Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
Phase I YC baseline. The full Phase I YC sample (n = 2,686 with controls) produced a treatment coefficient of −0.009 (SE = 0.014, p > 0.10) — a null result — against a White profile acceptance rate of 17.5% (regression constant = 0.175***). The corresponding raw gap was 1.1 pp (White 17.4%, Black 16.4%, p = 0.44). The most relevant Phase I signal for Phase II is the pro-bono mentoring interaction: Treatment × Mentoring = −0.120** to −0.122**, indicating a gap of approximately 11–12 pp among founders who signaled selective engagement. This pattern — discrimination concentrating when founders make deliberate, higher-commitment decisions — is the direct Phase I precedent for H2. Phase II power calculations use the manuscript regression baseline of 17.5% for the simple connection condition and assume a 40–50% reduction (to ~10%) for the advice request condition based on the literature on commitment-level effects.
Minimum detectable effects. With n = 1,369 per arm (α = 0.05, two-tailed, 80% power), MDEs are as follows. For H1, the pooled race main effect (n = 2,738 per race), MDE = 2.52 pp; power to detect a 2 pp gap is 60% and a 3 pp gap is 92%. For H1 (simple connection only, WS vs. BS, baseline ~17.5%), MDE = 4.1 pp two-sided; power to detect a 3 pp gap is 57% and a 5 pp gap is 96%. For H2, the race gap in the advice request condition (WA vs. BA, assumed baseline ~8.5%), MDE = 3.0 pp two-sided (2.65 pp one-sided); power to detect the expected 3 pp gap is 80%. H2 is thus the best-powered individual test in the design, consistent with the theoretical and Phase I pro-bono evidence that discrimination concentrates in selective, higher-commitment contexts.
Power for H3 (DiD/interaction). Under the expected scenario (a 3 pp gap in advice requests and a 1.1 pp gap in simple connections) the expected DiD is 1.9 pp. Two-sided power for this DiD is approximately 19% (28% one-sided). Power reaches 40% at a 3 pp DiD, 63% at 4 pp, and 83% at 5 pp. The study would require a much larger sample to achieve 80% power for the expected 1.9 pp DiD which is not feasible as we used the full YC sample. We address this limitation through three design choices. First, ANCOVA with pre-treatment covariates (followers, activity score, top university, elite employer) reduces residual variance by an estimated R² of 10–18%. This meaningfully improves power for H2 (lowering the advice-request race gap MDE from 3.0 pp to 2.70–2.83 pp and raising H2 power from 80% to 84–87%) but yields only modest gains for H3, raising DiD power from 19% to approximately 22% two-sided. Second, H3 is tested one-sided given the unambiguous directional prediction and Phase I precedent (Treatment × Mentoring = −0.120**, p < 0.01), raising power for the expected 1.9 pp DiD from 19% to 28%. Third, H2 is designated primary and H3 secondary, so the study’s inferential burden falls on H2 where power is adequate.
Informative null. If H2 is null, that result is substantially more informative than Phase I's null. Phase I could rule out gaps larger than approximately 3.6 pp in the YC network (a 20% relative effect at the 17.5% baseline). A null on H2 would rule out gaps larger than 3.0 pp in the advice request condition — a 35% relative effect at the ~8.5% advice-request baseline. Ruling out a 35% relative discrimination effect in a high-stakes context would constitute meaningful evidence that discrimination is not operating, directly addressing the AE's concern that Phase I's low-stakes context precluded conclusions about higher-stakes interactions.