### Scope Of This Appendix The plain-language version is on the [FoC Analysis Overview](/research/arrow-study-2026/methods/foc-analysis-overview/). This page is for readers who want the model definitions, the diagnostic checks, and the rules used to translate coefficients into inches. It is intentionally short and does not reproduce every regression table. ### Sample Size The 2026 FoC matrix has 35 configured builds. The primary analysis uses the 34 builds that share the same 100-grain field point and 100-grain QAD Exodus broadhead. The 35th build (Easton 5.0 / 340 spine with a 300-grain external point) is excluded because it breaks the constant-100 gr-tip constraint; it remains in the public data tables for reference. Some KPIs use slightly fewer than 34 builds when individual groups were flagged as bad data. The exact n per KPI is recorded in the regression output and in the [Derived Per-build Analysis Table](/research/arrow-study-2026/data/foc-derived-per-build/). ### Regression Model Definitions All three framings are weighted least squares regressions, one row per build, one model per KPI. Build weights are inverse-variance weights from each group’s standard error, so noisier builds are treated with less confidence. #### Strict / weight-controlled model ``` KPI = β₀ + β₁·foc_pct + β₂·total_weight + β₃·spine + β₄·is_5_0 ``` Treats total weight and spine as nuisance variables. Answers: what does FoC predict, isolated from total weight and spine? #### Dynamic-spine model ``` KPI = β₀ + β₁·foc_pct + β₂·measured_static_spine_26_in + β₃·insert_weight_total + β₄·is_5_0 ``` Replaces total weight with insert and FACT mass, and replaces the spine label with measured static spine deflection. Answers: at fixed FoC, what does the spine and front-mass tradeoff look like? #### Practical package (no-weight / no-speed) model ``` KPI = β₀ + β₁·foc_pct + β₂·spine + β₃·is_5_0 ``` Drops total weight and launch velocity from the adjustment set, supported by the speed result in [Are Fast Arrows Less Forgiving?](/research/arrow-study-2026/results/speed-vs-accuracy/), which compared the same vanes and broadheads at ~290 fps and ~325 fps and did not detect a meaningful accuracy or forgiveness penalty from speed across that range. Answers: what do high-FoC packages do when total weight and speed are treated as part of the package, not separately controlled? All three models use heteroscedasticity-consistent (HC3) standard errors with finite-sample t inference. The choice between z- and t-inference shifts a borderline torqued-broadhead p-value, which is called out in the strict-model report. ### Collinearity Note Several pairs of variables move together strongly across this matrix: - **FoC and total weight:** raising front mass raises both. - **Total weight and launch velocity:** heavier arrows leave the bow slower, so the two carry overlapping information. - **Insert mass, length, and balance point:** the available inserts span these three properties together; this matrix cannot separate them. Collinearity does not bias the model on average, but it widens the confidence intervals on the affected coefficients. ### Multiple-Outcomes Correction Each model is fit against six KPIs, so the chance of one p-value clearing 0.05 by accident is elevated. The reports apply two corrections: - **Benjamini-Hochberg FDR:** milder, reported as `q (FDR)` or `p_adj (BH)`. Controls the expected proportion of false discoveries among declared discoveries. - **Bonferroni:** stricter (raw p-value × number of KPIs in the model). The practical-effects table records whether each result survives Bonferroni (`bonferroni_survives`). "Detectable" in the reports means a result survives Bonferroni. "Borderline" means the raw p-value is below 0.05 but does not survive Bonferroni and / or the confidence interval is wide. ### Leave-One-Build-Out For each headline coefficient, the model is re-fit 34 times, dropping one build at a time. The check looks at whether any dropped build moves the coefficient outside the original 95% CI, and whether any dropped build flips the sign. "Leave-one-out stable" means no individual build dominated the result. "Leave-one-out fragile" means at least one build moves the estimate enough to change the conclusion; fragile cases are called out by name in the reports. ### Practical Effect Translation Coefficients are translated into inch-level predictions for fixed input steps: [Figure: div] The +5pp FoC effect is also reported divided by the observed spread of the metric across builds (`effect_5pp_foc_div_observed_metric_spread`), so the predicted FoC change can be compared to typical build-to-build variation in the same KPI. Full table in the [Practical Effects Summary](/research/arrow-study-2026/data/foc-practical-effects-summary/). ### Insert-Geometry Caveat Insert weight, insert length inside the shaft, and insert balance point are highly correlated across the inserts in this matrix. Diagnostic models that try to give insert length or balance point their own coefficient produce very wide confidence intervals. This matrix cannot prove or rule out an independent insert-geometry effect beyond insert mass. The raw measurements are still published in the [Insert Specs](/research/arrow-study-2026/data/foc-insert-specs/) table. ### Why No Sweet-Spot Claim Was Made A curvature term was added to the regression on the broadhead-quality KPIs. The curvature was not statistically detectable across the tested FoC range, so the reports describe the trend as "no detectable reversal" rather than asserting a sweet spot. The matrix can rule out a strong reversal within the tested range, but cannot describe behavior outside it. A sweet spot above 30% FoC, for example, cannot be claimed or ruled out from this data. ### The Other Methods Pages - [Front-of-Center Testing Overview](/research/arrow-study-2026/methods/foc/) - [Why FoC Is Hard To Test](/research/arrow-study-2026/methods/foc-why-this-is-hard/) - [FoC Analysis Overview](/research/arrow-study-2026/methods/foc-analysis-overview/)