Age curves – MetaLab

How do effect sizes change over development? Cao, Lewis, Tsuji, Bergmann, Cristia, & Frank (2025) fit developmental trajectories across 25 MetaLab datasets, comparing constant, linear, logarithmic, and quadratic functional forms with multilevel meta-regressions. The headline: most phenomena show no detectable age-related growth in effect size — and where growth exists, it is not linear. The figures below are interactive versions of the paper’s key analyses, drawn from the published model fits (repository).

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html`<style>
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    background-color: #eef2f6;
    padding: 0.2rem 0.5rem;
    border-radius: 0.4rem;
    margin-bottom: 0.25rem;
    width: auto;
  }

  .${ojs_ns} select,
  .${ojs_ns} input[type="text"],
  .${ojs_ns} input[type="file"] {
    background-color: #fff;
    border: 1px solid #d8dee4;
    border-radius: 0.4rem;
    padding: 0.2rem 0.4rem;
  }

  .${ojs_ns} input[type="checkbox"] {
    accent-color: #2780e3;
    margin-bottom: 0;
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html`<style>
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    background-color: #fff;
    color: #212529;
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    padding: 0.25rem 0.9rem;
    cursor: pointer;
  }

  .${ojs_ns} button:hover {
    background-color: #eef2f6;
  }
</style>`

curves = d3.json("slices/cao/curves.json")
aic = d3.json("slices/cao/aic.json")
slopes = d3.json("slices/cao/slopes.json")
es_points = d3.json("slices/cao/points.json")
points_by_ds = d3.group(es_points, (d) => d.dataset)

form_colors = ({ Constant: "#dc322f", Linear: "#859900", Log: "#2aa198",
  Quadratic: "#6c71c4" })

cao_datasets = [...new Set(curves.map((d) => d.dataset))].sort()

Which functional form fits development?

Each panel shows the fitted trajectory of effect size across age (with 95% CI) under each functional form, for one meta-analytic dataset. In 19 of 25 datasets the forms are statistically indistinguishable — mostly because there is little age-related change to describe.

viewof forms_shown = Inputs.checkbox(Object.keys(form_colors), {
  label: "Functional forms",
  value: Object.keys(form_colors)
})

viewof show_ribbons = Inputs.toggle({ label: "95% CI ribbons", value: false })

viewof show_points = Inputs.toggle({ label: "Individual effect sizes", value: true })

trajectory_panels = {
  clear_loading();
  const panels = cao_datasets.map((ds) => {
    const pts = points_by_ds.get(ds) ?? [];
    // clamp predictions to the dataset's own observed age range: the log
    // model diverges as age -> 0, and each dataset was only fit within its
    // own range (e.g. neonatal imitation lives entirely below 2 months)
    const amin = Math.max(d3.min(pts, (d) => d.age) ?? 1, 0.01);
    const amax = d3.max(pts, (d) => d.age) ?? 36;
    const cd = curves.filter((d) => d.dataset === ds &&
      forms_shown.includes(d.form) && d.age_months >= amin && d.age_months <= amax);
    let ymin = d3.min(cd, (d) => show_ribbons ? d.ci_lb : d.pred);
    let ymax = d3.max(cd, (d) => show_ribbons ? d.ci_ub : d.pred);
    if (show_points && pts.length) {
      // include the bulk of the points without letting outliers flatten the
      // curves; dots are clipped at the panel edge
      const ds_d = pts.map((p) => p.d).sort(d3.ascending);
      ymin = Math.min(ymin, d3.quantileSorted(ds_d, 0.02));
      ymax = Math.max(ymax, d3.quantileSorted(ds_d, 0.98));
    }
    const pad = (ymax - ymin) * 0.08 || 0.1;
    const plot = Plot.plot({
      width: 235, height: 160,
      marginLeft: 38, marginBottom: 28,
      x: { label: "age (mo)", ticks: 4, domain: [amin, amax] },
      y: { label: "d", ticks: 4,
           domain: [Math.min(ymin - pad, 0), ymax + pad] },
      marks: [
        Plot.ruleY([0], { strokeDasharray: "3,3", stroke: "#999" }),
        ...(show_points ? [Plot.dot(pts, { x: "age", y: "d", r: 1.6,
          fill: "#4a5568", fillOpacity: 0.3, clip: true })] : []),
        ...(show_ribbons ? [Plot.areaY(cd, { x: "age_months", y1: "ci_lb",
          y2: "ci_ub", z: "form", fill: (d) => form_colors[d.form],
          fillOpacity: 0.12 })] : []),
        Plot.line(cd, { x: "age_months", y: "pred", z: "form",
          stroke: (d) => form_colors[d.form], strokeWidth: 1.8 })
      ]
    });
    return html`<div style="flex: 0 0 auto;">
      <div style="font-size: 0.78rem; font-weight: 600; max-width: 235px;
        white-space: nowrap; overflow: hidden; text-overflow: ellipsis;"
        title="${ds}">${ds}</div>
      ${plot}</div>`;
  });
  return html`<div style="display:flex; flex-wrap:wrap; gap:0.6rem;">
    <div style="flex-basis:100%; display:flex; gap:1rem; font-size:0.85rem; margin-bottom:0.25rem;">
      ${Object.entries(form_colors).map(([f, c]) =>
        html`<span><span style="display:inline-block;width:0.8rem;height:0.8rem;background:${c};border-radius:2px;"></span> ${f}</span>`)}
    </div>
    ${panels}</div>`;
}

