Statistical validation linearity

Each of these two QSAR model searches led to pools of several thousands of statistically valid linear equations, expressing the estimate of the Cox2 pICso value as linear combinations of molecular descriptors selected by a Genetic Algorithm (GA) [57,... 

In equation 3.4-18, the right side is linear with respect to both the parameters and the variables, j/the variables are interpreted as 1/T, In cA, In cB,.. . . However, the transformation of the function from a nonlinear to a linear form may result in a poorer fit. For example, in the Arrhenius equation, it is usually better to estimate A and EA by nonlinear regression applied to k = A exp( —EJRT), equation 3.1-8, than by linear regression applied to Ini = In A — EJRT, equation 3.1-7. This is because the linearization is statistically valid only if the experimental data are subject to constant relative errors (i.e., measurements are subject to fixed percentage errors) if, as is more often the case, constant absolute errors are observed, linearization misrepresents the error distribution, and leads to incorrect parameter estimates. 

Compared with conventional impurities measurements, trace analyses caimot be expected to achieve the same linearity and precision values. This is due to the lower signal-to-noise ratios inevitable at low levels. Hence, while the same approaches can be used, greater latitude will be necessary in the acceptance criteria. What must be demonstrated is that the data is statistically valid to show that the levels of toxic analytes are below their specification limits. 

For chemical measurements with a linear calibration function, traceability of results can be formally established without great expenditure if the calibration is based on suitable reference standards and the linear regression is performed as shown above and (statistically) validated. The use of reference materials as samples make it possible to establish the traceability of a new analysis protocol by using an existing analysis method. 

An in vitro bioassay can be designed in several ways, but requires statistical validity. A one point assay is not valid. The bioassay should be designed to consider factors that introduce variability, and the analysis should test such variability. A measurement series of a test sample should be compared to an equivalent series of the reference material, carefully considering the comparisons between the linear portions of the dose-response curves (Mire-Sluis et al., 1996). To test validity of a bioassay inter- and intra-assay variability should be considered in both preparation, and in the case of multiwell plates, the variability between each plate. To reduce the positional effect in plate tests, it is advisable to distribute the points on the curves randomly and also to include a reference standard in each plate (Gaines-Das and Meager, 1995). One of the most widely used techniques to validate a bioassay s performance is to include internal duplicates. The data arising from the comparison can be important in assessing the test s variability. 

PCM modeling aims to find an empirical relation (a PCM equation or model) that describes the interaction activities of the biopolymer-molecule pairs as accurate as possible. To this end, various linear and nonlinear correlation methods can be used. Nonlinear methods have hitherto been used to only a limited extent. The method of prime choice has been partial least-squares projection to latent structures (PLS), which has been found to work very satisfactorily in PCM. PCA is also an important data-preprocessing tool in PCM modeling. Modeling includes statistical model-validation techniques such as cross validation, external prediction, and variable-selection and signal-correction methods to obtain statistically valid models. (For general overviews of modeling methods see [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]. 

Here rtm and are correlation coefficients between observed and calculated values of log kf for training set fits and cross-validated predictions, respectively. Correlations are the maximum ones observed for 10 independent trials, each with a different random number generator seed. Statistics for linear regression are available in Table V. 

The statistical validation of this linear regression for the validation of the typical errors was performed at the 95% confidential level. The Table 1 shown the figures of merit obtained for the developed of the methodology. The detection limit (DL, 3cj) was calculated for the diflFerent irradiation times (45, 60 and 75 min). 

The simplest method for generating a statistical QSAR model is to use multiple linear regression (MLR), employing a number of calculated data to explain the measured activity data. Metrics, such as the F value and can be used to describe the statistical validity of the result, and cross-validated (q ) is used to measure the predictivity of the model. For small numbers of descriptors compared to the number of molecules, this method can provide adequate results. However, the number of properties that can be generated far exceeds the number of molecules in most analyses. In these cases, the MLR approach will rarely provide a QSAR model that is statistically predictive outside of the molecules used to build the model. A number of other techniques are provided in Tsar to handle these large datasets. 

The results of a comparison between values of n estimated by the DRK and BET methods present a con. used picture. In a number of investigations linear DRK plots have been obtained over restricted ranges of the isotherm, and in some cases reasonable agreement has been reported between the DRK and BET values. Kiselev and his co-workers have pointed out, however, that since the DR and the DRK equations do not reduce to Henry s Law n = const x p) as n - 0, they are not readily susceptible of statistical-thermodynamic treatment. Moreover, it is not easy to see how exactly the same form of equation can apply to two quite diverse processes involving entirely diiferent mechanisms. We are obliged to conclude that the significance of the DRK plot is obscure, and its validity for surface area estimation very doubtful. 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

Another simple approach assumes temperature-dependent AH and AS and a nonlinear dependence of log k on T (123, 124, 130). When this dependence is assumed in a particular form, a linear relation between AH and AS can arise for a given temperature interval. This condition is met, for example, when ACp = aT" (124, 213). Further theoretical derivatives of general validity have also been attempted besides the early work (20, 29-32), particularly the treatment of Riietschi (96) in the framework of statistical mechanics and of Thorn (125) in thermodynamics are to be mentioned. All of the too general derivations in their utmost consequences predict isokinetic behavior for any reaction series, and this prediction is clearly at variance with the facts. Only Riietschi s theory makes allowance for nonisokinetic behavior (96), and Thorn first attempted to define the reaction series in terms of monotonicity of AS and AH (125, 209). It follows further from pure thermodynamics that a qualitative compensation effect (not exactly a linear dependence) is to be expected either for constant volume or for constant pressure parameters in all cases, when the free energy changes only slightly (214). The reaction series would thus be defined by small differences in reactivity. However, any more definite prediction, whether the isokinetic relationship will hold or not, seems not to be feasible at present. 

The necessity of the statistical approach has to be stressed once more. Any statement in this topic has a definitely statistical character and is valid only with a certain probability and in certain range of validity, limited as to the structural conditions and as to the temperature region. In fact, all chemical conceptions can break dovra when the temperature is changed too much. The isokinetic relationship, when significantly proved, can help in defining the term reaction series it can be considered a necessary but not sufficient condition of a common reaction mechanism and in any case is a necessary presumption for any linear free energy relationship. Hence, it does not at all detract from kinetic measurements at different temperatures on the contrary, it gives them still more importance. 

Because physicochemical cause-and-effect models are the basis of all measurements, statistics are used to optimize, validate, and calibrate the analytical method, and then interpolate the obtained measurements the models tend to be very simple (i.e., linear) in the concentration interval used. 

The main concept addressed in this new multi-part series is the idea of correlation. Correlation may be referred to as the apparent degree of relationship between variables. The term apparent is used because there is no true inference of cause-and-effect when two variables are highly correlated. One may assume that cause-and-effect exists, but this assumption cannot be validated using correlation alone as the test criteria. Correlation has often been referred to as a statistical parameter seeking to define how well a linear or other fitting function describes the relationship between variables however, two variables may be highly correlated under a specific set of test conditions, and not correlated under a different set of experimental conditions. In this case the correlation is conditional and so also is the cause-and-effect phenomenon. If two variables are always perfectly correlated under a variety of conditions, one may have a basis for cause-and-effect, and such a basic relationship permits a well-defined mathematical description. 

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