Multiple linear regression model

Gonzalez, A. G., TWo Level Factorial Experimental Designs Based on Multiple Linear Regression Models A Tutorial Digest Illustrated by Case Studies, Analytica Chimica Acta 360, 1998, 227-241. 

Two models of practical interest using quantum chemical parameters were developed by Clark et al. [26, 27]. Both studies were based on 1085 molecules and 36 descriptors calculated with the AMI method following structure optimization and electron density calculation. An initial set of descriptors was selected with a multiple linear regression model and further optimized by trial-and-error variation. The second study calculated a standard error of 0.56 for 1085 compounds and it also estimated the reliability of neural network prediction by analysis of the standard deviation error for an ensemble of 11 networks trained on different randomly selected subsets of the initial training set [27]. 

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. 

This problem corresponds to the multiple linear regression model with m= I, n=2 and p=3. In this case we take X =x2, x2=x and with Q,=l (all data points are weighed equally) Equations 3.19a and 3.19b become... 

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 . 

A possible relationship between DFR and the application rate, as well as the crop volume estimate (CrV), was investigated using a multiple linear regression model (ADFR = a + b AR + c CrV). No significant contribution of crop volume to the variation of ADFR was observed (p = 0.19 and p = 0.87 for high-volume applications and all applications, respectively). 

The factorial approach to the design of experiments allows all the tests involving several factors to be combined in the calculation of the main effects and their interactions. For a 23 design, there are 3 main effects, 3 two-factor interactions, and 1 three-factor interaction. Yates algorithm can be used to determine the main effects and their interactions (17). The data can also be represented as a multiple linear regression model... 

More specifically, let us consider the multiple linear regression model y = Xb + e, see Equation 4.36, which can be denoted for each object as... 

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

MLR is an inverse method that uses the multiple linear regression model that was discussed earlier [1,46] ... 

Like MLR, PCR [63] is an inverse calibration method. However, in PCR, the compressed variables (or PCs) from PCA are used as variables in the multiple linear regression model, rather than selected original X variables. In PCR, PCA is first done on the calibration x data, thus generating PCA scores (T) and loadings (P) (see Section 12.2.5), then a multiple linear regression is carried out according to the following model ... 

With the aid of multiple linear regression, model coefficients were calculated, which describe the effect of the variables pn the physical stability of the tablets. Since two levels of each variable were studied it was possible to calculate the linear contribution of the variables. The general form of the model which describes the effect of the variables is given by the following formula ... 

Empirical multiple linear regression models were developed to describe the foam capacity and stability data of Figures 2 and 4 as a function of pH and suspension concentration (Tables III and IV). These statistical analyses and foaming procedures were modeled after data published earlier (23, 24, 29, 30, 31). The multiple values of 0.9601 and 0.9563 for foam capacity and stability, respectively, were very high, indicating that approximately 96% of the variability contributing to both of these functional properties of foam was accounted for by the seven variables used in the equation. 

Table III. Empirical multiple linear regression model describing foaming capacity as a function of pH and suspension concentration.

M. Grotti, R. Leardi, C. Gnecco and R. Frache, Determination of manganese by graphite furnace atomic absorption spectrometry matrix eflects control by multiple linear regression model, Spectrochim. Acta, Part B., 54(5), 1999, 845-851. 

The multiple linear regression model assumes that in addition to the p independent x-variables, a response variable y is measured, which can be explained as an affine combination of the x-variables (also called the regressors). More precisely, the model says that for all observations (x, , y, ) with i = 1,. .., , it holds that... 

Figures 11 and 12 illustrate the performance of the pR2 compared with several of the currently popular criteria on a specific data set resulting from one of the drug hunting projects at Eli Lilly. This data set has IC50 values for 1289 molecules. There were 2317 descriptors (or covariates) and a multiple linear regression model was used with forward variable selection the linear model was trained on half the data (selected at random) and evaluated on the other (hold-out) half. The root mean squared error of prediction (RMSE) for the test hold-out set is minimized when the model has 21 parameters. Figure 11 shows the model size chosen by several criteria applied to the training set in a forward selection for example, the pR2 chose 22 descriptors, the Bayesian Information Criterion chose 49, Leave One Out cross-validation chose 308, the adjusted R2 chose 435, and the Akaike Information Criterion chose 512 descriptors in the model. Although the pR2 criterion selected considerably fewer descriptors than the other methods, it had the best prediction performance. Also, only pR2 and BIC had better prediction on the test data set than the null model.

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