AN-040: Does coverage deferral amplify sponsor bias *only* for leaders?

3-way sponsored × deferred × rank1 is NULL across all four specs (+2.03 / +4.09 / +0.46 / −12.51, SEs 3.3–10.5pp, sign-inconsistent). The AN-033 null does NOT hide leader-specific deferral amplification. Incidental finding: split-sample sponsored_by has a sharp rank gradient — +5.58pp (rank 1, p<0.01), +11.52pp (rank 2, p<0.001), −2.81pp (rank ≥3, null) — consistent with the AN-026/AN-027 coordination story but orthogonal to the deferral lever question.

Question

AN-033 found γ on sponsored × deferred ≈ 0 across all specs. That null is on the pooled candidate sample. A natural concern: the within-candidate average might hide a sharper rank-specific effect.

Leader polls have more room for coverage-style suppression — the leader's strength typically comes from broad reach across bairros, so a poll that restricts to favorable subareas can mechanically inflate their lead. A rank-1 sponsor pulling the deferred-coverage lever might amplify their bias more than a rank-3 sponsor doing the same. If so, the pooled γ averages an amplification effect for rank-1 with a null effect for trailing ranks, washing it out.

The 3-way sponsored × deferred × I(final_rank=1) is the rigorous test. The split-sample sp × def within each rank bin is the descriptive companion.

Design

Same analysis-table sample as AN-001 / AN-033 (n=27,907 candidate-poll rows after dropping NaN error / muni). Same deterministic deferral classifier from AN-024. final_rank from the analysis_table.

Specs (matched to AN-033 ladder):

• Spec 1 (pooled OLS): all 7 terms (sp, def, rank1, sp×def, sp×rank1, def×rank1, sp×def×rank1)
• Spec 2 (within-candidate FE): same, but rank1 main is absorbed
• Spec 3 (within-candidate + institute FE + methodology controls):
  • institute FE, log(sample_size), days_to_election, days²
• Spec 4 (race × week FE, strict clean comparator): independent-poll comparator restriction + race × week FE

Plus descriptive companion: estimate the simple error ~ sp + def + sp:def | candidate FE separately within rank bins {1, 2, ≥3}.

Headline: 3-way coefficient on sp:def:rank1 in Spec 2 and Spec 3.

Results

Table at build/table/an-040-deferral-rank-heterogeneity.csv.

Headline: 3-way interaction across specs

Spec	sp×def×rank1	sp×def	sp×rank1	def×rank1	sponsored_by	n
1. pooled OLS	+2.03 (3.32)	−0.84 (2.43)	−3.19 (2.05)	−0.72 (0.60)	+10.18 (1.57) ***	27,907
2. + candidate FE	+4.09 (4.53)	−0.96 (3.13)	−2.31 (3.25)	−0.74 (0.57)	+7.59 (2.69) **	27,907
3. + institute FE + controls	+0.46 (4.83)	+1.79 (3.65)	−1.12 (3.89)	−0.61 (0.70)	+6.50 (3.20) *	27,907
4. race × week FE clean	−12.51 (10.48)	−0.76 (5.92)	+2.54 (8.91)	+0.71 (1.04)	+8.15 (4.98)	19,576

Cluster-robust SE on muni_id in parentheses. *** = p<0.001, ** = p<0.01, * = p<0.05.

The 3-way sp × def × rank1 is null across all four specs — sign-inconsistent and never close to significance. Rank-specific deferral amplification is not in the data.

Split-sample companion: AN-033 spec within rank bin

Rank bin	n	sponsored_by	deferred	sp × def
rank 1 (winners)	8,416	**+5.58 (1.86) **	−0.87 (0.50)	+3.33 (3.18)
rank 2 (close losers)	8,058	**+11.52 (2.52) ***	−0.27 (0.50)	−5.51 (3.33)
rank ≥ 3	10,751	−2.81 (7.19)	−0.13 (0.20)	+11.41 (7.36)

The headline within-candidate sponsor bias coefficient has a sharp rank gradient that the +7pp pooled average obscures:

• Rank 2 candidates over-state by +11.5pp (close losers — exactly where the AN-026/AN-027 coordination demand is highest)
• Rank 1 winners over-state by +5.6pp (in line with the pooled average, smaller than rank 2)
• Rank 3+ stragglers don't over-state at all (−2.8pp, null) — noisy in the small sample but directionally absent

The sp × def cell coefficients are noisier (SEs 3.2–7.4pp) and bounce in sign (+3.3 in rank 1, −5.5 in rank 2, +11.4 in rank 3+); none reach significance individually. The pattern doesn't sharpen the deferral lever specifically.

Interpretation

The headline test of the AN-033 follow-up hypothesis (leader-specific deferral amplification) is null — the 3-way interaction is small, sign-inconsistent, and underpowered across every spec ladder rung. Combined with AN-024 and AN-033, the conclusion on deferral is now tight: it is neither selected by sponsors (AN-024 wrong-signed at universe scale) nor amplified by them (AN-033 pooled null, AN-040 no rank heterogeneity).

The incidental rank gradient on the sponsored_by main effect is the more interesting result: rank 2 candidates over-state by twice as much as rank 1, and rank 3+ candidates don't over-state at all. This matches AN-026/AN-027's selection finding — runners-up at the viability cutoff have the highest coordination demand and the strongest incentive to inflate. But the headline-decomposition story this implies (sponsor bias is concentrated in close-loser polls) is orthogonal to the deferral question and is best chased as a companion analysis to AN-026/AN-027, not as a deferral lever.

Follow-ups

1. AN-NNN — sponsor-bias by final_rank × rank-at-commission (extension, highest follow-up paper value). The +11.5pp rank-2 sponsor effect (this analysis) and the +12-19pp rank-2 over-commissioning (AN-027) are pointing at the same population. A regression decomposing total observed bias as error ~ sponsored × rank_at_commission × |race_margin| would localize the bias in the close-runner-up cell directly. Suggested script: source/analysis/an-NNN-sponsor-bias-by-rank.py.

2. AN-NNN — same deferral heterogeneity on the surviving levers (extension, held under main forward task). Re-run the AN-040 3-way structure on mixed_population and urban_only_resolved when those become universe-scale available. The deferral null ≠ the other-levers null; each needs its own rank-heterogeneity probe.

3. AN-040 deferred × rank3+ small-cell puzzle (puzzle, low priority). The split-sample sp × def for rank 3+ candidates is +11.4pp (large but noisy, p=0.12). Worth a quick look at whether this is the result of a small handful of high-leverage rank3+ sponsored polls — could be a data-quality flag or could be a real but tiny effect.