Bivariate data regression analysis

Beilken et al. [ 12] have applied a number of instrumental measuring methods to assess the mechanical strength of 12 different meat patties. In all, 20 different physical/chemical properties were measured. The products were tasted twice by 12 panellists divided over 4 sessions in which 6 products were evaluated for 9 textural attributes (rubberiness, chewiness, juiciness, etc.). Beilken etal. [12] subjected the two sets of data, viz. the instrumental data and the sensory data, to separate principal component analyses. The relation between the two data sets, mechanical measurements versus sensory attributes, was studied by their intercorrelations. Although useful information can be derived from such bivariate indicators, a truly multivariate regression analysis may give a simpler overall picture of the relation. 

Statistical indices are fundamental numerical quantities measuring some statistical property of one or more variables. They are applied in any statistical analysis of data and hence in most of Q S AR methods as well as in some algorithms for the calculation of molecular descriptors. The most important univariate statistical indices are indices of central tendency and indices of dispersion, the former measuring the center of a distribution, the latter the dispersion of data in a distribution. Among the bivariate statistical indices, the correlation measures play a fundamental role in all the sciences. Other important statistical indices are the diversity indices, which are related to the injbrmationcontentofavariahle,the —> regressiowparameters, used for regression model analysis, and the —> classification parameters, used for classification model analysis. 

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 peak-analysis is done by fitting a bivariate polynomial to the cap of the peak in order to determine its position and the standard deviations that describe the peak widths in equatorial and meridional directions [2, 9]. Let the y-direction be the meridian, in analogy to the correlation method, a ROI is defined by the user. Inside this ROI the algorithm searches for the peak. The 2D peak is fitted to a 2D function. Figure 3.6 demonstrates the fit of the long period peak by a bivariate quadratic polynomial. In order to assure numerical stability of the regression module on digital computers, the maximum intensity in the measured peak data is normalized to 1. 

The data were analysed in several stages. Descriptive statistics and bivariate correlations were calculated for independent, dependent and control variables. Control variables with significant bivariate correlations with outcome as measured by the NBAS scales were used in forward stepwise multiple regression analyses to determine the best joint predictors of the NBAS. After the best multiple regression model was constructed from the non-lead variables, lead measurements recorded at the three time points were added to the model to determine the relationship between each of the lead measurements and the adjusted NBAS scores. The overall plan of analysis follows Bellinger et al (1984 this volume). Analyses were performed with SAS programs. 

---