Small for gestational age: whose chart flags whom?

#The brief

A hospital network is replacing its newborn-assessment software. Every baby is weighed at birth, and a baby lighter than the 10th centile of birth weight for its gestational age and sex is flagged small for gestational age (SGA): it gets extra glucose checks, a paediatric review and closer follow-up, because SGA babies carry a higher risk of stillbirth, hypoglycaemia and later problems. The network's obstetric lead asks you:

"Build us the 10th-centile chart, by sex, from recent US births - with an honest statement of how sure you are of it, especially for the preterm weeks. And a question our midwives keep asking: should babies of mothers who smoked, or who had gestational hypertension, be judged against their own charts? Those babies are smaller anyway. How many babies per 1000 would that stop - or start - flagging, and would you recommend it?"

This is the sequel to example E83 (Bayesian quantile regression). Do E83 first: this challenge assumes you know the check loss, why the asymmetric Laplace is only a working likelihood, and the sandwich / learning-rate fixes. E83 asked what quantile regression can tell you; here a chart has to be used, by sex, at every week from 24 to 42, and a policy question rides on it.

#How this notebook works

• Each task states what to deliver, not how. Write your code in the YOUR CODE HERE cells.
• Stuck? h.hint("task2") reveals hints one level at a time: nudge → approach → code skeleton. Try to get by on nudges.
• h.check("task2", q10_40_girls=...) compares your numbers with the reference solution.
• A full worked solution lives in notebooks/solutions/. Open it only when you are done (or truly stuck).

#The data

One row per singleton birth. load only parses the raw NCHS codes; it does not clean them.

#Task 0 · Which gestational age? Which records?

A chart is only as good as the gestational ages it is built on, and the file gives you two.

Deliver

1. The unknown codes that matter for a weight-for-age chart, and how many records they remove.
2. Evidence for choosing between combgest and oegest_comb: how often do they disagree, by how much, and what would each do to a weekly 10th centile - at the preterm weeks and after 41 weeks? Report the share of births with a known weight that combgest places at 43 weeks or more.
3. A rule for records whose weight is implausible for their gestational age, and an argument for which side(s) of the distribution that rule may touch, given what the chart is for.
4. The range of weeks the chart can cover.

#Solution

Unknown codes first: 9999 g for weight and 99 weeks for gestation. A "99-week" baby would sit far off the right of any curve; a 9999 g baby is heavier than any newborn ever recorded.

The two estimates agree for about two births in three and differ by three weeks or more for about one in eleven. The LMP-based combgest puts about 3% of births at 43-47 weeks, which almost never happens in modern obstetrics (labour is induced well before that); the obstetric estimate puts three births there. The consequences for a chart:

• After 41 weeks the combgest 10th centile falls (middle panel): the "post-term" weeks are full of term babies whose LMP date was wrong. A chart built on them would be too lenient there.
• Before 34 weeks about one combgest baby in five weighs over 3 kg (right panel) - a term-sized baby dated as very preterm - against 0-3% from 29 to 33 weeks under the obstetric estimate (its 28-week point is two babies, both among the implausible records below).

The obstetric estimate is the better-dated measure, and it is the one NCHS itself has used as its standard measure of gestational age since the 2014 data year. We use oegest_comb from here on.

Implausible weight for age. A few births are still impossible as recorded - a 3.4 kg baby at 28 weeks. The usual rule is symmetric (drop anything more than k robust standard deviations from the weekly median). But the chart exists to find babies who are too small for their age. A 1.2 kg baby at 39 weeks is either misdated or severely growth-restricted, and it is exactly the second kind we must not delete. So we trim the upper side only, and only where heavy outliers are biologically implausible (preterm weeks): log weight more than 4 robust sds above the weekly, sex-specific median before 37 weeks. That costs a handful of records and barely moves a 10th centile - the point is not to touch the lower tail.

