On the correction for the number of observations in the calculation of Mahalanobis distances

Abstract

Introduction. Mahalanobis distances (D²) are used in craniological studies to provide a generalised assessment of differences between samples, taking into account trait variances and correlations. However, sample distances tend to be biased upwards compared to the true D² value, especially in the case of small sample sizes. An adjustment for the number of observations, proposed by D. Rightmire, is one way of compensating for this bias. However, the conditions for its application and even the methods for calculating the variables may vary according to the characteristics of the sample. The aim of this study is to evaluate the effective impact of the adjustment on Mahalanobis distances and to explore approaches to reduce bias in sample estimates.

Materials and methods. The analysis was based on three aggregated series of male skulls fr om Bashkirs, Chuvash and Latvians. D² was calculated using an averaged covariance matrix. Three methods of calculating Mahalanobis distances were considered: with no adjustment for the number of observations, with Rightmire's adjustment applied to all distances, and with an adjustment that considers only statistically significant distances. In addition, alternative approaches to adjustment were tested in cases wh ere there were large differences in the number of observations for individual traits: using the harmonic mean of the sample sizes and calculating D² separately for linear and angular traits, followed by summation.

Results. Rightmire's correction, when applied to all distances, generally provides accurate and unbiased estimates of D². When adjustments are applied, the D² value may approach zero or even become negative, even in cases where there are significant differences between populations. Since the medians of the sample D² values are closest to the true D² values, all negative D² values can be safely transformed to zero without compromising the calculations. A D² value of zero does not imply that there are no morphological differences between samples. To approximate the true distance between samples, confidence intervals can be calculated, e.g. using bootstrapping procedures.

Conclusion. Rightmire's correction allows for unbiased Mahalanobis distance estimates when using an averaged covariance matrix and small sample sizes. However, the choice of a specific correction method should take into account the sample size and the variability in the number of observations for different traits. When working with poorly preserved cranial series, it is advisable to use the harmonic mean of the sample sizes or separate distance calculations for angular and linear traits. © 2025. This work is licensed under a CC BY 4.0 license.

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