# Reproducing the reference classes

:::{admonition} The question
:class: note

Do the four data-driven classes of Litman et al. (2025) reproduce on the SPARK release held
here, and can they be named as the authors named them? The wider question is whether the four
classes are a stable property of the phenotype or an artefact of fitting one mixture model to a
pooled sample. The later analysis tests that by re-estimating the model within strata of age at
diagnosis and diagnostic era, which is only meaningful against a solid, named reference.
:::

:::{admonition} The result
:class: tip

The four classes reproduce. The model recovers proportions of about 39, 29, 18, and 15 per cent
(Social or behavioural, Moderate challenges, Mixed ASD with developmental delay, and Broadly
affected) against the published 37, 34, 19, and 10; every named-class anchor holds; and the
overall seven-category profile correlates with the published figure at $r = 0.902$, close to the
authors' own SSC-replication value of $r = 0.927$. This named fit is the fixed reference the
rest of the analysis is measured against.
:::

:::{figure} /_figures/reproduction.png
:alt: Recovered class signatures against the published figure-1b profile, one panel per class
:width: 100%
:align: center

The recovered class signatures against the values read from figure 1b of Litman et al. (dashed),
one panel per named class, ordered by published class size. Each panel draws two recovered
conditions, the full `2026-03-23` release and the cohort cut back to the records present at the
authors' V9 freeze (see
{doc}`subsetting the cohort to the V9 freeze <../guides/subsetting-to-the-v9-freeze>`), and notes
the two recovered proportions against the published one. The two conditions' signatures sit almost
on top of each other in every panel, so the V9 cut leaves the class shapes unchanged and moves only
the proportions. Mixed ASD with developmental delay tracks the published profile closely
($r = 0.97$); Social or behavioural ($r = 0.85$) shows the weaker self-injury, social-communication,
and restricted-or-repetitive enrichment noted below; the recovered Broadly affected profile dips on
the developmental category where the published one is saturated.
:::

## Reading the result

On SPARK 2026-03-23 the model recovers four classes whose proportions are about 39, 29, 18, and
15 per cent, against the published 37, 34, 19, and 10. Every anchor holds, so the four recovered
classes map cleanly onto the four named classes. The overall seven-category profile correlation
against the published figure is $r = 0.902$, taken over the full four-class, seven-category
matrix (28 points).

The per-class correlations are $r = 0.97$ for Mixed ASD with developmental delay and $r = 0.85$
for Social or behavioural. Each is taken over only the seven category points of one class, so it
is coarse, read for direction rather than as a precise value. Broadly affected and Moderate
challenges have saturated profiles (uniformly high and uniformly low respectively), so their
per-class correlation is undefined and they rest on the anchors instead. The main per-class
divergence from the published profile is that Social or behavioural shows weaker
social-communication and restricted-or-repetitive enrichment here than in the paper.

## How solid the reproduction is

Resampling the 11,704 probands with replacement (the fitted labels held fixed) and recomputing
the correlation over 500 resamples puts a 95 per cent interval of $[0.893, 0.916]$ on the
overall $r$, so the reproduction is precise to about $\pm 0.01$ from sampling alone. Holding the
fit fixed, the bootstrap captures the sampling variability in the signature, not the variability
from refitting the model; the figure-read resolution of the target is a further, separate source
of uncertainty.

The seven-category profile the correlation rests on is itself stable, which
[stability under refitting](stability-under-refitting) shows directly: across 200
re-initialisations and 50 random halves of the cohort, the profile reproduces against the
reference at about 0.91 to 0.92, and no fit ever collapsed a class. The reproduction is therefore
not an artefact of one fit or one sample.

## How the reproduction is built

:::{dropdown} Feature typing

StepMix fits a Gaussian density to each continuous feature, a Bernoulli to each binary feature,
and a multinomial to each categorical feature, so each feature has to be typed correctly. The
typing is derived three ways and reconciled:

1. from the data dictionary, through `dscat`: a calculated score or a dropdown age coding is
   continuous; a radio item with exactly two coded levels is binary, otherwise categorical;
2. from the authors' released typing files, which carry their own assignment;
3. from the observed number of distinct values in the cohort, as a cross-check.

