Testing the validity of the Montgomery–Koyama–Smith equation and the power law equation using 3231 tepals of a Magnolia species

Key message

The power-law equation provides marginally better accuracy than the Montgomery–Koyama–Smith equation for estimating total tepal area, with flexible definitions of maximum tepal length maintaining prediction reliability.

Abstract

Montgomery–Koyama–Smith equation (MKSE) and power law equation (PLE) were evaluated for estimating the total tepal area (A_T) of Magnolia × soulangeana flowers using 3231 tepals from 359 flowers. MKSE assumes an isometric relationship between the A_T and the product of summed tepal widths (L_KS) and maximum tepal length (W_KS), while PLE incorporates an allometric scaling exponent (α). Results showed α = 0.9561 (95% CI 0.9481–0.9641), confirming allometry. PLE exhibited slightly lower root-mean-square error (RMSE: 0.0149 vs. 0.0172) and mean absolute percentage error (MAPE: 1.18% vs. 1.35%) than MKSE. Redefining W_KS as a random selection from the largest 9, 6, or 3 tepal lengths per flower minimally affected model performance, with MAPE consistently below 5% even when sampling the entire length range. This flexibility simplifies field measurements without compromising accuracy. Variability in geometric series common ratios across flowers likely drives the observed allometric scaling. This study validates that A_T can be reliably estimated using summed widths and a flexibly defined maximum length, emphasizing PLE’s marginally superior fit. These findings advance methods for non-destructive floral trait quantification in species with fixed organ counts.