Linear regression biological data

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. 

The quantitative description of actual empirical data of the concentration-duration relationship can be expressed by any of a number of linear regression equations. In the assessment of empirical data reported by ten Berge et al. (1986), these workers quantified the exposure concentration-duration relationship by varying the concentration to the n power. Since raising c or t or both to a power can be used to define quantitatively the same relationship or slope of the curve and to be consistent with data and information presented in the peer-reviewed scientific literature, the equation C x t = k is used for extrapolation. It must be emphasized that the relationship between C and t is an empirical fit of the log transformed data to a line. No conclusions about specific biologic mechanisms of action can be drawn from this relationship. 

Motulsky, H., and Christopoulos, A. (2004). Fitting Models to Biological Data Using Linear and Nonlinear Regression A Practical Guide to Curve Fitting. Oxford University Press, New York. 

Quantitative structure-activity relationships QSAR. The QSAR approach pioneered by Hansch and co-workers relates biological data of congeneric structures to physical properties such as hydrophobicity, electronic, and steric effects using linear regression techniques to estimate the relative importance of each of those effects contributing to the biological effect. The molecular descriptors used can be 1-D or 3-D (3D-QSAR). A statistically sound QSAR regression equation can be used for lead optimization. 

In this chapter we presented two structural measures of molecular shape that can be used as predictor variables in MLR (multiple linear regression) analysis of structure-activity studies - cylindrical (8,G) and ovality ( in, i = 1,2,3) molecular descriptors - and two inexpensive overlapping methods useful for quick receptor mapping - MTD (minimal topological difference) and MVD (minimal volume difference). A subsequent statistical analysis of QSAR models developed with these shape molecular descriptors explained well the variance in the observed reactivity data (8 descriptor of cylindrical shape) and biological activity of retinoids (MTD) and sulfonamides (MVD). 

Non-linear models may be fitted to data sets by the inclusion of functions of physicochemical parameters in a linear regression model—for example, an equation in n and as shown in Fig. 6.5—or by the use of non-linear fitting methods. The latter topic is outside the scope of this book but is well covered in many statistical texts (e.g. Draper and Smith 1981). Construction of linear regression models containing non-linear terms is most often prompted when the data is clearly not well fitted by a linear model, e.g. Fig. 6.4e, but where regularity in the data suggests that some other model will fit. A very common example in the field of quantitative structure-activity relationship (QSAR) involves non-linear relationships with hydrophobic descriptors such as log P or n. Non-linear dependency of biological properties on these parameters became apparent early in the... 

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