The five preterm heavyweights (2.5-3.4 kg at 26-30 weeks) are far heavier than any plausible baby of their recorded week; two of them have a combined estimate of 38 and 41 weeks - misdated. The low outliers include babies of 1.2-1.8 kg at 37-39 weeks with both estimates agreeing on term - growth restriction, not a typo; one mother had gestational hypertension.

Range. Below 24 weeks there are a few dozen births in 20,000, most of them before viability; above 42 weeks three. The chart covers 24-42 weeks (obstetric estimate).

#The analysis data (everyone uses this from here on)

So that numbers can be compared, the rest of the notebook uses the rules above: obstetric gestational age, 24-42 weeks, unknown weight / gestation / smoking (first or second trimester) / hypertension dropped, and the upper-side-only, preterm-only trim. A smoker smoked in any trimester (a third-trimester "unknown" counts as no information, not as a reason to drop: there was no third trimester for many preterm babies, see E83). We model log birth weight: quantiles are equivariant under monotone transforms, so the 10th centile of log weight is the log of the 10th centile, and effects become percentages.

The births are split at random into a training set of 10,000 for fitting the chart and a held-out set for checking it and for the decision.

#Task 1 · The quick chart

The obvious chart: for each week and sex, the empirical 10th centile of the training births.

Deliver

1. The chart (both sexes, all weeks 24-42) on a scale where you can see the preterm weeks.
2. What is wrong with it, in words a neonatologist would accept: look at neighbouring weeks and at the difference between boys and girls where there are few births.
3. The share of held-out babies it flags, by gestational-age band and sex (24-31, 32-34, 35-36, 37, 38, 39, 40, 41-42 weeks). Is 10% what you get?

#Solution

From 36 to 41 weeks, with hundreds of births per week, the quick chart is fine: smooth, boys above girls by 2.5-5%, and the held-out flag rate is 8-13% at the term weeks. Before 34 weeks it is noise. With between 2 and 26 training births per week and sex, the "10th centile" is the first or second smallest baby: the curve goes down from one week to the next several times (girls at 29 weeks, boys at 26, 28 and 31), and the gap between boys and girls swings from boys 60% heavier (24 weeks) to girls heavier (26) and back to boys 27% heavier (30) - no biology does that. On held-out babies three of the four preterm cells flag 16-19%; for two of them (girls 24-31, boys 32-34) that is more than binomial noise explains. The chart needs to borrow strength: across neighbouring weeks (growth is smooth) and between the sexes (the sex difference is small and should change slowly).

#Task 2 · A smooth chart that borrows strength

Build a Bayesian model for the 10th centile of log birth weight at each week, for both sexes, that

• makes the curve smooth across weeks, with the amount of smoothing learned from the data;
• shares information between boys and girls: a common curve plus a sex difference that is small and changes slowly;
• uses the asymmetric Laplace working likelihood of E83 (pm.AsymmetricLaplace(..., q=0.1)).

Deliver

1. Your priors, and why the smoothing prior is on the scale you chose.
2. A fit with clean diagnostics.
3. The chart with 90% bands on top of Task 1's points, and the boy-girl difference by week.
4. The 10th centile at 40 weeks for girls and boys (posterior median, grams), and the held-out flag table of Task 1 for your new chart.

#Solution

Gestational age is recorded in whole weeks, so the "curve" is 19 numbers per sex and a smooth curve is one whose second differences are small: a second-order random walk (RW2) on the weekly log 10th centile, $f_{w+1} - 2f_w + f_{w-1} \sim N(0, s_f)$ - the discrete version of a P-spline penalty. Its scale $s_f$ is learned. What is a plausible curvature? The weekly increase of log weight falls from about 0.15 (15% a week) around 30 weeks to about 0.03 at 41 weeks, so the slope changes by a few hundredths per week: HalfNormal(0.1) on $s_f$ leaves room for that and pulls towards straighter curves where data are thin. Each week's level gets a vague Normal(0.5, 1) (log kg: 1.6 kg give or take a factor of 2.7), so the RW2 term does the smoothing.

The sexes: girls $= f - h/2$, boys $= f + h/2$, with the log ratio $h_w = h_0 + $ a first-order random walk whose step sd is HalfNormal(0.02) (non-centred). Boys are typically a few percent heavier; that prior says the difference drifts by a couple of percent across the whole range at most, unless the data insist. One function builds the model for the rest of the notebook: with w=None it is the asymmetric Laplace with a learned scale; with a number it becomes the generalised posterior of Task 3; X adds covariates (Task 4).

The sampler is happy (no divergences, $\hat R \le 1.01$), the curves are smooth, and in the data-rich weeks they sit on the empirical points. In the preterm weeks the model has pulled the jagged points onto a smooth, increasing curve, and the sex difference is a slowly varying 3-9% instead of Task 1's swings (the model also smooths away the boys' last point at 42 weeks, 13 births). The held-out flag rates in the preterm cells are now 8-15% instead of 10-19%, though with 47-112 held-out babies per cell the binomial noise alone is 3-4.5 points (one sd).

The bands look reassuringly narrow - at 40 weeks about 40 g wide. Before anybody prints this chart, Task 3 asks whether those bands mean what they say.

#Task 3 · Can you trust the bands?

The asymmetric Laplace is a working likelihood (E83). Find out what that does to this chart.

Deliver

1. At the data-rich weeks (37-41), compare the posterior sd of each sex's 10th centile with an honest frequentist yardstick. Report the median ratio (posterior sd / yardstick sd).
2. The held-out flag rates of Task 2 with an interval for each cell that includes both the binomial noise of the held-out cell and your posterior's uncertainty about the curve. How many cells fall outside their 90% interval? What can this check see, and what can it not?
3. Fix the bands - one of E83's fixes is enough - refit, and redo 1 and 2. Did anything besides the band width change? (Look at the learned smoothness.)

#Solution

The yardstick. At 37-41 weeks there are hundreds to thousands of training babies per week and sex, so the smoothing prior hardly matters there and the posterior should be about as wide as the sampling distribution of the plain weekly 10th centile. The bootstrap of the training births gives that.

The posterior is about half as wide as the bootstrap at the weeks where the bootstrap is reliable: a nominal 90% band ($\pm 1.645$ posterior sds) is only about $\pm 0.9$ honest sds wide - roughly a 63% band. This is E83's sandwich result at work: the asymmetric Laplace gets the curvature of the check loss right but not its variance.

The held-out check. For a cell with $n$ held-out babies, a chart that is the 10th centile flags a Binomial$(n, 0.1)$ count. The chart's own uncertainty adds to that: evaluate every posterior draw of the curve on the held-out babies of the cell and see how much its flag rate moves. The interval is $0.1 \pm 1.645\sqrt{0.09/n + \operatorname{var}_{\text{draws}}(\text{rate})}$.

Read the two sd columns. In every cell the binomial noise of the held-out babies is larger than the curve's posterior uncertainty, usually two to three times larger: the held-out check is dominated by noise that has nothing to do with the bands. It can catch a curve in the wrong place (a cell flagging 25% or 2%), but a band half as wide as it should be hardly changes the intervals. With these overconfident bands 14 of 16 cells are inside their 90% intervals - about what a correct model would give (14.4 expected). A held-out check is only as sharp as the held-out sample is large; here the bootstrap, not the held-out set, exposes the bands.

The fix. A generalised (Gibbs) posterior, $\pi(\theta)\exp\{-w\sum_i\rho_{0.1}(y_i - \mu_i)\}$. The asymmetric Laplace is the special case $w = 1/\sigma$. Posterior variance scales roughly like $1/w$, so we calibrate $w = \frac{1}{\hat\sigma}\times$ (geometric mean over the 10 term cells of posterior variance / bootstrap variance), and refit - E83's moment-matching recipe, applied where the bootstrap is reliable.

