Transformed regression ranges

This observation is expected from theory, as the observed thickness distributions are exactly the functions by which one-dimensional short-range order is theoretically described in early literature models (Zernike and Prins [116] J. J. Hermans [128]). From the transformed experimental data we can determine, whether the principal thickness distributions are symmetrical or asymmetrical, whether they should be modeled by Gaussians, gamma distributions, truncated exponentials, or other analytical functions. Finally only a model that describes the arrangement of domains is missing - i.e., how the higher thickness distributions are computed from two principal thickness distributions (cf. Sect. 8.7). Experimental data are fitted by means of such models. Unsuitable models are sorted out by insufficient quality of the fit. Fit quality is assessed by means of the tools of nonlinear regression (Chap. 11). 

Ordinary least squares regression requires constant variance across the range of data. This has typically not been satisfied with chromatographic data ( 4,9,10 ). Some have adjusted data to constant variance by a weighted least squares method ( ) The other general adjustment method has been by transformation of data. The log-log transformation is commonly used ( 9,10 ). One author compares the robustness of nonweighted, weighted linear, and maximum likelihood estimation methods ( ). Another has... 

Two primary methods exist for constraining the search to values considered reasonable by the analyst transformation of the independent variables and penalty functions (5). The main point to note is that with nonlinear regression the analyst should have this very significant power over the search. Then, instead of trying to find a solution frctti an infinite number of possible values for the unknown coefficients, the search will focus on a physically reasonable range. 

The linearity of an analytical method is its ability to elicit test results that are directly, or by means of well-defined mathematical transformation, proportional to the concentration of analytes in samples within a given range. Linearity is determined by a series of three to six injections of five or more standards whose concentrations span 80-120% of the expected concentration range. The response should be—directly or by means of a well-defined mathematical calculation—proportional to the concentrations of the analytes. A linear regression equation applied to the results should have an intercept not significantly differ-... 

Transformations Transformation of the variables X and Y are used to correct for nonlinearity, but are also used to correct for heteroskedasticity (i.e., the variance is not constant over the concentration range, see below). Transformations of the X variable (i.e., X = log10X or X = /X ) are used to linearize concave regression curves, while transformations of X = X2 or X = ex are used for convex curves. 

Determination of Linearity and Range Determine the linearity of an analytical method by mathematically treating test results obtained from analysis of samples with analyte concentrations across the claimed range of the method. The treatment is normally a calculation of a regression line by the method of least squares of test results versus analyte concentrations. In some cases, to obtain proportionality between assays and sample concentrations, the test data may have to be subjected to a mathematical transformation before the regression analysis. The slope of the regression line and its variance (correlation coefficient) provide a mathematical measure of linearity the y-intercept is a measure of the potential assay bias. 

The problem in both methods is the error propagation. If an error exists in the measurement, this error will be submitted to the transformation as well. A second problem arises in the variances. Usually the variances of measurement in TLC are constant within the calibration range. The transformation of data will lead to inhomogeneous variances and this is the reason for unreliable regression analysis. 

The second method is an acceptable alternative. The laboratory must identify an established laboratory that is willing to share its reliable median values. Then, 25 to 50 specimens are assayed at each laboratory. These specimens are selected so that their results span the analytical measurement range. The two sets of values can then be compared using linear regression analysis (after appropriate transformations) to establish the relationship between the two assays. The regression equation can then be applied to the reliable set of median values to derive a set of medians appropriate for the laboratory. These median values can be used temporarily until values from 300 to 500 patients are available for the analysis provided earlier. 

One method to obtain initial estimates is linearization of the problem. For example, the 1-compartment model with bolus intravenous administration can be reformulated to a linear problem by taking the log-transform on both sides of the equation. This is a trivial example, but is often reported in texts on nonlinear regression. Another commonly used method is by eyeballing the data. For example, in an Emax model, Emax can be started at the maximal of all the observed effect values. EC50 can be started at the middle of the range of concentration values. Alternatively, the data can be plot-... 

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