The Classical Multiple Regression Model
The two line model of the previous chapter was a first example of the classical multiple regression model. In general, in the classical multiple regression model the expected value of an outcome variable Y is modeled as a function of the covariates X1,X2, . . . , Xp. Denoting the expected value of Y given the covariate values x1,x2, . . . , xp by μ(x1,x2, . . . , xp), we can express this model as
μ(x1,x2, . . . , xp) = β0 +β1x1 +β2x2 + . . . +βpxp . The regression coefficients β j in such a model have a clear and unique interpretation: If we compare two subjects with identical covariate values except for a difference in covariate Xj, and if the subjects differ by an amount of Δ in this covariate, then the expected values of Y differ between these two subjects by Δ×β j. This follows immediately from the model above, because
μ(x1,x2, . . . ,x j +Δ, . . . ,xp)−μ(x1,x2, . . . ,x j, . . . ,xp) = β0 +β1x1 +β2x2 + . . .+β j(x j +Δ)+ . . .+βpxp
− β0−β1x1−β2x2− . . .−β jx j− . . .−βpxp = β j(x j +Δ)−β jx j = β jΔ
In particular, if the subjects differ by an amount of 1 in Xj, then the expected values of Y differ by β j. This applies, of course, also to a binary variable Xj, for which β j is just the difference in the expected values between two subjects differing only in this covariate, but sharing all other covariate values.