The calibrated posterior is much closer to the bootstrap at term: median ratio 0.85 against 0.54, with cells scattering from 0.6 to 1.2 (the bootstrap of a quantile is itself noisy). The remaining shortfall is expected in part: the smooth model borrows from neighbouring weeks, which the weekly bootstrap does not, so a ratio somewhat below 1 is honest. At 40 weeks the 90% band widened from about 40 g to about 60 g (left panels; the black squares are the bootstrap's 90% widths). What else changed:

• The curve hardly moved: at 37-41 weeks the two posterior-median charts differ by at most 0.2%, in the preterm weeks by up to about 4% (girls, 24 weeks). The learned smoothness $s_f$ is the same (0.024) in both fits. The sex-difference step $s_h$ shrank from 0.020 to 0.009: once the likelihood stops overstating the data, the wiggles of the preterm sex difference are no longer worth a rougher curve.
• In the preterm weeks the bands widened less (about 1.25x at 24 weeks, 1.35-1.5x at 30): there the band is set partly by the smoothing prior, which the learning rate does not touch. At 42 weeks (24 training births) both bands are about 300 g wide.
• The held-out check passes before and after, as predicted.

Two caveats for the obstetric lead. The calibration is done at term, where the bootstrap works; in the preterm weeks one learning rate is an extrapolation, and the band there is as much the prior's as the data's. And a 10th centile at 24-27 weeks estimated from about 20 training births per sex is not something to print without a "few data" warning, whatever the model.

#Task 4 · Charts for smokers and for gestational hypertension

The midwives' question needs conditional 10th centiles: the 10th centile of birth weight given week, sex, smoking and gestational hypertension.

Deliver

1. A model for that, built on Task 3's (calibrated) chart. Think about whether the hypertension effect should be the same before and after 37 weeks - and about why it might not be.
2. The effects at the 10th centile as percentages with 90% intervals, and a check that the intervals of the effect you will use in Task 5 are honest.
3. Explain why the hypertension effect at the 10th centile, given gestational age, is so much smaller than the roughly -290 g E83 found at the 10th centile without gestational age in the model.

#Solution

Add shifts on the log scale to Task 3's model: one for smoking, and for hypertension one for term births and an extra one for preterm births. Hypertension is a reason to deliver early: severe pre-eclampsia, often with a growth-restricted baby, ends in a preterm birth. Among preterm births, then, hypertension selects the sickest pregnancies, and a single shift would average two different things. The learning rate from Task 3 is a starting point; whether it also calibrates these coefficients is a question for the bootstrap (E83: one scalar cannot fix every coefficient).

Smoking lowers the 10th centile by about 5% at every week (about 150 g at term; 90% interval roughly -7% to -3%). Before quoting intervals, read the sd ratio column: the learning rate calibrated on the weekly curve gives the smoking and term-hypertension coefficients posterior sds within 10% of their bootstrap sds, so their intervals can be used. The extra preterm hypertension shift is at 0.77 - somewhat overconfident - and its bootstrap only sees 32-36 weeks (the linear programme cannot use the sparse earlier cells), so treat that interval as a lower bound on the uncertainty.

Hypertension at term lowers the 10th centile by less than 1%, with an interval that includes zero; before 37 weeks the total shift is about -4.6% (-8.5% to -0.9%), from 171 hypertensive preterm births in training. The contrast with E83's -290 g at the 10th centile is the gestational age in the model: most of what hypertension does to birth weight it does by ending pregnancies early. Given the week, a hypertensive mother's baby at term is barely lighter. For a weight-for-age chart, that is the relevant comparison.

#Task 5 · The decision

Deliver, on the held-out births, comparing the conditional chart (Task 4) with the population chart (Task 3), with 90% intervals that include both charts' uncertainty and the sampling of the babies:

1. Per 1000 babies of smokers, and per 1000 babies of hypertensive mothers: how many fewer are flagged under the conditional chart?
2. Per 1000 births overall: how many babies stop being flagged, how many start being flagged, and the net change. Explain the sign of each.
3. A recommendation to the obstetric lead in a few sentences - including a reason that does not come out of the posterior.

