Spearman

Overview

Spearman's rank correlation coefficient ρ measures the strength of a monotonic relationship between a parameter x and an objective y. It works on ranks rather than raw values, so it handles non-linear but monotonic relationships and is robust to outliers.

The Importance Chart displays |ρ| (absolute value): it captures how strongly a parameter is related to the objective, regardless of direction.

Formula

Step 1: Convert values to ranks

$$\mathbf{x} = [3.1, 1.2, 4.5, 2.0]$$

$$\operatorname{rank}(\mathbf{x}) = [3, 1, 4, 2]$$

Ties receive the average rank:

$$\mathbf{x} = [1.0, 2.0, 2.0, 3.0] \implies \operatorname{rank}(\mathbf{x}) = [1, 2.5, 2.5, 4]$$

Here "values" refers to each of the two series being compared (parameter $x_i$ and objective $y$). Ranking is done independently for each series, based on that series' own values: the ranks of $x$ come from the values of $x$, and the ranks of $y$ from the values of $y$. This does not mean reordering the parameters according to the objective values.

The trial correspondence (rows) stays fixed; the $x$ column and $y$ column are each converted to ranks independently. This measures whether the rank of $x$ moves together with the rank of $y$ (i.e. whether a monotonic relationship exists).

For example, suppose four trials yield the following values (rows — the trial ordering — are fixed):

trial	$x$	$y$	$\operatorname{rank}(x)$	$\operatorname{rank}(y)$
A	3.1	10.0	3	2
B	1.2	30.0	1	4
C	4.5	5.0	4	1
D	2.0	20.0	2	3
$\operatorname{rank}(x)$ is determined by the magnitudes in the $x$ column and $\operatorname{rank}(y)$ by those in the $y$ column, each independently.

Step 2: Apply Pearson correlation to ranks

Let $R_x = \operatorname{rank}(x)$ and $R_y = \operatorname{rank}(y)$ denote the rank series from Step 1. That is, $R_{x_i}$ is the rank of the $x$ value of the $i$-th trial, and $R_{y_i}$ is the rank of the $y$ value of the same trial. $\bar{R}_x$ and $\bar{R}_y$ are the means of those rank series (equal to $\frac{n+1}{2}$ when there are no ties).

Pearson's correlation is then computed on these rank series $R_x$ and $R_y$:

$$\rho = \operatorname{Corr}(R_x, R_y) = \frac{\sum (R_{x_i} - \bar{R}$$x)(R{y_i} - \bar{R}y)}{\sqrt{\sum (R{x_i} - \bar{R}x)^2 \cdot \sum (R{y_i} - \bar{R}_y)^2}}

For ties-free data this simplifies to:

$$\rho = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}$$

where $d_i = R_{x_i} - R_{y_i}$ (difference of ranks).

Aside: What is Pearson's product-moment correlation?

For the definition and properties of the Pearson product-moment correlation used in Step 2, see Pearson product-moment correlation.

Spearman's correlation is simply this measure applied to the rank series instead of the raw values. Because ranks are evenly spaced, looking at the linear co-movement of ranks corresponds to looking at the monotonic co-movement of the original values.

Multiple Objectives

Spearman's ρ is computed independently for each objective $y_k$:

$$\rho_k(p_j) = \rho(p_j, y_k)$$

There is no cross-objective averaging: the dashboard keeps one score per (parameter, objective) pair. The Importance Chart displays $|\rho_k(p_j)|$ for the currently selected objective $k$ only — switching the objective selector recomputes and redraws the scores for that objective.

Characteristics

Strengths:

• Robust to outliers (uses ranks, not raw values).
• Detects any monotonic relationship, not just linear.
• Non-parametric — no distributional assumptions.
• Very fast: O(n log n).

Limitations:

• Cannot detect non-monotonic (U-shaped, multi-modal) relationships.
• Does not capture parameter interactions.
• No magnitude scale (tells you strength, not direction of effect size).

When to Use

• Small datasets (n < 50) — still reliable.
• First pass to screen all parameters.
• When the shape of the objective function is unknown.

References

• Spearman, C. (1904). The proof and measurement of association between two things. American Journal of Psychology, 15(1), 72–101. https://doi.org/10.2307/1412159