Download Sex- and Age-Specific Reference Curves for Serum CrossLaps and

Statistical analysis: additional information concerning statistical modelling
Age-specific reference intervals were established for all biological parameters, by the
parametric method described by Royston and Wright (24, 25) taking into account two, three
or four moments of the distribution: mean, standard deviation and, if necessary, skewness and
kurtosis. The parametric model from which centile curves were estimated has two basic
components: a parametric density for the measurement of interest Y according to age T, and
an age-related regression model for each of its parameters.
The normal distribution is usually a good starting point for model building because it is
mathematically convenient. Normal-based centile curves are given by C pN  T  u p T ,
where T and  T are the age-specific mean and standard deviation of Y|T, respectively, and u
is the corresponding centile of the standard Gaussian distribution. The Z-score, which
transforms the observation to a derived standard normal distribution is calculated as
Z _ score 
Y  T
T
. The positive skewness and heteroscedasticity of measurements of
interest can be reduced by applying an initial Box-Cox transformation, of which natural
logarithmic transformation is a particular case. If Y is normally distributed, the distribution of
Z is normal. If Z does not follow a standard normal distribution, Manly’s exponential
transformation is applied to generate an age-related skewness curve for Z and, therefore, also
for Y. This gives an exponential normal (EN) model, for which centiles are estimated as
C pEN  T   T
log( 1   T u p )
T
with  T inversely related to skewness.
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We used two different approaches to model the mean T , scale  T and skewness  T curves.
The first was the approach recommended by Royston and Wright, in which fractional
polynomials (FPs) for age are fitted simultaneously to each of the three parameters by
maximum likelihood methods, with powers as numbers chosen from the set {-2, -1, -0.5, 0,
0.5, 1, 2, 3}. FPs provide a wide range of functional forms that are potentially useful in
parametric modelling. However, the values for measurement Y were highly dispersed around
puberty, so we then used natural cubic splines to model mean and scale. FPs were used to
model skewness only when necessary. The splines were defined with two, three or four knots,
giving a set of knots {kmin, k1, …, km, kmax} with (kmin, kmax) at the extreme of the age
distribution. The number of knots was determined visually, to ensure that the curve obtained
was consistent with the clinical evidence and did not display too much wiggling or board
effects. We selected the 33rd and 67th percentiles of age for two knots or the 1st quartile,
median and 3rd quartile for 3 knots. A natural cubic spline for
T  0  1 (T )   21 (T )  ...   m1m (T )
j 
k max  k j
k max  k min
where
T may be written as
 j (T )  (T  k j )3   j (T  kmin )3 ,
and (T  k )   max( 0, T  k ) . A similar equation can be written for  T .
The goodness of fit of the model fit was evaluated by calculating the SDS (Z-score). The
ordered Z-scores were plotted for the graphical checking of normality (QQ-plots). The
goodness-of-fit methods employed were determination of the proportions of observations
above and below the 90th reference intervals, Q-tests exploring the moments of the Z-scores,
and permutation bands, a graphical method for assessing the appropriateness of models (26).
Estimated centile and reference intervals were calculated by introducing the fitted curves of
the mean and standard deviation into the equation for CpN or CpEN, depending on whether it
was necessary to model skewness. When the variable being modelled, Y, was initially Box-
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Cox transformed, centiles curves on the original scale were obtained by applying a back-
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
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transformation to the calculated curves, Coriginalscale  1   C p (t )  with  being the parameter
of the Box-Cox transformation. Sex-specific curves were generated and the effect of puberty
was explored by adding the variable to the reference interval model.
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