Multi-level modelling of longitudinal child growth data from the Birth-to-Twenty Cohort: a comparison of growth models

Figures & data

Table 1. Summary of some studies that have used structural and non-structural models to describe physical growth*.

	Reference	Study population	Model(s) used	Period of growth	Variable
1	Black and Krishnakumar (1999)	US (92% African-American)	Quadratic	0–6 years	Height, weight
2	Ehrenkranz et al. (1999)	US	Piece-wise quadratic	0–6 months	Weight
3	Martin-Gonzalez et al. (2012)	Spanish and Siberian	Kouchi	Birth–6 years	Height
4	Grimm et al. (2011)	US	Linear; Quadratic; Latent basis model, Preece-Baines	3–19 years	Height
5	Johnson et al. (2012b)	Indian	Berkey-Reed 1st order; Count; Quadratic	0–15 months	Weight
6	Botton et al. (2008)	French	Adapted Jenns-Bayley model (with a quadratic term)	0–12 years	Weight, height
7	Simondon et al. (1992)	Congolese (African)	Berkey-Reed 1st order; Count; Karlberg; Berkey-Reed 2nd order; Kouchi	0–13 months	Weight
8	Tilling et al. (2011)	Belarus	Fractional Polynomial	0–6.5 years	Weight, height
9	Flexeder et al. (2012)	German	Berkey-Reed 1st order	0–2 years	Weight, height
10	Steele (2008)	British	3rd Order Polynomial (cubic)	11–14 years	Height

* Publications found using Pubmed and Google-Scholar search.

Search terms used: mathematical growth curve, child growth models, human growth model.

Table 2. Goodness-of-fit statistics from fitting growth models to weight measurements (in kg) from birth to 10 years.

		Models fitted
		Berkey-Reed 1	Count	Jenss-Bayley	Adapted Jenss-Bayley	2nd Order Polynomial	3rd Order Polynomial
Variance components^a	Random Intercept	0.160 (0.072, 0.355)	0.117 (0.035,0.389)	0.008 (−0.173, 0.188)	0.008 (−0.17, 0.181)	0.005 (0.0001, 0.166)	0.249 (0.142, 0.436)
	Random Slope	0.0013 (0.001, 0.002)	0.0013 (0.001, 0.002)	0.0013 (0.001, 0.002)	0.0013 (0.001, 0.0014)	0.001 (0.0010, 0.002)	0.0013 (0.001. 0.002)
	Covariance	−0.002 (−0.006, 0.001)	−0.003 (−0.007, 0.001)	−0.002 (−0.006, 0.003)	−0.002 (−0.006, 0.003)	−0.002 (−0.007, 0.002)	−0.005 (−0.01, −0.0006)
	Random Residuals	1.98 (1.86, 2.10)	2.35 (2.02, 2.50)	3.06 (2.87, 3.25)	3.05 (2.87, 3.23)	3.051 (2.87, 3.24)	1.95 (1.83, 2.07)
Information criterion	AIC	10 758	11 127	11 704	11 693	11 684	10 758
	BIC	10 811	11 175	11 739	11 724	11 708	10 811
Absolute residuals^b,^c	Median	0.62 (0.28, 1.20)	0.72 (0.33, 1.35)	0.99 (0.44, 1.89)	0.99 (0.44, 1.84)	0.99 (0.44, 1.84)	0.86 (0.42, 1.37)
	Maximum	8.46	8.80	8.53	8.66	8.66	7.16
