Simple linear regression

While simple linear regression uses only one independent variable for modeling, multiple linear regression uses more variables. 

There are instances where it is important to know if a given regression line is linear. For example, simple competitive antagonism should yield a linear Schild regression (see Chapter 6). A statistical method used to assess whether or not a regression is linear utilizes analysis of covariance. A prerequisite to this approach is that there... 

The analysis of experimental results by simple linear regression provide an equation from which the estimation is straightforward. Nevertheless, to obtain an accurate model, an equation for each structural type is needed. Thus, for hydrocarbons, which are one of the best examples for this approach, an equation for linear saturated hydrocarbons is required, one for the branched ones, and one for the cyclic compounds. The same is needed for unsaturated, then aromatic compounds etc. The more the study is based on a precise structural type, the better the linear adjustment and the better the forecast standard deviation but at the same time there will be fewer points with which to calculate the model and the forecast standard deviation will be higher. It is not simple to find a compromise and it was decided to give up on this approach as soon as the relevance of the Hass model was noted. 

The intensity of this effect was estimated by simple linear regression. 

There are many different methods for selecting those descriptors of a molecule that capture the information that somehow encodes the compounds solubility. Currently, the most often used are multiple linear regression (MLR), partial least squares (PLS) or neural networks (NN). The former two methods provide a simple linear relationship between several independent descriptors and the solubility, as given in Eq. (14). This equation yields the independent contribution, hi, of each descriptor, Di, to the solubility ... 

The offset a, and the multiplication constant bj are estimated by simple linear regression of the ith individual spectrum on the reference spectrum z. For the latter one may take the average of all spectra. The deviation e, from this fit carries the unique information. This deviation, after division by the multiplication constant, is used in the subsequent multivariate calibration. For the above correction it is not mandatory to use the entire spectral region. In fact, it is better to compute the offset and the slope from those parts of the wavelength range that contain no relevant chemical information. However, this requires spectroscopic knowledge that is not always available. 

All other approaches try and relate the child spectra to the parent spectra. In the patented method of Shenk and Westerhaus [41 Sh], in its simplest form, one first applies a wavelength correction and then a correction for the absorbance. Each wavelength channel i of the parent instrument is linked to a nearby wavelength channel j(i) in the child instrument, namely the one to which it is maximally correlated. Then, for each pair of wavelengths, i for the parent and j i) for the child, a simple linear regression is carried out, linking the pair of measured absorbances... 

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

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... 

The results from the simple linear regression of (1/r) versus (1/S) are shown graphically in Figure 17.1. 

Equation 17.7 suggests that the oxygen uptake rate (q02xv) can be obtained by simple linear regression of Co2(t) versus time. This is shown in Figure 17.5 where the oxygen uptake rate has been estimated to be 0.0813 nig/L-min. 

Having performed all the above computer runs, a simple linear correlation was found relating the required cpu time as model equivalent runs (MER) with the number of unknown parameters regressed. The correlation was... 

More on Simple linear least squares regression (SLLSR), also known as Simple least squares regression (SLSR) or univariate least squares regression... 

The scope of this chapter-formatted mini-series is to provide statistical tools for comparing two columns of data, X and Y. With respect to analytical applications such data may be represented for simple linear regression as the concentration of a sample (X) versus an instrument response when measuring the sample (Y). X and Y may also denote a comparison of the reference analytical results (X) versus predicted results (Y) from a calibrated instrument. At other times one may use X and Y to represent the instrument response (X) to a reference value (Y). Whatever data pairs one is comparing as X and Y, there are several statistical tools that are useful to assess the meaning of a change in... 

Principal component analysis (PCA) of the soil physico-chemical or the antibiotic resistance data set was performed with the SPSS software. Before PCA, the row MPN values were log-ratio transformed (ter Braak and Smilauer 1998) each MPN was logio -transformed, then, divided by sum of the 16 log-transformed values. Simple linear regression analysis between scores on PCs based on the antibiotic resistance profiles and the soil physico-chemical characteristics was also performed using the SPSS software. To find the PCs that significantly explain variation of SFI or SEF value, multiple regression analysis between SFI or SEF values and PC scores was also performed using the SPSS software. The stepwise method at the default criteria (p=0.05 for inclusion and 0.10 for removal) was chosen. 

Once suitable parameters are available the values of g can be correlated with them by means of either simple linear regression analysis if the model requires only a single variable, or multiple linear regression analysis if it requires two or more variables. Such a correlation results in a SPQR. In this work we consider only those parameters that are defined directly or indirectly from suitable reference sets or, in the case of steric parameters, calculated from molecular geometries. 

The traditional approach to grain-size normalization - simple linear regression of raw data - is likely to result in dubious interpretation ... 

Predominantly, Freundlich s fitted adsorption isotherms computed by means of simple linear regression were proposed for the mathematical description of the process studied. Unlike the Langmuir equation, the Freundlich model did not reduce to a linear adsorption expression at very low nor very high solute concentrations, as above resulted. 

Due to the orthonormality of V, this is a particularly simple linear regression calculation. The vector b is computed as ... 

The subsequent computation of the absorption spectra A from C and Y is a simple linear regression. This is followed by the normalisation of the concentration profiles to a maximum of one, as has been outlined already in the preceding chapter Iterative Refinement of the Concentration Profiles. The normalisation is done using the routine norm max. m (p.275). 

Both simple linear regression (SLR) and PLS equations were tested for accuracy and precision. The SEP values for methyl oleate, glucose, glutamate, and ammonium were, respectively, 0.65 (and 0.68, 0.57 three models), 1.35, 0.61, and 0.016. For an immensely complex system, these values are outstanding. 

Comparison of the regressions based on a simple linear equation P = - c.t (where P is the initial trifluralin... 

The transformed response values were regressed on the transformed amount values using the simple linear regression model and ordinary least squares estimation. The standard deviation of the response values (about the regression line) was calculated, and plots were formed of the transformed response values and of the residuals versus transformed amounts. 

The simple linear regression model will be assumed throughout this section That is. 

Amount Transformation. Step 2. The amount transformation was performed in a way similar to that of response by use of a power series but for a different reason. In this case linearity was desired in order to use a simple linear regression model. This transformation therefore required a test for satisfactory conformity. One can use a variety of criteria including the correlation coefficient or visual examination of the plot of rgsiduals verses amount. We chose the F test for lack of fit,... 

Assessing linearity is an important aspect in calibration work since lack-of-fit will usually lead to biased results. When a simple linear regression model is chosen, the more general test of goodness-of-fit becomes a test of linearity. 

Calibration Most process analyzers are designed to monitor concentration and/or composition. This requires a calibration of the analyzer with a set of prepared standards or from well-characterized reference materials. The simple approach must always be adopted first. For relatively simple systems the standard approach is to use a simple linear relationship between the instrument response and the analyte/ standard concentration [27]. In more complex chemical systems, it is necessary to adopt either a matrix approach to the calibration (still relying on the linearity of the Beer-Lambert law) using simple regression techniques, or to model the concentration and/or composition with one or more multivariate methods, an approach known as chemometrics [28-30]. 

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