. Author manuscript; available in PMC: 2022 Jun 1.

Published in final edited form as: J Biomed Inform. 2021 Apr 20;118:103788. doi: 10.1016/j.jbi.2021.103788

Table 3.

Comparison of distance metrics.

Distance	Data type	Mathematical Expression	Method
Euclidean	Continuous	$d (i, j) = {({\| x_{i 1} - x_{j 1} \|}^{2} + {\| x_{i 2} - x_{j 2} \|}^{2} + \dots + {\| x_{i p} - x_{j p} \|}^{2})}^{1 / 2}$ $d_{B I N A R Y} = \sqrt{{(b + c)}^{2}}$	Distance in real space
Manhattan	Continuous	$d (i, j) = \| x_{i 1} - x_{j 1} \| + \| x_{i 2} - x_{j 2} \| + \dots + \| x_{i p} - x_{j p} \|$ $d_{B I N A R Y} = b + c$	Distance in real space
Jaccard Index	Asymmetric binary	$d = \frac{a}{a + b + c}$	Negative match exclusive
Hamming	Symmetric binary	$d = b + c$	Hamming-like
Gower	Nominal, ordinal; binary, continuous¹	$s (i, j) = \sum_{k = 1}^{n} s_{i j k} / \sum_{k = 1}^{n} δ_{i j k}$ $s_{i j k; B I N A R Y} = \frac{a}{a + b + c}$ $s_{i j k; N O M I N A L} = 1 i f x_{i k} = x_{j k}; s_{i j k; N O M I N A L} = 0 i f x_{i k} \neq x_{j k}$ $s_{i j k; Q U A N T I T A T I V E^{2}} = 1 - \| x_{i k} - x_{j k} \| / r_{k}$	Simple matching

Although the Gower coefficient can be implemented for multiple data types, in this study it is implemented for only nominal and ordinal data.

“Quantitative” = ordinal or continuous.