Multiple linear regression definition

Note that the lipophilicity parameter log P is defined as a decimal logarithm. The parabolic equation is only non-linear in the variable log P, but is linear in the coefficients. Hence, it can be solved by multiple linear regression (see Section 10.8). The bilinear equation, however, is non-linear in both the variable P and the coefficients, and can only be solved by means of non-linear regression techniques (see Chapter 11). It is approximately linear with a positive slope (/ ,) for small values of log P, while it is also approximately linear with a negative slope b + b for large values of log P. The term bilinear is used in this context to indicate that the QSAR model can be resolved into two linear relations for small and for large values of P, respectively. This definition differs from the one which has been introduced in the context of principal components analysis in Chapter 17. 

It should be noted that the above definition of Xj is different from the one often found in linear regression books. There X is defined for the simple or multiple linear regression model and it contains all the measurements. In our case, index i explicitly denotes the i"1 measurement and we do not group our measurements. Matrix X, represents the values of the independent variables from the i,h experiment. 

Corrosion rate was evaluated with respect to, 1) flux of pollutants (sulfur oxides, nitrogen oxides, oxidants, and particles) to the steel during both wet and dry periods, 2) temperature, and 3) exposure history. Different definitions of when the steel was wet were evaluated to determine the most likely "critical relative humidity." Non-linear multiple regression techniques were used to determine the statistical significance of each factor and develop a theoretically consistent environmental damage function. 

The regression coefficient, R, has the same sign as the slope of the relationship. These definitions are applicable for two variable linear or nonlinear correlations as well as for multiple variable correlation. ... 

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