Statistical analysis least-square regression

After an alignment of a set of molecules known to bind to the same receptor a comparative molecular field analysis CoMFA) makes it possible to determine and visuahze molecular interaction regions involved in hgand-receptor binding [51]. Further on, statistical methods such as partial least squares regression PLS) are applied to search for a correlation between CoMFA descriptors and biological activity. The CoMFA descriptors have been one of the most widely used set of descriptors. However, their apex has been reached. 

The quantities AUMC and AUSC can be regarded as the first and second statistical moments of the plasma concentration curve. These two moments have an equivalent in descriptive statistics, where they define the mean and variance, respectively, in the case of a stochastic distribution of frequencies (Section 3.2). From the above considerations it appears that the statistical moment method strongly depends on numerical integration of the plasma concentration curve Cp(r) and its product with t and (r-MRT). Multiplication by t and (r-MRT) tends to amplify the errors in the plasma concentration Cp(r) at larger values of t. As a consequence, the estimation of the statistical moments critically depends on the precision of the measurement process that is used in the determination of the plasma concentration values. This contrasts with compartmental analysis, where the parameters of the model are estimated by means of least squares regression. 

Human perception of flavor occurs from the combined sensory responses elicited by the proteins, lipids, carbohydrates, and Maillard reaction products in the food. Proteins Chapters 6, 10, 11, 12) and their constituents and sugars Chapter 12) are the primary effects of taste, whereas the lipids Chapters 5, 9) and Maillard products Chapter 4) effect primarily the sense of smell (olfaction). Therefore, when studying a particular food or when designing a new food, it is important to understand the structure-activity relationship of all the variables in the food. To this end, several powerful multivariate statistical techniques have been developed such as factor analysis Chapter 6) and partial least squares regression analysis Chapter 7), to relate a set of independent or "causative" variables to a set of dependent or "effect" variables. Statistical results obtained via these methods are valuable, since they will permit the food... 

It may be possible to use an array of electrodes containing various enzymes in combination with multivariate statistical analyses (principal component analysis, discriminant analysis, partial least-squares regression) to determine which pesticide(s) the SPCE has been exposed to and possibly even how much, provided sufficient training sets of standards have been measured. The construction methods for such arrays would be the same as described in this protocol, with variations in the amounts of enzyme depending on the inhibition constants of other cholinesterases for the various pesticides of interest. 

The online statistical calculations can be performed at http //members.aol.com/ johnp71/javastat.html. To carry out linear regression analysis as an example, select Regression, correlation, least squares curve-fitting, nonparametric correlation, and then select any one of the methods (e.g., Least squares regression line, Least squares straight line). Enter number of data points to be analyzed, then data, x and y . Click the Calculate Now button. The analytical results, a (intercept), b (slope), f (degrees of freedom), and r (correlation coefficient) are returned. 

The following protocol was proposed and consisted of 4 measuring days. Each day, four (or six at day 1) standards and four samples are analyzed. The calibration curves are constructed by least squares regression analysis and statistically tested for nonlinearity by means of an F-test on the residuals. The amount of cortisol in the serum samples is obtained by linear interpolation on the daily calibration curve. Preliminary experiments were also set up to determine the influence of the use of peak height or peak area ratios. For the cortisol measurement, some separation takes place between syn and anti isomers, therefore the use of peak heights is less favorable. 

Standard statistical packages for computing models by least-squares regression typically perform an analysis of variance (ANOVA) based upon the relationship shown in Equation 5.15 and report these results in a table. An example of a table is shown in Table 5.3 for the water model computed by least squares at 1932 nm. 

Structure-activity correlations were carried out using least-squares regression analysis techniques on an IBM 360 computer. As in the accompanying publication (6), the data in Tables I and II were fitted to Equation 3 in stepwise fashion. Standard statistical tests were carried out at each stage of fitting to determine the over-all goodness of fit of the x and o- data to the various equational forms examined. As in our previous study (6), the most statistically significant correlations were always obtained when activity data for meta-substituted and para-sub-stituted TFMS herbicides were divided into two discrete series and fitted separately. 

