The LMS method for constructing normalized growth standards


Cole, T.J.

European Journal of Clinical Nutrition 44(1): 45-60

1990


It is now common practice to express child growth status in the form of SD scores. The LMS method provides a way of obtaining normalized growth centile standards which simplifies this assessment, and which deals quite generally with skewness which may be present in the distribution of the measurement (eg height, weight, circumferences or skinfolds). It assumes that the data can be normalized by using a power transformation, which stretches one tail of the distribution and shrinks the other, removing the skewness. The optimal power to obtain normality is calculated for each of a series of age groups and the trend summarized by a smooth (L) curve. Trends in the mean (M) and coefficient of variation (S) are similarly smoothed. The resulting L, M and S curves contain the information to draw any centile curve, and to convert measurements (even extreme values) into exact SD scores. A table giving approximate standard errors for the smoothed centiles is provided. The method, which is illustrated with US girls' weight data, should prove useful both for the construction and application of growth standards.

European
Journal
of
Clinical
Nutrition
(1990)
44,
45-60
©
Macmillan
Press
Ltd.
1990
Received
1
September
1989:
accepted
22
September
1989.
The
LMS
method
for
constructing
normalized
growth
standards
T.
J.
Cole
MRC
Dunn
Nutrition
Unit,
Downhams
Lane,
Milton
Road,
Cambridge
CB4
IXJ,
UK
It
is
now
common
practice
to
express
child
growth
status
in
the
form
of
SD
scores.
The
LMS
method
provides
a
way
of
obtaining
normalized
growth
centile
standards
which
simplifies
this
assessment,
and
which
deals
quite
generally
with
skewness
which
may
be
present
in
the
distribution
of
the
measurement
(eg
height,
weight,
circumferences
or
skinfolds).
It
assumes
that
the
data
can
be
normalized
by
using
a
power
transformation,
which
stretches
one
tail
of
the
distribution
and
shrinks
the
other,
removing
the
skewness.
The
optimal
power
to
obtain
normality
is
calculated
for
each
of
a
series
of
age
groups
and
the
trend
summarized
by
a
smooth
(L)
curve.
Trends
in
the
mean
(M)
and
coefficient
of
variation
(S)
are
similarly
smoothed.
The
resulting
L,
M
and
S
curves
contain
the
information
to
draw
any
centile
curve,
and
to
convert
measurements
(even
extreme
values)
into
exact
SD
scores.
A
table
giving
approximate
standard
errors
for
the
smoothed
centiles
is
provided.
The
method,
which
is
illustrated
with
US
girls'
weight
data,
should
prove
useful
both
for
the
construction
and
application
of
growth
standards.
Anthropometric
data
are
used
worldwide
to
assess
the
growth
status
of
children,
and
an
important
requirement
for
this
is
a
reference
growth
standard
against
which
to
compare
them.
Britain,
the
USA
and
the
Netherlands
have,
among
other
countries,
produced
growth
standards
suitable
for
their
national
populations
(Tanner,
Whitehouse
&
Taka-
ishi,
1966;
Hamill
et
al.,
1977;
Roede
&
Van
Wieringen,
1985),
and
these
are
widely
used
in
the
form
of
centile
charts.
A
disadvantage
of
such
charts
is
that
when
they
are
applied
to
relatively
deprived
popu-
lations,
a
large
proportion
of
children
are
found
to
lie
below
the
lowest
centile
(usually
the
3rd
or
5th).
This
is
clearly
an
inefficient
way
of
quantifying
the
degree
of
malnutrition
in
the
population,
and
Waterlow
et
al.
(1977)
proposed
that
SD
scores
rather
than
centiles
should
be
used
to
quantify
growth
status.
An
SD
score
(SDS
or
Z
score)
is
a
normally
distributed
variable
with
mean
zero
and
stan-
dard
deviation
(SD)
1.
Because
it
is
normally
distributed,
an
SD
score
can
be
converted
to
a
centile
and
vice
versa,
with
the
use
of
normal
distribution
tables.
Table
1
gives
some
exam-
ples
of
centiles
and
their
corresponding
SD
scores.
Any
normally
distributed
anthropometric
measurement,
for
example
height,
can
be
con-
verted
to
an
SD
score
by
subtracting
from
it
the
mean
height
of
reference
children
of
the
same
age
and
sex,
and
dividing
by
the
corre-
sponding
height
SD.
If
the
mean
and
SD
increase
smoothly
with
age,
then
the
height
corresponding
to
a
particular
SD
score
or
centile
also
increases
smoothly.
The
British
height
standard
of
Tanner
et
al.
(1966)
was
constructed
in
this
way,
by
smoothing
the
mean
and
SDs
of
height
across
age
groups.
Table
1.
Values
for
the
SD
score
(Z)
corresponding
to
some
commonly
used
centiles.
Centile
2
2.5
3
5
10
25
50
SD
score
(Z)
—2.326
—2.054
1.960
—1.881
—1.645
—1.282
—0.674
0
Centile
99
98
97.5
97
95
90
75
50
SD
score
(Z)
2.326
2.054
1.960
1.881
1.645
1.282
0.674
0
46
Ti.
Cole
Unfortunately
anthropometric
data
are
not
often
normally
distributed
instead
they
tend
to
be
skew,
usually
with
the
right
tail
of
the
distribution
longer
than
the
left.
Weight
and
skinfold
thickness
are
obvious
examples.
The
problem
posed
by
Waterlow
et
al.
(1977)
was
to
construct
a
reference
standard
that
would
provide
SD
scores
when
the
reference
data
were
non-normal.
One
result
was
a
normalized
version
of
the
NCHS
weight-for-age
and
height-for-age
standard
(WHO,
1978),
the
derivation
of
which
has
recently
been
described
(Dibley
et
al.,
1987).
Weight
was
assumed
to
be
normal-
ly
distributed
above
and
below
the
median
at
each
age,
but
with
a
larger
SD
above
than
below.
The
two
SDs
were
calculated
at
vari-
ous
ages
from
the
four
upper
and
four
lower
centiles
respectively,
and
97th
and
3rd
centiles
were
obtained
as
the
median
plus
and
minus
1.88
SDs.
The
3rd,
50th
and
97th
centiles
were
then
published
as
smooth
cubic
spline
curves.
The
NCHS
height-for-age
standard
was
sum-
marized
in
the
same
way
except
that
a
single
SD
was
calculated
at
each
age,
ie
height
was
assumed
to
be
normally
distributed.
It
was
then
possible
to
convert
individual
heights
and
weights
to
SD
scores,
by
interpolating
or
extrapolating
between
the
3rd,
50th
and
97th
centiles
for
sex
and
age.
Healy,
Rasbash
&
Yang
(1988)
proposed
a
radically
different
method
for
constructing
growth
standards.
It
involves
first
smoothing
the
selected
centiles
non-parametrically
and
then
fitting
polynomial
curves
to
them.
The
coefficients
of
the
curves
for
each
centile
are
related
to
each
other
in
a
way
which
forces
the
centile
curves
to
be
of
similar
shape.
The
main
disadvantage
of
the
method,
which
is
otherwise
very
flexible
and
powerful,
is
that
it
requires
a
dedicated
computer
program
to
do
the
calculations.
A
different
approach
again
to
dealing
with
non-normal
anthropometry
was
suggested
by
Van't
Hof,
Wit
&
Roede
(1985).
They
pro-
posed
using
a
power
transformation
(Box
&
Cox,
1964)
of
the
data
at
each
age
to
remove
the
skewness,
making
the
data
close
to
a
normal
distribution.
They
also
suggested
that
the
power
used
for
the
transformation
should
change
smoothly
with
age,
allowing
the
amount
of
skewness
in
the
distribution
to
vary
from
one
age
to
another.
If
the
trans-
formed
distribution
is
assumed
to
be
normal,
and
if
the
mean
and
SD
of
the
distribution
are
also
made
to
change
smoothly
with
age,
this
allows
smooth
c.entile
curves
to
be
con-
structed.
