### Overview The previous page, [Why FoC Is Hard To Test](/research/arrow-study-2026/methods/foc-why-this-is-hard/), covered why FoC cannot be isolated and why a matrix design was used instead. This page covers what happens after the data is collected: how the matrix is turned into trends, and how to read those trends. The main tool is regression. Full model definitions live on the [FoC Analysis Appendix](/research/arrow-study-2026/methods/foc-analysis-appendix/). This page is intentionally light on math. ### What Is A Regression? A regression estimates how one variable relates to an outcome after accounting for other variables that changed at the same time. A non-arrow example: does adding a bedroom raise the price of a house? Larger houses tend to have more bedrooms, but they are also bigger overall, so a scatter plot of bedrooms against price cannot tell whether the price rose because of bedrooms or square footage. A regression of: ``` price = bedrooms + square footage + neighborhood ``` estimates the bedroom effect among houses that are otherwise similar on square footage and neighborhood. Same logic for this study. A scatter plot of FoC against broadhead group size mixes FoC with everything that moved with FoC. A regression of: ``` broadhead group size = FoC + total weight + spine + shaft family ``` estimates the FoC effect among builds that are otherwise similar on total weight, spine, and shaft family. Regression is not magic. If two variables move together tightly enough, the model cannot fully separate them. That limit shows up as wider uncertainty, not a hidden answer. ### What Is A Coefficient? For each variable in the regression, the model produces a coefficient: how much the outcome is predicted to move per one-unit change in that variable, after the other variables are accounted for. Worked example using one of the actual FoC results: ``` FoC coefficient on broadhead mean radius ≈ -0.44 in / pp ``` A 1 percentage-point increase in FoC is associated with about 0.44 inches smaller broadhead mean radius, holding total weight, spine, and shaft constant. Smaller is the better direction for groups. A +5 percentage-point FoC change therefore predicts roughly 5 × 0.44 = 2.2 inches of mean-radius improvement. The reports lead with this kind of inch-level translation rather than the raw coefficient. ### What Is A Confidence Interval? Every coefficient comes with a 95% confidence interval. A narrow interval means the data pinned the estimate down tightly; a wide interval means the data is less certain about the size or even the sign of the effect. Continuing the example: ``` FoC coefficient = -0.44 in / pp, 95% CI [-0.61, -0.20] ``` The whole interval is on the negative side, so the model is fairly confident the effect points toward smaller groups. If the interval crossed zero, the direction itself would be uncertain and the result would be called borderline. ### What Does “Borderline” Mean? Borderline does not mean wrong. It means the estimate has a clear direction, but at least one of the following is true: - The 95% confidence interval is wide. - The result depends visibly on which builds are included (leave-one-out check). - The result does not survive the stricter multiple-outcomes threshold (described below). Borderline results are reported, but not promoted to clean claims without further qualification. ### What Regression Does Not Do Regression does not turn the matrix into a perfect controlled experiment. Two limits matter for reading the FoC results. First, collinearity is real. If two variables move together very tightly, the model cannot fully separate them. The estimates remain unbiased on average but get more uncertain. The reports flag this where it matters. Second, regression cannot fix coverage gaps. Variables that did not vary in this matrix (very weak shafts outside the tested range, heavier external points, different bow classes, different draw lengths) are outside what this matrix can answer. ### Three Model Framings The same matrix can be analyzed in more than one way, depending on the question. The reports use three framings: [Figure: div] These are different questions. The reports keep them labeled separately so the reader can see when a result holds across framings and when it depends on which framing is used. The practical-package framing drops total weight and launch velocity because a separate study, [Are Fast Arrows Less Forgiving?](/research/arrow-study-2026/results/speed-vs-accuracy/), 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. That result supports, but does not prove, treating velocity as part of the package instead of a confound to control for. ### Leave-One-Build-Out For each headline result, the model is re-fit with one build removed at a time, cycling through every build. If a result depends on a single odd build, the estimate moves significantly when that build is dropped. The reports summarize this as “leave-one-out stable” or “leave-one-out fragile,” without listing every individual re-fit. ### Why “Corrected For Multiple Outcomes” Matters The matrix produces several KPIs (untorqued broadhead mean radius, torqued broadhead mean radius, broadhead extra drift, and so on). When several outcomes are tested at once, the chance of one looking statistically significant by accident is higher than the standard p-value suggests. The reports apply a stricter threshold to the strongest candidates. When a result is described as "survives the multiple-outcomes correction" or "detectable," that stricter threshold has been applied. Mechanics live in the [FoC Analysis Appendix](/research/arrow-study-2026/methods/foc-analysis-appendix/). ### How To Read A Forest Plot Several reports summarize results as forest plots, in the same format every time: - **Dot:** the estimated effect for one variable on one outcome. - **Horizontal bar:** the 95% confidence interval around that estimate. - **Vertical zero line:** the “no effect” reference. If the bar crosses the zero line, the direction is uncertain. The bar matters more than the dot. A tight bar that sits clearly on one side of zero is a confident effect; a wide bar that straddles zero is not. ### Why Effect Sizes Are Reported In Inches A coefficient like “-0.44 in / pp” is precise but hard to picture. The reports translate every headline coefficient into an inches-at-target prediction, so the question becomes “is two inches a meaningful change for me?” instead of “is -0.44 a meaningful coefficient?” The published table uses three FoC step sizes (+3, +5, and +10 percentage points) and a 100-unit spine stiffening step, with confidence intervals translated alongside. See the [Practical Effects Summary](/research/arrow-study-2026/data/foc-practical-effects-summary/) data table. ### Where To Read Next - [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 Appendix](/research/arrow-study-2026/methods/foc-analysis-appendix/)