Median absolute residuals^c,d	Birth (n = 365)	0.32 (0.15, 0.54)	0.54 (0.27, 0.83)	2.03 (1.70, 2.37)	1.90 (1.59, 2.26)	1.91 (1.59, 2.25)	0.99 (0.67, 1.30)
	3 m (n = 126)	1.14 (0.76, 1.57)	0.61 (0.30, 1.05)	0.58 (0.31, 0.90)	0.58 (0.34, 0.99)	0.58 (0.34, 0.99)	0.72 (0.47, 1.14)
	6 m (n = 87)	0.55 (0.31, 0.89)	1.18 (0.55, 1.66)	1.34 (0.77, 1.96)	1.41 (0.84, 2.03)	1.42 (0.84, 2.03)	1.47 (0.82, 1.82)
	Year 1 (n = 270)	0.85 (0.34, 1.44)	1.19 (0.55, 1.82)	1.74 (1.00, 2.53)	1.77 (1.02, 2.55)	1.77 (1.02, 2.56)	1.20 (0.55, 1.72)
	Year 2(n = 264)	0.79 (0.38, 1.43)	0.82 (0.39, 1.41)	1.16 (0.52, 1.94)	1.13 (0.51, 1.90)	1.13 (0.51, 1.90)	0.77 (0.38, 1.22)
	Year 4 (n = 345)	0.65 (0.32, 1.18)	0.76 (0.32, 1.30)	0.74 (0.35, 1.34)	0.73 (0.35, 1.31)	0.73 (0.35, 1.31)	0.76 (0.35, 1.33)
	Year 5 (n = 306)	0.63 (0.25, 1.09)	0.73 (0.34, 1.26)	0.60 (0.25, 1.12)	0.66 (0.25, 1.11)	0.66 (0.25, 1.11)	0.59 (0.30, 1.04)
	Year 7/8 (n = 322)	0.76 (0.35, 1.74)	0.83 (0.39, 1.83)	0.87 (0.40, 1.85)	0.92 (0.44, 1.89)	0.92 (0.44, 1.89)	0.93 (0.48, 1.41)
	Year 9 (n = 299)	0.48 (0.23, 0.89)	0.45 (0.23, 0.87)	0.63 (0.29, 0.99)	0.60 (0.29, 0.97)	0.60 (0.29, 0.97)	0.51 (0.23, 0.90)
	Year 10 (n = 313)	0.79 (0.32, 1.68)	0.96 (0.34, 1.94)	0.71 (0.35, 1.51)	0.76 (0.35, 1.61)	0.76 (0.35, 1.61)	1.06 (0.55, 1.57)
Maximum absolute residuals^d	Birth (n = 365)	1.27	1.89	3.44	3.32	3.32	2.32
	3 m (n = 126)	3.54	2.62	2.60	2.70	2.71	2.61
	6 m (n = 87)	2.60	3.60	3.94	4.03	4.02	3.75
	Year 1 (n = 270)	5.57	6.25	6.97	7.01	7.01	6.00
	Year 2 (n = 264)	4.14	4.03	4.99	4.96	4.95	3.51
	Year 4 (n = 345)	6.33	6.77	6.10	6.24	6.24	6.85
	Year 5 (n = 306)	4.37	4.81	4.28	4.45	4.45	3.81
	Year 7/8 (n = 322)	7.31	7.40	7.25	7.32	7.32	6.18
	Year 9 (n = 299)	5.05	5.31	5.18	5.24	5.24	4.78
	Year 10 (n = 313)	8.45	8.80	8.53	8.66	8.66	7.16
Sum of the rank		56	89.5	96	109	107	68.5
Kruskal--Wallis Rank Sum	p < 0.001	1076	1956	2096	2423	2361	1412
% positive raw residuals	Birth (n = 365)	174 (48)	45 (12)	0 (0)	0 (0)	0 (0)	3 (1)
	3 m (n = 126)	6 (5)	96 (76)	79 (63)	89 (71)	89 (71)	110 (87)
	6 m (n = 87)	51 (59)	79 (91)	81 (93)	82 (94)	82 (94)	85 (98)
	Year 1 (n = 270)	215 (80)	241 (89)	252 (93)	257 (95)	257 (95)	243 (90)
	Year 2 (n = 264)	166 (63)	149 (56)	214 (81)	212 (80)	212 (80)	118 (45)
	Year 4 (n = 345)	160 (46)	106 (31)	203 (59)	182 (53)	182 (53)	97 (28)
	Year 5 (n = 306)	139 (45)	78 (25)	163 (53)	139 (45)	139 (45)	182 (59)
	Year 7/8 (n = 322)	50 (16)	41 (13)	42 (13)	35 (11)	35 (11)	211 (66)
	Year 9 (n = 299)	115 (38)	131 (44)	88 (29)	96 (32)	96 (32)	180 (60)
	Year 10 (n = 313)	226 (72)	264 (84)	195 (62)	205 (65)	205 (65)	103 (33)