The
next
section
explains
how
the
power
transformation
works.
Cole
(1988,
1989a)
has
recently
modified
and
generalized
the
method
proposed
by
Van't
Hof
et
al.
(1985),
calling
it
the
LMS
method
(for
reasons
which
are
explained
be-
low).
The
LMS
method
fits
growth
standards
to
all
forms
of
anthropometry
by
making
the
simple
assumption
of
a
skew
normal
distribu-
tion
in
this
respect
the
method
is
an
improvement
over
that
of
Dibley
et
al.
(1987).
In
addition,
the
way
that
standard
centile
curves
are
fitted
can
be
reversed
in
a
natural
way
to
convert
individual
measurements
to
SD
scores.
The
method
summarizes
each
standard
with
three
smooth
curves,
of
which
one
is
the
50th
centile.
The
other
two
curves
represent
the
power
needed
to
normalize
the
data
and
the
coefficient
of
variation
of
the
distribution
at
each
age.
Taken
together
the
three
curves
allow
any
required
centiles
to
be
drawn.
Fol-
lowing
Cole
(1988)
the
curves
are
called
L
(for
lambda,
the
power
transformation),
M
(mu
the
median)
and
S
(sigma
the
coefficient
of
variation)
respectively
hence
the
method's
name.
In
practice
the
LMS
method
is
very
simple
to
apply.
The
aim
of
this
paper
is
to
describe
its
application
in
detail,
using
two
worked
examples,
in
the
hope
that
it
will
encourage
research
workers
to
use
it
for
generating
smooth
reference
centile
curves
from
anthro-
pometry
data.
Statistical
methods
1.
Skewness
and
power
transformations
The
purpose
of
applying
a
power
transforma-
tion
to
the
data
(weight,
say)
is
to
treat
the
two
tails
of
the
distribution
differently.
Weight
by
itself,
ie
raised
to
the
power
one,
handles
the
two
tails
equivalently,
whereas
the
transformations
log
weight
or
weight'
stretch
one
tail
relative
to
the
other.
Figure
1
illustrates
four
hypothetical
distributions
of
weight-for-age
with
varying
degrees
of
skew-
LMS
method
for
growth
standards
47
EFFECT
OF
SKEWNESS
ON
WEIGHT
DISTRIBUTION
C.V.
12%
POWER
2
POWER
1
POWER
0
POWER
-1
40
70
100
130
180
WEIGHT
-FOR-
AGE
Figure
1.
Hypothetical
distributions
of
weight-for-age
(%)
which
can
be
made
normal
by
a
suitable
power
transformation.
The
seven
marks
on
each
axis
indicate
the
positions
of
the
3rd,
10th,
25th,
50th,
75th,
90th
and
97th
centiles
of
the
distribution.
Left
skewness
is
removed
with
a
power
greater
than
1,
while
right
skewness
requires
a
power
less
than
1.
Anthropometry
is
usually
right
rather
than
left
skewed.
ness,
shown
by
the
differing
tail
lengths.
Weight-for-age
(WA)
is
defined
as
weight
scaled
to
a
median
of
unity,
or
equivalently
100
per
cent,
and
in
this
example
the
coeffi-
cient
of
variation
(CV)
is
chosen
to
be
0.12
or
12
per
cent.
In
each
case,
the
observed
distri-
bution
can
be
converted
to
a
normal
distribu-
tion
by
raising
WA
to
the
appropriate
power.
The
seven
marks
on
the
axis
beneath
each
distribution
represent
the
positions
of
7
con-
ventional
centiles
of
the
distribution
ranging
from
the
3rd
through to
the
97th.
In
the
top
distribution
of
Fig.
1,
the
slight
left
skewness
can
be
removed
by
replotting
the
distribution
against
the
square
of
WA
(power
2).
The
second
example,
where
the
power
is
1,
leaves
WA
unchanged
after
trans-
formation.
The
distribution
here
is
already
normal,
and
needs
no
transformation.
The
third
and
fourth
examples,
where
the
powers
are
0
and
1,
allow
for
increasing
amounts
of
right
skewness
in
the
distribution.
These
cor-
respond
to
log
WA
and
reciprocal
WA
(or
1/
WA)
transformations
respectively.
Note
that
the
log
transformation
used
throughout
is
logs
to
base
e,
or
natural
logs.
The
power
transformation
reduces
the
asymmetry
of
the
seven
centiles.
However,
the
amount
of
asymmetry
present
depends
on
the
CV
of
the
distribution.
In
Fig.
1
the
CV
is
12
per
cent,
but
if
the
CV
were
smaller
(say
4
per
cent,
a
typical
CV
for
height)
then
the
effect
would
be
less
obvious.
Conversely
for
larger
CVs
weight
for
example
reaches
a
CV
of
48
T.J.
Cole
over
20
per
cent
during
puberty
the
effect
would
be
more
striking.
In
general,
anthropometry
tends
to
be
right
skew
rather
than
left
skew,
which
is
why
a
log
transform
is
often
suggested
to
cope
with
it.
However
this
may
not
be
the
best
power
transform
to
use
the
best
power
is
the
Box-
Cox
power
(Box
&
Cox,
1964),
which
is
calculated
from
the
data
to
completely
re-
move
the
skewness
in
the
distribution.
There
are
two
important
things
to
realize
about
the
Box-Cox
power:
unlike
the
examples
in
Fig.
1
it
is
not
usually
a
convenient
whole
number,
and
it
does
not
remain
the
same
over
the
age
range
of
the
data
it
changes
with
age.
2.
Fitting
the
LMS
curves
The
idea
of
weight
raised
to
a
power
is
intuitively
difficult
to
grasp
if
the
power
is
non-integral
and
it
changes
with
age
this
makes
things
worse.
In
particular,
the
mean
and
SD
of
power-transformed
weight
are
in
unfamiliar
units
which
cannot
easily
be
inter-
preted.
The
LMS
method
works
with
power-trans-
formed
weight,
but
converts
the
mean
back
to
original
units
(ie
kg
for
weight)
and
uses
the
CV
rather
than
the
SD
of
the
data.
In
this
way
the
results
for
different
power
transforma-
tions
can
be
compared,
and
the
best
(Box-
Cox)
power
can
be
identified
as
the
one
which
gives
the
smallest
CV
(Cole,
1988).
The
LMS
method
calculates
the
best
power
(L
for
lambda),
mean
(M
for
mu)
and
CV
(S
for
sigma)
in
each
of
a
series
of
age
groups,
and
then
causes
them
to
change
smoothly
with
age
hence
the
name
`LMS
method'.
The
separate
stages
of
the
method
are
described
here,
and
two
worked
examples
are
then
used
to
illustrate
the
method
in
practice.
2.1
Defining
the
age
groups
The
data
first
need
to
be
divided
into
distinct
age
groups.
The
age
range
of
each
group
should
be
as
narrow
as
practicable,
given
the
available
data.
Ideally
it
should
be
1
year
or
less,
but
at
the
same
time
the
numbers
in
each
group
should
be
adequate
at
least
100
for
the
best
results.
For
ages
with
a
high
growth
rate
the
groups
need
to
be
even
narrower
in
age,
down
to
only
1
month
or
so
in
early
life.
2.2
Calculating
L,
M
and
S
The
second
stage
is
to
calculate
the
power
L,
mean
M
and
coefficient
of
variation
S
for
each
age
group
in
turn.
This
is
described
step
by
step
in
Appendix
A,
and
formulae
are
also
given
for
the
standard
errors
of
L,
M
and
S.
2.3
Drawing
the
L,
M
and
S
curves
As
the
next
stage,
plot
the
values
of
L,
M
and
S
obtained
for
each
age
group
against
the
group
mean
ages,
to
give
separate
graphs
for
L,
M
and
S.
Then
draw
smooth
curves
through
each
set
of
points,
to
give
the
L
curve,
the
M
curve
and
the
S
curve
respectively.
