What's the one-line summary?

On the 195-release CAD CPI m/m × USDCAD 15m sample, the 5-bucket surprise assignment explains 19.52% of the total variance in move_pips (SS_between = 33,031 pips² out of SS_total = 169,211 pips²). One-way F-test statistic is 11.52 on (4, 190) degrees of freedom, giving p = 2.14×10⁻⁸ — the between-bucket differences are massively significant. Bias-corrected ω² = 0.1775 shrinks the estimate slightly because R² is upward-biased by the number of degrees of freedom the bucket assignment spends (k-1 = 4). Cohen's f = 0.492 is 'large' by Cohen's own 1988 convention (>0.40). The 80.48% within-bucket noise is what Stats #23's partial correlation collapsed to near-zero (the identity r_XY|Z = 0 when Z is the categorical bucket assignment).

Why compute R² AND ω²? Aren't they the same thing?

R² (equivalently η² for one-way ANOVA) is SS_between / SS_total = 0.1952 — the observed sample proportion of variance explained by bucket. It's upward-biased as an estimator of the true population effect because with only 5 groups and 195 observations, the bucket means themselves are noisy estimates that soak up variance beyond what a hypothetical infinite sample would attribute. Hays (1963) proposed the ω² correction: (SS_between - (k-1)·MS_within) / (SS_total + MS_within). On this sample ω² = 0.1775 — about 1.8 percentage points below R². The gap is small here because n=195 is comfortable relative to k=5, so R² is only mildly optimistic. On a sample with n=30 and k=5 the gap would be larger (a heuristic: R²-ω² ≈ (k-1)·(1-R²)/n). For inference, quote ω²; for descriptive fit, quote R².

F(4, 190) = 11.52 is far above any critical value — is the p-value trustworthy at that magnitude?

The F-test's p-value derivation assumes homogeneity of variance (equal within-bucket σ²) and Normally-distributed residuals. Within-bucket SDs on this sample are 24.4, 26.1, 26.4, 27.5, 27.7 — a spread of at most 14% max-to-min, which passes any reasonable Levene / Brown-Forsythe test. Residuals are approximately Normal per Stats #6 (fat-tails discussion) with the caveat that the in_line bucket has fatter tails than the others. When variance homogeneity holds and n is moderate (n/k ≈ 40 here), F-test p-values under the null are known to be extremely accurate down to at least p = 10⁻⁶. At p = 2.14×10⁻⁸ we're 2-3 orders of magnitude past 'moderately violated F assumptions could distort the p-value by 10×'. Even a 100× coverage error would leave p safely below 0.001. The between-bucket differences are real.

How does Cohen's f = 0.492 map to 'small / medium / large'?

Cohen 1988 proposed three conventional thresholds for the f effect size: 0.10 small, 0.25 medium, 0.40 large. Cohen's f is defined as sqrt(η²/(1-η²)) — a monotone transformation of R² onto the (0, ∞) scale. Today's 0.492 is comfortably 'large' by Cohen's convention. Practically: the bucket assignment carries the same predictive signal as an ordinary least-squares model with an R² of about 20% on a continuous predictor. For the retail-trader interpretation, that's about the ceiling any single bucket-filter strategy on any Vantage News-Impact table would reach — the residual 80% within-bucket noise is what makes bucket-filter Sharpe ratios cap out around 1.0-2.0 rather than exploding.

How does this connect to Stats #23's partial correlation?

Stats #23 said: total Pearson r(surprise_z, move_pips) = -0.4256, and controlling for bucket collapses it to r_partial = -0.05 (with 95% CI crossing zero). The SS decomposition side of that (which Stats #23 mentioned but didn't derive) is what Stats #24 formalizes: 19.5% of move_pips variance is BETWEEN buckets, 80.5% is WITHIN. Total r's -0.4256 is picking up the between-bucket component (sqrt(0.1952) ≈ 0.442, close to the observed r of 0.4256). Partial r after removing bucket is near-zero because 80.5% within-bucket variance has essentially no linear surprise_z structure inside each bucket. Stats #23's slogan was 'buckets carry the signal'; Stats #24 quantifies it as η² = 0.20. The two posts describe the same underlying fact in two languages: partial correlation and ANOVA are algebraically the same variance decomposition, just expressed differently.

Can I reproduce this in the tool?

Yes for the raw data (event = Canada CPI m/m, instrument = USDCAD, window = 15m in Vantage News Impact). The ANOVA computation itself needs offline arithmetic — bucket means and SDs are visible in the tool's 5-bucket stats table; grand mean, SS_between, SS_within, F, and p need Python (numpy.mean, scipy.stats.f_oneway) or the equations spelled out in the post. Cross-verified against numpy on the 195-row release table pulled today.