#Solution

For each posterior draw, flag every held-out baby under both charts. Draws from the two fits are paired by index (they are independent fits, so this ignores their positive correlation and makes intervals of differences a little conservative). To include the sampling of the babies - the hospital's future patients are not these babies - each draw also gets Bayesian-bootstrap weights over the held-out births.

One trap: which babies switch cannot be counted by comparing two independent draws. Two draws of the same chart disagree about babies sitting near the line, so draw-by-draw "stop" and "start" counts are inflated by Monte Carlo jitter (their net is not). The network will deploy one chart of each kind - the posterior medians - so the switches are counted between those, with the Bayesian bootstrap for the babies.

What the numbers say.

• Among babies of smokers, the population chart flags about one in six; a smoking-specific chart would flag one in ten - about 60 fewer per 1000 babies of smokers, with an interval that stays well away from zero.
• Among babies of hypertensive mothers the change is smaller and uncertain - about 13 per 1000, with an interval from about -11 to +37 - because, given the week, hypertension moves the 10th centile little except before 37 weeks.
• Over all births the net change is about zero (-0.4 per 1000, interval about -8 to +7): a conditional chart is a 10th centile too, so it flags about 10% of births overall. It does not flag fewer babies; it flags different babies. Between the two deployed charts, about 3 babies per 1000 births stop being flagged - all of them babies of exposed mothers - and about 2 per 1000 start, all babies of unexposed mothers, whose own 10th centile is a little higher than the population's. (Counting switches draw by draw would have reported 6 per 1000 stopping: jitter, not policy.)

Recommendation. Keep one chart per sex (the calibrated population chart of Task 3) and do not introduce smoking- or hypertension-specific charts.

1. The normative reason, which is not in the posterior. A centile chart encodes what counts as normal growth. Smoking restricts fetal growth: babies of smokers are smaller because something is wrong, and that growth restriction carries the risks SGA flagging is meant to catch. A smoking chart would declare a smoking-restricted baby "normal for a smoker" and take it off the list - about 60 babies per 1000 smokers. The same holds for hypertension, a disease of the placenta. Customised growth charts (Gardosi's GROW approach) adjust for physiological determinants of size (sex, maternal height, weight, parity, ethnic origin) and deliberately not for pathological ones such as smoking.
2. The numbers do not make up for it. The conditional chart does not reduce the workload (net change about zero); it moves surveillance from exposed babies to unexposed ones, which is the wrong direction.
3. If the midwives' worry is over-flagging, the honest answer is that among babies of smokers a higher flag rate is expected and appropriate. Smoking and hypertension belong in the risk assessment alongside the SGA flag, not inside the definition of normal.

Caveats. The chart is fitted on US births of 2023; the network should check its own flag rate by week after a few months (Task 3's check, on their data). The preterm part rests on about 50 training births per sex below 32 weeks; it is shown with its band and should carry a "few data" note.

#Going further

• A 3rd centile ("severe SGA") line. Fit it separately with the same model and check whether it crosses the 10th anywhere in the posterior draws; then fit both jointly with $\log q_{0.03} = \log q_{0.1} - e^{g}$ and a smooth $g$, which cannot cross. Does the 3rd centile need its own learning rate?
• Does overconfidence roughen the curve? Refit both charts on a few other random training sets. Does the asymmetric Laplace sometimes learn a larger $s_f$ (a rougher curve) than the calibrated posterior? Why would an overconfident likelihood push the smoothing prior towards more curvature?
• Measurement error in the week. Even the obstetric estimate is uncertain by about a week. Treat the true gestational age as latent (a mixture over neighbouring weeks) and see how much the preterm 10th centiles move.
• A mixture for the lower tail. The 10th centile at 37 weeks is partly a statement about which babies are delivered early because they are small. Model birth weight at each week as a mixture of "normally grown" and "growth-restricted" components, and compare the flag it implies with the 10th centile.
• Customised charts done right. Add maternal height and parity (physiological), not smoking or hypertension (pathological), and redo Task 5. Which babies change status now?