## 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 to fit age-conditional references of the D-score for boys and girls combined by the LMS method. The LMS method by 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 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.

 age M S L 0.0383 8.81 0.3126 1.3917 0.0575 10.59 0.2801 1.4418 0.0767 12.27 0.2526 1.4891 0.0958 13.87 0.2291 1.5331 0.1150 15.39 0.2089 1.5722 0.1342 16.83 0.1916 1.6049 0.1533 18.20 0.1767 1.6304 0.1725 19.50 0.1640 1.6487 0.1916 20.75 0.1531 1.6607 0.2108 21.94 0.1436 1.6676 0.2300 23.07 0.1354 1.6706 0.2491 24.16 0.1283 1.6711 0.2683 25.21 0.1220 1.6698 0.2875 26.21 0.1165 1.6673 0.3066 27.17 0.1117 1.6636 0.3258 28.10 0.1074 1.6589 0.3450 28.99 0.1035 1.6533 0.3641 29.86 0.1001 1.6471 0.3833 30.70 0.0970 1.6403 0.4025 31.50 0.0942 1.6330 0.4216 32.29 0.0917 1.6255 0.4408 33.05 0.0894 1.6178 0.4600 33.79 0.0873 1.6100 0.4791 34.51 0.0854 1.6022 0.4983 35.21 0.0837 1.5946 0.5175 35.89 0.0821 1.5870 0.5366 36.55 0.0807 1.5797 0.5558 37.20 0.0793 1.5725 0.5749 37.83 0.0781 1.5656 0.5941 38.44 0.0770 1.5588 0.6133 39.04 0.0759 1.5523 0.6324 39.63 0.0749 1.5460 0.6516 40.21 0.0740 1.5399 0.6708 40.77 0.0731 1.5340 0.6899 41.32 0.0723 1.5284 0.7091 41.86 0.0715 1.5230 0.7283 42.39 0.0707 1.5178 0.7474 42.91 0.0700 1.5128 0.7666 43.42 0.0693 1.5081 0.7858 43.92 0.0687 1.5036 0.8049 44.40 0.0681 1.4993 0.8241 44.88 0.0674 1.4952 0.8433 45.36 0.0669 1.4913 0.8624 45.82 0.0663 1.4876 0.8816 46.27 0.0657 1.4841 0.9008 46.72 0.0652 1.4809 0.9199 47.16 0.0647 1.4778 0.9391 47.59 0.0642 1.4749 0.9582 48.01 0.0637 1.4723 0.9774 48.43 0.0632 1.4698 0.9966 48.84 0.0627 1.4676 1.0157 49.24 0.0622 1.4655 1.0349 49.64 0.0618 1.4637 1.0541 50.03 0.0613 1.4620 1.0732 50.41 0.0608 1.4605 1.0924 50.79 0.0604 1.4592 1.1116 51.16 0.0600 1.4580 1.1307 51.53 0.0595 1.4570 1.1499 51.89 0.0591 1.4561 1.1691 52.24 0.0587 1.4553 1.1882 52.59 0.0583 1.4547 1.2074 52.94 0.0578 1.4542 1.2266 53.27 0.0574 1.4538 1.2457 53.61 0.0570 1.4535 1.2649 53.94 0.0566 1.4534 1.2841 54.26 0.0562 1.4533 1.3032 54.58 0.0559 1.4533 1.3224 54.89 0.0555 1.4533 1.3415 55.20 0.0551 1.4535 1.3607 55.50 0.0547 1.4537 1.3799 55.81 0.0544 1.4539 1.3990 56.10 0.0540 1.4542 1.4182 56.39 0.0536 1.4546 1.4374 56.68 0.0533 1.4551 1.4565 56.97 0.0530 1.4555 1.4757 57.25 0.0526 1.4561 1.4949 57.52 0.0523 1.4567 1.5140 57.80 0.0520 1.4573 1.5332 58.06 0.0517 1.4580 1.5524 58.33 0.0514 1.4587 1.5715 58.59 0.0510 1.4595 1.5907 58.85 0.0508 1.4603 1.6099 59.11 0.0505 1.4612 1.6290 59.36 0.0502 1.4620 1.6482 59.61 0.0499 1.4630 1.6674 59.86 0.0496 1.4639 1.6865 60.11 0.0494 1.4649 1.7057 60.35 0.0491 1.4660 1.7248 60.59 0.0488 1.4670 1.7440 60.82 0.0486 1.4681 1.7632 61.06 0.0483 1.4692 1.7823 61.29 0.0481 1.4704 1.8015 61.52 0.0478 1.4716 1.8207 61.75 0.0476 1.4728 1.8398 61.97 0.0474 1.4740 1.8590 62.20 0.0471 1.4752 1.8782 62.42 0.0469 1.4765 1.8973 62.64 0.0467 1.4778 1.9165 62.85 0.0465 1.4791 1.9357 63.07 0.0463 1.4805 1.9548 63.28 0.0461 1.4818 1.9740 63.49 0.0459 1.4832 1.9932 63.70 0.0457 1.4846 2.0123 63.91 0.0455 1.4861 2.0315 64.11 0.0453 1.4875 2.0507 64.32 0.0451 1.4890 2.0698 64.52 0.0449 1.4904 2.0890 64.72 0.0447 1.4919 2.1081 64.92 0.0445 1.4934 2.1273 65.11 0.0443 1.4949 2.1465 65.31 0.0441 1.4964 2.1656 65.50 0.0440 1.4979 2.1848 65.70 0.0438 1.4994 2.2040 65.89 0.0436 1.5009 2.2231 66.08 0.0434 1.5024 2.2423 66.26 0.0433 1.5039 2.2615 66.45 0.0431 1.5054 2.2806 66.64 0.0429 1.5069 2.2998 66.82 0.0428 1.5084 2.3190 67.00 0.0426 1.5098 2.3381 67.18 0.0425 1.5113 2.3573 67.36 0.0423 1.5127 2.3765 67.54 0.0421 1.5142 2.3956 67.72 0.0420 1.5156 2.4148 67.89 0.0418 1.5170 2.4339 68.07 0.0417 1.5185 2.4531 68.24 0.0415 1.5199 2.4723 68.41 0.0414 1.5213 2.4914 68.59 0.0412 1.5226 2.5106 68.75 0.0411 1.5240 2.5298 68.92 0.0410 1.5254 2.5489 69.09 0.0408 1.5267 2.5681 69.26 0.0407 1.5281 2.5873 69.42 0.0405 1.5294 2.6064 69.59 0.0404 1.5308 2.6256 69.75 0.0403 1.5321 2.6448 69.91 0.0401 1.5334 2.6639 70.07 0.0400 1.5347 2.6831 70.23 0.0399 1.5360 2.7023 70.39 0.0397 1.5373 2.7214 70.55 0.0396 1.5386 2.7406 70.71 0.0395 1.5398 2.7598 70.86 0.0394 1.5411 2.7789 71.02 0.0392 1.5423

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

$D_t(\alpha)= M_t \exp(S_t Z_\alpha).$