Single-response linear regression

We first discuss the regression, or estimation, of parameters in linear models tfom singleresponse data. Let us say that we have performed a set of TV experiments in which for each experiment A = 1, 2. W, the set of predictor variables xfis known a priori, and a measurement is made of the single-response variable We assume that this single-response variable depends linearly upon the predictors, 

In some instances, we wish to fit a model with a zeroy-intercept fio = 0, such that 

We introduce a common notation for both cases by defining for each experiment the vectors of predictors and parameters, 

Because we do not know the true values of the parameters, we must consider 

We further define for our set of experiments the design matrix X to contain for each experiment, the row vector of the predictor values, 

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For the single response linear regression model (w=l), Equations (3.17a) and (3.17b) reduce to... 

The simple linear regression model which has a single response variable, a single independent variable and two unknown parameters. 

Problems that can be described by a multiple linear regression model (i.e., they have a single response variable, 1) can be readily solved by available software. We will demonstrate such problems can be solved by using Microsoft Excel and SigmaPlot . 

An extension of linear regression, multiple linear regression (MLR) involves the use of more than one independent variable. Such a technique can be very effective if it is suspected that the information contained in a single dependent variable (x) is insufficient to explain the variation in the independent variable (y). In PAT, such a situation often occurs because of the inability to find a single analyzer response variable that is affected solely by the property of interest, without interference from other properties or effects. In such cases, it is necessary to use more than one response variable from the analyzer to build an effective calibration model, so that the effects of such interferences can be compensated. 

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