Working with CDI vocabulary data

The MacArthur-Bates Communicative Development Inventories are parent-report checklists: a caregiver marks which of ~400–700 words their child produces (or understands). Words & Gestures (WG, ~8–18 mo, comprehension + production) and Words & Sentences (WS, ~16–30 mo, production) are the two English forms, with adaptations in dozens of languages. Every number you compute inherits the instrument's properties, and most disputes about vocabulary findings are really disputes about those properties. Start here, not with the model.

The literature you actually need

Measurement properties that change your analysis

• Ceiling and floor. WS ceilings out well before 30 months for advanced children; WG floors near 8 months. Any growth model fitted across the full age range is fitting censored data at both ends. Check the proportion at 0 and at max before interpreting a slope.
• The scale is ordinal, not interval. A 100-word gain from 50→150 is not the same learning as 500→600. This is the single most common source of bogus findings — including much of the historical "vocabulary spurt" literature, which Mayor & Plunkett (2010) argue falls out of the Zipfian difficulty distribution rather than a change in the child. If a claim depends on the shape of a raw-count trajectory, it is probably an artifact. Model item responses, or work in a latent ability space, or at minimum check the claim survives a monotone rescaling.
• Form differences are not child differences. WG and WS have different item sets; the overlap is partial. Never pool raw counts across forms.
• Parent report is a proxy with known biases (education, birth order, recency). It correlates well with lab measures in aggregate; it is noisier per child. Treat single administrations accordingly.

IRT on CDI items: use the Rasch model unless you need discrimination

The natural model is a Rasch/1PL: P(produce word j) = logit^-1(theta_i - delta_j), child ability minus item difficulty. The obvious upgrade is a 2PL adding a per-item discrimination lambda_j.

The 2PL fits better and changes nothing you care about. Fitted on five longitudinal CDI samples in three languages:

• 2PL wins LOO in every dataset, decisively (ΔELPD −1,400 to −4,400).
• Discrimination genuinely varies — SD of log lambda ≈ 0.24–0.34, a roughly 2× spread from the 10th to 90th percentile — and is only weakly related to difficulty (r ≈ 0.05–0.22).
• But age of acquisition is essentially identical: r(1PL, 2PL) = 0.986–0.999, median absolute difference 0.13–1.47 months. Predicted trajectories move by ≤4 percentage points in four of five samples.

The trap: the 2PL is much harder to sample. In the same runs, R-hat was 1.05–1.07 for the three English samples but 1.83 for Norwegian — badly unconverged — and Japanese, the smallest sample, was the one where results actually shifted (κ +28% vs +5–9% elsewhere, trajectory difference 0.15 vs 0.02–0.04). Larger item sets and sparser data make discrimination weakly identified. Always check R-hat per dataset on a 2PL; do not report a pooled diagnostic. If you only need AoA or a growth parameter, the Rasch model gives the same answer and converges.

The efficiency/acceleration decomposition

For growth modelling, a useful reparameterization of ability over time:

theta_i(t) = xi_i + kappa_i * log(t / a_0) + log H

• xi_i — efficiency: the child's level. Where the trajectory sits.
• kappa_i — acceleration: returns to experience. How much ability grows per log unit of accumulated input.
• kappa = 1 is the meaningful null — the pure accumulator, in which learning is proportional to input and nothing about the learner changes. Fitted CDI data lands around 11–13.

Two things to know before using this:

1. Cross-sectional data cannot separate xi from kappa. One observation per child makes "children differ in level" and "children differ in slope" generate identical clouds. You need repeated administrations per child. This is not a power problem; it is non-identifiability.

2. Per-child kappa_i needs 3+ administrations. With two, the per-child slope has zero residual degrees of freedom and sigma_b is inflated by sparse data masquerading as real variance. Use a 3+ filter for anything about individual differences in slope; 2+ is fine for population-level parameters.

The three artifact objections, and how they are answered

Anyone presenting CDI-derived growth parameters will get these. Each has a concrete empirical answer; know which one you have run.

• "It's the instrument." If kappa only reflects how the CDI spaces its items, compressing the difficulty range should shrink it proportionally. Refitting on deliberately narrowed item sets (middle 50%, middle 25%, easy/hard halves) compresses difficulty SD up to 6.5× — a pure rescaling account predicts kappa falls to 1.6–9.1, i.e. most of the way to the null. Observed across 20 narrowed fits: −3% to +29%, i.e. it does not fall, and mostly rises. The control matters as much as the test: 10 size-matched random subsets, which compress item count without compressing difficulty, move −3% to +4% and carry a ruler prediction of 10.4–13.5 — so the design separates "fewer items" from "narrower difficulty range". Run this before you need it.
• "It's the estimator." Fit the comparison system's own estimator to the child data and see whether the contrast survives. For the LM comparison this means a per-word 4-parameter logistic on the proportion of children not yet producing each word, with kappa = 0.434/scale — no IRT model anywhere. Child estimates come out robustly above the LM value (~1.2) but attenuated relative to the IRT fit, because pooling across children averages over heterogeneous acquisition curves. Correct for that with a recovery factor calibrated by simulation: generate data from the fitted model, run the identical pipeline, and take the ratio of recovered to true population kappa. Corrected estimates land close to the IRT values. Do not quote the raw pooled estimate as if it were comparable — the attenuation is the whole reason the correction exists.
• "It's a power law, so the result is definitional." Fractional return gamma = alpha is an identity for a power law; concede it. The content is not that returns are constant in the exponent but that the exponent is ~12 rather than ~1, and that a fitted kappa = 1 model cannot reproduce the observed range of acquisition ages.

Data access

• wordbankr — R client for Wordbank; get_instrument_data(), get_administration_data(), get_item_data(). Cite Frank et al. (2017).
• CHILDES via childesr for input-side frequency estimates. Word frequencies for CDI items usually come from parental speech in CHILDES, not from corpora of adult-directed language — the difference matters for AoA-vs-frequency analyses.
• Longitudinal samples are rarer than they look. Wordbank is mostly cross-sectional; the longitudinal subsets (Thal, Smith, Marchman in English, Simonsen's Norwegian, Hagihara's Japanese) are what support any slope claim. Some are unpublished datasets available through Wordbank and have no citable publication — label them like "Marchman (Wordbank)" rather than inventing a reference.
• Child IDs do not always link across forms. Verify that a longitudinal extraction actually joins WG to WS by child before trusting it; a silent failure here produces a dataset that looks longitudinal and is not, and it will show up as an inexplicable model anomaly rather than an error.

Quality filtering

Longitudinal CDI series contain impossible transitions — parents re-taking a form report large decreases, or implausible jumps. A defensible filter: exclude administrations showing a decrease of more than 25% of productive vocabulary, or an increase exceeding 40% of total vocabulary per month. On five samples this removed ~1.9% of children, but was very unevenly distributed (16% in one dataset, 0.5% in another). Report the per-dataset exclusion rate, not the pooled one — an uneven rate is itself a finding about data quality.