Linearity bivariate data

Figure 5 Contour plots of two groups of bivariate data with each group having identical variance-covariance matrices. Such groups are linearly separable...

Figure 12 A simple two-group, bivariate data set that is not linearly separable by a single function. The lines shown are the linear classifiersfrom the two units in the first layer of the multilayer system shown in Figure 13...

Statistical methods for assessing bivariate data (two dimensional data where one parameter is measured as a function of another) are used to performr ression analysis of calibration data and to determine the goodness of fit of the calibration curve. A simple, linear equation is desirable when fitting quantitative calibration data for several reasons i) only two fitting parameters (A = intercept and B = slope) need to be calculated ii) it is straightforward to invert the equation so as to calculate an unknown value of x (e.g., analyte concentration) from a measured value of Y (e.g. mass spectrometer response), i.e. Xj = (Yj —A)/B and iii) relatively few experimental measurements (Xj,Yj) are required to establish reliable values of A and B in the catibration equation... 

From Figure 17.1 consider the situation where S and T have a bivariate normal distribution and the data are obtained from a single study. One can model the relationship between S and T and Z as three distinct linear regressions ... 

If data are collected from a random population (X, Y) from a bivariate normal distribution and predictions about Y given X are desired, then from the previous paragraphs it may be apparent that the linear model assuming fixed x is applicable because the observations are independent, normally distributed, and have constant variance with mean 0o + 0iX. Similar arguments can be made if inferences are to be made on X given Y. Thus, if X and Y are random, all calculations and inferential methods remain the same as if X were fixed. 

Inversion of the calibration equation is used to determine the unknown amount of the analyte in the sample, X, and is simplest for caUhration equations that give Y as a linear function of x, although a quadratic function is only slightly less convenient. Confidence limits (CLs) for X can be determined from those measured for Y to establish the main sources of uncertainly in x. CLs can be regarded as the bivariate equivalents of the RSD and other measures of uncertainty for univariate data. 

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