Ordinary least-squares linear regression coefficients

The most widely used techmque for deriving QSAR equations is linear regression, which uses least-squares fitting to find the best combination of coefficients in the QSAR equation (the technique is also referred to as ordinary least-squares). We can illustrate the least-squares technique using the simple case where the activity is a function of just one property (when the technique is known as simple linear regression). We therefore want to derive an equation of the form ... 

With this transformation, the linear regression model, using the ordinary least-squares method of determination, is valid. However, to employ it, we need to know the population serial correlation coefficient, P. We estimate it by r. The population Equation 3.9 through Equation 3.11 will be changed to population estimates ... 

Y = XB -I- F (see Figure 5), with y (7 x 1) a vector of response values for many samples, Y (7 x M) a matrix of response values for many responses, X (7 x K) the spectral data for the samples, b (Kxl) a vector of regression coefficients, B (X X M) a matrix of regression coefficients for the M responses, f I x 1) a vector of residuals and F (7 X M) a vector of residuals for many responses. One may consider percentages of fat, water and protein as three different responses (M = 3). The traditional least-squares solution for b B) is called multiple linear regression or ordinary least squares. The method was used in the pioneering days of NIR, when there were only few wavelength bands available. 

Table 2 offers a skeleton relationship of assumptions to goals in a hierarchical order (for a more complete discussion, see Eisenhart 1947). Random residual variation in T is associated with a host of small, unimportant (in context) contributions. Notice that the usual assumptions of homoscedas-ticity and normality are not imposed for specification and estimation. The least-squares estimates of the regression coefficients provided by MLR are the most efficient, unbiased linear estimates among all linear estimates for uniform error variance and are still unbiased for nonuniform error variance. The central limit theorem will give very good protection— just as with ordinary averaging—allowing... 

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