The dictionary inference and the released typing agree on 237 of the 238 features. The one
disagreement is `repeat_grade`, a yes/no item the authors placed with the continuous features
rather than the binary ones. The run defers to the released typing there, since the aim is to
reproduce the authors' model, and records the disagreement in a reconciliation report alongside
the cohort matrix. The reconciled typing is 38 continuous, 33 binary, and 167 categorical
features.
:::

:::{dropdown} The mixture model

The `fit` stage trains a StepMix general finite mixture model with four components: a one-step
joint estimation, the measurement densities set by the reconciled typing, and sex and age at
evaluation as structural covariates. The 200 random restarts are delegated to StepMix, as in the
released code, and the best restart by log-likelihood is kept. The released rounding of the
feature matrix to integers is applied at fit time, so the cached cohort matrix stays unrounded
while the model sees the values the authors fit on. The fit takes around ten minutes and predicts
a hard class label per proband.
:::

:::{dropdown} Per-class enrichment and the class signature

A feature is *enriched* in a class when probands there carry it more often, or score higher on
it, than the rest of the cohort, and *depleted* when they carry or score less. Each feature is
tested for this in every class against the rest, in both directions: a binomial test for binary
features and a Welch $t$-test for the others, Benjamini-Hochberg corrected within each class and
direction. A corrected $p$-value below $0.05$ marks the feature enriched or depleted. The 24
reverse-coded SCQ social items have their direction flipped, and the features are summarised into
the seven literature-defined categories (anxiety or mood, attention, disruptive behaviour,
self-injury, social or communication, restricted or repetitive, and developmental) as the signed
proportion enriched minus depleted. That seven-category vector is each class's *signature*, and
it is the currency every later investigation compares against.
:::

:::{dropdown} Naming the classes

The authors' three routes to naming a recovered class are mostly closed here: there are no shared
probands to match on, and no reference model was released. The classes are therefore aligned on
the published class signatures, the same currency the authors used to declare replication in the
SSC.

The primary mechanism is the set of named-class anchors, the substantive characteristics that
define each class:

- the highest-difficulty class overall, which is also the smallest, is Broadly affected;
- the most developmental of the rest is Mixed ASD with developmental delay;
- the largest of the rest, high on core, attention, and anxiety with no developmental delay, is
  Social or behavioural;
- the uniformly lowest is Moderate challenges.

The assignment is cross-checked for mutual consistency: the class named Broadly affected should
be both the highest overall and the smallest, the class named Social or behavioural should be the
largest, and so on. The published seven-category signature, read from the paper's figure, gives a
profile correlation for each class and overall as a second measure.
:::

:::{dropdown} The values read from figure 1b

The published seven-category signatures are not released as a numeric table, so they are read
from figure 1b of Litman et al. (2025), the per-category proportion-and-direction plot, at the
figure's resolution. The signed values run from $-1$ (depleted across that category) to $+1$
(enriched), in the seven-category order:

| Class | anxiety/mood | attention | disruptive | self-injury | social/comm | restricted/rep | developmental |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Social or behavioural | +1.0 | +1.0 | +0.95 | +0.50 | +0.50 | +0.45 | -0.90 |
| Moderate challenges | -0.90 | -0.95 | -1.0 | -0.95 | -1.0 | -1.0 | -1.0 |
| Mixed ASD with developmental delay | -0.90 | -0.45 | -0.65 | -0.10 | +0.10 | +0.05 | +0.45 |
| Broadly affected | +1.0 | +1.0 | +1.0 | +1.0 | +1.0 | +1.0 | +1.0 |

Broadly affected sits near $+1$ and Moderate challenges near $-1$ across every category, so both
profiles are saturated and their per-class correlation is undefined; they rest on the anchors
instead. The published class proportions the anchors use, also from the paper, are about 37, 34,
19, and 10 per cent for Social or behavioural, Moderate challenges, Mixed ASD with developmental
delay, and Broadly affected. Every value here is read to the figure's resolution, so it is
approximate; the numeric supplementary tables, if obtained, would replace them and let the
alignment use the published profile directly.
:::

## Caveats

The reproduction is benchmarked on the published profile and proportions, not on a per-proband
agreement, for two reasons stated up front. The authors' SPARK v9 release and their per-proband
labels are not available, so an exact reproduction of their cohort and a label-level comparison
are out of reach. And the published seven-category profile is read from the paper's figure at the
figure's resolution, not from a numeric supplementary table, so the correlation is read against
the figure. The cohort is also larger than the authors' (11,704 against 5,392), because the
release is later and broader.

With those caveats, the four classes reproduce with the correct structure and clean naming, and
proportions of the same order as the published ones. The smallest, Broadly affected, is about 15
per cent here against the published 10, the one proportion that stands apart. Cutting the cohort
back to the records present at the authors' V9 freeze (the third condition in the figure) shifts the
two largest classes toward the published split, Social or behavioural from 39 to 34 per cent and
Moderate challenges from 29 to 31, against the published 37 and 34, while leaving the smallest class
at 15 and the recovered signatures unchanged. A size-matched random subsample of the full release
does not reproduce those shifts, so they track the V9 records rather than the smaller sample, while
the inflated smallest class is unmoved by the cut. This named fit is the fixed reference the
stratified analysis is measured
against: that work asks whether the same four classes survive re-estimation within strata of age
at diagnosis and diagnostic era. Reproducibility is necessary for that test but is not the same
as validity. A partition can reproduce across samples and still reflect parent-reported,
deficit-framed measurement rather than a biological kind, a distinction the genetics arm and the
construct-validity checks speak to and this investigation does not.