The
curves
can
be
fitted
by
formal
statistical
meth-
ods,
for
example
cubic
splines
or
polynomial
equations,
or
they
can
be
drawn
by
eye.
In
practice
the
M
curve,
being
the
median
curve,
is
relatively
complex
in
shape,
so
that
some
care
is
needed
in
smoothing
it.
Parametric
forms
of
curve
may
be
useful,
for
example
the
Jenss-Bayley
in
infancy
(Berkey,
1982)
or
the
Preece-Baines
in
puberty
(Preece
&
Baines,
1978).
The
L
and
S
curves
by
contrast
are
usually
less
well
defined,
particularly
the
L
curve,
so
that
very
simple
shapes
of
curve
may
be
adequate.
The
simplest
possible
curve
is
a
constant
value
for
example
if
there
is
no
obvious
trend
L
or
S
can
be
summarized
as
the
mean
across
all
the
age
groups.
Alternati-
vely
a
simple
curve
like
a
straight
line
or
a
quadratic
curve
can
be
fitted
using
linear
regression
analysis.
Such
simple
curves
will
be
appropriate
if
the
standard
errors
are
large
or
the
age
range
is
narrow,
so
that
the
L
and/or
S
values
show
no
obvious
trends.
Conversely,
if
the
standard
errors
are
small
and
the
age
range
is
wide,
the
L
and
S
curves
may
need
a
more
complex
shape
of
curve.
The
second
example
described
in
the
Results
is
such
a
case,
where
cubic
splines
are
used.
Whichever
method
is
used
to
smooth
the
curves,
it
is
important
to
take
into
account
the
relative
precision
of
the
estimates
at
each
age.
This
is
done
by
using
a
form
of
analysis,
for
example
weighted
regression,
which
weights
each
estimate
according
to
the
reci-
procal
of
the
square
of
its
standard
error.
This
ensures
that
age
groups
with
large
numbers
or
little
variation
are
given
a
greater
weight-
ing
in
the
analysis.
LMS
method
for
growth
standards
49
2.4
Obtaining
the
centile
curves
After
the
three
curves
have
been
fitted,
they
can
be
used
to
calculate
the
required
centiles.
The
50th
centile,
or
median,
is
exactly
equiva-
lent
to
the
M
curve;
centiles
other
than
the
50th
involve
the
L
and
S
curves
in
addition
to
the
M
curve.
First
decide
which
particular
centiles
are
required,
and
obtain
from
Table
1
the
SD
scores
corresponding
to
them.
Also
choose
at
what
ages
the
centiles
are
to
be
plotted;
to
ensure
that
the
curves
are
smooth,
these
ages
need
to
be
closer
together
than
the
age
group
means.
Then,
for
each
age
and
SD
score
(Z)
in
turn,
read
values
for
L,
M
and
S
from
the
curves
and
substitute
them
into
the
equation
C
=
M(1
+
LSZ)
1
/
1-
(1)
to
give
the
required
centiles
C.
This
gives
a
table
of
centiles
at
each
age.
Then
plot
the
values
for
each
centile
at
each
age,
and
join
up
the
points
for
each
centile
to
give
a
set
of
smooth
centile
curves.
Appendix
B
gives
an
approximate
formula
for
the
standard
error
of
C,
and
Table
2
gives
the
calculated
standard
errors
for
different
combinations
of
L,
S
and
N
the
sample
size.
3.
Using
the
LMS
growth
standard
The
prime
purpose
of
the
LMS
method
is
to
fit
centile
standards.
However,
it
is
equally
important
to
use
the
standards
to
assess
indi-
vidual
subjects,
by
expressing
measured
anth-
ropometry
in
centile
or
SD
score
terms.
With
most
existing
standards
this
can
be
done
only
Table
2.
Approximate
standard
errors
for
centiles,
expressed
as
percentages,
tabulated
for
various
values
of
the
Box-Cox
power
L.
the
coefficient
of
variation
S,
and
the
'notional'
sample
size
N.
See
Appendix
B
for
details.
N
L
S
3
10
25
Centile
50
75
90
97
100
1
0.05
1.2
0.8
0.6
0.5
0.6
0.7
1.0
0.10
2.6
1.6
1.1
1.0
1.1
1.4
1.9
0.20
6.9
3.7
2.3
2.0
2.2
2.7
3.3
0
0.05
1.1
0.7
0.6
0.5
0.6
0.7
1.1
0.10
2.2
1.5
1.1
1.0
1.1
1.5
2.2
0.20
4.4
3.0
2.3
2.0
2.3
3.0
4.4
-
1
0.05
1.0
0.7
0.6
0.5
0.6
0.8
1.2
0.10
1.9
1.4
1.1
1.0
1.1
1.6
2.6
0.20
3.3
2.7
2.2
2.0
2.3
3.7
6.9
400
1
0.05
0.6
0.4
0.3
0.3
0.3
0.4
0.5
0.10
1.3
0.8
0.6
0.5
0.6
0.7
0.9
0.20
3.5
1.9
1.2
1.0
1.1
1.3
1.7
0
0.05
0.5
0.4
0.3
0.3
0.3
0.4
0.5
0.10
1.1
0.7
0.6
0.5
0.6
0.7
1.1
0.20
2.2
1.5
1.1
1.0
1.1
1.5
2.2
-
1
0.05
0.5
0.4
0.3
0.3
0.3
0.4
0.6
0.10
0.9
0.7
0.6
0.5
0.6
0.8
1.3
0.20
1.7
1.3
1.1
1.0
1.2
1.9
3.5
1600
1
0.05
0.3
0.2
0.1
0.1
0.1
0.2
0.2
0.10
0.7
0.4
0.3
0.2
0.3
0.3
0.5
0.20
1.7
0.9
0.6
0.5
0.6
0.7
0.8
0
0.05
0.3
0.2
0.1
0.1
0.1
0.2
0.3
0.10
0.5
0.4
0.3
0.2
0.3
0.4
0.5
0.20
1.1
0.8
0.6
0.5
0.6
0.8
1.1
-1
0.05
0.2 0.2
0.1
0.1
0.1
0.2
0.3
0.10
0.5
0.3 0.3
0.2
0.3
0.4
0.7
0.20
0.8
0.7
0.6
0.5
0.6
0.9
1.7
50
T.J.
Cole
approximately,
by
observing
the
centiles
be-
tween
which
the
measurement
falls,
and
in
the
cases
where
it
is
below
the
lowest
centile
or
above
the
highest
the
assessment
is
particu-
larly
crude.
Standards
constructed
by
the
LMS
method
are
different
they
convert
measurements
on
individual
children
into
exact
SD
scores
(and
thus
exact
centiles),
and
this
holds
true
even
for
measurements
lying
outside
the
con-
ventional
range
of
centile
curves.
To
calculate
the
SD
score
for
an
individual
child,
the
appropriate
values
of
L,
M
and
S
for
the
child's
age
and
sex
need
to
be
read
off
the
three
curves.
Because
of
this,
it
is
impor-
tant
for
standards
calculated
by
the
LMS
method
to
tabulate
the
values
for
L
and
S
at
each
age
as
well
as
the
conventional
seven
centiles
(M
is
the
same
as
the
median,
and
so
is
tabulated
anyway).
If
Anth
is
the
measured
anthropometry
for
the
child,
then
the
SD
score
is
calculated
as:
(Anth/M
)[-
1
Z
=
LS
(2)
The
SD
score
(Z)
can
be
converted
to
an
exact
centile
using
normal
distribution
tables,
or
a
rough
estimate
can
be
found
using
Table
1.
4.
Extensions
of
the
LMS
method
This
section
discusses
some
extra
details
of
the
method
which
are
not
important
for
a
first
reading.
The
fitting
of
the
LMS
method
as
described
treats
all
the
subjects
in
each
age
group
as
being
of
the
same
age.
In
practice
this
is
rarely
the
case.
