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 )
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.
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 0. 0.075 0.04 Indian -0.56 0.3 < 0.0005 Pakistani -0.57 0.0 < 0.0005 Bangladeshi -0.38 0.05 0.063 Caribbean 0.45 0.37 0.89 African 0.034 0.5 0.8 Other -0.344 0.39 0.03 birth weight 0.334 0.035 < 0.0005 zheight intercept on age -0.4 0.085 0.0 Indian 0.46 0.8 < 0.0005 Pakistani 0.83 0.089 < 0.0005 Bangladeshi -0.083 0.88 0.659 Caribbean 0.349 0. < 0.0005 African 0.76 0.097 < 0.0005 Other -0.5 0.3 0.8 birth weight 0.49 0.06 < 0.0005 zbmi slope 3-5 on Indian 0.07 0.043 0.06 Pakistani 0.008 0.04 0.847 Bangladeshi 0.03 0.09 0.63 Caribbean 0.03 0.06 0.697 African -0.05 0.05 0.77 Other 0.0 0.05 0.658 birth weight 0.009 0.06 0.585 zbmi slope 5-7 on Indian 0.45 0.044 < 0.0005 Pakistani 0. 0.06 0.06 Bangladeshi 0.08 0.067 0.787 Caribbean 0.007 0.055 0.898 African 0. 0.048 0.0 Other 0.04 0.04 0.56 birth weight -0.05 0.0 0.95 zheight slope 3-5 on Indian 0.0 0.07 0.675 Pakistani 0.03 0.0 0.5 Bangladeshi 0.087 0.058 0.3 Caribbean 0.038 0.034 0.7 African 0.0 0.03 0.699 Other 0.05 0.03 0.09 birth weight -0.007 0.008 0.346 zheight slope 5-7 on Indian 0 0.03 0.994 Pakistani -0.035 0.0 0.095 Bangladeshi -0.06 0.035 0.08 Caribbean 0.07 0.0 0.4 African -0.046 0.09 0. Other 0.0 0.0 0.305 birth weight -0.007 0.005 0.7 3
zheight 3 on HH income3-0.04 0.0 0.05 zbmi 3 on HH income3-0.09 0.09 0.3 zheight 5 on HH income5 0.0 0.0 0.047 zbmi 5 on HH income5-0.043 0.0 0.033 zheight 7 on HH income7 0.04 0.0 < 0.0005 zbmi 7 on HH income7-0.084 0.0 < 0.0005 Covariances zbmi intercept with zbmi slope 3-5 -0.087 0.058 0.38 zbmi slope 5-7 -0.06 0.0 0.0 zheight intercept -0.00 0.05 0.964 zheight intercept with zheight slope 3-5 -0.0 0.06 0.5 zheight slope 5-7 -0.009 0.004 0.0 zbmi slope 3-5 with zbmi slope 5-7 -0.0 0.0 0.53 zheight slope 3-5 -0.037 0.005 < 0.0005 zheight intercept 0.069 0.0 < 0.0005 zbmi slope 5-7 with zheight slope 5-7 -0.09 0.004 < 0.0005 zheight intercept 0.06 0.007 < 0.0005 zheight slope 3-5 0.04 0.004 0.07 zheight slope 3-5 with zheight slope 5-7 -0.006 0.005 0. zbmi intercept 0.065 0.008 < 0.0005 zheight slope 5-7 with zbmi inercept -0.0 0.005 0.043 zbmi slope 3-5 0.04 0.003 < 0.0005 Intercepts HH income3-0.005 0.008 0.548 HH income5-0.005 0.007 0.4 HH income7-0.00 0.007 0.76 zbmi 3 0.593 0.04 < 0.0005 zheight 3-0.006 0.0 0.778 zbmi 3-5 -0.03 0.0 < 0.0005 zbmi 5-7 -0.06 0.007 < 0.0005 zheight 3-5 0.005 0.006 0.48 zheight 5-7 0.063 0.004 < 0.0005 Variances zheight 3 0.65 0.03 < 0.0005 zheight 5-7 0.035 0.05 0.07 zheight 7 0.08 0.0 0.0 zbmi 3 0.46 0.086 < 0.0005 zbmi 5-7 0.7 0.07 0.05 zbmi 7 0. 0.07 < 0.0005 zbmi intercept 0.93 0. < 0.0005 zheight intercept 0.83 0.04 < 0.0005 zbmi slope 3-5 0.04 0.03 < 0.0005 zbmi slope 5-7 0.06 0.034 0.067 zheight slope 3-5 0.09 0.008 < 0.0005 zheight slope 5-7 0.09 0.007 < 0.0005 Table S. Estimates from the joint trajectory model of zheight and zbmi for 5640 girls in the MCS 4
