By Tormod Naes
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Additional info for A user-friendly guide to multivariate calibration and classification
14) C(x , - x>2- 3. Let us assume that we are interested in estimating a linear combination of the parameters, C C , ~=, c’p. Then it can be shown that among all linear unbiased estimators, the least squares estimator c$ will be the one with the smallest variance. , P,)’ has the smallest variance among all linear unbiased estimators. In other words, any other linear unbiased estimate will have a larger variance. This property, commonly known under the name GaussMarkoo theorem, is not proved here; the interested reader is referred to Rao (1965, p.
These indicator or dummy variables are defined as IND,, = 1 if the observation comes from level i (for 1 G i d k - l), and 0 otherwise. Combining the effects of these indicators with the effects of p quantitative variables XI,. , Xp,we can write the model as If the observations are from level k, the effect of the quantitative independent variables is given by E ( Y , ) = Po + If the observations are from level i (1 Q P c I= I PIX,, i G k - l), their effect is The parameter 8, is thus the effect of level i relative to level k.
X p l X l , .. , X q , . 28) To test the hypothesis that none of the additional variables Xq+ I , . . , Xp has 29 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING an influence on the dependent variable Y ( H , : flq+ I consider the F statistic = * .. - flP = 0), we Under H,,this statistic follows an F distribution with p - q and n - p - 1 degrees of freedom. Thus the test of H,:S,, = . - = bp = 0 against H,: at least one pi 0 (for 4 + 1 d i < p) is gwen by , If P > Fu(p - q, n - p - I), we reject H, in favor of HI at level a.
A user-friendly guide to multivariate calibration and classification by Tormod Naes