Regression and Multivariate Data Analysis Summary

(Note that this is technically a regression hyperplane, not a regression line.) This model is estimated by choosing estimates β̂0, ..., β̂p to minimize sum of squared residuals, ∑n i=1(yi − (β̂0 + β̂1x1i + ... + β̂pxpi)). In this model, β̂j is the estimated expected change in the response variable associated with a one-unit change in the j predictor, holding all the other predictors constant (this is a partial association, so the estimates will depend on all the variables in the model). β̂0 is the estimated expected value of the response variable when all the predictors equal zero (this may be irrelevant). The fitted values are ŷi = β̂0 + β̂1x1i + ...+ β̂pxpi. Note that the fitted values may also be written in matrix form as Ŷ = Xβ̂ = X(XTX)−1XTY = HY . We make the following assumptions about the model: