Multiple Regression

The theory discussed in the previous chapter is readily extended to cover the case of regression on any number of variables. The theory of least squares provides estimates of unknown parameters, tests of significance, etc., and the methods are relatively simple provided the model can be expressed in the form (11.1) % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe % qaaiaadweacaGGOaGaamyEaiaacMcacqGH9aqpcqaHXoqycqGHRaWk % cqaHYoGydaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiEamaaBaaale % aacaaIXaaabeaakiabgkHiTiqadIhagaqeaiaacMcacqGHRaWkcqaH % YoGydaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiEamaaBaaaleaaca % aIYaaabeaakiabgkHiTiqadIhagaqeamaaBaaaleaacaaIYaaabeaa % kiaacMcacqGHRaWkcqWIVlctcqGHRaWkcqaHYoGydaWgaaWcbaGaam % 4AaaqabaGccaGGOaGaamiEamaaBaaaleaacaWGRbaabeaakiabgkHi % TiqadIhagaqeamaaBaaaleaacaWGRbaabeaakiaacMcaaeaacaWGwb % GaaiikaiaadMhacaGGPaGaeyypa0Jaeq4Wdm3aaWbaaSqabeaacaaI % Yaaaaaaakiaaw2haaaaa!6355! $$ \left. \begin{array}{l}E(y) = \alpha + {\beta _1}({x_1} - \bar x) + {\beta _2}({x_2} - {{\bar x}_2}) + \cdots + {\beta _k}({x_k} - {{\bar x}_k}) \\V(y) = {\sigma ^2} \\\end{array} \right\} $$ where all the y’s are independent, and where x̄ 1, x̄ 2, etc., are the means of the x’s. The quantities β 1,..., β k, are called partial regression coefficients, and they each measure the variation in y due directly to the variation in the respective x i , the other variables being fixed. The essential features of this model are: (i) Only y is a random variable. The x’s are assumed to be either under the control of the experimenter, or else to be measurable with negligible error. (ii) The expectation of y is a linear function of the unknown parameters β j . (iii) The observations y i , i = 1, 2,..., n, are uncorrelated and have constant variance.