The Linear Regression Model

As we have already mentioned in Chapter 2, assuming linearity with respect to the unknown parameters, the general algebraic model can be reduced to the following form 

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

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. 

---

Often, it is not quite feasible to control the calibration variables at will. When the process under study is complex, e.g. a sewage system, it is impossible to produce realistic samples that are representative of the process and at the same time optimally designed for calibration. Often, one may at best collect representative samples from the population of interest and measure both the dependent properties Y and the predictor variables X. In that case, both Y and X are random, and one may just as well model the concentrations X, given the observed Y. This case of natural calibration (also known as random calibration) is compatible with the linear regression model... 

A valuable inference that can be made to infer the quality of the model predictions is the (l-a)I00% confidence interval of the predicted mean response at x0. It should be noted that the predicted mean response of the linear regression model at x0 is y0 = F(x0)k or simply y0 = X0k. Although the error term e0 is not included, there is some uncertainty in the predicted mean response due to the uncertainty in k. Under the usual assumptions of normality and independence, the covariance matrix of the predicted mean response is given by... 

Compared with the artificial neural network (ANN) approach used in previous work to predict CN12 the linear regression model by QSAR is as good or better and easier to implement. The predicted CN values, some of which are tabulated in Table 1, will be employed below to evaluate the different catalytic strategies to optimize the fuel. 

A model with a perfect fit will yield an RMSEE of 0. The r and RMSEE of the linear regression model built from the example data are shown in Eigure 12.1. 

The multiple linear regression model is simply an extension of the linear regression model (Equation 12.7), and is given below ... 

This is particularly important if the linear regression model is used rather than a mechanistic model. 

Prove that Newton s method for minimizing the sum of squared residuals in the linear regression model will converge to the minimum in one iteration. 

As for sample selection, I will submit two different methods for variable selection one that is relatively simple and one that is more computationally intensive. The simpler method68 involves a series of linear regressions of each X-variable to the property of interest. The relevance of an X-variable is then expressed by the ratio of the linear regression slope (b) to the variance of the elements in the linear regression model residual... 

Determine coefficients in the linear regression model and do analysis of variance taking into account check of lack of fit of the obtained regression model. 

In Example 2.26, we have obtained the linear regression model for dynamic viscosity y, P, as a function of mixing speed X3, min"1 and mixing time X2, min of composite rocket propellant. To determine the conditions of minimal viscosity, a method of steepest ascent has been applied. This method has defined the local optimum region that has to be described by a second-order model. Conditions of the factor variations are shown in Table 2.146. 

Values for all variation levels are shown in Table 2.154. Select FUFE 23 as a basic design of experiment. Determine the linear regression model from experimental outcomes, Table 2.155. Assume that the obtained linear model is inadequate and that there is curvature of the response surface. To check these assumptions, additional design points were done in the experimental center so that their average is y0=0.1097 (y0—estimate of free member in linear regression, i.e. y0 — 30). Since h0 — y0 = 3 is the measure... 

When drawing conclusions we must take into account lack of fit or inadequacy of the linear regression model. 

The linear regression model is inadequate with 95% confidence. Since the linear model is neither symmetrical nor adequate and since the application of the method of steepest ascent would lead to a one-factor optimization (b2 is by far the greatest), a new FRFE 24 1 has been designed with doubled variation intervals for X3 X3 and X4. 

Earlier research proved that the linear regression model adequately describes the octane number as a function of component ratio. According to XVERT algorithm, design core and candidate subgroups are shown in Table 3.10. 

It is interesting to note that the XVERT algorithm includes in a design of experiment the first two points from candidate groups. The linear regression model has this form ... 

Fig. 5. Dependence of paclitaxel production on headspace gas concentrations of ethylene, carbon dioxide and oxygen according to the linear regression model...

This section gives a review of the linear regression model, including an augmented form that allows specification of the subset of active effects. Prior distributions are introduced and relevant posterior distributions given. 

The kinds of calculations described above are done for all the molecules under investigation and then all the data (combinations of 3-point pharmacophores) are stored in an X-matrix of descriptors suitable to be submitted for statistical analysis. In theory, every kind of statistical analysis and regression tool could be applied, however in this study we decided to focus on the linear regression model using principal component analysis (PCA) and partial least squares (PLS) (Fig. 4.9). PCA and PLS actually work very well in all those cases in which there are data with strongly collinear, noisy and numerous X-variables (Fig. 4.9). 

One structure, ester VI, is more active than its carbonyl oxygen separation would suggest. However, even if this point is included in the linear regression model, the r2 value is still excellent (0.946). Several possibilities may explain this apparent anomaly and further studies on VI are in progress. 

Table 2 Errors and goodness of fit calculations associated with the linear regression model for nickel AAS data from Table 1...

Table 5 ANO VA table for the linear regression model applied to thefluorescence data, emission intensity as a function of concentration...

FIGURE 47.9 Diagnostic plots used in the model evaluation of the linear regression model that related log(AUC) to log(dose) incorporating subject type (1 for diseased subjects (patient), and 0 for healthy subject) as covariate. Top row The left and right panels are residuals showing that the adequacy of the model fit. Bottom row The left panel plot reinforces the fact that the model adequately describes the data. The right-hand plot shows the adequacy of the error model. 

The output statement in the first program creates a file with the data, the predicted masses by the linear regression model (Y ), and the residuals of the observed values minus the predicted values for each replicate at each expected mass. These deviations can be assessed for meaningful lack-of-fit. 

---