2.4. Measurements of F0-induced biases

We report three measurements of F0-induced biases in this study: (1) token error, (2) central tendency error, and (3) formant variability.

Token error is calculated as |e_i| = | f^i − f|, where f^i is the LPC-estimated formant frequency for the ith sample and f is the true formant frequency.

Central tendency error is calculated by subtracting the true formant frequency f from the mean of multiple formant estimations: ē = [(1/n) · Σ f^i] − f, where n is 1000, the number of samples for each group. Positive error values indicate over-estimation of formant frequency over multiple measurements; negative values, under-estimation. If token errors are random, then central tendency error is expected to be zero. This calculation is used to simulate the error of the estimated central tendency when a large number of tokens are averaged. For each sample, we also calculated the Boolean value denoting whether the nearest harmonic is on the same side of the true formant as the estimated formant f^i, as b_i = u(hf_i − f) XNOR u(f^i − f), where hf_i is the harmonic nearest to f for the ith sample, u(x) is the unit step function [u(x) is equal to 1 if x ≥ 0, and is zero elsewhere]. XNOR is the exclusive-nor operation; if both operands are greater than or both are less than zero, b_i = 1; otherwise, b_i = 0. The mean b¯i = (1/n) · Σ b_i denotes the proportion of error tokens in each group that can be explained by harmonic attraction.

Formant variability is calculated by dividing the SD of the measured formant frequencies for each age-gender-vowel group by the corresponding group mean. This ratio is known as relative standard deviation (RSD), or coefficient of variation, and has been used in Eguchi and Hirsh (1969) and Lee et al. (1999) to indicate normalized variabilities.