Correlation and regression - relationships between measured values

Correlation and regression -relationships between measured values 

Essential Statistics for the Pharmaceutical Sciences Philip Rowe 

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CH14 CORRELATION AND REGRESSION - RELATIONSHIPS BETWEEN MEASURED VALUES... 

CH14 CORRELATION AND REGRESSION - RELATIONSHIPS BETWEEN MEASURED VALUES Table 14.9 Meteorological data for two potential growing sites... 

However, the expected value of s is zero with variance a2 + O o. Thus the variance of the measurement errors are propagated to the error variance term, thereby inflating it. An increase in the residual variance is not the only effect on the OLS model. If X is a random variable due to measurement error such that when there is a linear relationship between xk and Y, then X is negatively correlated with the model error term. If OLS estimation procedures are then used, the regression parameter estimates are both biased and inconsistent (Neter et al., 1996). 

To summarize, when one wishes to evaluate the relationship between two variables, one of which is fixed (the independent variable) and one which is allowed to vary (the dependent variable), the analysis is termed a regression problem. A special type of regression problem is called calibration. In the calibration problem, one wishes to predict (future) values of the independent variable for given values of the independent variable. If, on the other hand, both variables are measured with error, correlation analysis and error in variables analyses are two approaches which one can use in the evaluation of the data. More extensive details of regression and correlation analysis are found in the references cited at the end of this paper. The calibration problem is discussed in 2, 8, 9 and 10. 

All developments of quantitative structure activity relationships (QSARs)/ quantitative structure-property relationships (QSPRs)/QSDRs go through similar steps (1) collection of a database of measured values for model development and validation/evaluation, (2) selection of chemical descriptors (can include connection indices, atom, bond, or functional groups, molecular orbital calculations), (3) development of the model (develop a correlation between the chemical descriptors and the activity/property/degradation values) using a variety of statistical approaches (linear and non-linear regression, neural networks, partial least squares (PLS), etc. [9]), and (4) validate/evaluate the model for predictability (usually try to use a separate set of chemicals other than the ones used to train the model external validation) [10]. 

Although not stated as such, the discussion thus far has implicitly concerned univariate data, i.e., replicate measurements of a single parameter under closely controlled conditions. A simple example might be a series of weighings to determine the mass of an object. Of course, the fact that a spread of experimental values is always obtained indicates that some of the experimental conditions are not completely under control. However, this class of measurements is usefully contrasted with bivariate and multivariate data (we shall be mainly concerned with the bivariate case. Section 8.3). Experimental measurements become two-dimensional under various sets of circumstances (Meier 2000). The case of main interest in this book corresponds to cases in which measured values (e.g., mass spectrometric signal intensities) are considered as functions of an experimental parameter (e.g., concentration or amount of a specified analyte injected into the instrument), as in acquisition of a calibration determination of the functional relationship between the two parameters is called regression. A related but somewhat different case concerns correlation analysis between two experimentally observable quantities (e.g., signals from a mass spectrometer and from a UV absorbance detector). The correlation behaviour is tested... 

The toxicity of 44 metals to the biofilms and planktonic cells of Pseudomonas fluorescens was measured and expressed as minimum inhibitory concentration, minimum bactericidal concentration, and minimum biofilm eradication concentration. Linear regression analyses wa-e conducted to determine the relationships between the measured toxicity values and the following physicochemical parameters standard reduction-oxidation potential, electronegativity, the solubility product of the corresponding metal-sulfide complex, the Pearson softness index, electron density, and the covalent index. Each of the physicochemical parameters was significantly (P < 0.05) correlated with one or more of the toxicity measurements. Heavy metal ions were found to show the strongest correlations between toxicity and physicochemical parameters. 

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