Linear age slopes across datasets

The linear model’s age coefficient (d per month, 95% CI) for every dataset. Red intervals exclude zero (9 of 25); even there, growth is gradual — at most 0.17 d per month.

slope_forest = {
  const rows = d3.sort(slopes, (d) => -d.estimate);
  return Plot.plot({
    width: 780, height: rows.length * 24 + 60,
    marginLeft: 240,
    x: { label: "Age slope (d per month)" },
    y: { domain: rows.map((d) => d.dataset), label: null },
    marks: [
      Plot.ruleX([0], { strokeDasharray: "3,3", stroke: "#999" }),
      Plot.ruleY(rows, { y: "dataset", x1: "ci_lb", x2: "ci_ub",
        stroke: (d) => d.significant ? "#dc322f" : "#333", strokeWidth: 1.6 }),
      Plot.dot(rows, { y: "dataset", x: "estimate",
        fill: (d) => d.significant ? "#dc322f" : "#333", r: 4,
        title: (d) => `${d.dataset}\n${d.estimate.toFixed(3)} [${d.ci_lb.toFixed(3)}, ${d.ci_ub.toFixed(3)}]\np = ${d.p.toFixed(4)}`,
        tip: true })
    ]
  });
}

Model comparison (ΔAICc)

Each row is a dataset; each column is one of the four candidate trajectory shapes. A cell shows how much worse that shape fits than the dataset’s best-fitting shape, in AICc units: 0.00 (bold) marks the winner, and bigger numbers mean worse fits. Shapes within ~4 units of the winner are statistically indistinguishable from it; shaded cells (Δ > 4) are meaningfully worse. In 19 of 25 datasets nothing is shaded — the four shapes can’t be told apart, usually because there is little age-related change to describe. In the six datasets with a meaningful contrast, the winner is logarithmic or quadratic, never linear.

aic_table = {
  const forms = ["Constant", "Linear", "Log", "Quadratic"];
  const byds = d3.group(aic, (d) => d.dataset);
  const rows = [...byds.entries()].map(([ds, vals]) => {
    const r = { dataset: ds };
    for (const f of forms) r[f] = vals.find((v) => v.form === f)?.delta;
    return r;
  });
  const cell = (v) => v === undefined ? "" :
    html`<td style="text-align:right; padding:0.25rem 0.75rem;
      ${v === 0 ? "font-weight:700;" : ""}
      ${v > 4 ? "background:#fdecea;" : ""}">${v.toFixed(2)}</td>`;
  return html`<div style="overflow-x:auto;"><table style="border-collapse:collapse; font-size:0.9rem;">
    <thead><tr style="border-bottom:2px solid #ccc;">
      <th style="text-align:left; padding:0.25rem 0.75rem;">dataset</th>
      ${forms.map((f) => html`<th style="text-align:right; padding:0.25rem 0.75rem;">${f}</th>`)}
    </tr></thead>
    <tbody>${rows.map((r) => html`<tr style="border-bottom:1px solid #eee;">
      <td style="padding:0.25rem 0.75rem;">${r.dataset}</td>
      ${forms.map((f) => cell(r[f]))}</tr>`)}</tbody>
  </table></div>`;
}

