SUPPLEMENTAL MATERIAL
Article title: Bias related to body mass index in pediatric echocardiographic Z scores
Journal name: Pediatric Cardiology
Authors: Frederic Dallaire, Jean-Luc Bigras, Milan Prsa, and Nagib Dahdah
Affiliation and email address of the corresponding author : Division of Pediatric Cardiology,
Department of Pediatrics, Faculty of Medicine, University of Sherbrooke, email: frederic.a.dallaire@usherbrooke.ca.
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SUPPLEMENTAL METHODS
Modelization of the mean - bivariate models
Bivariate allometric models (y = axb) were first computed using four body size measurements:
height, weight, BSA, and LBM. Allometric models were chosen based on the work by Sluysmans and Colan [1]. Other types of bivariate models were also explored to optimize the fit of the
regressions. These included a linear model (y = ax + b), a gamma function model [y = axb
exp(−cx)], a polynomial model with the square root of the dependent variable (y = ax + bx + c),
and a regular polynomial model up to the third order (y = ax3 + bx2 + cx + d) [2-4]. Only subjects
with a BMI within the 2nd and 98th percentiles were used to derive the new Z score equations.
Modelization of residual values
For each model described above, residual values (observed diameter minus the mean diameter
predicted from the model) were calculated. Heteroscedasticity was present in most regression
models: that is, the variance of the residual values increased with body size. To account for this,
the slope of increasing standard deviation (SD) was calculated by computing a linear regression
of the absolute value of the residual values against the independent variable used for normalization (height was used when more than one independent variable was present in the model). In
preliminary analyses, more complex modelization of the SD of the residual values did not significantly improve goodness-of-fit. Residual values are assumed to be normally distributed (see
below). Consequently, the absolute values of the residuals values will adopt a half-normal distribution. Since the mean of a half standard normal distribution is √(2/π), the predicted mean of the
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absolute residuals multiplied by √(π/2) is an estimate of the SD of the residuals according to the
independent variable [5].
Assessment of the validity of newly computed Z scores
Z scores are used to evaluate if a dimension is within the limits (usually +/- two SDs from the
mean) expected for a given individual. Because the aim is to obtain a Z score that is independent
of body size, there should not be any residual association with the dependent variables used for
the normalization. Furthermore, one should also ascertain that a strong association is not present
with other body size measurements. To examine this, the Z scores of all the models were plotted
against age, weight, height, and BSA to visually assess the absence of residual association and to
appreciate the symmetry of the distribution of the residual values around the mean. Linear regression was also used to detect statistically significant slopes.
Finally, because Z scores need to have a normal distribution in order to validly estimate percentiles, each Z score estimation was evaluated for departure from a normal distribution with a mean
of 0 and a SD of 1 by visual assessment (normal probability plot) and by using the AndersonDarling test. This was done for all subjects, and for three subgroups corresponding to the lowest,
middle and highest tertiles of the independent variable. Obvious outliers with significant influence on regression coefficients were deleted from the database and the Z scores were recomputed.
SUPPLEMENTAL TABLES
Table 3: Z score equations for ascending aortic diameter according to the independent variables used for normalization.
Description
Equation for the predicted mean
Equation for the standard deviation
Polynomial models with square root transformation (males)
Weight
AscAomean = (−0,220 × wt) + (4.906 × √wt) + (−0.347)
AscAoSD = 0.0161 × wt + 1.303
Height
AscAomean = (−8.364 × ht 2 ) + (69.469 × ht) + (−78.841 × √ht) + 33.221
AscAoSD = 0.830 × ht + 0.731
LBM
AscAomean = (−0.00221 × LBM 2 ) + (0.0205 × LBM) + (3.747 × √LBM) + 3.300
AscAoSD = 0.0219 × LBM + 1.263
BSA
AoVmean = (−4.591 × BSA) + (27.966 × √BSA) + (−4.033)
AscAoSD = 0.716 × BSA + 1.057
Polynomial models with square root transformation (females)
Weight
AscAomean = (−0.142 × wt) + (3.986 × √wt) + 1.705
AscAoSD = 0.0207 × wt + 0.996
Height
AscAomean = (12.588 × ht 2 ) + (−59.142 × ht) + (89.827 × √ht) + (−28.118)
AscAoSD = 1.030 × ht + 0.302
LBM
AscAomean = (−0.0000158 × LBM 2 ) + (−0.0861 × LBM) + (3.718 × √LBM) + 4.0436
AscAoSD = 0.0269 × LBM + 0.999
BSA
AscAomean = (−0.0593 × BSA) + (18.933 × √BSA) + (−0.133)
AscAoSD = 0.958 × BSA + 0.354
Males
AscAomean = (−10.202 × ht 2 ) + (74.192 × ht) + (−86.798 × √ht) + (0.717 × √wt) + 35.442
AscAoSD = 0.836 × ht + 0.707
Females
AscAomean = (9.160 × ht 2 ) + (−44.664 × ht) + (69.321 × √ht) + (0.653 × √wt) + (−21.262)
AscAoSD = 0.949 × ht + 4.170
Multivariable stepwise model
AscAo = ascending aortic diameter, BSA = body surface area, ht = height; LBM = lean body mass, wt = weight.