Coefficient S.E. P Regressions zbmi intercept on Age 0.05 0.08 0.547 Indian -0.46 0.05 < 0.0005 Pakistani -0.96 0.097 0.044 Bangladeshi -0.097 0.38 0.759 Caribbean 0.5 0.4 0.075 African 0.85 0.57 0.069 Other -0.006 0.3 0.957 birth weight 0.399 0.03 < 0.0005 zheight intercept on Age 0.006 0.069 0.93 Indian 0.36 0.34 0.007 Pakistani 0.93 0.09 < 0.0005 Bangladeshi 0.305 0.3 0.09 Caribbean 0.33 0.6 0.064 African 0.968 0.06 < 0.0005 Other 0.4 0. 0.98 birth weight 0.508 0.09 < 0.0005 zbmi slope 3-5 on Indian 0.07 0.039 0.07 Pakistani 0.049 0.04 0. Bangladeshi 0.09 0.36 0.83 Caribbean -0.03 0.048 0.78 African 0.05 0.054 0.348 Other -0.043 0.034 0.97 birth weight -0.07 0.0 0.08 zbmi slope 5-7 on Indian 0.9 0.04 < 0.0005 Pakistani -0.09 0.056 0.099 Bangladeshi -0.0 0.06 0.87 Caribbean 0.076 0.033 0.03 African 0.076 0.05 0.4 Other -0.038 0.037 0.33 birth weight -0.09 0.0 0.055 zheight slope 3-5 on Indian -0.0 0.03 0.48 Pakistani -0.04 0.0 0.046 Bangladeshi -0.048 0.057 0.405 Caribbean 0.059 0.033 0.075 African 0.00 0.03 0.939 Other -0.0 0.07 0.4 birth weight -0.05 0.007 0.047 zheight slope 5-7 on Indian -0.036 0.0 0.065 Pakistani -0.05 0.03 < 0.0005 Bangladeshi -0.077 0.03 < 0.0005 Caribbean -0.08 0.09 0.34 African -0.034 0.043 0.48 Other -0.04 0.08 0.07 birth weight -0.0 0.006 < 0.0005 zheight 3 on HH income3-0.0 0.0 0.3 zbmi 3 on HH income3 0.0 0.0 0.59 5
Coefficient S.E. P zheight 5 on HH income5 0.04 0.009 0.008 zbmi 5 on HH income5-0.038 0.04 0.006 zheight 7 on HH income7 0.09 0.0 0.044 zbmi 7 on HH income7-0.044 0.03 < 0.0005 Covariances zbmi intercept with zbmi slope 3-5 -0.07 0.048 0.38 zbmi slope 5-7 -0.005 0.007 0.45 zheight intercept 0.053 0.0 0.007 zheight intercept with zheight slope 3-5 0.009 0.04 0.53 zheight slope 5-7 -0.06 0.004 < 0.0005 zbmi slope 3-5 wirh zbmi slope 5-7 -0.005 0.0 0.64 zheight slope 3-5 -0.04 0.003 < 0.0005 zheight intercept 0.045 0.008 < 0.0005 zbmi slope 5-7 with zheight slope 5-7 -0.006 0.00 < 0.0005 zheight intercept 0.0 0.006 < 0.0005 zheight slope 3-5 0.03 0.00 < 0.0005 zheight slope 3-5 with zheight slope 5-7 -0.005 0.003 0. zbmi intercept 0.053 0.006 < 0.0005 zheight slope 5-7 with zbmi inercept -0.0 0.003 < 0.0005 zbmi slope 3-5 0.08 0.00 < 0.0005 Intercepts HH income3-0.004 0.007 0.63 HH income5-0.003 0.006 0.595 HH income7-0.00 0.007 0.79 zbmi 3 0.483 0.0 < 0.0005 zheight 3-0.38 0.08 < 0.0005 zbmi 3-5 -0.036 0.008 < 0.0005 zbmi 5-7 -0.049 0.006 < 0.0005 zheight 3-5 0.03 0.005 < 0.0005 zheight 5-7 0.067 0.004 < 0.0005 Variances zheight 3 0.73 0.08 < 0.0005 zheight 5-7 0.08 0.0 0. zheight 7 0.087 0.09 < 0.0005 zbmi 3 0.3 0.065 < 0.0005 zbmi 5-7 0.09 0.05 0.07 zbmi 7 0.8 0.085 < 0.0005 zbmi intercept 0.846 0. < 0.0005 zheight intercept 0.809 0.037 < 0.0005 zbmi slope 3-5 0.07 0.04 < 0.0005 zbmi slope 5-7 0.0 0.03 0.494 zheight slope 3-5 0.0 0.007 0.005 zheight slope 5-7 0.0 0.006 0.057 6
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 5.00-0. (0.56) 0.4 (0.08)*** 0.7 (0.)** 0.03 (0.04) zwaist 5-7.00-0.05 (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 5.00 0.8 (.) 0.46 (0.3)*** 0.6 (0.5)! 0.03 (0.05) zwaist 5-7.00-0.0 (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 7.00 0.3 (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 7.00 0.8 (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;44:45-60 7