•We now consider the general linear model:? = 𝛽0+ 𝛽1?1+ 𝛽2?2+ ⋯ + 𝛽𝑝?𝑝+ 𝜀•We will discuss how to estimate the model parameters,𝛽0, 𝛽1, 𝛽2, … , 𝛽𝑝, and how to test various hypotheses about them.•To start, suppose we have information onncases, or subjects𝑖 = 1, 2, … , 𝑛•Let??be the observed response value for subject𝑖and let??1, ??2, … , ??𝑝be the values on the explanatory or predictorvariables.
•Recall, the values of theppredictor variables are treated as fixedconstants; however, the responses are subject to variability.•Hence the model forsubject𝒊iswritten as??= 𝛽0+ 𝛽1??1+ 𝛽2??2+ ⋯ + 𝛽𝑝??𝑝+ 𝜀?•Also assume, as before that𝜀?is a random variable having a mean of0 and constant variance,𝜎2. We suppose that they are normallydistributed and that errors for different cases (𝜀?, 𝜀?) are assumedindependent.•Recall these points also imply that the responses?1, ?2, … , ?𝑛areindependent normal random variables with mean𝐸??= 𝜇?andvariance𝑉𝑎𝑟??= 𝜎2.
This model can be expressed in vector form, we write??= ??′𝜷 + 𝜀?where??′=1??1⋯??𝑝and𝜷 =𝛽0𝛽1⋮𝛽𝑝
Combining the vectors we obtain the following model:?1?2⋮?𝑛=1?11⋯?1𝑝1⋮⋯⋯⋱⋱⋮⋮1⋯⋯?𝑛𝑝𝛽0𝛽1⋮𝛽𝑝+𝜀1𝜀2⋮𝜀𝑛𝒚 = 𝑋𝜷 + 𝝐•𝒚is known as the response vector•X is a non-random matrix of our explanatory (predictor) variables known asthe design matrix•βis a vector of unknown parameters•𝝐is a random vector of errors