Statistical regression method

Statistical regression methods, as described in this text, require objective observational data that result from measuring specific events or phenomena under controlled conditions in which as many extraneous influences as possible, other than the variable(s) under consideration, are eliminated. To be valid, regression methods employed in experimentation require at least four conditions to be satisfied ... 

This approach can be used for multicomponent mixtures by applying matrix algebra. This is generally done with a software program and even nonlinear calibrations can be handled with statistical regression methods. 

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. 

Hoskuldsson A (1988) PLS regression methods. Chemom 2 211 Huber PJ (1981) Robust statistics. Wiley, New York... 

All regression methods aim at the minimization of residuals, for instance minimization of the sum of the squared residuals. It is essential to focus on minimal prediction errors for new cases—the test set—but not (only) for the calibration set from which the model has been created. It is relatively easy to create a model— especially with many variables and eventually nonlinear features—that very well fits the calibration data however, it may be useless for new cases. This effect of overfitting is a crucial topic in model creation. Definition of appropriate criteria for the performance of regression models is not trivial. About a dozen different criteria— sometimes under different names—are used in chemometrics, and some others are waiting in the statistical literature for being detected by chemometricians a basic treatment of the criteria and the methods how to estimate them is given in Section 4.2. 

Differences in calibration graph results were found in amount and amount interval estimations in the use of three common data sets of the chemical pesticide fenvalerate by the individual methods of three researchers. Differences in the methods included constant variance treatments by weighting or transforming response values. Linear single and multiple curve functions and cubic spline functions were used to fit the data. Amount differences were found between three hand plotted methods and between the hand plotted and three different statistical regression line methods. Significant differences in the calculated amount interval estimates were found with the cubic spline function due to its limited scope of inference. Smaller differences were produced by the use of local versus global variance estimators and a simple Bonferroni adjustment. 

Data Processing and Statistics. Linear, power, and exponential lines were fitted to the dust, trash, and reflectance data by standard regression methods. 

While computers are a substantial aid in statistical analysis, it is also true that statistical methods have helped in certain computer applications. In Section V the subject of mathematical models will be discussed. These are in many cases based on empirical correlations. When these have been obtained by regression methods, not only is the significance of the results better understood, but also the correlation is expressed directly in a mathematical form suitable for programming. 

In addition to the choice of QSAR descriptors, many statistical learning methods are available to relate the descriptors to the predicted value. There are two main categories of prediction to answer two different questions regression models (for predicting the activity of a peptide as a continuous variable such as MIC or a surrogate such as in the luminescence assay) or classification where the model is trained to classify as simply active or inactive. Historically, linear... 

J. N. Miller, Basic statistical methods for analytical chemistry. Part 2. Calibration and regression methods. A review, Analyst, 116 (1991), 3-14. 

The problem consists in seeking such a combination of the values of constants k which gives the minimum value of Q ( >mm). Before computers became commonly available, the kinetic equations had usually been transformed into a linear form and the linear regression ( least-squares method ) had been applied to find the best set of constants. This procedure is not statistically correct in most cases. Therefore, only the nonlinear regression method can be recommended to optimize constants in kinetic equations that have a nonlinear form [48-51]. 

Fairly sophisticated and specialized statistical software is needed in the practical application of nonlinear regression methods to mechanistic modeling for industrial chemistry applications. The techniques implemented in such software are discussed in Seber and Wild,32 Bates and Watts,30 Bard,31 and Riley and Blau.29... 

A few comments about the method are warranted. The controlled (dominant) variables, Ycd, should be measured such that they belong to the set Yd for rapid control. Similarly, the manipulators in the feedback control loops should belong to the set, Ud. The feedback controllers should have integral action (PI controllers). These can be tuned with minimal information (e.g., ultimate gain and frequency from a relay test). The model Ms is usually quite simple and can be developed from operating data using statistical regressions. This works because the model includes all the dominant variables of the system, Y d, as independent variables by way of their setpoints, Y. The definition of domi-... 

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