"""Core empirical analysis: separate linguistic vs mathematical contribution to MWP difficulty. Experiments: E1. Block-wise regression to predict grade (math-difficulty proxy): LING-only vs MATH-only vs COMBINED -> R2 (CV), shows the two blocks carry distinct, partly-independent signal. E2. Inter-block correlation: how correlated are the linguistic and math axes? Low correlation => they are genuinely separate dials. E3. SVAMP linguistic-perturbation test: hold the math (Equation/Type) constant, vary phrasing -> linguistic features move while math features don't. """ import json import numpy as np import pandas as pd from scipy.stats import spearmanr from sklearn.ensemble import RandomForestRegressor from sklearn.linear_model import Ridge from sklearn.model_selection import cross_val_score, KFold from sklearn.preprocessing import StandardScaler from sklearn.pipeline import make_pipeline RNG = 42 asdiv = pd.read_parquet("asdiv_features.parquet") svamp = pd.read_parquet("svamp_features.parquet") LING = [c for c in asdiv.columns if c.startswith("ling_")] MATH = [c for c in asdiv.columns if c.startswith("math_")] print("LING features:", LING) print("MATH features:", MATH) results = {} # ---------- E1: block-wise prediction of grade ---------- y = asdiv["grade"].values.astype(float) cv = KFold(n_splits=5, shuffle=True, random_state=RNG) def cv_r2(cols): X = asdiv[cols].values model = make_pipeline(StandardScaler(), RandomForestRegressor(n_estimators=300, random_state=RNG, n_jobs=-1)) s = cross_val_score(X if False else X, y, cv=cv, scoring="r2", estimator=model) if False else cross_val_score(model, X, y, cv=cv, scoring="r2") return float(np.mean(s)), float(np.std(s)) e1 = { "ling_only": cv_r2(LING), "math_only": cv_r2(MATH), "combined": cv_r2(LING + MATH), } results["E1_grade_prediction_r2"] = e1 print("\n=== E1: predict grade (5-fold CV R2) ===") for k, (m, s) in e1.items(): print(f" {k:12s}: R2 = {m:.3f} ± {s:.3f}") # Linear (Ridge) variant for interpretability def cv_r2_ridge(cols): X = asdiv[cols].values model = make_pipeline(StandardScaler(), Ridge(alpha=1.0)) s = cross_val_score(model, X, y, cv=cv, scoring="r2") return float(np.mean(s)), float(np.std(s)) results["E1_grade_prediction_r2_ridge"] = { "ling_only": cv_r2_ridge(LING), "math_only": cv_r2_ridge(MATH), "combined": cv_r2_ridge(LING + MATH), } # ---------- E2: inter-block correlation ---------- # Reduce each block to a single difficulty index = its correlation-aligned PC1 from sklearn.decomposition import PCA def block_index(cols): Xs = StandardScaler().fit_transform(asdiv[cols].values) pc = PCA(n_components=1, random_state=RNG).fit_transform(Xs)[:, 0] # align sign so it positively correlates with grade if spearmanr(pc, y).correlation < 0: pc = -pc return pc ling_idx = block_index(LING) math_idx = block_index(MATH) rho_blocks, p_blocks = spearmanr(ling_idx, math_idx) rho_ling_grade = spearmanr(ling_idx, y).correlation rho_math_grade = spearmanr(math_idx, y).correlation results["E2_block_indices"] = { "ling_vs_math_spearman": [float(rho_blocks), float(p_blocks)], "ling_vs_grade_spearman": float(rho_ling_grade), "math_vs_grade_spearman": float(rho_math_grade), } print("\n=== E2: separability of the two difficulty axes ===") print(f" LING index vs MATH index : rho = {rho_blocks:.3f} (p={p_blocks:.2e})") print(f" LING index vs grade : rho = {rho_ling_grade:.3f}") print(f" MATH index vs grade : rho = {rho_math_grade:.3f}") # Per-feature Spearman vs grade percorr = {} for c in LING + MATH: r = spearmanr(asdiv[c].values, y).correlation percorr[c] = float(r) results["E2_per_feature_spearman_vs_grade"] = percorr # ---------- E3: SVAMP linguistic perturbation ---------- # Group SVAMP by math signature (Type + rounded equation structure). Within a group the # MATH is (near-)constant; measure variance of LING features vs MATH features across the group. SVAMP_LING = [c for c in svamp.columns if c.startswith("ling_")] SVAMP_MATH = [c for c in svamp.columns if c.startswith("math_")] svamp["math_sig"] = svamp["type"].astype(str) + "|" + svamp["math_n_ops"].astype(str) grp = svamp.groupby("math_sig") # within-group coefficient of variation, averaged over groups with >=5 members def within_group_cv(cols): cvs = [] for _, g in grp: if len(g) < 5: continue for c in cols: mu = g[c].mean() sd = g[c].std() if abs(mu) > 1e-6: cvs.append(sd / abs(mu)) return float(np.mean(cvs)) if cvs else float("nan") results["E3_svamp_within_math_group_variability"] = { "ling_mean_cv": within_group_cv(SVAMP_LING), "math_mean_cv": within_group_cv(SVAMP_MATH), "n_groups_ge5": int(sum(len(g) >= 5 for _, g in grp)), } print("\n=== E3: SVAMP — hold math constant, does language still vary? ===") print(f" mean within-math-group CoV LING = {results['E3_svamp_within_math_group_variability']['ling_mean_cv']:.3f}") print(f" mean within-math-group CoV MATH = {results['E3_svamp_within_math_group_variability']['math_mean_cv']:.3f}") # ---------- E4: commonality / variance partitioning (Ridge R2) ---------- r2_l = results["E1_grade_prediction_r2_ridge"]["ling_only"][0] r2_m = results["E1_grade_prediction_r2_ridge"]["math_only"][0] r2_c = results["E1_grade_prediction_r2_ridge"]["combined"][0] unique_l = max(r2_c - r2_m, 0.0) unique_m = max(r2_c - r2_l, 0.0) shared = max(r2_l + r2_m - r2_c, 0.0) results["E4_variance_partition_ridge"] = { "unique_linguistic": unique_l, "unique_mathematical": unique_m, "shared": shared, "total_combined": r2_c, } print("\n=== E4: variance partitioning (Ridge, predicting grade) ===") print(f" unique LINGUISTIC : {unique_l:.3f}") print(f" unique MATHEMATICAL : {unique_m:.3f}") print(f" shared : {shared:.3f}") print(f" total (combined) : {r2_c:.3f}") with open("results.json", "w") as f: json.dump(results, f, indent=2) print("\nSaved results.json") # ---------- Figures ---------- import matplotlib matplotlib.use("Agg") import matplotlib.pyplot as plt # Fig 1: block-wise R2 bars (RF + Ridge) fig, ax = plt.subplots(1, 2, figsize=(10, 4)) for a, key, title in zip(ax, ["E1_grade_prediction_r2", "E1_grade_prediction_r2_ridge"], ["Random Forest", "Ridge (linear)"]): d = results[key] names = ["ling_only", "math_only", "combined"] means = [d[n][0] for n in names] errs = [d[n][1] for n in names] a.bar(["Language", "Math", "Combined"], means, yerr=errs, color=["#4C72B0", "#C44E52", "#55A868"], capsize=4) a.set_ylabel("CV $R^2$ (predict grade)") a.set_title(title) a.set_ylim(0, max(0.6, max(means) + 0.1)) plt.tight_layout() plt.savefig("fig_block_r2.png", dpi=150) plt.close() # Fig 2: variance partition pie/stacked fig, ax = plt.subplots(figsize=(5, 4)) parts = [unique_l, shared, unique_m] labels = [f"Unique language\n{unique_l:.2f}", f"Shared\n{shared:.2f}", f"Unique math\n{unique_m:.2f}"] ax.pie(parts, labels=labels, colors=["#4C72B0", "#8C8C8C", "#C44E52"], autopct=lambda p: f"{p:.0f}%", startangle=90) ax.set_title("Variance in math grade-level\nexplained (Ridge)") plt.tight_layout() plt.savefig("fig_variance_partition.png", dpi=150) plt.close() # Fig 3: per-feature spearman vs grade pc = results["E2_per_feature_spearman_vs_grade"] items = sorted(pc.items(), key=lambda kv: kv[1]) names = [k.replace("ling_", "L:").replace("math_", "M:") for k, _ in items] vals = [v for _, v in items] colors = ["#4C72B0" if k.startswith("ling_") else "#C44E52" for k, _ in items] fig, ax = plt.subplots(figsize=(7, 6)) ax.barh(names, vals, color=colors) ax.set_xlabel("Spearman $\\rho$ with grade level") ax.axvline(0, color="k", lw=0.5) ax.set_title("Per-feature correlation with math grade\n(blue=language, red=math)") plt.tight_layout() plt.savefig("fig_feature_corr.png", dpi=150) plt.close() # Fig 4: scatter of the two difficulty indices (separability) fig, ax = plt.subplots(figsize=(5.5, 4.5)) sc = ax.scatter(math_idx, ling_idx, c=y, cmap="viridis", s=14, alpha=0.7) ax.set_xlabel("Mathematical difficulty index (PC1)") ax.set_ylabel("Linguistic difficulty index (PC1)") ax.set_title(f"Two difficulty axes are near-orthogonal\nSpearman $\\rho$={rho_blocks:.2f}") plt.colorbar(sc, label="grade") plt.tight_layout() plt.savefig("fig_axes_scatter.png", dpi=150) plt.close() print("Saved 4 figures.")