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Statistics24 August 2026· 5 min read

Statistics for Traders #24: One-way ANOVA and R² on yesterday's Stats #23 sample — 19.52% of move_pips variance is between-bucket, F(4, 190) = 11.52, p = 2.14×10⁻⁸

Yesterday’s Stats #23 reported that on the CAD CPI m/m × USDCAD 15m sample, the surprise-move variance splits 19.5% between-bucket / 80.5% within-bucket, and stopped there. Today’s Stats #24 formalizes the same decomposition as one-way ANOVA: R² = 0.1952, F(4, 190) = 11.52, p = 2.14×10⁻⁸, Cohen’s f = 0.492 (large). Bias-corrected ω² = 0.1775 trims R² by ~2 points for the k-1 degrees of freedom the bucket assignment spends.

A single stacked horizontal bar sized to SS_total = 169,211 pips². The left segment is coral labelled SS_between 33,031 (19.52%). The right segment is muted grey labelled SS_within 136,180 (80.48%). Below the bar, five statistic tiles: R^2 (eta^2) 0.1952, omega^2 (unbiased) 0.1775, F(4, 190) 11.52, p-value 2.14e-8, Cohen's f 0.492. — One-way ANOVA decomposition of move_pips variance on CAD CPI m/m × USDCAD 15m, n=195. Between-bucket carries 19.5%; within-bucket carries 80.5%.

The bucket-level table

Bucket	n	Mean (pips)	SD (pips)	Contribution to SS_between (pips²)
big_miss	13	+24.315	27.450	6,257
small_miss	46	+18.470	27.699	11,502
in_line	84	+0.423	26.388	471
small_beat	32	-6.806	24.383	2,928
big_beat	20	-21.990	26.056	12,254

Grand mean = +2.7877p. SS_between = Σ n_g·(μ_g - μ_grand)² = 33,031 pips². SS_within = Σ_g Σ_i (x_i - μ_g)² = 136,180 pips². SS_total = SS_between + SS_within = 169,211 pips². Big_beat (n=20) and big_miss (n=13) both punch above their sample-share weight in the SS_between column because their bucket means sit far from the grand mean — that’s the tail-bucket pattern.

The five equations

The whole computation traces to five lines of arithmetic on the n=195 sample:

Quantity	Formula	Value
R² = η²	SS_between / SS_total	0.1952
MS_between	SS_between / (k−1) = 33,031 / 4	8,257.76
MS_within	SS_within / (n−k) = 136,180 / 190	716.74
F(4, 190)	MS_between / MS_within	11.5213
ω² (Hays 1963)	(SS_between − (k−1)·MS_within) / (SS_total + MS_within)	0.1775

Cohen’s f (a monotone re-scaling of R² onto the [0, ∞) interval) = √(0.1952 / 0.8048) = 0.4925. Cohen 1988 conventional threshold for "large" f is 0.40.

Same 19.5% number as yesterday, derived properly today

Stats #23 said "SS decomposition: surprise_z 90.9% between-bucket / 9.1% within (mechanical, since bucket boundaries are quantiles of z); move_pips 19.5% between-bucket / 80.5% within." That 19.5% number is today’s R² = 0.1952, the between-bucket share of move_pips variance. Yesterday’s post computed it as a byproduct of the partial-correlation identity r_XY|Z ≈ 0. Today’s post computes it directly via the ANOVA equations and extends to the F-statistic (test whether the bucket means truly differ), the p-value (extremely small), the unbiased ω² (a 2-point shrinkage of R²), and Cohen’s f (a "large" effect on the standard convention). Same quantity, three lenses.

Practical takeaway for bucket-filter strategy sizing

20% of the move_pips variance is explained by the bucket assignment. That’s a large effect by academic conventions and it’s the reason Vantage’s News Impact tables are worth staring at. It’s also the ceiling: no bucket-filter strategy on this event-pair can explain more than 20% of the variance, because 80% of it lives WITHIN each bucket where surprise_z has essentially zero remaining linear signal (Stats #23). If you filter by bucket and then hope a further z-continuous overlay adds edge, you run into the r_XY|Z ≈ 0 wall the partial correlation post just described.

The 80% within-bucket noise is also what caps bucket-filter Sharpe ratios in the 1-2 range — see Stats #13 (information ratio) for the two-bucket variant, which collapses to Welch’s t-statistic (a one-degree-of-freedom special case of today’s ANOVA).

Verification note

Sample: 195 non-contaminated Canada CPI m/m × USDCAD 15-minute releases (same as Stats #23), pulled from /api/v1/news-impact/releaseson 2026-08-24. Every number above cross-verified against Python numpy to 4 decimal places (bucket means and SDs, SS_between/within/total, F, R², ω², Cohen’s f). The p-value = 2.14 × 10⁻⁸ computed via the regularized incomplete-beta representation of the F-distribution CDF (no scipy dependency needed for the sanity check). Chart via a one-off script reusing scripts/insights-charts/svg.ts + theme.ts primitives + sharprasterization; not committed under scripts/insights-charts/ since the single-bar layout is single-use.

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