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Unbiased mixed variables distance

This paper addresses the bias in existing mixed variable distance measures caused by differing variable types and scales by defining relevant concepts to quantify such biases and proposing a general formulation for constructing unbiased distances where individual variable contributions are independent of measurement types or units.

Original authors: Michel van de Velden, Alfonso Iodice D'Enza, Angelos Markos, Carlo Cavicchia

Published 2026-06-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Michel van de Velden, Alfonso Iodice D'Enza, Angelos Markos, Carlo Cavicchia

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to measure how different two people are from each other. You have a list of traits: their height (a number), their favorite color (a category), their age (a number), and their job title (a category).

In the world of data science, this is called a "mixed variable" problem. You are trying to mix apples (numbers) and oranges (categories) into a single fruit salad of "difference."

The paper by Van de Velden and colleagues argues that the standard recipes for making this salad are biased. They tend to make the oranges taste way stronger than the apples, or vice versa, simply because of how the ingredients are measured, not because one trait is actually more important than the other.

Here is a simple breakdown of their argument and solution:

1. The Problem: The "Volume Knob" is Broken

When we calculate how "far apart" two people are, we usually add up the differences for every trait.

The Issue: If you measure height in meters, the difference might be 0.5. If you measure it in millimeters, the difference is 500. If you have a job title with 50 options, the "distance" between titles might be huge compared to a color with only 3 options.
The Bias: The paper shows that standard methods (like the famous "Gower's distance") accidentally turn up the volume on categorical variables (like job titles or colors) and turn down the volume on numerical ones (like height or salary). It's like trying to compare a whisper to a shout just because the microphone settings are different. The result isn't a true measure of difference; it's a measure of how the data was formatted.

2. The Two Rules for a Fair Mix

To fix this, the authors propose two rules for a "fair" distance calculator:

Rule #1: Additivity (The Sum of Parts)
Imagine you are building a tower out of blocks. The total height of the tower should just be the sum of the heights of the individual blocks. The paper insists that the total "distance" between two people should simply be the sum of the differences for each specific trait. No fancy square roots or hidden math that changes how much each block counts.
Rule #2: Commensurability (The "Apples-to-Apples" Rule)
This is the most important part. "Commensurable" means "measurable on the same scale."
- The Analogy: Imagine you are paying a bill. You have a pile of pennies and a pile of dollar bills. If you just count the number of coins and bills without converting them to dollars, the pennies will seem to matter way more than the dollars.
- The Fix: You need to convert everything to a standard unit (like "average difference"). The paper demands that, on average, every single variable (whether it's a number or a category) should contribute the same amount to the total distance. If a job title has 50 options, we shouldn't just count "different = 1." We need to adjust the math so that the "weight" of that job title matches the weight of a height difference.

3. The Solution: The "Fairness Weight"

The authors provide a general formula to fix this. Think of it as a smart scale.

How it works: Before you add up the differences, the scale looks at every variable. It asks: "On average, how much does this variable usually change?"
The Adjustment: If a variable usually changes a lot (like a job title with 50 options), the scale puts a tiny weight on it. If a variable usually changes very little (like a color with only 3 options), the scale puts a heavy weight on it.
The Result: When you finally add them all up, every variable gets an equal say. The distance is now "unbiased." It doesn't matter if the data is a number or a word; the math ensures they contribute equally to the final score.

4. Testing the Theory

The authors didn't just write a theory; they tested it.

The Simulation: They created fake data with a hidden "true" pattern (like a secret shape). They then tried to find that shape using different distance methods.
- The old, biased methods got the shape wrong because they were too distracted by the categorical variables (the job titles).
- The new "unbiased" methods found the secret shape much more accurately because they treated all variables fairly.
The Real World Test: They used data from FIFA soccer players (2021). They looked at things like height, weight, and position. They showed that standard methods made the "position" variable dominate the results, while their new method balanced the influence of position against physical stats like height and weight.

The Bottom Line

The paper concludes that when we mix numbers and categories, we often accidentally let the type of data dictate the results, rather than the actual data itself.

Their solution is a new way to calculate distance that acts like a leveling field. It ensures that whether you are comparing a person's salary or their favorite sport, neither one gets a "free pass" to dominate the comparison just because of how it's measured. This gives researchers a much more honest starting point for grouping data or visualizing it.

