4. Pearson and Spearman correlation

In occupant safety CAE, we routinely need to answer a deceptively simple question: does the simulation track the test? Before any regression model is fitted, the first quantitative step is usually a correlation analysis between paired CAE and test response channels — HIC, chest deflection, femur force, head acceleration, belt force, and so on. Two coefficients dominate practice: Pearson's r and Spearman's ρ. They answer different questions, rest on different assumptions, and are frequently misapplied. This section establishes when each is appropriate, how to compute them from paired CAE/test data, how to test significance, and how to interpret the result honestly.

What Each Coefficient Actually Measures

Pearson's correlation coefficient quantifies the strength of a linear relationship between two continuous variables. For paired observations (x_i, y_i), i = 1…n:

r = Σ(x_i − x̄)(y_i − ȳ) / sqrt[ Σ(x_i − x̄)² · Σ(y_i − ȳ)² ]

It is bounded in [−1, 1], is symmetric, and is invariant to positive affine transformations of either variable (adding a constant or multiplying by a positive scale does not change r). That last property matters: if CAE outputs are in kN and test data in N, Pearson's r is unaffected by the unit mismatch, provided the scaling is consistent across all pairs.

Spearman's ρ is the Pearson correlation of the ranks of the two variables. It measures monotonic association — whether y tends to increase whenever x increases, without requiring that the increase be linear. It is computed by ranking x and y separately (average ranks for ties), then applying the Pearson formula to the ranks. Spearman's ρ is robust to monotone transformations (log, square root, exponential) and to outliers in the sense that a single extreme value cannot dominate the coefficient the way it can with Pearson's r.

Assumptions and When They Break in CAE/Test Data

Pearson's r assumes: (1) both variables are continuous and measured on an interval or ratio scale; (2) the relationship is approximately linear; (3) observations are independent; (4) for significance testing, the pairs are approximately bivariate normal, or n is large enough for the asymptotic distribution to hold. In occupant CAE, assumption (2) is the frequent offender. Head acceleration versus HIC, or belt force versus chest deflection, often shows a curved or saturating relationship — Pearson's r will then understate the true association and can even flip sign if the curve is non-monotone.

Spearman's ρ assumes only that the pairs are independent and that the relationship is monotone. It is the safer default when the scatter plot shows curvature, when a few high-severity tests dominate the range, or when the response is ordinal (e.g., injury severity bins, star-rating categories). It is not appropriate when the relationship is non-monotone (e.g., an inverted-U), because rank correlation will collapse toward zero even when a strong deterministic relationship exists.

Computation on a CAE/Test Dataset

Consider a worked example with n = 8 paired sled-test responses (values are illustrative, not measured):

Applying the Pearson formula to the raw columns gives r ≈ 0.94; applying it to the rank columns gives ρ ≈ 0.79. The gap is diagnostic: the last pair (T08) is a high-leverage point that inflates the linear coefficient while the rank coefficient reflects the more modest overall ordering agreement. Neither number is "wrong" — they answer different questions. Pearson's r says the linear fit is tight; Spearman's ρ says the monotone ordering is good but not excellent.

Significance Testing

For Pearson's r, the standard test statistic is:

t = r · sqrt( (n − 2) / (1 − r²) ),  df = n − 2

Under the null hypothesis H₀: ρ = 0 (no linear association) and bivariate normality, t follows a Student's t distribution with n − 2 degrees of freedom. For the example above, r = 0.94, n = 8, so t ≈ 6.75 with df = 6, giving p < 0.001 — a highly significant linear association. For Spearman's ρ, the same t approximation is commonly used for n ≥ 10; for smaller n, consult exact tables or use a permutation test. With n = 8 and ρ = 0.79, the exact permutation p-value is approximately 0.02, still significant at α = 0.05.

Two cautions apply. First, significance is not the same as agreement: with large n, even a modest r can be statistically significant while being practically useless for prediction. Second, the bivariate normality assumption is often violated in CAE/test data because the response distribution is skewed (HIC, for instance, is bounded below and right-skewed). For n below roughly 30, prefer a permutation or bootstrap confidence interval over the t approximation.

Interpreting Correlation in the CAE/Test Context

Correlation measures association, not agreement. A CAE model can correlate strongly with test (r = 0.95) while systematically over-predicting by 15% — the points lie tightly along a line, but that line is not y = x. For agreement, use Bland–Altman analysis or the concordance correlation coefficient (CCC), which penalizes both scatter and bias. Correlation is a necessary but not sufficient condition for a validated model.

Also distinguish population correlation from sample correlation. The r you compute is an estimate; report it with a confidence interval (Fisher z-transform: z = arctanh(r), SE = 1/√(n − 3), back-transform the interval). And remember that correlation is not causation — a strong CAE/test correlation across a fleet of load cases does not prove the model captures the underlying physics; it may simply reflect that both are driven by the same input severity.

Practical Workflow and Common Failure Modes

A defensible correlation analysis for a CAE/test comparison follows a short, repeatable sequence. First, assemble the paired dataset with a stable key (test ID, load case, channel) and verify that each CAE value corresponds to the same physical channel and the same time window as the test value — mismatched windows are the single most common source of spurious correlation. Second, plot the scatter with the identity line. Third, choose Pearson if the cloud is linear and roughly homoscedastic; choose Spearman if it is monotone but curved, or if outliers dominate. Fourth, compute the coefficient, its confidence interval, and a significance test appropriate to n. Fifth, and most importantly, interpret the number alongside an agreement metric.

Typical failure symptoms and their remedies: a high r with visible curvature indicates Pearson is being fooled by a few extreme points — switch to Spearman or fit a nonlinear model. A near-zero ρ with an obvious relationship suggests a non-monotone pattern — inspect the scatter and consider a segmented analysis. A significant p-value with a small r means the sample is large but the association is weak — do not claim validation. A large r that disappears when one test is removed signals a high-leverage outlier — report the sensitivity explicitly.

Used with these safeguards, Pearson and Spearman correlation provide a fast, transparent first read on CAE/test consistency and a principled basis for deciding whether a regression model is worth fitting at all. They are diagnostic instruments, not verdicts; the verdict comes from agreement analysis and physical review of the load cases that drive the residuals.