There
is
usually
a
spread
of
ages,
and
Healy
(1962)
gave
a
formula
to
adjust
for
this
in
calculating
the
standard
deviation.
An
equivalent
adjustment
can
be
made
using
the
LMS
method,
and
it
is
just
one
example
of
a
whole
set
of
adjustments
that
can
be
made.
In
Stage
2
of
the
method,
where
the
SDs
of
weight,
log
weight
and
reciprocal
weight
are
calculated
(Appendix
A),
it
is
quite
permissi-
ble
to
use
either
linear
regression
or
analysis
of
covariance
to
adjust
for
relevant
covar-
iates.
The
required
SDs
are
then
the
residual
standard
deviations
for
each
of
the
three
forms
of
weight
after
making
the
adjustment.
To
adjust
for
age
changes
within
each
age
group,
the
linear
regression
of
weight
(and
log
weight
and
reciprocal
weight)
on
age
is
calcu-
lated
for
each
age
group
separately.
The
resi-
dual
SDs
obtained
are
slightly
smaller
than
the
crude
SDs,
reflecting
the
presence
of
an
age
trend
in
weight.
Other
covariates,
for
example
birth
order,
parental
height
or
geo-
graphic
location,
could
be
adjusted
for
in
the
same
way
if
required
(Cole,
1988).
The
two
key
equations
of
the
method
are
equations
(1)
and
(2),
which
allow
for
quite
general
power
transformations.
Now,
it
has
already
been
pointed
out
that
the
log
trans-
form
is
equivalent
to
a
power
of
zero.
In
practice
this
causes
difficulties,
as
both
equa-
tions
(1)
and
(2)
involve
dividing
by
L,
which
is
impossible
if
L
=
0.
If
L
is
ever
close
to
zero,
then
modified
forms
of
the
two
equa-
tions
need
to
be
used:
C
=
M[exp(SZ)
L/2(SZ)
2
]
(3)
Z
=
log
(Anth/M)/S
(4)
They
can
be
used
for
values
of
L
between
0.01
and
+
0.01
including
zero,
and
are
accurate
enough
to
ensure
smooth
centiles
throughout.
Data
To
illustrate
the
method
two
datasets
are
used
a
set
of
just
10
weights
on
which
Stage
2
of
the
method
is
described
in
detail
(see
Table
3),
and
10
649
weights
of
girls
from
the
three
US
health
examination
surveys
HES2,
HES3
and
HANESI
(Hamill
et
a/.,
1977).
HES2
and
HES3
have
data
on
about
550
children
for
each
year
of
age
from
6
+
to
17
+
years,
while
HANES1
has
about
200
per
year
from
age
1
+
to
20
+
.
Results
The
complicated
part
of
the
LMS
method
is
Stage
2,
as
described
in
Appendix
A,
which
derives
L,
M
and
S
for
each
age
group.
Before
describing
the
method
in
full
on
the
US
girls'
data,
Stage
2
is
explained
with
the
aid
of
the
worked
example
set
out
in
Table
3.
1.
Calculating
L,
M
and
S
Ten
weights
are
given
in
the
first
column
of
Table
3,
sorted
into
order
and
showing
a
twofold
range
from
31.0
to
62.2
kg.
The
sec-
ond
column
gives
the
corresponding
natural
LMS
method
for
growth
standards
51
Table
3.
A
worked
example
of
the
LMS
method
based
on
ten
weights.
Tra,,sfi'rm:
Power.
none
logarithmic
0
reciprocal
-/
L
-
0.64
31.0
3.434
0.03226
34.3
3.535
0.02915
36.6
3.600
0.02732
38.8
3.658
0.02577
40.9
3.711
0.02445
43.2
3.766
0.02315
45.7
3.822
0.02188
48.9
3.890
0.02045
53.4
3.978
0.01873
62.2
4.130
0.01608
Mean
43.50
3.7525
0.023924
SD
9.42
0.2108
0.004916
Mean:
M
43.50
42.63
41.80
42.09
CV:
S
0.2211
0.2108
0.2095
0.2090
Type:
arithmetic
geometric
harmonic
generalized
The
weights
in
the
first
column
are
transformed
to
logarithms
and
reciprocals,
and
the
means
and
SDs
are
calculated.
These
are
then
converted
back
to
means
and
CVs
on
the
original
scale
(see
text
for
details).
From
the
CV:S
the
Box-Cox
power
L
and
the
generalized
CVs
are
calculated,
and
the
means
give
the
generalized
mean
M.
The
three
quantities
L,
M
and
S
summarize
the
weight
distribution.
logarithms
of
weight
and
column
3
the
reci-
procals
of
weight
(ie
1/weight).
Beneath
each
column
is
the
mean
and
SD
of
the
ten
values.
The
transformations
used
in
columns
1
to
3
each
correspond
to
a
particular
power
trans-
form;
untransformed
weight
is
weight',
log
weight
can
be
thought
of
as
weight
°
,
and
reciproca'
weight
is
weight
-I
.
The
means
are
also
shown
converted
back
to
original
(kg)
units
(labelled
M
in
Table
3)
by
reversing
the
transformation
applied
to
the
original
measurements.
That
is,
the
log
mean
is
antilogged
and
the
reciprocal
mean
is
converted
to
its
reciprocal.
Note
that
as
the
power
gets
smaller
(ie
moves
from
+
1
to
-
1)
the
mean
M
of
weight
also
gets
smaller.
This
is
generally
true,
because
to
compensate
for
the
right
skewness
the
decreasing
power
makes
the
left
tail
progressively
longer
than
the
right
(see
Fig.
1),
and
so
the
mean
is
pulled
to
the
left.
The
SDs
are
converted
to
CVs
by
dividing
the
SD
in
the
first
column
by
the
geometric
mean,
multiplying
the
SD
in
the
third
column
by
the
geometric
mean,
and
leaving
the
SD
in
the
second
column
unchanged.
The
resulting
CVs
are
called
S
in
Table
3,
where
0.2211
=
9.42/42.63
and
0.2095
=
0.004916
x
42.63.
Like
the
Ms
the
Ss
are
very
similar
to
each
other,
and
they
also
show
a
trend
downwards
across
the
Table.
The
value
of
S
is
at
its
smallest
when
there
is
no
skewness
in
the
distribution,
so
that
the
reciprocal
transform
is
less
skew
than
the
log,
and
much
less
skew
than
untransformed
weight.
To
find
the
'best'
power
L,
which
makes
S
as
small
as
possible,
the
arithmetic,
geometric
and
harmonic
S
values
are
first
put
into
equation
(Al)
to
obtain
the
two
ratios
A
and
B,
and
then
these
are
substituted
into
equation
(A2).
Equation
(A3)
gives
the
standard
error
of
L.
The
values
of
A
and
B
are
0.054
and
0.042
respectively.
L
is
then
-0.054/(2
X
0.042)
or
-0.64,
and
its
standard
error
is
1/V(10
X
0.042)
or
1.55.
This
is
obviously
enormous
due
to
the
tiny
sample
size.
So
the
best
(Box-Cox)
power
is
-
0.64,
which
is
nearer
to
the
reciprocal
than
the
log
transform.
Substituting
the
geometric
CV
S,
and
the
ratio
A
into
equation
(A4)
now
gives
S,
the
CV
for
power
transform
L.
The
result,
S
=
0.2090,
is
smaller
than
S„,
S,
and
S
h
as
expected.
Its
standard
error
is
0.0487,
as
defined
in
equation
(A5).
52
T.J.
Cole
The
last
part
of
Stage
2
is
to
obtain
M
from
M
a
,
M
g
and
M
h
,
interpolating
to
power
L
using
equation
(A6).
This
makes
M
42.09
kg,
with
a
standard
error
of
2.78
kg
from
equa-
tion
(A7).
So
the
distribution
of
weight
in
this
small
sample
is
summarized
by
the
three
quantities
L
=
-0.64,
M
=
42.1
and
S
=
0.209.
