An updated estimate of the lower speed boundary of preferred stride ratio constancy

The preferred stride ratio (PSR), defined as the ratio of step length to cadence, is approximately invariant across a wide range of walking speeds in healthy adults but breaks down at slow speeds. The lower speed boundary at which this constancy is broken was estimated by Murakami and Otaka (2017) to be approximately 62 m min-1 ({approx} 1.03 m s-1) on the basis of unstandardised K-means cluster analysis applied to data from 21 healthy adults at five speed conditions. The present report re-examines this estimate using the digitised individual-level scatter of Fig. 1-A and the published group-level statistics of Table 1 of that study, applying three breakpoint estimators in parallel: (i) unstandardised K-means (replicating the original method), (ii) a Gaussian mean-and-variance changepoint estimator, and (iii) a piecewise-linear regression on PSR. Applied directly to the digitised scatter (n = 84 resolved markers from a total of 105; 44 of 44 slow-walk markers, 40 of 61 normal-walk markers), the unstandardised K-means estimator returned 62.0 m min-1, matching the originally reported value to the reported precision; the mean-and-variance changepoint estimator returned 55 m min-1; and the piecewise-linear estimator was numerically unstable on the raw heteroscedastic data. To quantify uncertainty, 5 000 Monte Carlo realisations of synthetic individual-level data were generated from a bivariate truncated-normal model conditioned on the published condition means and standard deviations and on the published within-cluster speed-PSR correlations. The Monte Carlo distributions gave median estimates of 61 m min-1 (95 % MC interval 55-67) for unstandardised K-means, 39 m min-1 (29-53) for the mean-and-variance changepoint estimator, and 35 m min-1 (19-49) for piecewise-linear regression. Under a log-normal sensitivity model the corresponding intervals were 60 [55, 66], 34 [20, 58], and 19 [5, 42] m min-1. The likelihood-based estimator placed the central tendency substantially below 62 m min-1, and its Monte Carlo intervals did not include the original boundary under either marginal-distribution model. An additional robust heteroscedastic segmented profile-likelihood analysis on log-PSR yielded lower Monte Carlo median breakpoints across all model specifications, although the full-variance intervals overlapped the original K-means boundary. The qualitative finding of Murakami and Otaka -- that PSR constancy breaks down at slow walking speeds -- is supported by the present reanalysis. The original 62 m min-1 boundary is reproduced under the unstandardised K-means estimator, where it reflects the location of the largest density gap in the published five-condition speed sampling rather than a formally estimated changepoint; estimators formally designed for changepoint detection localise the joint PSR mean-and-variance transition substantially below this value. O_FIG O_LINKSMALLFIG WIDTH=162 HEIGHT=200 SRC="FIGDIR/small/720900v2_fig1.gif" ALT="Figure 1"> View larger version (41K): org.highwire.dtl.DTLVardef@2bb53dorg.highwire.dtl.DTLVardef@187d9bborg.highwire.dtl.DTLVardef@1e7a6a0org.highwire.dtl.DTLVardef@16c587b_HPS_FORMAT_FIGEXP M_FIG O_FLOATNOFigure 1.C_FLOATNO Reproduction and likelihood-based extension of the boundary reported in Murakami and Otaka [5]. (A) Digitised individual-level scatter from Fig. 1-A of [5] (n = 84 resolved markers from a total of 105: 44 of 44 slow-walk markers and 40 of 61 normal-walk markers). The dashed vertical line marks the value 62.4 m min-1 as drawn in the original figure. (B) PSR variance amplification across the five speed conditions, expressed as Var(PSR)/Var(PSR)Preferred, on a logarithmic vertical axis. (C) Distributions of the breakpoint estimates over N = 5 000 Monte Carlo realisations under the bivariate truncated-normal model with cluster-specific within-cluster correlations: unstandardised K-means (median 61 m min-1), the Gaussian mean-and-variance changepoint estimator (median 39 m min-1), and piecewise-linear regression on PSR (median 35 m min-1). The dashed vertical line marks 62.4 m min-1. (D) Sensitivity of each estimator to the choice of marginal-distribution model (truncated normal vs. log-normal); error bars are 95 % Monte Carlo simulation intervals. (E) PSR mean {+/-} SD across the five speed conditions (Table 1 of [5], height-adjusted). C_FIG O_TBL View this table: org.highwire.dtl.DTLVardef@24fe39org.highwire.dtl.DTLVardef@ae8fdborg.highwire.dtl.DTLVardef@66a473org.highwire.dtl.DTLVardef@b6ad84org.highwire.dtl.DTLVardef@139bca7_HPS_FORMAT_FIGEXP M_TBL O_FLOATNOTable 1.C_FLOATNO O_TABLECAPTIONSource data reproduced from Murakami and Otaka [5], height-adjusted, n = 21 per condition. C_TABLECAPTION C_TBL