Why might trajectories look flat? An interactive explainer

The paper tested four explanations for flat trajectories. Two of them can be felt directly: publication bias that is stronger for younger infants props up early effect sizes, and method adaptation (harder tasks for older infants) pushes later effect sizes down — both flatten an underlying increase. Set a true developmental slope, then apply each distortion and watch what the observed literature would show.

viewof true_slope = Inputs.range([0, 0.1], { label: "True slope (d/month)",
  step: 0.005, value: 0.05 })

viewof bias_strength = Inputs.range([0, 1], {
  label: "Age-graded publication bias", step: 0.05, value: 0 })

viewof adaptation = Inputs.range([0, 0.1], {
  label: "Method adaptation (d/month penalty)", step: 0.005, value: 0 })

viewof resample = Inputs.button("Resample studies")

explainer = {
  resample;
  const n = 120;
  const rand_norm = () => {
    const u = Math.random() || 1e-12, v = Math.random();
    return Math.sqrt(-2 * Math.log(u)) * Math.cos(2 * Math.PI * v);
  };
  // simulate studies across 0-36 months
  let pts = d3.range(n).map(() => {
    const age = Math.random() * 36;
    const se = 0.15 + Math.random() * 0.25;
    const true_es = 0.1 + true_slope * age - adaptation * age;
    const obs = true_es + rand_norm() * se;
    // age-graded publication bias: young + null results get suppressed
    const sig = obs / se > 1.64;
    const suppress_p = !sig ? bias_strength * Math.max(0, (36 - age) / 36) : 0;
    return { age, obs, se, published: Math.random() > suppress_p };
  });
  const pub = pts.filter((d) => d.published);
  const fit = (rows) => {
    const sw = d3.sum(rows, (d) => 1 / (d.se * d.se));
    const mx = d3.sum(rows, (d) => d.age / (d.se * d.se)) / sw;
    const my = d3.sum(rows, (d) => d.obs / (d.se * d.se)) / sw;
    const b = d3.sum(rows, (d) => (d.age - mx) * (d.obs - my) / (d.se * d.se)) /
      d3.sum(rows, (d) => (d.age - mx) ** 2 / (d.se * d.se));
    return { a: my - b * mx, b };
  };
  const f = fit(pub);
  const lines = [
    { x1: 0, y1: 0.1, x2: 36, y2: 0.1 + true_slope * 36, kind: "true slope (no distortion)" },
    { x1: 0, y1: f.a, x2: 36, y2: f.a + f.b * 36, kind: "what the published literature shows" }
  ];
  return Plot.plot({
    width: 780, height: 420,
    marginLeft: 55,
    x: { label: "Mean age (months)" },
    y: { label: "Effect size (d)" },
    color: { legend: true,
             domain: ["true slope (no distortion)", "what the published literature shows"],
             range: ["#999", "#dc322f"] },
    marks: [
      Plot.ruleY([0], { strokeDasharray: "3,3", stroke: "#ccc" }),
      Plot.dot(pts, { x: "age", y: "obs", r: 3,
        fill: (d) => d.published ? "#268bd2" : "#bbb",
        fillOpacity: (d) => d.published ? 0.6 : 0.35,
        title: (d) => d.published ? "published" : "unpublished (suppressed)" }),
      ...lines.map((l) => Plot.link([l], { x1: "x1", y1: "y1", x2: "x2", y2: "y2",
        stroke: "kind", strokeWidth: 2.5,
        strokeDasharray: l.kind.startsWith("true") ? "6,4" : null }))
    ]
  });
}

Grey points are studies that were run but never published. The paper found that neither distortion consistently explains the observed flatness — see the full analyses.

Data and models: Cao et al. (2025), fitted objects from the paper’s repository. The paper’s corpus merges some MetaLab datasets (gaze following, word segmentation), splits language discrimination/preference, and uses the updated IDS-preference data of Zettersten et al. (2024) — so panels here can differ from the same datasets on the explorer.