5.4 Age-conditional references

5.4.1 Motivation

The last step involves estimation an age-conditional reference distribution for the D-score. This distribution can be used to construct growth charts that portray the normal variation in development. Also, the references can be used to calculate age-standardized D-scores, called DAZ, that emphasize the location of the measurement in comparison to age peers.

Estimation of reference centiles is reasonably standard. Here we follow van Buuren (2014) to fit age-conditional references of the D-score for boys and girls combined by the LMS method. The LMS method by Cole and Green (1992) assumes that the outcome has a normal distribution after a Box-Cox transformation. The reference distribution has three parameters, which model respectively the location (\(M\)), the spread (\(S\)), and the skewness (\(L\)) of the distribution. Each of the three parameters can vary smoothly with age.

5.4.2 Estimation of the reference distribution

The parameters are estimated using the BCCG distribution of gamlss 5.1-3 (Stasinopoulos and Rigby 2007) using cubic splines smoothers. The final solution used a log-transformed age scale and fitted the model with smoothing parameters \(\mathrm{df}(M)=2\), \(\mathrm{df}(S)=2\) and \(\mathrm{df}(L)=1\).

Figure 4.3 plots the D-scores together with five grey lines, corresponding to the centiles -2SD (P2), -1SD (P16), 0SD (P50), +1SD (P84) and +2SD (P98). The area between the -2SD and +2SD lines delineates the D-score expected if development is healthy. Note that the shape of the reference is quite similar to that of weight and height, with rapid growth occurring in the first few months.

Table 5.4 defines age-conditional references for Dutch children as the \(M\)-curve (median), \(S\)-curve (spread) and \(L\)-curve (skewness) by age. This table can be used to calculate centile lines and \(Z\)-scores.

The references are purely cross-sectional and do not account for the correlation structure between ages. For prediction purposes, it is useful to extend the modelling to include velocities and change scores.

5.4.3 Conversion of \(D\) to DAZ, and vice versa

Suppose that \(M_t\), \(S_t\) and \(L_t\) are the parameter values at age \(t\). Cole (1988) shows that the transformation

\[Z=\frac{(D_t/M_t)^{L_t}-1}{L_t S_t}\]

converts measurement \(D_t\) into its normal equivalent deviate \(Z\). If \(L_t\) is close to zero, we use

\[Z=\frac{\ln(D_t/M_t)}{S_t}\]

We may derive any required centile curve from Table 5.4. First, choose \(Z_\alpha\) as the \(Z\)-score that delineates \(100\alpha\) per cent of the distribution, for example,  \(Z_{0.05}=-1.64\). The D-score that defines the \(100\alpha\) centile is equal to

\[D_t(\alpha) = M_t (1+L_t S_t Z_\alpha)^{1/L_t}\]

If \(L_t\) is close to zero, we use

\[D_t(\alpha)= M_t \exp(S_t Z_\alpha).\]