Methods for Constructing Normalised Reference Scores: An Application for Assessing Child Development at 24 Months of Age

Figures & data

Table 2. Distribution of z-scores derived from the best selected models from each distribution family, in boys (n = 3,321) and girls (n = 3,081).

	Boys			Girls
	BCCG	BCPE	BEINF01	BCCG	BCPE	BEINF01
	M1.3	M2.6	M3.3	M1.3	M2.8	M3.1
Min.	−3.11	−4.23	−3.89	−4.17	−4.35	−4.21
P50 (P25; P75)	−0.02 (−0.75; 0.77)	−0.07 (−0.68; 0.59)	−0.11 (−0.72; 0.63)	−0.08 (−0.77; 0.77)	−0.08 (−0.65; 0.60)	−0.09 (−0.71; 0.67)
Mean (SD)	0.004 (0.99)	−0.03 (0.99)	−0.008 (1.00)	−0.02 (1.01)	−0.02 (0.99)	−0.004 (1.00)
Max.	2.08	2.57	3.41	2.09	2.91	4.02

Table 1. Information criteria for the development of the GAMLSS models for total PRC score in boys (n = 3,321) and girls (n = 3,081), with separate predictors for median mu $(\mu ),$ variability sigma $(\sigma ),$ skewness lamdba $(\lambda )$ and kurtosis tau $(\tau ).$

	Linear predictor for										GAIC
	μ		σ		λ		τ		df	Deviance	k = 2	k = 5	k = 8
	edf	age	edf	age	edf	age	edf	age
BOYS
Distribution
Continuous
BCCG
M1.1	2.9	age	1.0	constant	1.0	constant	–	–	4.9	241	251	266	281
M1.2	3.0	age	2.0	age	1.0	constant	–	–	6.0	237	249	266	285
M1.3	3.2	age	1.0	constant	2.0	age	–	–	6.2	223	235	254	273
M1.4	3.1	age	2.0	age	2.0	age	–	–	7.1	222	236	257	280
BCPE
M2.1	2.0	age	1.0	constant	1.0	constant	1.0	constant	5.0	28	38	53	69
M2.2	2.0	age	1.0	constant	2.0	age	1.0	constant	6.0	19	31	49	67
M2.3	2.0	age	1.0	constant	1.0	constant	2.0	age	6.0	26	38	56	75
M2.4	2.0	age	1.0	constant	2.0	age	2.0	age	7.0	19	33	54	76
M2.5	2.0	age	2.0	age	1.0	constant	1.0	constant	6.0	27	39	57	76
M2.6	2.0	age	2.0	age	2.0	age	1.0	constant	7.0	0	14	35	57
M2.7	2.0	age	2.0	age	1.0	constant	2.0	age	7.0	23	37	58	80
M2.8	2.0	age	2.0	age	2.0	age	2.0	age	8.0	0	16	40	65
Mixed
BEINF01
M3.1	2.9	age	1.0	constant	1.0	constant	1.0	constant	5.9	8	19	37	55
M3.2	2.9	age	1.0	constant	2.0	age	1.0	constant	6.9	8	21	42	63
M3.3	2.9	age	1.0	constant	1.0	constant	2.0	age	6.9	2	15	36	57
M3.4	2.9	age	1.0	constant	2.0	age	2.0	age	7.9	2	17	41	66
M3.5	2.9	age	2.0	age	1.0	constant	1.0	constant	6.9	6	20	41	62
M3.6	2.9	age	2.0	age	2.0	age	1.0	constant	7.9	6	22	46	70
M3.7	2.9	age	2.0	age	1.0	constant	2.0	age	7.9	0	16	40	64
M3.8	2.9	age	2.0	age	2.0	age	2.0	age	8.9	0	18	45	72
GIRLS
Distribution
Continuous
BCCG
M1.1	2.0	age	1.0	constant	1.0	constant	–	–	4.0	301	309	321	334
M1.2	2.3	age	20	age	1.0	constant	–	–	23.7	781	829	900	972
M1.3	2.0	age	1.0	constant	2.0	age	–	–	5.0	282	292	307	322
M1.4	2.4	age	20	age	2.0	age	–	–	24.8	767	817	891	966
BCPE
M2.1	2.0	age	1.0	constant	1.0	constant	1.0	constant	5.0	41	51	66	82
M2.2	2.0	age	1.0	constant	2.0	age	1.0	constant	6.0	41	53	71	89
M2.3	2.0	age	1.0	constant	1.0	constant	2.0	age	6.0	35	47	65	83
M2.4	2.0	age	1.0	constant	2.0	age	2.0	age	7.0	29	43	64	86
M2.5	2.0	age	2.0	age	1.0	constant	1.0	constant	6.0	29	41	60	78
M2.6	2.0	age	18	age	2.0	age	1.0	constant	22.8	0	46	114	183
M2.7	2.0	age	2.0	age	1.0	constant	2.0	age	7.0	35	49	70	91
M2.8	2.0	age	3.9	age	2.0	age	2.0	age	9.9	3	23	53	82
Mixed
BEINF01
M3.1	2.0	age	1.0	constant	1.0	constant	1.0	constant	5.0	2	12	27	43
M3.2	2.0	age	1.0	constant	2.0	age	1.0	constant	6.0	2	14	32	51
M3.3	2.0	age	1.0	constant	1.0	constant	2.0	age	6.0	1	13	31	49
M3.4	2.0	age	1.0	constant	2.0	age	2.0	age	7.0	1	15	36	57
M3.5	2.0	age	2.0	age	1.0	constant	1.0	constant	6.0	1	13	31	50
M3.6	2.0	age	2.0	age	2.0	age	1.0	constant	7.0	1	15	36	58
M3.7	2.0	age	2.0	age	1.0	constant	2.0	age	7.0	0	14	35	56
M3.8	2.0	age	2.0	age	2.0	age	2.0	age	8.0	0	16	40	64