^aVariance components are given with their 95% CI.

^bDistribution of absolute residuals for each model.

^cMedians are given with their Interquartile Range (IQR).

^dDistribution of absolute residuals for each model at each measurement occasion.

AIC, Akaike Information Criterion; BIC, Bayesian Information Criterion.

Figure 1. Graphs of weight profiles and growth models fitted to weight from birth to 10 years.

Figure 2. Graphs of height profiles and growth models fitted to height from 3 months to 10 years.

Table 3. Goodness-of-fit statistics from fitting growth models to height measurements (in centimetres) from 3 months to 10 years.

		Models fitted
		Berkey-Reed 1	Count	Jenss-Bayley	Adapted Jenss-Bayley	2nd Order Polynomial	3rd Order Polynomial
Variance components^a	Random Intercept	6.32 (5.14, 7.78)	6.25 (5.08, 7.70)	6.45 (5.10, 7.80)	6.21 (4.91, 7.51)	6.78 (5.40, 8.52)	6.65 (5.39, 8.20)
	Random Slope	0.002 (0.001, 0.0021)	0.002 (0.001, 0.0021)	0.002 (0.001, 0.0022)	0.002 (0.001, 0.0022)	0.002 (0.001, 0.0022)	0.002 (0.001, 0.0021)
	Covariance	0.007 (−0.008, 0.022)	0.007 (−0.007, 0.022)	0.007 (−0.008, 0.022)	0.007 (−0.007, 0.023)	0.005 (−0.011, 0.022)	0.006 (−0.009, 0.022)
	Random Residuals	4.30 (4.01, 4.60)	4.33 (4.04, 4.64)	4.48 (4.17, 4.78)	4.38 (4.07, 4.68)	6.53 (6.10, 7.00)	4.75 (4.44, 5.09)
Information criterion	AIC	11510	11519	11586	11539	12259	11696
	BIC	11562	11565	11620	11577	12305	11747
Absolute residuals^b,^c	Median	0.94 (0.41, 1.75)	0.97 (0.43, 1.80)	0.95 (0.42, 1.80)	0.90 (0.40, 1.69)	1.11 (0.51, 2.09)	1.00 (0.44, 1.86)
	Maximum	12.87	12.95	12.64	12.46	11.16	11.76
Median absolute residuals^c,^d	3 m (n = 119)	1.37 (0.48, 2.37)	1.38 (0.64, 2.37)	1.24 (0.66, 2.50)	1.39 (0.52, 2.36)	3.58 (2.37, 5.20)	1.86 (1.07, 3.44)
	6 m (n = 87)	1.31 (0.68, 2.29)	1.23 (0.51, 2.40)	1.34 (0.78, 2.25)	1.20 (0.52, 2.59)	1.29 (0.57, 2.92)	1.24 (0.56, 2.33)
	Year 1 (n = 259)	1.18 (0.57, 1.95)	1.17 (0.58, 1.90)	1.54 (0.68, 2.40)	1.09 (0.58, 2.01)	1.65 (0.76, 2.81)	1.79 (0.85, 2.83)
	Year 2 (n = 303)	1.32 (0.59, 2.19)	1.34 (0.62, 2.21)	1.19 (0.54, 2.12)	1.08 (0.51, 1.96)	1.22 (0.55, 2.05)	0.95 (0.38, 1.75)
	Year 4 (n = 336)	0.94 (0.43, 1.59)	0.93 (0.44, 1.64)	0.90 (0.40, 1.58)	0.96 (0.47, 1.78)	1.06 (0.53, 1.80)	0.91 (0.43, 1.66)
	Year 5 (n = 303)	1.09 (0.45, 1.77)	1.17 (0.51, 1.91)	0.98 (0.43, 1.62)	0.96 (0.44, 1.67)	0.83 (0.37, 1.47)	0.80 (0.39, 1.42)
	Year 7/8 (n = 314)	0.84 (0.42, 2.14)	0.86 (0.42, 2.13)	0.96 (0.41, 2.20)	0.88 (0.36, 2.33)	1.48 (0.71, 2.86)	1.08 (0.43, 2.37)
	Year 9 (n = 299)	0.52 (0.25, 1.03)	0.53 (0.25, 1.06)	0.54 (0.24,1.06)	0.52 (0.21, 0.99)	0.57 (0.25, 1.01)	0.71 (0.37, 1.32)
	Year 10 (n = 308)	0.81 (0.31, 1.39)	0.84 (0.33, 1.37)	0.82 (0.31,1.38)	0.77 (0.42, 1.35)	1.10 (0.52, 1.85)	0.92 (0.36, 1.58)
Maximum absolute residuals^d	3 m (n = 119)	6.98	6.53
	6 m (n = 87)	7.46	7.63	7.30	7.83	8.65	7.70
	Year 1 (n = 259)	5.32	5.05	6.15	4.96	6.89	6.71
	Year 2 (n = 303)	12.87	12.95	12.64	12.46	10.44	11.76
	Year 4 (n = 336)	5.80	5.67	6.01	5.71	6.24	6.58
	Year 5 (n = 303)	10.12	10.27	9.88	9.99	9.16	9.17
	Year 7/8 (n = 314)	6.78	6.73	6.64	7.47	8.27	6.33
	Year 9 (n = 299)	4.76	4.78	4.69	4.88	4.83	4.32
	Year 10 (n = 308)	9.55	9.47	9.45	10.26	11.09	8.89
Sum of the Ranks		70	75	76.5	68	111	84.5
Kruskal--Wallis Rank Sum	p = 0.002	1348	1463	1496	1300	2297	1686
% positive raw residuals	3 m (n = 119)	66 (55)	74 (62)	37 (31)	69 (58)	1 (1)	16 (13)
	6 m (n = 87)	49 (56)	45 (52)	50 (57)	43 (49)	26 (30)	45 (52)
	Year 1 (n = 259)	168 (65)	150 (58)	202 (78)	143 (55)	202 (78)	215 (83)
	Year 2 (n = 303)	62 (20)	62 (20)	66 (22)	96 (32)	218 (72)	134 (44)
	Year 4 (n = 336)	186 (55)	199 (59)	154 (46)	224 (67)	230 (68)	126 (38)
	Year 5 (n = 303)	234 (77)	248 (82)	222 (73)	222 (73)	162 (53)	167 (55)
	Year 7/8 (n = 314)	152 (48)	154 (49)	165 (52)	86 (27)	29 (9)	181 (58)
	Year 9 (n = 299)	189 (63)	184 (62)	202 (67)	156 (52)	141 (47)	237 (79)
	Year 10 (n = 308)	137 (44)	125 (41)	133 (43)	188 (61)	245 (80)	106 (34)

^aVariance components are given with their 95% CI.

^bDistribution of absolute residuals for each model.

^cMedians are given with their Interquartile Range (IQR).

^dDistribution of absolute residuals for each model at each measurement occasion.

AIC, Akaike Information Criterion; BIC, Bayesian Information Criterion.

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