# Power-Cadence Relationships in Cycling: Building Models from a Limited Number of Data Points
## 2 Materials and Methods
Considering this, we applied two residual-based filtering strategies to select subsets of data for modeling the power-cadence relationship of each participant. These were compared against a model built from the complete dataset. All models were generated using third-order polynomial regression with the intercept fixed at zero [5,11,12], because power equals zero when there is no rotation of the cranks. The three models differed only in the number and selection criteria of the data points used (Figure 2):  
1. High-Number (HN) model: Included all recorded experimental data points, with an average of 81 \(\pm\) 7 points per participant. This model incorporated many submaximal data points (Figure 2b, Table 1). 2. Moderate-Number (MN) model: Included only the point with the highest positive residual (i.e., furthest above the fitted curve) in each 5-RPM cadence interval. This method, previously described [5], is intended to eliminate submaximal points and retain the most likely maximal point per interval (Figures 1b and 2c, Table 1). On average, MN models included 34 \(\pm\) 2 data points per participant (about 40% of the HN dataset). 3. Low-Number (LN) model: Included only five data points selected from strategic portions of the power-cadence relationship to ensure proper representation of three  
Figure 2: Illustration of experimental data points and models for a representative participant. **(a)** All data points recorded during testing. (b) High-Number (HN) model constructed using all experimental data points. (c) Moderate-Number (MN) model constructed using one point per 5-RPM cadence bin. **(d)** Low-Number (LN) model constructed using five maximal data points strategically distributed along the power-cadence curve. The experimental data corresponding to Ppeak (peak power), as well as the x- and y-coordinates of the apex of the power vs. cadence relationships are shown for each model. critical regions: (i) low cadences to inform the linear coefficient (_ax_), (ii) near the apex to capture peak curvature (_bx_2), and (iii) high cadences to define the descending portion of the curve (_cx_3). These data points were selected as (i) the point with the highest residual in the low-cadence range (0-60 rpm), (ii) the point of peak power output (Ppeak), the point with the highest residual 5-50 rpm below Ppeak cadence (left side of the apex), and the point with the highest residual 5-50 rpm above Ppeak cadence (right side of the apex), and (iii) the point with the highest residual within 10 rpm of the maximal cadence recorded during the flywheel-only sprint. This method was designed to capture strategically located maximal values for optimal estimation of Equation (1) coefficients (Figures 1c and 2d, Table 1). In total, 69 third-order polynomial models were generated (23 participants \(\times\) 3 datasets). For each model, curve goodness of fit was evaluated using the coefficient of determination (R\({}^{2}\)) and the standard error of the estimate (SEE). From each fitted equation, the x- and y-coordinates of the apex were extracted, corresponding to optimal cadence (Copt) and maximal power (Pmax), respectively. To further assess model accuracy, we calculated additional error metrics, including the mean absolute error (MAE), the mean absolute percentage error (MAPE), and the standard deviation (SD) of the prediction error. We also generated a Bland-Altman plot to visualize the agreement between modeled and experimentally measured peak power (Ppeak), reporting bias, and 95% limits of agreement. We also assessed the robustness of the LN model to data point selection by conducting a leave-one-out (LOO) sensitivity analysis. Specifically, we generated three additional LN models using only four of the five selected data points by systematically omitting: (i) the low-cadence point (0-60 rpm), (ii) one of the medium-cadence points (Ppeak, or the points 5-50 rpm below or above Ppeak cadence), or (iii) the high-cadence point (within 10 rpm of maximal cadence).