Correlation and Regression

Correlation and Regression

1. Definition & overview

• Correlation and regression are the two complementary techniques for studying the relationship (association) between two continuous (quantitative) variables measured on the same set of individuals — e.g. height and anatomical dead space in children, dose and response, age and creatinine clearance (Swinscow & Campbell 10e Ch.11, p.111).
• The two methods answer different questions and must not be confused (Swinscow & Campbell 10e Ch.11, pp.111, 119):
  • Correlation measures the degree of association — how closely the two variables move together — and returns a single dimensionless number, the correlation coefficient r (it makes the two variables symmetric: neither is privileged as cause).
  • Regression estimates the mathematical relationship — the equation of the line that best predicts one variable (the dependent / outcome / response variable, y) from the other (the independent / predictor / explanatory variable, x) — and is directional (regressing y on x ≠ regressing x on y).
• The data for both are first explored by plotting a scatter diagram (scatter plot): the predictor x is conventionally placed on the horizontal axis and the outcome y on the vertical axis; each individual contributes one point (Swinscow & Campbell 10e Ch.11, p.111).
  • The scatter is inspected before any calculation — to judge whether the relationship is plausibly linear (a straight-line pattern), to detect curvature, outliers, and clustering, all of which invalidate a naïve r (Swinscow & Campbell 10e Ch.11, p.111).
• Within clinical/drug research these tools sit inside the larger framework of measures of association and multivariable analysis of cross-sectional and cohort data, where the choice of regression model is dictated by the type of outcome variable (continuous → linear regression; binary → logistic regression; rate / time-to-event → Poisson or Cox regression) (Browner — Designing Clinical Research 5e Ch.8, pp.~9–10, Table 8.3).