Pearson Correlation

The Pearson coefficient is invariant to changes in the location and scale of the variables. Adding a constant to all values or multiplying by a positive constant does not alter the correlation.

• Formula basis: Standardization of values into z-scores (mean=0, std=1)
• Practical implication: Comparing TPM-normalized vs. raw count data yields the same correlation
• Exception: Non-linear transformations (log, square root) will change the coefficient

Pearson correlation is highly sensitive to extreme values. A single outlier can dramatically inflate or deflate the coefficient, making it an unreliable measure for noisy genomic data without preprocessing.

• A single data point far from the regression line can dominate the sum of squared deviations
• Mitigation: Winsorization or robust alternatives like Spearman's rank correlation
• In RNA-seq, lowly expressed genes with high variance often act as influential outliers

Pearson correlation only captures linear relationships. It will fail to detect strong non-linear dependencies, returning a value near zero even when a clear functional relationship exists.

• Example: A perfect parabolic relationship (y = x²) can yield r ≈ 0
• Diagnostic: Always inspect scatter plots before relying on r alone
• Genomic context: Gene regulatory networks often exhibit sigmoidal or switch-like behavior that Pearson misses

The coefficient is strictly bounded between -1 and +1, providing an intuitive, normalized measure of association strength and direction.

• +1: Perfect positive linear relationship
• 0: No linear relationship
• -1: Perfect negative linear relationship
• Interpretation benchmarks: |r| > 0.7 is generally considered strong, 0.3-0.7 moderate, < 0.3 weak in biological contexts

Pearson correlation is symmetric: the correlation between X and Y is identical to the correlation between Y and X. It does not imply directionality or causation.

• Corr(X, Y) = Corr(Y, X)
• This distinguishes it from regression coefficients, which are asymmetric
• Critical note: In gene expression prediction, high correlation between predicted and observed values does not mean the model captures causal regulatory mechanisms

The square of the Pearson correlation coefficient equals the coefficient of determination (R²) in simple linear regression, representing the proportion of variance explained.

• r = 0.8 implies R² = 0.64, meaning 64% of variance is explained
• This squared relationship penalizes moderate correlations: r = 0.5 explains only 25% of variance
• Model evaluation: R² provides a more interpretable measure of predictive power than r alone

A non-parametric measure of monotonic relationship between predicted and observed expression values. Unlike Pearson, it operates on ranked data rather than raw values, making it robust to outliers and non-linear associations. Spearman's correlation assesses whether the relationship between variables can be described by any monotonic function, not just a straight line. It is often reported alongside Pearson to validate that high correlation is not driven by a few extreme expression values.

The square of the Pearson correlation coefficient, representing the proportion of variance in the observed expression that is predictable from the model's predictions. An R² of 0.90 indicates the model explains 90% of the variability in the experimental data. This metric is particularly sensitive to scale and bias errors; a model can have a high Pearson correlation but a low R² if predictions are systematically shifted from the diagonal identity line.

A foundational loss function that measures the average squared difference between predicted and observed expression levels. MSE heavily penalizes large prediction errors due to the squaring operation, making it sensitive to outliers. In gene expression prediction, MSE is often computed in log-transformed space (e.g., log(TPM+1)) to prevent highly expressed genes from dominating the error signal and to better reflect fold-change accuracy.

A reproducibility index that evaluates the agreement between two measurements by simultaneously assessing precision (how close points are to the fitted line) and accuracy (how close the fitted line is to the 45-degree identity line). Unlike Pearson, which only measures precision around any straight line, CCC penalizes systematic bias. This makes it a stringent metric for expression prediction where both the relative ranking and absolute value calibration matter.

A critical preprocessing step where raw expression counts are transformed using log2(TPM+1) or similar functions. This transformation stabilizes variance across the wide dynamic range of gene expression, making the data more normally distributed and reducing the disproportionate influence of highly expressed genes. Pearson correlation computed on log-transformed values reflects fold-change agreement rather than absolute count agreement, aligning with biological interpretation.

A validation strategy where Pearson correlation is computed per tissue type or aggregated across multiple tissues from resources like GTEx. This reveals whether a model captures tissue-specific regulatory programs or merely predicts a generic average expression profile. A model might achieve high genome-wide correlation by accurately predicting housekeeping genes while failing on tissue-specific genes, which are often of greater biological interest.