The multivariate piecewise linear growth model for ZHeight and zbmi can be expressed as:

Transcription

1 Bi-directional relationships between body mass index and height from three to seven years of age: an analysis of children in the United Kingdom Millennium Cohort Study Supplementary material The multivariate piecewise linear growth model for ZHeight and zbmi can be expressed as: zheight ij = b h 0j + bh j t ij + bh j t ij + Σbh 3kj x kij + eh ij zbmi ij = b w 0j + bw j t ij + bw j t ij + bw 3kj x kij + ew ij where zheight ij and zbmi ij are the observed values of the repeated measures of zheight and zbmi for individual i at time j, b h 0j and b w 0j are the latent intercept factors, and t ij and t ij are the latent slope factors (for before and after the knot at 5 years, respectively) representing a linear change in zheight or zbmi when b h j and b w j (pre-knot) and b h j and b w j (post-knot) change by one unit. The timevarying covariates are represented by x kij with coefficients b h 3kj and b w 3kj. The individual and time specific measurement error is represented by e h ij and e w ij, assumed to be normally distributed with variance σ h e and σ w e.the measurement error terms e h ij and e w ij are assumed to be independent given the covariance between the level- error terms u h j and u w j, defined below. The intercept and slopes are treated as random variables, which can be expressed as: b h 0j = bh 0 + Σbh 0k z kj + u h 0j b h j = bh + Σ bh k z kj + uh j b h j = bh + Σ bh k z kj + uh 3j b w 0j = bw 0 + Σbw 0k z kj + u w 0j b w j = bw + Σ bw k z kj + uw j b w j = bw + Σ bw k z kj + uw 3j j ~ N(0, σh e j ~ N(0, σw u j ~ N(0, σw e u h j ~ N(0, σh u ); e h ); u w ); e w )

2 The intercepts b h 0j and b w 0j have fixed components b h 0 and b w 0, respectively, adjusted for time-invariant covariates z kj with coefficients b h k and b w k, and individual (random) components u h 0j and u w 0j. Similarly, the slopes b h j, b h j b w j and b w j have fixed components b h, b h, b w and b w, respectively, adjusted for time-invariant covariates z kj with coefficients b h k, b h k, b w k and b w k, and individual (random) components u h j, u h j, uw j, and u w j. The random components are not estimated directly. Instead they are assumed to be normally distributed with variance σ h u and σ w u and allowed to covary.

3 Table S. Estimates from the joint trajectory model of zheight and zbmi for 577 boys in the MCS Coefficient S.E. P Regressions zbmi intercept on age Indian < Pakistani < Bangladeshi Caribbean African Other birth weight < zheight intercept on age Indian < Pakistani < Bangladeshi Caribbean < African < Other birth weight < zbmi slope 3-5 on Indian Pakistani Bangladeshi Caribbean African Other birth weight zbmi slope 5-7 on Indian < Pakistani Bangladeshi Caribbean African Other birth weight zheight slope 3-5 on Indian Pakistani Bangladeshi Caribbean African Other birth weight zheight slope 5-7 on Indian Pakistani Bangladeshi Caribbean African Other birth weight

4 zheight 3 on HH income zbmi 3 on HH income zheight 5 on HH income zbmi 5 on HH income zheight 7 on HH income < zbmi 7 on HH income < Covariances zbmi intercept with zbmi slope zbmi slope zheight intercept zheight intercept with zheight slope zheight slope zbmi slope 3-5 with zbmi slope zheight slope < zheight intercept < zbmi slope 5-7 with zheight slope < zheight intercept < zheight slope zheight slope 3-5 with zheight slope zbmi intercept < zheight slope 5-7 with zbmi inercept zbmi slope < Intercepts HH income HH income HH income zbmi < zheight zbmi < zbmi < zheight zheight < Variances zheight < zheight zheight zbmi < zbmi zbmi < zbmi intercept < zheight intercept < zbmi slope < zbmi slope zheight slope < zheight slope < Table S. Estimates from the joint trajectory model of zheight and zbmi for 5640 girls in the MCS 4

5 Coefficient S.E. P Regressions zbmi intercept on Age Indian < Pakistani Bangladeshi Caribbean African Other birth weight < zheight intercept on Age Indian Pakistani < Bangladeshi Caribbean African < Other birth weight < zbmi slope 3-5 on Indian Pakistani Bangladeshi Caribbean African Other birth weight zbmi slope 5-7 on Indian < Pakistani Bangladeshi Caribbean African Other birth weight zheight slope 3-5 on Indian Pakistani Bangladeshi Caribbean African Other birth weight zheight slope 5-7 on Indian Pakistani < Bangladeshi < Caribbean African Other birth weight < zheight 3 on HH income zbmi 3 on HH income

6 Coefficient S.E. P zheight 5 on HH income zbmi 5 on HH income zheight 7 on HH income zbmi 7 on HH income < Covariances zbmi intercept with zbmi slope zbmi slope zheight intercept zheight intercept with zheight slope zheight slope < zbmi slope 3-5 wirh zbmi slope zheight slope < zheight intercept < zbmi slope 5-7 with zheight slope < zheight intercept < zheight slope < zheight slope 3-5 with zheight slope zbmi intercept < zheight slope 5-7 with zbmi inercept < zbmi slope < Intercepts HH income HH income HH income zbmi < zheight < zbmi < zbmi < zheight < zheight < Variances zheight < zheight zheight < zbmi < zbmi zbmi < zbmi intercept < zheight intercept < zbmi slope < zbmi slope zheight slope zheight slope

7 Table S3. Correlations (standard errors) between baseline zheight and zwaist and changes over time for 575 boys in the MCS zwaist 5 zwaist 5-7 zheight 3 zheight 3-5 zheight 5-7 zwaist (0.56) 0.4 (0.08)*** 0.7 (0.)** 0.03 (0.04) zwaist (0.03) 0.6 (0.) 0. (0.)!! P < 0.0, * P < 0.05, ** P < 0.0, *** P < 0.00 zheight and zwaist expressed in standard deviate scores, derived using LMS standardization [] Table S4. Correlations (standard errors) between baseline zheight and zwaist and changes over time for 566 girls in the MCS zwaist 5 zwaist 5-7 zheight 3 zheight 3-5 zheight 5-7 zwaist (.) 0.46 (0.3)*** 0.6 (0.5)! 0.03 (0.05) zwaist (0.04) 0.37 (0.3) 0.55 (0.50)! P < 0.0, * P < 0.05, ** P < 0.0, *** P < 0.00 zheight and zwaist expressed in standard deviate scores, derived using LMS standardization [] Table S5. Correlations (standard errors) between % Fat at age seven and zheight trajectories for 5803 boys in the MCS %Fat 7 zheight 3 zheight 3-5 zheight 5-7 %Fat (0.0)*** 0.8 (0.0)** 0.05 (0.03)!! P < 0.0, * P < 0.05, ** P < 0.0, *** P < 0.00 zheight expressed in standard deviate scores, derived using LMS standardization [] Table S6. Correlations (standard errors) between % Fat at age seven and zheight trajectories for 5700 girls in the MCS %Fat 7 zheight 3 zheight 3-5 zheight 5-7 %Fat (0.0)*** 0.35 (0.7)* 0.7 (0.) * P < 0.05, ** P < 0.0, *** P < 0.00 zheight expressed in standard deviate scores, derived using LMS standardization [] Reference. Cole TJ. The LMS method for constructing normalized growth standards. Eur. J. Clin. Nutr. 990;44: