Application of Regression Analysis

Most published method evaluations fail to apply regression analysis in a rigorous fashion. This section considers both the use of OLR instead of Deming regression and the use of unweighted analysis in the setting of proportional random errors. 

Application of OLR in Case of Random Errors Associated with Both xl and x2 

Application of Unweighted Forms of Regression Analysis in Case of Proportional Random Errors 

According to current practice in method comparison studies, it is usual to apply unweighted forms of regression analysis (i.e., OLR and the Deming procedure), even though the SDs vary with the measured concentration, as occurs 

In conclusion, planning a method comparison study to achieve a given power for detection of medically notable differences should be considered. In this way, a method comparison study is likely to be conclusive either the null hypothesis of no difference is accepted, or the presence of a relevant difference is established. Otherwise, a statistically nonsignificant slope deviation from unity or intercept deviation from zero or both may either imply that the null hypothesis is true, or be an example of a Type II error (i.e., an overlooked real difference of medical importance). 

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At this point it was clear that the highest potential for increased activity was by substitution in the 2-position of the biphenyl alcohol. We prepared the sequence of compounds shown in Table 1. Substituents were again chosen to maximize the parameter space covered within the relatively stringent synthetic limitations of the biphenyl substitution pattern. The application of regression analysis to the data for these compounds provided no clear relationship between structure and activity when the parameters in our standard data base were used. The best linear fit was found for B4, the STERIMOL maximum radius. However, the correlation coefficient was only 0.625. 

The application of regression analysis involves four steps ... 

Devillers, J. and Lipnick, R.L. (1990). Practical Applications of Regression Analysis in Environmental QSAR Studies. In Practical Applications of Quantitative Structure-Activity Relationships (QSAR) in Environmental Chemistry and Toxicology (Karcher, W. and Devillers, J., eds.), Kluwer, Dordrecht (The Netherlands), pp. 129-144. 

Devillers J, Lipnick RL. Practical applications of regression analysis in environmental QSAR studies. In Karcher W, Devillers J, editors, Practical applications of quantitative structure-activity relationships (QSAR) in environmental chemistry and toxicology. Dordrecht Kluwer Academic, 1990. p. 129 13. 

Example of Application of Regression Analysis (Weighted Deming Analysis)... 

Some problems related to the proper application of regression analysis and of other multivariate statistical methods in QSAR studies and concerning the validity of the obtained results have recently been reviewed [403, 408, 409] (compare chapter 4.1). 

The statistical analysis of data requires a proper design of experiments to prove or disprove a certain hypothesis which has been formulated in advance. From the viewpoint of a puritanical statistician most QSAR analyses are forbidden , because they are retrospective studies and, in addition, many different hypotheses (i.e. combinations of independent variables) are tested sequentially. Indeed, many problems arise from the application of regression analysis in ill-conditioned data sets. Only in later stages of lead structure optimization are certain hypotheses, e.g. on the influence of more lipophilic, electronegative, polar, or bulky substituents in a certain position, systematically tested, now fulfilling the requirements for the proper application of statistical methods. 

Although not as demanding in its requirements as the additivity model, multiple-parameter analysis also requires a closely related series of compounds for application. Usually, at least five analogs should be present for each term in the resulting equation, in order for the application of regression analysis to result... 

Figure 3.2 Example of the inappropriate application of regression analysis to dichotomous activity data (A) as compared to continuous activity data (B). Data distributed as in A are unsuitable for regression analysis because it results in a two-point correlation.

Application of regression analysis, using partition coefficients, to the binding of organic compounds by synthetic polymers has been made by Hansch and Helmer. Based on prior work,their studies of the nylon and rayon binding of derivatives of aniline and acetanilide from an aqueous solution have shown that the amount of compound bound is related... 

From a theoretical point of view, the proper application of regression analysis requires the formulation of a working hypothesis, the design of experiments (i.e., compounds to be tested), the selection of a mathematical model, and the test of statistical significance of the obtained result. In QSAR studies, this is pure theory. Reality is different QSAR studies are most often retrospective studies and in several cases many different variables are tested to find out whether some of them, alone or in combination, are able to describe the data. In principle, there are no objections against this method because QSAR equations should be used to derive new hypotheses and to design new experiments, based on these hypotheses. Then the requirements for the application of statistical methods are fulfilled. 

A reported application of canonical analysis involved a novel combination of the canonical form of the regression equation with a computer-aided grid search technique to optimize controlled drug release from a pellet system prepared by extrusion and spheronization [28,29]. Formulation factors were used as independent variables, and in vitro dissolution was the main response, or dependent variable. Both a minimum and a maximum drug release rate was predicted and verified by preparation and testing of the predicted formulations. Excellent agreement between the predicted values and the actual values was evident for the four-component pellet system in this study. 

In a review article entitled How to get wrong results from good experimental data a survey of incorrect applications of regression , Exner offered some trenchant warnings which should be heeded by all those who engage in correlation analysis.133 Numerous examples are given from the literature, in which experimental data were processed in an incorrect way from the point of view of statistics. The results were more or less biased and sometimes completely wrong. 

The use and benefits of regression analysis can be appreciable, particularly in the evaluation of process data. In these applications, processes having as many as fifty variables, which are continuously changing over months of operation, can be evaluated by this technique. For these, the daily log records for say 400 to 500 data points are analyzed through the selected model (usually linear as a first approximation) to determine the relative effects of each variable on the response. This analysis in many cases has led to qualitative and often to quantitative determination of key operating variables whose effect had been masked on individual data point comparisons by the simultaneous changes in other less important, but unknown, variables. 

Erom an operational standpoint, the LFER, LSER, QSAR, and QSPR approaches can be quite similar, with distinctions based on their applications. QSAR is usually applied to biological properties, especially those important to pharmacology and toxicology. QSPR usually dwells on physicochemical properties in general. LSER focuses on solute-solvent systems. For organizational purposes, we like to view LSER and some applications of QSAR and QSPR (along with related methods) as subsets of LFER. Each approach typically uses some form of regression analysis (statistics) to help find a mathematical relationship between a property and a set of descriptors. 

Before proceeding, it is as well to test the significance of the two regression coefficients byw.a and bya.w. This we can do with an application of the analysis of variance. 

To this point, the discussion of regression analysis and its applications has been limited to modelling the association between a dependent variable and a... 

To this point, the discussion of regression analysis and its applications has been limited to modelling the association between a dependent variable and a single independent variable. Chemometrics is more often concerned with multivariate measures. Thus it is necessary to extend our account of regression to include cases in which several or many independent variables contribute to the measured response. It is important to realize at the outset that the term independent variables as used here does not imply statistical independence, as the response variables may be highly correlated. 

Before leaving the subject of regression analysis, and in particular the use of PCR and PLS algorithms, it is instructive to examine some of the diagnostic statistics often available from their application. 

Calibration and response surface methods are indeed the most important applications of regression methods in analytics. Other applications are seen in environmental analysis, where receptor modek are developed on the basis of multivariate relationships... 

This chapter has described some of the more commonly used supervised learning methods for the analysis of data discriminant analysis and its relatives for classified dependent data, variants of regression analysis for continuous dependent data. Supervised methods have the advantage that they produce predictions, but they have the disadvantage that they can suffer from chance effects. Careful selection of variables and test/training sets, the use of more than one technique where possible, and the application of common sense will all help to ensure that the results obtained from supervised learning are useful. 

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