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Table 4: Z score equations for Sinus of Valsalva diameter according to the independent variables used for normalization.
Description
Equation for the predicted mean
Equation for the standard deviation
Polynomial models with square root transformation (males)
Weight
SoVmean = (−0.239 × wt) + (5.439 × √wt) + 0.00511
SoVSD = 0.0249 × wt + 1.232
Height
SoVmean = (4.906 × ht 2 ) + (−20.711 × ht) + (49.276 × √ht) + (−15.549)
SoVSD = 1.133 × ht + 0.540
LBM
SoVmean = (0.00153 × LBM 2 ) + (−0.484 × LBM) + (6.988 × √LBM) + 0.0352
SoVSD = 0.0282 × LBM + 1.316
BSA
SoVmean = (−5.390 × BSA) + (31.987 × √BSA) + (−4.597)
SoVSD = 0.992 × BSA + 0.958
Polynomial models with square root transformation (females)
Weight
SoVmean = (−0.210 × wt) + (4.908 × √wt) + 0.987
SoVSD = 0.0289 × wt + 1.00104
Height
SoVmean = (18.0511 × ht 2 ) + (−98.937 × ht) + (149.277 × √ht) + (−51.260)
SoVSD = 1.288 × ht + 0.212
LBM
SoVmean = (0.00285 × LBM 2 ) + (0.476 × LBM) + (6.205 × √LBM) + 1.748
SoVSD = 0.0414 × LBM + 0.984
BSA
SoVmean = (−3.331 × BSA) + (26.751 × √BSA) + (−2.430)
SoVSD = 1.197 × BSA + 0.646
SoVmean = (3.100 × ht 2 ) + (−15.734 × ht) + (40.991 × √ht) + (0.684 × √wt) + (−13.197)
SoVSD = 1.141 × ht + 0.513
SoVmean = (14.478 × ht 2 ) + (−83.380 × ht) + (127.421 × √ht) + (0.637 × √wt) + (−43.906)
SoVSD = 1.316 × ht + 0.172
Multivariable stepwise model
Males
Females
BSA = body surface area, ht = height; LBM = lean body mass, wt = weight, SoV = Sinus of Valsalva diameter.
SUPPLEMENTAL REFERENCES
1.
Sluysmans T and Colan SD,(2005) Theoretical and empirical derivation of cardiovascular
allometric relationships in children. J Appl Physiol 99(2): 445-457.
2.
Dallaire F and Dahdah N,(2011) New Equations and a Critical Appraisal of Coronary
Artery Z Scores in Healthy Children. J Am Soc Echocardiogr 24(1): 60-74.
3.
Mawad W, Drolet C, Dahdah N, and Dallaire F,(2013) A Review and Critique of the
Statistical Methods used to Generate Reference Values in Pediatric Echocardiography. J
Am Soc Echocardiogr 26(1): 29-37.
4.
Nevill AM, Bate S, and Holder RL,(2005) Modeling physiological and anthropometric
variables known to vary with body size and other confounding variables. Am J Phys
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5.
Altman DG,(1993) Construction of age-related reference centiles using absolute
residuals. Stat Med 12(10): 917-924.