Values shown in table were rounded up or down to the closest whole number. Bold fonts indicate the lowest deviance or GAIC value. Box indicate the final selected model. To facilitate the comparison values 32,410 in boys and 29,512 in girls (i.e. the minimum value noticed in the continuous models for boys and girls respectively) was subtracted from the continuous distributions and 1150 in boys and 1566 in girls (i.e. the minimum value noticed in the mixed models for boys and girls respectively) was added to the mixed ones. Abbreviations: edf: Effective degrees of freedom for the spline function which have been superimposed on the linear trend. These contain the constant, the linear part and the additive part of the smoothing function (if included in the modeling); df: degrees of freedom in the model, which represent the sum of the edf for all the parameters in the model; GAIC: generalized Akaike information criterion.

Figure 1. Q-Q plot of the normalized residuals (z-scores), derived from modeling PRC against age using three distribution families in (A) boys (n = 3,321; top panel) and (B) girls (n = 3,081; bottom panel).

Legend: the shaded area from -2 to +2 z-scores represents the range that z-scores derived from each distribution are similar.

Figure 2. Density distributions of the total PRC score (upper left) and the fitted BCPE (lower left) and BEINF01 (lower right) distributions, using the fitted parameters of the best-selected models (BCPE, M2.6: mu = 90.22, sigma = 0.3711, lambda = 1.0512, tau = 5.3556 at the 0–158 range; BEINF01, M3.3: mu = 0.5664, sigma = 0.4284, lambda = 8.9e-09, tau = 0.0006 at the 0–1 range), in boys (n = 3,321).

Figure 3. Density distributions of the total PRC score (upper left) and the fitted BCPE (lower left) and BEINF01 (lower right) distributions, using the fitted parameters of the best-selected models (BCPE, M2.8: mu = 108.4, sigma = 0.2774, nu = 1.733, tau = 10.50 at the 0–158 range; BEINF01, M3.1: mu = 0.6606, sigma = 0.4261, lambda = 3.3e-09, tau = 0.0033 at the 0–1 range), in girls (n = 3,081).

Figure 4. Trends between the standardized and PRC scores for each distribution family over age for (A) boys (n = 3,321; upper panel), and (B) girls (n = 3,081; lower panel).

Table 3. Predicted vs raw PRC scores, by age group, using the best selected model from each distribution family, in boys (n = 3,321) and girls (n = 3,081).

	Raw PRC score
	Sample	Predicted
		BCCG	BCPE	BEINF01
Boys		M1.3	M2.6	M3.3
Age group, months
23 mo 16 d to 24 mo 15d	84 (33)	81 (32)	84 (34)	84 (36)
24 mo 16 d to 25 mo 15d	89 (33)	88 (34)	90 (33)	90 (39)
25 mo 16 d to 26 mo 15d	93 (33)	92 (36)	95 (33)	94 (40)
26 mo 16 d to 27 mo 15d	93 (34)	95 (37)	101 (32)	96 (41)
Girls		M1.3	M2.8	M3.1
Age group, months
23 mo 16 d to 24 mo 15d	102 (32)	90 (50)	105 (31)	100 (43)
24 mo 16 d to 25 mo 15d	105 (31)	93 (53)	109 (30)	105 (45)
25 mo 16 d to 26 mo 15d	109 (32)	97 (55)	114 (29)	110 (47)
26 mo 16 d to 27 mo 15d	113 (32)	102 (57)	119 (27)	115 (49)

Predicted scores were calculated using the mid-point of each age group, i.e. at 24 (i.e. 730 days), 25 (i.e. 760 days), 26 (i.e. 790 days) and 27 months (i.e. 822 days) respectively.

Data availability statement

Data sharing is not applicable as secondary data were analyzed in this study, hence no new data were created in this study. The code used to analyze the current data is available as an Appendix in the online supplementary material. The parameters obtained from the equations derived from the mixed distributions are also freely available on the PARCA-R website (www.parca-r.info) to facilitate its use in research and clinical practice.