There
is
little
point
in
recording
more
significant
figures
than
these
as
the
values
are
subsequently
smoothed.
2.
US
girls'
weights
The
first
stage
in
the
analysis
of
the
US
girls'
weights
is
to
divide
them
into
age
groups
of
1
year
width.
The
HES2/HES3
data
are
suffi-
ciently
numerous
to
consider
narrower
limits,
but
1
year
is
more
suitable
for
the
HANES1
data.
Stage
2
generates
L,
M
and
S
values
for
each
age
group
as
described
above,
and
these
are
given
in
Tables
4
and
5.
The
SDs
for
each
age
group
are
calculated
here
adjusted
for
age,
as
described
under
'Extensions
of
the
LMS
method',
so
that
the
values
of
S
in
Tables
4
and
5
reflect
the
'instantaneous'
rather
than
the
grouped
variation
in
weight.
Stage
3
then
plots
the
L,
M
and
S
values
against
the
mean
age
for
each
group,
and
smooth
curves
are
drawn
through
the
points.
Figure
2
shows
the
plot
for
the
power
L,
with
the
smooth
L
curve
drawn
through
the
points.
In
this
and
the
next
two
Figures
each
point
is
shown
bracketed
with
its
standard
error,
which
corresponds
to
a
68
per
cent
confidence
interval.
The
points
for
HES2
and
HES3
are
shown
as
asterisks,
while
HANESI,
with
its
smaller
numbers,
appears
as
plus
signs.
The
smooth
curve
is
a
weighted
cubic
spline
curve
fitted
using
the
Bathspline
pack-
age
of
Silverman
(1985),
and
plotted
with
its
95
per
cent
confidence
interval.
Figure
2
shows
that
the
Box-Cox
power
is
greater
than
zero
only
in
the
1
+
year
group,
and
at
most
subsequent
ages
it
is
nearer
to
-
1.
This
means
that
for
most
of
childhood
a
transformation
more
extreme
than
logarith-
mic
is
needed
to
remove
the
skewness
in
weight.
There
is
a
rise
in
L
during
puberty,
indicating
a
reduction
in
skewness,
but
the
skewness
increases
again
as
adulthood
ap-
proaches.
The
95
per
cent
confidence
interval
(ie,
plus/minus
2
standard
errors)
around
the
L
curve
is
similar
in
size
to
the
plus/minus
1
standard
error
for
each
age
group
showing
that
the
smoothing
process
roughly
halves
the
variability.
Figure
3
gives
the
corresponding
estimates
of
M
for
each
age
group
in
the
HES
and
HANES
data,
plus
the
fitted
weighted
cubic
spline
M
curve
and
its
confidence
interval.
This
is
the
same
as
the
median
curve
for
the
growth
chart,
and
so
is
of
a
familiar
shape.
The
confidence
interval
is
much
narrower
than
for
Figure
2.
Figure
4
gives
the
CVs
for
each
group,
and
the
fitted
S
curve
with
its
confidence
interval.
Table
4.
Numbers,
values
and
standard
errors
for
the
generalized
power
L,
mean
M
and
coefficient
of
variation
S
for
girls'
weight
by
year
of
age.
Age
(years)
N
Power
Mean
weight
(kg)
Coefficient
of
variation
6+
538
-1.02
(0.19)
21.0
(0.1)
0.154
(0.005)
7+
608
-1.03
(0.19)
23.5
(0.2)
0.160
(0.005)
8+
611
-0.90
(0.18)
26.7
(0.2)
0.181
(0.005)
9+
584
-
0.82
(0.17)
30.3
(0.2)
0.197
(0.006)
10
+
578
-
0.68
(0.15)
34.0
(0.3)
0.211
(0.006)
11
+
558
-
0.64
(0.16)
38.5
(0.3)
0.214
(0.007)
12
+
549
-
0.13
(0.16)
45.6
(0.4)
0.209
(0.007)
13
+
581
-
0.45
(0.16)
49.3
(0.4)
0.198
(0.006)
14
+
585
-
0.62
(0.16)
52.5
(0.4)
0.177
(0.005)
15
+
504
-
0.97
(0.18)
54.4
(0.4)
0.181
(0.006)
16
+
535
-
1.40
(0.17)
55.5
(0.4)
0.169
(0.005)
17
+
412
-
1.17
(0.18)
55.9
(0.5)
0.168
(0.006)
Data
obtained
from
the
US
health
examination
surveys
HES2
and
HES3.
LMS
method
for
growth
standards
53
Table
5.
Numbers,
values
and
standard
errors
for
the
generalised
power
L,
mean
M
and
coefficient
of
variation
S
for
girls'
weight
by
year
of
age.
Age
(years)
N
Power
Mean
weight
(kg)
Coefficient
of
variation
1
+
262
0.54
(0.34)
10.8
(0.1)
0.110
(0.005)
2
+
264
-
0.71
(0.30)
12.8
(0.1)
0.116
(0.005)
3
+
289
-
0.80
(0.39)
14.7
(0.1)
0.120
(0.005)
4
+
280
-
1.27
(0.36)
16.7
(0.1)
0.119
(0.005)
5
+
311
-
1.35
(0.23)
19.1
(0.2)
0.152
(0.006)
6
+
177
-
0.28
(0.34)
21.2
(0.2)
0.153
(0.008)
7
+
168
-
1.42
(0.40)
23.7
(0.2)
0.137
(0.008)
8
+
151
-
1.08
(0.33)
26.4
(0.4)
0.192
(0.011)
9
+
171
-
1.04
(0.33)
30.5
(0.4)
0.189
(0.011)
10
+
197
-
1.05
(0.27)
33.0
(0.5)
0.192
(0.010)
11
+
165
-
0.50
(0.31)
39.7
(0.7)
0.222
(0.013)
12
+
175
-
0.40
(0.28)
45.4
(0.8)
0.233
(0.013)
13
+
197
-
0.90
(0.26)
49.4
(0.7)
0.207
(0.011)
14
+
184
-
0.79
(0.29)
52.6
(0.8)
0.196
(0.011)
15
+
172
-
1.60
(0.29)
54.8
(0.7)
0.174
(0.010)
16
+
174
-
1.33
(0.29)
55.2
(0.8)
0.195
(0.011)
17
+
155
-
1.17
(0.32)
56.9
(0.9)
0.189
(0.011)
18
+
144
-
0.88
(0.38)
56.7
(0.8)
0.166
(0.010)
19
+
134
-
1.68
(0.30)
57.7
(0.8)
0.165
(0.010)
20
+
236
-
0.98
(0.28)
58.0
(0.7)
0.177
(0.008)
Data
obtained
from
the
US
health
examination
survey
HANES].
1
0.5
Age
in
years
8
10
12
14
16
18
20
-1.5
-2
.
Figure
2.
HES/HANES
Gi
r
l
s
we
i
g
h
t
-
L
curve.
C
a
l
cu
l
a
t
e
d
values
for
the
Box-Cox
power
L
of
girls'
weight
by
age
in
the
US
h
ea
l
t
h
examination
surveys
HES2/HES3
(*)
and
HANES!
(+).
Error
b
ars
are
p
lus
and
minus
one
standard
error,
corresponding
to
a
68%
confidence
interval.
The
smooth
curve
through
th
e
p
oints
is
fitted
by
the
Bathspline
package
of
Silverman
(1985)
and
is
drawn
with
its
95%
confidence
interval.
54
T.J.
Cole
55
50
45
0)
40
35
30
co
a)
2
25
20
15
10
2
4
6
8
10
12
14
16
18
20
Age
in
years
Figure
3.
HES/HANES
Girls
weight
M
curve.
Calculated
values
for
the
generalized
mean
M
of
girls'
weight
(kg)
by
age
in
the
US
health
examination
surveys
HES2/HES3
(*)
and
HANESI
(+).
Error
bars
and
curve
drawn
as
in
Figure
2.
0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
2
4
6
8
10
12
14
16
18
20
Age
in
years
Figure
4.
