Testing for non-linearity

Figure 65-1 shows a schematic representation of the F-test for linearity. Note that there are some similarities to the Durbin-Watson test. The key difference between this test and the Durbin-Watson test is that in order to use the F-test as a test for (non) linearity, you must have measured many repeat samples at each value of the analyte. The variabilities of the readings for each sample are pooled, providing an estimate of the within-sample variance. This is indicated by the label Operative difference for denominator . By Analysis of Variance, we know that the total variation of residuals around the calibration line is the sum of the within-sample variance (52within) plus the variance of the means around the calibration line. Now, if the residuals are truly random, unbiased, and in particular the model is linear, then we know that the means for each sample will cluster... 

Table 65-1 Various tests for (non) linearity that have been proposed and a summary of their characteristics...

Such an approach is justified if the analyst is sure that the analytical calibration is linear over the relevant concentration range, since tests for non-linearity require a large number of measurements to have a useful degree of statistical power [9, 10]. 

Multidimensional Data Intercomparisons. Estimation of reliable uncertainty intervals becomes quite complex for non-linear operations and for some of the more sophisticated multidimensional models. For this reason, "chemometric" validation, using common, carefully-constructed test data sets, is of increasing importance. Data evaluation intercomparison exercises are thus analogous to Standard Reference Material (SRM) laboratory intercomparisons, except that the final, data evaluation step of the chemical measurement process is being tested. 

The exact solutions of the linear elasticity theory only apply for small strains, and under idealised loading conditions, so that they should at best only be treated as approximations to the real behaviour of materials under test conditions. In order to describe a material fully we need to know all the elastic constants and, in the case of linear viscoelastic materials, relaxed and unrelaxed values of each, a distribution of relaxation times and an activation energy. While for non-linear viscoelastic materials we cannot obtain a full description of the mechanical properties. 

Bardsley, W. G. McGinlay, P. B. (1989). Optimal design for model discrimination using the F-test with non-linear biochemical models. Criteria for choosing the number and spacing of experimental points. J. Theoret. Biol. 139, 85—102. 

In this paper time-temperature equivalence for rocks is investigated for non-linear behaviours of rocks, based upon inner-variable theory of irreversible process and creep tests. 

Complex behavior, uncertainty, many coupled processes, and massive non-linearities lead to difficulty in formulating a reasonable model. Functionalities for non-linear coupled processes must be based on tests, and the experimental effort that this requires, even for one shale type, is enormous. All aspects of shale behavior are not fully understood at present one example is the possible large effect of capillarity on test results (Schmidt et al. 1994). Even with extensive data, fundamental uncertainties exist, particularly with shales, as compared with other rocks. 

Non-linearity between ESR signal and dose. Statistical tests for the linearity of the dose curve can be applied. 

The SBI test was designed for two-dimensional products and it has still not been established how to adapt or apply such a test for non-planar parts such as pipes, cables and linear seals. Indeed, the cable industry is moving towards total unacceptance of the new system, and is proposing its own system that would feature completely different test methods. 

J0RGENSEN, E., Framework program for non-linear regression analysis with tests, Danish Institute of Computing Machinery, November 1963. 

Load-displacement data were monitored during many of the bend tests and fracture toughness tests, A high accuracy LVDT-type gage, calibrated to a maximum displacement of 25 to 50-microns, was used to measure displacement of the center of the bend bar with respect to the upper loading pins. Non-linearities were observed in the load-displacement records, particularly for the notched bend bars used in the fracture toughness tests. Such non-linearities may have been the result of subcritical crack growth. 

It should be noted that the results of the low impedance test or excitation level test may differ from those for the test carried out under higher seismic levels for non-linear systems. To be of use in seismically qualifying equipment, low impedance tests require the response of the equipment to be essentially linear up to potential failure mode levels of exdtation so as to be able to determine design margins. 

Mandel s test was summarized as a comparison of the residual standard deviation of the linear model with that of the nonlinear mode . Such a definition results in the well-known conceptual Fischer-Snedecor s F test (Fexperimentai = s ylx,Mn / s y x,nof), whcrc s Stands for standard error of the regression, lin for straight line model and non for non-linear model (here a quadratic one, although this is not mandatory). Note that although the term variance should be used senso stricto instead of standard error, this is not relevant for our discussions here. The definition was then resolved to eqn A 1.1 below (numbered 51 in ref. 8) ... 

L. Bruggermann, W. Quapp and R. Wennrich, Test for non-hnearity concerning linear cahbrated chemical measurements, Accred. Qual. Assur., 2006,11, 625 31. 

The value of the eorrelation eoeffieient using the least squares teehnique and the use of goodness-of-fit tests (in the non-linear domain) together probably provide the means to determine whieh distribution is the most appropriate (Keeeeioglu, 1991). However, a more intuitive assessment about the nature of the data must also be made when seleeting the eorreet type of distribution, for example when there is likely to be a zero threshold. 

Occasionally, materials are tested in tension by applying the loads in increments. If this method is used for plastics then special caution is needed because during the delay between applying the load and recording the strain, the material creeps. Therefore if the delay is not uniform there may appear to be excessive scatter or non-linearity in the material. In addition, the way in which the loads are applied constitutes a loading history which can affect the performance of the material. A test in which the increments are large would quite probably give results which are different from those obtained from a test in which the increments were small or variable. 

This lack of sharpness of the 1-way F-test on REV s is sometimes seen when there is information spanned by some eigenvectors that is at or below the level of the noise spanned by those eigenvectors. Our data sets are a good example of such data. Here we have a 4 component system that contains some nonlinearities. This means that, to span the information in our data, we should expect to need at least 4 eigenvectors — one for each of the components, plus at least one additional eigenvector to span the additional variance in the data caused by the non-linearity. But the F-test on the reduced eigenvalues only... 

The few remaining discrepancies are probably due to error in the assumed relative reflecting powers. To test this, we made use of an F-curve for OF obtained by linear extrapolation from Na+ and Cf, and one for Tii+ from CF and K+. These F-curves (which are not reproduced here because of uncertainty in their derivation) lead to structure factors which are, for the same final parameter values, also in good but not complete agreement with the observed intensities. Possibly somewhat different F-curves (corresponding to non-linear extrapolation) would give better agreement, but because of the arbitrariness of this procedure no attempt was made to utilize it. 

---