Chemometrics is the discipline concerned with the application of statistical and mathematical methods to chemical data [2.18], Multiple linear regression, partial least squares regression and the analysis of the main components are the methods that can be used to design or select optimal measurement procedures and experiments, or to provide maximum relevant chemical information from chemical data analysis. Common areas addressed by chemometrics include multivariate calibration, visualisation of data and pattern recognition. Biometrics is concerned with the application of statistical and mathematical methods to biological or biochemical data. 

Statistical analysis Clustering based on correlations to identify group relationships. Partial least square regression analysis to track pathways of signal flow. Principal component analysis to identify significant signaling components. (82-85)... 

Near-infrared (NIR) spectroscopy is becoming an important technique for pharmaceutical analysis. This spectroscopy is simple and easy because no sample preparation is required and samples are not destroyed. In the pharmaceutical industry, NIR spectroscopy has been used to determine several pharmaceutical properties, and a growing literature exists in this area. A variety of chemoinfometric and statistical techniques have been used to extract pharmaceutical information from raw spectroscopic data. Calibration models generated by multiple linear regression (MLR) analysis, principal component analysis, and partial least squares regression analysis have been used to evaluate various parameters. 

Within each of the assays A, B, C, and D, least squares linear regression of observed mass will be regressed on expected mass. The linear regression statistics of intercept, slope, correlation coefficient (r), coefficient of determination (r ), sum of squares error, and root mean square error will be reported. Lack-of-fit analysis will be performed and reported. For each assay, scatter plots of the data and the least squares regression line will be presented. 

An alternative approach to linearity analysis is to analyze the linearity of all the data. The SAS code for the overall analysis is the same as shown before except that the line By Assay is deleted from the code. Table 13 shows the summary statistics for the least squares regression, Figure 7 shows a scatter plot... 

The availability of spreadsheets makes it unnecessary to plot data on graph paper and do hand calculations for the least-squares regression analysis and statistics. We will use the data in Example 3.21 to prepare the plot shown in Figure 3.8, using Excel. 

The second and preferred method is to apply appropriate statistical analysis to the dataset, based on linear regression. Both EU and USFDA authorities assume log-linear decline of residue concentrations and apply least-squares regression to derive the fitted depletion line. Then the one-sided upper tolerance limit (95% in EU and 99% in USA) with a 95% confidence level is computed. The WhT is the time when this upper one-sided 95% tolerance limit for the residue is below the MRL with 95% confidence. In other words, this definition of the WhT says that at least 95% of the population in EU (or 99% in USA) is covered in an average of 95% of cases. It should be stressed that the nominal statistical risk that is fixed by regulatory authorities should be viewed as a statistical protection of farmers who actually observe the WhT and not a supplementary safety factor to protect the consumer even if consumers indirectly benefit from this rather conservative statistical approach. 

The emphasis in this chapter is on problems that are common in, or unique to, computer simulation relative to statistical experiments as a whole. The reader should refer to Chapters 83 to 87 of the Handbook for basic statistical methods. No specific computer hardware or software is assumed other than a language for programming the simulation experiment and possibly a statistical-analysis package capable of standard procedures such as least-squares regression. 

The use of Eq. (2.18) to quantitatively estimate the IDL for chromatographs and for spectrometers has been roundly criticized for more than 10 years. There have been reported numerous attempts to find alternative ways to calculate IDEs. This author will comment on this most controversial topic in the following manner. The approach encompassed in Eq. (2.18) clearly lacks a statistical basis for evaluation and, hence, is mathematically found to be inadequate. As if this indictment is not enough, IDEs calculated based on Eq. (2.18) also ignore the uncertainty inherent in the least squares regression analysis of the experimental calibration as presented earlier. In other words, what if the analyte is reported to be absent when, in fact, it is present (a false negative) In the subsections that follow, a more contemporary approach to the determination of IDEs is presented and starts first with the concept of confidence intervals about the regression line. 

Include all calibration data, ICVs, and sample unknowns for both instrumental methods. Perform a statistical evaluation in a manner that is similar to previous experiments. Use EXCEL or LSQUARES (refer to Appendix C) or other computer programs to conduct a least squares regression analysis of the calibration data. Calculate the accuracy (expressed as a percent relative error for the ICV) and the precision (relative standard deviation for the ICV) from both instrumental methods. Calculate the percent recovery for the matrix spike and matrix spike duplicate. Report on the concentration of Cr in the unknown soil samples. Be aware of all dflution factors and concentrations as you perform calculations ... 

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