Correlation and Regression Analysis
Correlation and Regression Analysis tests measurement of the strength and direction of a linear relationship and the use of regression lines for prediction.
Correlation
Karl Pearson's coefficient is r = Σ(x − x̄)(y − ȳ)/√(Σ(x − x̄)²·Σ(y − ȳ)²), with −1 ≤ r ≤ 1. It is unchanged by a change of origin or by multiplying by a positive constant; multiplying one variable by a negative constant reverses its sign. Spearman's rank correlation is ρ = 1 − 6Σd²/(n(n² − 1)), where d is the difference of ranks; ties need a mid-rank correction.
Regression
The line of y on x is y − ȳ = b_(yx)(x − x̄), with b_(yx) = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)² = r·σ_y/σ_x. The line of x on y uses b_(xy) = r·σ_x/σ_y. Both lines pass through (x̄, ȳ), the two coefficients share the same sign, and r = ±√(b_(yx)·b_(xy)).
Worked example
If b_(yx) = 0.8 and b_(xy) = 0.45, then r = √(0.8 × 0.45) = √0.36 = 0.6, taken positive because both coefficients are positive. The correlation is moderate and positive.
Traps
Interchanging b_(yx) and b_(xy); taking √(b_(yx)b_(xy)) without checking the common sign; applying the rank formula with ties unadjusted; concluding causation from a high r; using the y-on-x line to predict x.