Linear regression model

Linear regression models a linear relationship between two variables or vectors, x and y Thus, in two dimensions this relationship can be described by a straight line given by tJic equation y = ax + b, where a is the slope of tJie line and b is the intercept of the line on the y-axis. 

Determine the relationship between S eas and Ca using a weighted linear regression model. ... 

In the following the standard unweighted linear regression model is introduced. All necessary equations are found in Table 2.1 and are used in program LINREG. In a later section (2.2.10) nonuniform weighting will be dealt with. 

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. 

Almost all widely used, reliable prediction models for logarithmic partition coefficients, and especially for the octanol-water partition coefficient log P w, are linear regression models with respect to fragment counts, atom types, bond types or... 

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]. 

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... 

The soil residue level is determined from the relative responses of the analytes to the internal standards. A five-point calibration curve is analyzed in triplicate, and the data are analyzed by a weighted 1 /x linear regression model. The calculated slope and intercept from the linear regression are used to calculate the residue levels in the soil samples. A 20% aliquot of the sample extract receives 10 ng of each internal standard... 

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. 

Let us consider first the most general case of the multiresponse linear regression model represented by Equation 3.2. Namely, we assume that we have N measurements of the m-dimensional output vector (response variables), y , M.N. 

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

This problem corresponds to the simple linear regression model (w= 1, n= 1, p=2). Taking as Q,=l (all data points are weighed equally) Equations 3.19a and 3.19b become... 

Once we have estimated the unknown parameter values in a linear regression model and the underlying assumptions appear to be reasonable, we can proceed and make statistical inferences about the parameter estimates and the response variables. 

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... 

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). 

Parczewski A, Danzer K, Singer R (1986) Investigation of the homogeneity of solids with a linear regression model. Anal Chim Acta 191 461... 

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... 

For 5 min, the type of burner (lower limit of 250 kW (series B and C) compared to 500 kW (series A)) also had a significant effect. For a linear regression model with interior finish and the burner level as the variables, the R = 0.96. Because the burner level primarily affected the total heat release from the plywood, the cross-product of burner level and type of interior finish is also a significant factor. At 10 min, the presence of insulation was more significant than the burner level. For a model with interior finish, insulation, and burner level as variables, the R = 0.95 for 10-min data. At 15 min, insulation was no longer a significant factor. This is consistent with the visual observations that the insulations in the plywood tests were gone after approximately 10 min. 

In an excellent paper, Zhao et al. [29] assembled a carefully reviewed literature set of human absorption data on 241 drugs. They showed that a linear regression model built with 5 Abraham descriptors could fit percent human absorption data reasonably well (r2 = 0.83, RMSE = 14%). The descriptors are excess molar refraction (E), polarizability (S), hydrogen bond acidity (A), hydrogen bond basicity (B), and McGowan volume (V), all related to lipophilicity, hydrophilicity, and size. In a follow-on paper, data on rat absorption for 151 drugs was collected from the literature and modeled using the Abraham descriptors [30]. A model with only descriptors A and B had r2 = 0.66, RMSE = 15%. 

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. 

---