass to control for developmental increases in body size is unlikely to be the most
appropriate body size covariate. The height exponents,
1.23 and 1.06 for the strength in the quadriceps and
biceps, respectively, may simply reflect a mechanical
advantage of being taller, as reflected in the approximate linear contribution of the term ‘‘height’’ when
strength is predicted.
The multilevel regression analysis in Table 5 identified a significant increase in quadriceps strength explained by the developmental component in body size
(weight and height) plus a sex difference and an
additional contribution identified by the age and age2
components. Interpreting the sex and age terms in
Table 5 suggests that at PHV the sex differences in the
strength of the quadriceps and biceps would appear to
be equivalent to ,3 yr of developmental growth for
boys. The age contribution in the model for the girls’
quadriceps strength was negligible at the age of PHV,
identified by the value of the age * sex interaction term
(20.0441) that must be subtracted from the boys’ age
969
970
MODELING GROWTH CHANGES IN STRENGTH AND AEROBIC POWER
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