HES/HANES
Girls
weight
S
curve.
Calculated
values
for
the
generalized
coefficient
of
variation
S
of
girls'
weight
by
age
in
the
US
health
examination
surveys
HES2/HES3
(*)
and
HANES!
(+).
Error
bars
and
curve
drawn
as
in
Figs.
2
and
3.
Co
e
ff
ic
ie
n
t
o
f
v
a
r
ia
t
io
n
LMS
method
for
growth
standards
55
95
90
75
10
0
2
4
6
8
10
12
14
16
18
Age
in
years
Figure
5.
HES/HANES
Girls
weight
undies.
US
girls'
weight
centiles
as
constructed
from
the
curves
of
L,
M
and
S
by
age
(solid
lines),
and
as
published
by
Hamill
et
al.
(1977)
(dotted
lines).
80
70
60
50
40
30
20
10
This
has
a
characteristic
shape
with
its
puber-
tal
peak,
and
shows
the
CV
increasing
from
about
11
per
cent
(ie
0.11)
at
age
2
to
a
peak
of
21
per
cent
at
age
11
or
12,
and
then
dropping
back
to
17
per
cent
in
adulthood.
It
is
now
possible
using
the
L,
M
and
S
curves
to
read
off
smoothed
values
for
L,
M
and
S
at
a
series
of
ages
through
childhood.
Particular
centiles
C
of
weight
at
each
age
can
then
be
obtained
from
equation
(1),
where
Z
is
the
SD
score
corresponding
to
the
required
centile
obtained
from
Table
1.
As
an
example
take
HANESI
girls
aged
11
+
years.
Smoothed
values
for
L,
M
and
S
at
age
11.5
are
obtained
from
Figs.
2
to
4
as
0.53,
39.03
and
0.2102
respectively.
These
values,
being
smoothed,
are
not
quite
the
same
as
the
calculated
values
in
Tables
4
and
5,
but
they
are
similar.
To
obtain
the
10th
centile
at
this
age,
substitute
L,
M
and
S
with
Z
=
1.282
into
equation
(1);
the
result
is
30.34
kg.
Note
the
need
to
work
to
greater
accuracy
at
this
stage,
to
ensure
that
the
plotted
centiles
are
smooth.
Appendix
B
discusses
how
to
obtain
an
approximate
standard
error
for
the
centile.
The
formula
given
there
involves
a
notional
'smoothed'
sample
size,
which
is
the
actual
sample
size
scaled
up
by
factor
of
2
or
3
to
re-
flea
the
inclusion
of
adjacent
age
groups
by
smoothing.
The
actual
sample
size
at
age
11+
is
723
(Tables
4
and
5).
which
is
thus
equival-
ent
to
1600
or
more
after
smoothing.
Table
2
gives
a
standard
error
of
0.8
per
cent
for
this
case
(N
1600,
L
between
0
and
—1.
and
S
0.2
on
the
10th
centile),
equivalent
to
0.2
kg.
Repeating
the
calculations
for
seven
cen-
tiles
from
the
5th
to
the
95th
and
for
ages
from
1
to
18
years
generates
the
complete
growth
chart
for
girls'
weight
shown
in
Fig.
5.
Also
shown,
with
dotted
lines,
is
the
publ-
ished
NCHS
standard,
which
was
based
on
the
same
data
but
calculated
in
a
quite
differ-
ent
way
(Hamill
et
al.,
1977).
The
agreement
between
the
two
standards
is
on
the
whole
reasonable,
although
the
two
median
curves
are
slightly
different
in
shape.
Discussion
The
LMS
method
provides
a
way
of
fitting
reference
standards
which
should
prove
use-
ful
in
many
areas
of
application.
Including
a
smoothly
varying
Box-Cox
power
provides
four
main
benefits
over
current
fitting
meth-
ods:
(a)
it
copes
with
non-normality
in
the
data,
which
is
a
perennial
problem
in
anthro-
pometry;
(b)
the
estimates
for
each
centile
have
smaller
standard
errors,
being
based
on
56
T.J.
Cole
the
transformed
mean
and
SD
(Healy,
1974;
Appendix
B);
(c)
the
method
emphasizes
that
the
shape
of
the
standard
depends
on
just
three
quantities
the
power,
the
mean
and
the
CV
which
focuses
attention
on
their
changing
pattern
during
childhood;
in
this
respect
the
shapes
of
the
three
curves
are
interesting
in
their
own
right;
and
(d)
mea-
surements
on
individuals
can
be
expressed
relative
to
the
standard
in
the
form
of
SD
scores,
which
simplifies
the
analysis
and
pre-
sentation
of
growth
survey
data.
A
common
question
when
constructing
centile
charts
is:
"How
many
subjects
do
I
need?"
The
approximate
standard
errors
gi-
ven
in
Table
2
provide
a
direct
answer
to
this
question,
by
indicating
the
precision
of
indi-
vidual
centiles
according
to
the
sample
size
and
the
distribution
of
the
measurement.
Thus
height,
with
a
CV
of
less
than
0.05,
has
a
precision
of
about
1
per
cent
on
the
3rd
and
97th
centiles
when
the
sample
size
is
100.
Conversely
weight,
with
a
CV
of
0.2
or
more
during
puberty,
needs
1600
subjects
to
achieve
this
precision
on
the
3rd
centile
(as-
suming
L
is
1)
on
the
97th
centile
it
needs
coniderably
more.
The
sample
size
quoted
is
the
'notional'
sample
size,
which
is
2
to
3
times
greater
than
the
actual
number
of
subjects
seen
in
each
age
group
(see
Appendix
B).
Even
so,
it
indicates
that
500
is
the
actual
number
required
per
year
to
achieve
standard
errors
of
1
per
cent
on
the
extreme
weight
centiles.
This
is
the
size
of
the
HES
surveys
(Table
4).
The
significance
of
the
shape
of
the
LMS
curves
in
Figs.
2
to
4
is
emphasized
by
com-
paring
them
with
the
LMS
curves
of
publ-
ished
growth
standards
(Cole,
1989a).
Although
the
technique
as
described
here
is
designed
to
calculate
LMS
curves
from
raw
data,
it
can
be
modified
to
work
with
publ-
ished
centiles
(Cole,
1988).
As
an
example,
Fig.
6
shows
the
L,
M
and
S
curves
for
the
girls'
weight
standards
of
Tanner
et
al.
(1966).
The
curves
are
similar
in
general
shape
to
those
for
the
HES/HANES
girls
(Figs.
2
to
4),
with
pubertal
peaks
on
the
L
and
S
curves.
Notice
too
that
the
L
curve
peaks
at
a
later
age
than
the
S
curve.
These
features
are
common
to
several
standards
(Cole,
1989a),
even
though
the
populations
on
which
they
are
based
are
quite
different.
To
explain
why
the
peaks
in
the
L
and
S
curves
occur,
Fig.
7
shows
the
LMS-fitted
centiles
of
the
HES-HANES
girls
(Fig.
5)
replotted
as
percentages
of
the
median.
The
form
of
each
centile
is
given
by
equation
(1)
divided
by
the
median
M,
which
emphasizes
that
the
shape
depends
only
on
the
L
and
S
curves.
The
changing
skewness
of
the
centiles
and
the
increasing
variability
up
to
puberty
are
very
obvious,
but
in
addition
it
can
be
seen
that
the
peak
variability
on
the
95th
centile
(at
11
years)
occurs
1
year
earlier
than
for
the
5th
centile
(at
12
years).
This
is
because
the
95th
centile
during
puberty
is
advanced
(ie
pushed
up)
by
girls
with
an
early
growth
spurt,
while
the
5th
centile
1
year
later
is
pulled
down
by
late
developers.
In
statistical
terms
the
distribution
is
skew
to
the
right,
ie
towards
the
95th
centile,
until
the
95th
centile
has
passed
its
peak
of
variabi-
lity,
and
then
the
skewness
shifts
towards
the
5th
centile.
