Regression analysis

• Regression fits a line, or equation, that lets you predict one variable's value (the outcome) from another (the predictor), turning an association into an actual prediction rather than just stating how strongly the two move together. • Worked example: predicted blood pressure = 100 + 0.5 × age. A 40-year-old's predicted blood pressure would be 100 + (0.5 × 40) = 120. • The slope is the predicted CHANGE in the outcome for each one-unit rise in the predictor. It is not the strength of association (that is r or r²) and not the predicted value when the predictor is zero (that is the intercept). • Regression is directional: a predictor (independent) variable and an outcome (dependent) variable are named explicitly, and the equation predicting Y from X differs from the equation predicting X from Y. Correlation, by contrast, is symmetric, it does not matter which variable comes 'first'. • Multivariable regression extends simple (one-predictor) regression by adjusting for several predictors, and possible confounders, at the same time, rather than examining one predictor in isolation. • Stepwise regression builds a multiple regression equation by entering predictors one at a time, in order from most to least statistically significant based on their correlation with the outcome, rather than entering them all at once. • Fitting a regression line presupposes some correlation between predictor and outcome, but that correlation does not itself need to be statistically significant, causal, or temporally ordered for a line to be fitted, fitting a line and trusting what it predicts are different steps. • Beyond simple and multiple regression sit three related techniques. MANOVA (multivariate analysis of variance) compares groups on several outcome variables at once, rather than on one. Factor analysis finds a small number of underlying 'factors' that explain the pattern of correlations across many measured variables; it is built directly on a correlation matrix. Cluster analysis groups cases (not variables) by similarity, with no outcome variable at all. None of the three is a syllabus leaf in its own right; all three generalise the idea multiple regression already introduces, using more than one variable at once. • In practice, this means once a linear relationship is confirmed appropriate, regression is used to build predictive equations, for example in clinical risk prediction models, and multivariable regression is the standard way to adjust an association for known confounders before drawing any causal-sounding conclusion.