Pearson correlation statistical testing

Pearson Correlation Statistical Testing

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Summary

Compute the Pearson R coefficient between two co-registered LA-ICP-MS element channels and assess its statistical significance by comparing the observed R value to a null distribution generated from random shuffles of one input channel. This quantifies spatial correlation between element distributions and provides a probability estimate (ρ) that the observed correlation arose by chance.

When to use

When you have two co-registered LA-ICP-MS element images and need to determine whether the spatial distribution of one element correlates significantly with another. Use this skill after image alignment and co-registration, particularly when interpreting whether observed spatial clustering of two elements is statistically meaningful or attributable to random chance.

When NOT to use

• Input images are not co-registered or have misaligned spatial coordinates; alignment must be verified before correlation analysis.
• One or both channels contain predominantly NaN or missing values in the region of interest, as this will bias the R and ρ estimates.
• You are comparing more than two element channels; use alternative multivariate methods (e.g., principal component analysis) instead.

Inputs

• Two co-registered LA-ICP-MS element channel images (numeric intensity arrays)
• Image or selected region of interest (optional; defaults to entire image)

Outputs

• Pearson R coefficient (real number, range −1 to +1)
• Probability value ρ (p-value; real number, range 0 to 1)
• Shuffled R distribution (array of R values from permutation tests)

How to apply

Load two co-registered element images (channels) into the Colocalisation Dialog via the image or selection context menu in pewpew. Compute the Pearson R coefficient over the entire image region or a user-defined region of interest. To assess statistical significance, calculate the probability (ρ) by comparing the observed R value to a distribution of R values obtained from random shuffles (permutations) of one of the input channels. Display and export the resulting R coefficient and ρ-value. Interpret R values close to +1 as strong positive correlation, close to −1 as strong negative correlation, and values near 0 as weak or no correlation. Use the ρ-value (e.g., ρ < 0.05) to reject the null hypothesis that the correlation is random.

Related tools

• pewpew (GUI application providing the Colocalisation Dialog for computing Pearson R and managing co-registered LA-ICP-MS image pairs) — https://github.com/djdt/pewpew
• pewlib (Python library underlying pewpew that implements LA-ICP-MS data import, image alignment, and correlation coefficient computation) — https://github.com/djdt/pewlib

Evaluation signals

• Pearson R coefficient is in the valid range [−1, +1] and matches manual computation on the input arrays.
• Probability value ρ is in the valid range [0, 1] and reflects the fraction of shuffled R values that equal or exceed the observed R in magnitude.
• Shuffled R distribution is approximately symmetric around zero, indicating that the permutation test is unbiased under the null hypothesis of no correlation.
• When applied to synthetic data with known correlation (e.g., identical channels), R approaches 1.0 and ρ approaches 0; when applied to independent random noise, R approaches 0 and ρ approaches 1.0.
• Results exported from the dialog via copy-to-clipboard or context menu match the displayed R and ρ values with full precision.

Limitations

• Pearson R assumes linear relationships; nonlinear spatial associations between channels may be missed or underestimated.
• The permutation-based ρ-value is computationally expensive for large images and may require significant runtime or memory; the number of shuffles influences precision of the p-value estimate.
• Spatial autocorrelation within each channel can inflate the significance of apparent correlations; the method does not account for non-independence of neighboring pixels.
• The presence of outliers or extreme intensity values in either channel can substantially affect R; no robust correlation variant (e.g., Spearman rank) is described in the workflow.

Evidence

• [other] Compute Pearson R coefficient over the entire image region and optionally calculate its probability (ρ) by comparing the R value to those of random shuffles of the input.: "Compute Pearson R coefficient over the entire image region and optionally calculate its probability (ρ) by comparing the R value to those of random shuffles of the input."
• [other] The Colocalisation Dialog computes Pearson R, Li ICQ, and Manders coefficients (via Costes thresholding method) to quantify spatial relationships between two element channels in co-registered images.: "The Colocalisation Dialog computes Pearson R, Li ICQ, and Manders coefficients (via Costes thresholding method) to quantify spatial relationships between two element channels in co-registered images."
• [other] Load two co-registered element images (channels) into the Colocalisation Dialog via image or selection context menu.: "Load two co-registered element images (channels) into the Colocalisation Dialog via image or selection context menu."
• [readme] Pew² is a GUI for importing and processing line-by-line, spot-wise and ablation-time-aligned LA-ICP-MS data using the python library pewlib.: "Pew² is a GUI for importing and processing line-by-line, spot-wise and ablation-time-aligned LA-ICP-MS data using the python library pewlib."