Once
past
the
5th
centile
peak,
the
skewness
returns
to
its
previous
value.
This
is
catered
for
by
the
L
curve
rising
and
then
fall-
ing
again
once
the
S
curve
has
passed
its
peak.
The
two
versions
of
the
NCHS
girls'
weight
standard
in
Fig.
5
are
obviously
broadly
similar,
but
they
differ
in
detail.
One
reason
for
this
may
be
that
the
original
data
were
obtained
by
a
stratified
sampling
procedure,
and
the
NCHS
analysis
took
this
into
ac-
count
by
weighting
the
subjects
appropriate-
ly.
In
the
present
analysis
all
the
subjects
were
given
equal
weightings,
and
this
will
inevita-
bly
have
affected
the
results
to
some
extent.
One
strength
of
the
LMS
method
is
its
ability
to
express
growth
data
in
SD
score
terms
this
is
best
seen
with
an
example.
Consider
a
girl
aged
10.5
years
weighing
23
kg,
who
a
year
later
weighs
27
kg.
These
weights
are
both
well
below
the
5th
centile
of
the
NCHS
standard.
The
LMS
values
from
Figs.
2
to
4
are
0.72,
33.99
and
0.2053
at
10.5
years,
and
—0.53,
.39.03
and
0.2102
at
11.5
years.
So
the
distribution
becomes
slightly
less
skew
over
the
year,
while
the
CV
increases
slightly
and
the
mean
increases
by
about
5
kg.
Putting
these
values
into
equation
(2),
the
weights
expressed
as
SD
scores
are
2.20
at
age
10.5
and
1.94
a
year
later
a
shift
up
10
Age
(Years)
15
20
L
curve
10
Age
(Years)
IS
20
0.6
0.4
0.2
0.0
—0.2
—0.4
—0.6
—0.8
Bo
x
Cox
Pow
er
LMS
method
for
growth
standards
57
0.22
0.20
0.18
O
0.16
0
w
0.14
0.12
0.10
10
Age
(Years)
15
20
60
M
curve
01
50
_Nc
40
rn
47;
30
,
20
0
10
Figure
6.
L,
M
and
S
curves
derived
from
the
UK
girls'
weight
standard
of
Tanner,
Whitehouse
&
Takaishi
(1966).
The
pubertal
peak
on
the
S
curve
occurs
earlier
than
the
pubertal
peak
on
the
L
curve.
from
the
1.4th
to
the
2.7th
centile.
This
repre-
sents
an
improvement
in
growth
status
which
comparison
with
the
conventional
5th
or
3rd
centile
would
fail
to
detect.
In
situations
where
the
calculated
SD
score
is
outside
the
range
3
to
+
3
(say),
it
needs
to
be
treated
with
some
caution.
The
LMS
method's
assumption
of
a
normal
distribu-
tion
is
based
almost
entirely
on
data
within
this
range,
so
that
values
falling
outside
it
involve
extrapolation
beyond
the
reference
data.
Although
the
assumption
of
a
normal
distribution
is
a
reasonable
one,
there
are
no
data
in
the
extreme
tails
to
judge
whether
or
not
it is
appropriate.
The
LMS
method
has
been
shown
here
applied
only
to
weight
data,
but
it
works
equally
well
with
other
anthropometry
in-
cluding
circumferences
and
skinfolds.
Publ-
ished
examples
include
four
skinfolds
from
the
Nymegen
Growth
Study
(Van't
Hof
et
al.,
1985),
body
mass
index
(weight/length
2
),
head
and
arm
circumferences
and
triceps
plus
subscapular
skinfolds
from
the
Cambridge
Infant
Growth
Study
(Cole
et
al.,
1989),
and
triceps
skinfolds
from
the
Nigerian
Growth
58
T.J.
Cole
150
140
n"-'
130
E
0
C
a.
90
a)
a,
70
60
0
Age
in
years
Figure
7.
HEA/HANES
Girls
weight
centiles.
Centiles
of
the
HES-HANES
LMS-fitted
girl's
weight
standard
(Fig.
5)
expressed
as
percentages
of
the
median.
The
peak
on
the
95th
centile
occurs
earlier
than
the
trough
on
the
5th
centile.
Study
(Macfarlane,
1988).
It
has
also
been
applied
to
height
and
length
(Cole,
1988;
Cole
et
al.,
1989;
Macfarlane,
1988),
but
with
the
small
CV
for
height
very
large
sample
sizes
are
needed
to
fit
the
L
curve
at
all
reliably.
So
it
is
perfectly
possible
to
obtain
SD
scores
for
several
measurements
simultan-
eously
on
the
same
child
or
group
of
children.
In
addition,
if
there
is
a
whole
series
of
such
measurements
available.
all
the
SD
scores
can
be
displayed
on
one
graph,
with
each
series
joined
up
across
ages,
giving
a
multi-dimen-
sional
view
of
growth
over
the
period.
Cole
et
al.
(1989)
use
this
technique
to
illustrate
7-
dimensional
growth
in
children
over
a
period
of
a
year
or
more.
The
examples
here
show
that
the
LMS
method
is
not
only
a
simple
procedure
to
carry
out,
it
is
also
flexible
and
powerful
for
fitting
growth
standards
to
anthropometric
data.
If
the
method
is
to
be
used
in
the
future,
it
is
important
to
exploit
fully
its
ability
to
express
anthropometry
as
SD
scores.
This
requires
that
published
standards
include
the
L
and
S
curves
along
with
the
fitted
centiles.
A
start
to
this
has
been
made
by
Cole
(1989a,
b)
in
providing
tables
of
LMS
values
by
age
and
sex
for
the
NCHS
height
and
weight
standards.
Acknowledgements-1
should
like
to
thank
the
anony-
mous
referee,
plus
Peter
Davies,
Dot
Jackson,
Nick
Masce-Taylor,
Andrew
Prentice,
Ann
Prentice,
Stanley
Ulijaszek
and
Lawrence
Weaver,
for
their
comments
on.
previous
drafts
of
the
paper.
I
also
thank
the
NCHS
for
making
available
their
anthropometry
data.
120
110
100
80
2
4
6
8
10
12
14
16
18
References
Berkey
CS
(1982):
Comparison
of
two
longitudinal
growth
models
for
preschool
children.
Biometrics
38,
221-234.
Box
GEP
&
Cox
DR
(1964):
An
analysis
of
transforma-
tions.
J.
R.
Statist.
Soc.
B.
26,
211-252.
Cole
TJ
(1988):
Fitting
smoothed
centile
curves
to
refer-
ence
data
(with
Discussion).
J.
R.
Statist.
Soc.
A
151,
385-418.
Cole
Ti
(1989a):
Using
the
LMS
method
to
measure
skewness
in
the
NCHS
and
Dutch
national
height
standards.
Ann.
Hunt.
Biol.
16,
407-419.
Cole
TJ
(1989b):
The
British,
American
NCHS
and
Dutch
weight
standards compared
using
the
LMS
method.
Am.
J.
Hum.
Biol.
1,
397-408.
Cole
TJ,
Paul
AA,
Eccles
M
&
Whitehead
RG
(1989):
The
use
of
a
multiple
growth
standard
to
highlight
the
effects
of
diet
and
infection
on
growth.
In
Auxo-
logy
'88:
Perspectives
in
the
science
of
growth
and
development
ed
JM
Tanner
pp.
91-100.
London:
Smith-Gordon.
Dibley
MJ,
Goldsby
JB,
Staehling
NW
&
Trowbridge
FL
(1987):
Development
of
normalized
curves
for
the
international
growth
reference:
historical
and
techni-
cal
considerations.
Am.
J.
Clin.
Nutr.
46,
736-748.
Hamill
PVV,
Drizd
TA,
Johnson
CL,
Reed
RB
&
Roche
AF
(1977):
NCHS
growth
curves
for
children
birth-18
years.
Washington
DC:
National
Center
for
Health
Statistics.