Technical Summary: Unbiased Mixed Variables Distance

Problem Statement
In data analysis tasks such as clustering, classification, and visualization, measuring the distance between observations is fundamental. A significant challenge arises when datasets contain mixed variables—combinations of numerical (continuous or discrete) and categorical (binary, ordinal, or nominal) variables. Existing methods for calculating mixed variable distances, most notably Gower's general coefficient of similarity, often suffer from "bias." This bias manifests when the contribution of individual variables to the overall distance is unduly influenced by their measurement type, scale, or the number of categories, rather than their actual observed differences. Consequently, certain variable types may dominate the distance calculation, leading to distorted results in downstream analyses.

Methodology
The authors propose a general formulation for constructing "unbiased" mixed variable distances. Their approach is grounded in two theoretical properties:

Multivariate Additivity: The overall multivariate distance is defined as the sum of univariate, variable-specific distances. This ensures that the contribution of each variable is additive and explicitly defined, avoiding the implicit geometric assumptions of non-additive metrics like Euclidean distance (which decreases per-variable contribution as dimensionality increases).
Commensurability: The univariate contributions to the overall distance must be commensurable, meaning their expected values are constant across all variables, regardless of variable type, scale, or underlying distribution.

The authors derive a general distance formulation:
$d(x_i, x_l) = \sum_{j=1}^{p} w_j \delta_j(x_i, x_l)$
where $\delta_j$ quantifies the dissimilarity for variable $j$ , and $w_j$ is a weight. To achieve commensurability, the weights are defined as the inverse of the expected distance for that variable:
$w_j = 1 / E[d_j(X_i, X_l)]$

The paper systematically analyzes how common scaling and distance choices violate these properties:

Numerical Variables: The authors evaluate standard deviation scaling, range scaling (used in Gower's distance), robust range scaling, and principal component scaling. They demonstrate that while these methods standardize scales, they do not guarantee commensurability across different underlying distributions (e.g., Normal vs. Uniform vs. Skewed).
Categorical Variables: The authors review various category dissimilarity matrices (e.g., simple matching, Eskin, Occurrence Frequency, Hennig-Liao scaling, and association-based measures like Kullback-Leibler and Total Variance). They show that the mean distance for categorical variables is heavily dependent on the number of categories and the distribution of category probabilities.

To construct an unbiased distance, the authors propose estimating the expected mean distance for each variable from the observed data (or theoretical distributions) and applying the inverse of this value as a weight. This creates a family of unbiased distances where the "unit" of measurement is effectively the "mean distance."

Key Contributions

General Formulation: The paper provides a flexible, general framework for mixed variable distances that satisfies multivariate additivity and commensurability, allowing for various implementations of $\delta_j$ and scaling methods.
Bias Quantification: The authors derive the magnitude of biases introduced by popular methods (e.g., Gower's distance, Hennig-Liao scaling) and demonstrate how these biases vary with distribution types and the number of categories.
Unbiased Construction: They provide a concrete method to correct these biases by weighting variables based on their empirical or theoretical mean distances, ensuring that variable types and scales do not trivially dictate the distance structure.
Evaluation: The study evaluates both independent (variable-specific) and association-based (joint distribution) approaches for categorical variables, showing how association-based methods can further refine distance calculations in mixed settings.

Results
The authors validate their findings through a simulation study and an empirical application using FIFA player data:

Simulation: In a controlled environment with known underlying structures, the authors show that non-commensurable methods (like standard Gower or Naive Euclidean approaches) lead to variable contributions that are skewed by the number of categories or variable type. For instance, Gower's distance tends to over-weight categorical variables, while Hennig-Liao scaling with Euclidean distance can over-weight numerical variables.
Recovery of Structure: When retrieving a low-dimensional configuration via Multidimensional Scaling (MDS), the unbiased variants (specifically the "Unbiased Dependent" variant using association-based measures) demonstrated superior performance in recovering the true underlying structure, particularly as the number of categories increased.
Empirical Application: Analysis of the FIFA dataset confirmed that biased methods produce predictable distortions in variable importance and MDS solutions. The unbiased methods provided a more objective baseline where variable contributions were balanced according to the commensurability principle.

Significance and Claims
The authors claim that their work addresses a fundamental issue in distance-based data analysis: the lack of objectivity in how different variable types contribute to overall dissimilarity. By defining and constructing unbiased distances, they provide a more neutral starting point for exploratory analysis.

The paper explicitly states modesty regarding optimality. The authors do not claim that their proposed method is the "optimal" distance for all scenarios. Instead, they argue that an "unbiased" distance is one where variable types, scales, and measurement levels do not trivially impact the distance. If domain knowledge suggests specific variables should be weighted more heavily, this can be done intentionally; however, the proposed method ensures that such weighting is a deliberate choice rather than an artifact of the distance calculation method. The significance lies in providing a theoretically sound, general formulation that allows researchers to construct mixed variable distances that are free from inherent structural biases.