(Vital
and
health
statistics.
Series
11:
#
165
[DHEW
publication
#78-1650].).
Healy
MJR
(1962):
The
effect
of
age-grouping
on
the
distribution
of
a
measurement
affected
by
growth.
Am.
J.
Phys.
Anthropo.
20,
49-50.
Healy
MJR
(1974):
Notes
on
the
statistics
of
growth
standards.
Ann.
Hum.
Biol.
1,
41-46.
Healy
MJR,
Rasbash
J
&
Yang
M
(1988).
Distribution-
free
estimation
of
age-related
centiles.
Ann.
Hum.
Biol.
15,
17-22.
Macfarlane
SBJ
(1988):
In
Discussion
of
Cole
(1988):
J.
R.
Statist.
Soc.
A.
151,
413-414.
Preece
MA
&
Baines
MJ
(1978):
A
new
family
of
mathematical
models
describing
the
human
growth
curve.
Ann.
Hum.
Biol.
5,
1-24.
Roede
MJ
&
Van
Wieringen
JC
(1985):
Growth
dia-
grams
1980.
Netherlands
third
nation-wide
survey.
71jdschrift
voor
Sociale
Gezondheidszorg
63,
Suppl.,
1-34.
Silverman
BW
(1985):
Some
aspects
of
the
spline
smoothing
approach
to
non-parametric
regression
LMS
method
for
growth
standards
59
curve
fitting.
J.
R.
Statist.
Soc.
B.
47,
1
-
52.
Tanner
JM,
Whitehouse
RH
&
Takaishi
M
(1966):
Standards
from
birth
to
maturity
for
height,
weight,
height
velocity,
and
weight
velocity.
British
children
1965-1.
Arch.
Dis.
Childh.
41,
454-471.
Van't
Hof
M,
Wit
JM
&
Roede
MJ
(1985):
A
method
to
construct
age
references
for
skewed
skinfold
data,
using
Box-Cox
transformations
to
normality.
Hum.
Biol.
57,
131-139.
Waterlow
JC,
Buzina
R,
Keller
W,
Lane
JM,
Nichaman
MZ
&
Tanner
JM
(1977):
The
presentation
and
use
of
height
and
weight
data
for
comparing
nutritional
status
of
groups
of
children
under
the
age
of
10
years.
Bull.
WHO
55,
489
-498.
World
Health
Organization
(1978):
A
growth
chart
for
international
use
in
maternal
and
child
care:
guidelines
for
primary
health
care
personnel.
Geneva:
WHO.
Appendix
A
This
gives
a
step
by
step
account
of
how
L,
M
and
S
are
calculated
for
each
age
group,
including
formulae
for
the
standard
errors.
Table
2
gives
a
small
worked
example
of
the
calculations
involved.
(1)
Calculate
the
mean
and
SD
of
the
natural
logarithms
of
the
measurements.
The
antilog
of
the
mean
is
the
geometric
mean
of
the
measurement
(called
M
g
here
for
convenience).
By
analogy
the
SD
is
called
the
'geometric'
CV,
or
Sg.
(2)
Calculate
the
mean
and
SD
of
the
original measurements.
This
is
the
arithmetic
mean
M.
of
the
measurement.
Divide
the
SD
by
the
geometric
mean
M
g
to
give
the
'arithmetic'
CV,
S..
(3)
Calculate
the
mean
and
SD
of
the
reciprocals
of
the
measurements.
The
reciprocal
of
the
mean
is
the
harmonic
mean
M
b
of
the
measurement.
Multiply
the
SD
by
the
geometric
mean
M
g
to
give
the
'harmonic'
CV,
S
h
.
(4)
The
values
for
S.,
S
g
and
S
h
should
be
very
similar.
Now
substitute
them
into
the
equations:
A
=
log(S./S
h
)
B
=
log(S.S
h
/S
g
2
)
where
A
and
B
should
be
very
small.
The
estimate
of
the
Box-Cox
power
L
is
then
given
by
L
=
—A/(2B)
(A2)
and
its
standard
error
is
1/
\
/(nB)
(A3)
where
n
is
the
number
of
measurements
in
the
age
group
(Cole,
1988).
(5)
Next
calculate
the
generalized
coefficient
of
variation
S
at
this
value
of
L,
S
=
S
g
exp(AL/4).
(A4).
This
interpolates
between
the
three
CVs
to
find
the
minimum
value,
so
that
S
is
slightly
smaller
than
S.,
S
g
and
Sh.
The
approximate
standard
error
of
S
is
then
S,./([S
2
+
0.5]/n).
(A5).
(6)
Finally,
the
generalized
mean
M
for
power
L
is
obtained
by
interpolating
between
M.,
M
g
and
M
b
to
give
M
=
M
g
+
(M
a
M
h
)L/2
+
(M
a
2M
g
+
M
h
)L
2
/
2
,
(A6).
with
standard
error
MS/
,/n.
(A7).
Note
that
the
standard
errors
of
L,
M
and
S
are
all
inversely
proportional
to
the
square
root
of
n,
the
number
of
children
in
the
age
group.
Appendix
B
Equation
(B1)
gives
an
approximate
formula
for
the
variance
of
a
centile,
expressed
as
a
fraction
of
the
centile's
value.
The
standard
error
is
the
square
root
of
this
quantity,
which
can
be
expressed
as
a
(Al)
60
T.J.
Cole
percentage
by
multiplying
by
100.
(S
2
+
0.5)Z
2
.
S
2
(LSZ
3)
2
Z
4
N
1
+
(1
+
LSZ)
2
36B(1
+
LSZ)
2
In
addition
to
Z,
which
defines
the
centile
in
question,
the
formula
depends
on
L
the
Box-Cox
power,
S
the
coefficient
of
variation,
B
the
quantity
described
in
Appendix
A
(equation
Al),
and
N
the
'notional'
sample
size.
Normally
the
values
for
L
and
S
in
(B1)
are
smoothed
prior
to
substitution,
so
that
their
precision
is
rather
greater
than
implied
by
the
sample
size
for
the
age
group.
This
can
be
adjusted
for
by
scaling
the
sample
size
up
by
a
suitable
factor
to
give
the
'notional'
sample
size
N.
the
exact
figure
to
use
depends
on
how
the
smoothing
is
done,
and
whether
the
particular
age
group
is
near
to
the
centre
or
the
extremes
of
the
age
range
studied.
For
this
reason
a
precise
value
cannot
be
given,
but
using
a
scaling
factor
of
2
or
3
ought
to
be
safe,
if
conservative.
Table
2
provides
values
for
the
percentage
standard
errors
of
specified
centiles
as
calculated
from
(B1),
tabulated
by
L,
S
and
N.
(B
tends
to
increase
as
S
2
increases,
so
appropriate
values
for
B
have
been
assumed.)
Standard
errors
for
intermediate
values
of
L,
S
or
N
can
be
obtained
by
interpolation.
The
standard
errors
increase
with
increasing
S,
increasing
SD
score
and
decreasing
N.
When
L
is
zero,
the
errors
for
equivalent
centiles
in
the
two
tails
of
the
distribution
are
the
same,
whereas
when
L
is
1
or
—1
the
errors
are
different.
When
L
is
negative
the
higher
centiles
are
more
variable
than
the
lower
centiles,
while
for
L
positive
the
opposite
is
true.
The
extreme
case
is
for
L
=
—1
and
S
=
0.2,
where
the
97th
centile
for
N
=
100
has
an
error
of
6.9
per
cent,
whereas
the
3rd
centile
error
is
only
3.3
per
cent.
It
is
counter-intuitive
for
the
error
on
the
3rd
centile
to
be
larger
than
for
the
97th
centile
when
L
=
1.
it
is
true
that
the
absolute
error
is
the
same
in
the
two
tails,
but
when
expressed
as
a
percentage
(as
Table
2
is),
the
error
is
larger
on
the
3rd
than
on
the
97th
centile.
(B1)