Mandel’s test

One approach to decide on the extent of linearity is to delete successively the highest concentrations of the standardization and apply the lack of fit or Mandel s test to the remaining points see section 2.4.4. This yields a concentration range in which it has been shown statistically that lack of fit to the straight line function could not be detected and that quadratic terms were not required. 

Testing for a non-significant value of r is irrelevant in analytical calibration, as such a poor correlation between signals and concentrations is almost inconceivable." Our suggestion is to use the lack-of-fit or Mandel s tests (see next section) as discussed elsewhere (e.g. refs. 6, 35, 41). 

In analytical chemistry we expect the calibration (standardization) not to depart much from a straight line. Should a slight deviation occur, a quadratic trend will usually describe the trend acceptably well. This means that we can make a decision by seeing whether a linear or a quadratic curve fits the data better. Thus, Mandel s test" compares the residual standard error of both models by an test [see eqn (2.18)] and helps us in deciding whether additional terms should be added to the straight line fit (see more details in Appendix 1). The practical equation to be used is ... 

Mandel s test was also suggested by ISO and lUPAC, although the degrees of freedom indicated by the latter are not correct. This is studied in more detail in Appendix 1. 

In Section 2.4.4 various tests were discussed that can be used to check for nonlinearity in a least squares fit. Among them Mandel s test was presented and recommended although it was also said that the definition given by the lUPAC was wrong. In the following paragraphs this will be explained in some detail. 

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) ... 

Another interesting application of Mandel s test is to decide on the linear calibration range. It consists of the deletion of the highest concentration levels of the standardization and application of Mandel s test repeatedly. This will yield a concentration range where it can be proved statistically that the function is linear and that quadratic terms are not required. This can also be done by the successive application of the lack of fit test. ... 

A third conclusion is that the relative magnitude of the residual variances of the models is a major issue when detecting differences between them. When a model is clearly different from its alternative (the model is much worse because its residual variance is 900%, 100% or 50% higher), Mandel s test will statistically detect it, providing we have a reasonable number of calibrators, around 20 standards (at the 99% confidence level). Recall that the simplified test is independent of the dof and, therefore, it will always yield wrong conclusions. 

The trend is that the closer the residual variances of the two models are, the more calibrators are required by Mandel s test to exceed the critical F values (to reject the null hypothesis i.e. to conclude that the variance explained by the additional term of the alternative model is significant). This is due to the similarity between the models. 

J. M. Andrade and M. P. Gomez-Carracedo, Notes on the use of the Mandel s test to check for nonlinearity in laboratory calibrations. Anal. Meth., 2013, 5, 1145-1149. 

The linearity of (a part of) the range should be evaluated to check the appropriateness of the straight-line model. This can be achieved by a graphical evaluation of the residual plots or by using statistical tests. It is strongly recommended to use the residual plots in addition to the statistical tests. Mostly, the lack-of-fit test and Mandel s fitting test are used to evaluate the linearity of the regression line [8, 10]. The ISO 8466 describes in detail the statistical evaluation of the linear calibration function [11]. 

Figure 6.3. (a) Plot of r versus the lack-of-fit (LOF) test. The bold line represents the critical F-value at the 95 percent confidence level. (h) Plot of r versus Mandel s fitting test. The bold line represents the critical / -value at the 95 percent confidence level. 

The linear range of most detectors is limited. In case of rejection of linearity by the LOF- or Mandel s fitting test, LoDs can be estimated with linearized... 

Standard ISO 5725-2 recommends that suitable procedures be used to detect and remove outliers in data. The procedures used within the ISO 5725-2 document include Mandel s h and k statistics for overall assessment and comparison of between-laboratory and within-laboratory consistency, respectively Cochran s test for evaluating within-laboratory consistency and Grubb s outlier tests for evaluating data. These procedures will also be used for this example. See Equations (9.23)-(9.45) for relevant definitions and equations. MandeTs h and k statistics, given in Tables 9.11 and 9.12, respectively, were calculated using Equations (9.23) and (9.24). 

Outlier tests for cell means as well as for laboratory 5 observations are statistically insignificant. However, the fact that Mandel s k statistics for laboratory 5 were inconsistent with the findings from the other laboratories, and that the Cochran test statistics for all concentrations in laboratory 5 were statistically significant at the 5% but not at the 1% critical value, raises the question as to whether there is a problem with the results reported by laboratory 5. Although... 

The important issue here is to realize that Mandel did not define the variance explained by the additional term(s) as a mere subtraction of the variances of the two models (z.e. yix, m y/x,non) but as a subtraction of the sum of squares of the linear and quadratic fits, divided by the difference on their degrees of freedom. Equation (A 1.3) presents Mandel s formal definition for the F test, which can be developed further to eqn (A1.4) and eqn (A1.5), which is used currently. Equation (A1.5) stems from eqn (A1.4) by simply introducing eqn (A 1.2) into it. Accordingly, the difference between the definitions given by lUPAC and Mandel lies in the use of the dof to weight the residual variances of the fits. 

Following Mandel," if the null hypothesis cannot be rejected, z.e. the straight line is a correct model and the alternative does not improve its fit, both the numerator and denominator estimate the true pure residual variance, the experimental Fisher-Snedecor s test will be lower than the critical value for a given probability level, and 1 and —3 dof for the numerator and denominator (in case a polynomial other than the quadratic one is considered, n — k dof should be considered instead of — 3, with k the order of the polynomial plus 1). Otherwise, the alternative hypothesis must be accepted z.e. the numerator contains structured variance which is larger than the pure residual variance. 

Each of the required three individual values for each nuclide was corrected with the factor that resulted from the weights of labelled and imlabelled spinach powder. The arithmetic mean and standard deviation for each laboratory was calculated. The data were visually inspected for outlier elimination. In addition, outliers were identified using Mandel s wilhin- and between-laboratory consistency test statistic k and A-values, respectively) and Grubbs I and II tests according to DIN ISO 5125-2 The outher-free data sets were used to calculate repeatability and reproducibility. Individual z-scores were used as a measure of performance characteristic of the participating laboratories. ... 

Other than the auxiliary reagents described by Riguera s group in the preceding pages, very few papers have been devoted to the determination by NMR of absolute configuration of carboxylic acids. In one such example, however, (5)-methyl mandelate was tested as a chiral auxiliary with racemic and... 

The use of surface bound triflate ions has been exploited by Raja et al. to immobilize the complexes [Rh(COD) fSj-(-i-)-PMP ], [Pd(allyl) fSj-(-i-)-PMP ], [Rh(COD) fSj-(-)-AEP rand[Rh(COD) flR,2Rj-(-t)-DED ]"in the pores of silicas possessing various pore sizes with narrow distributions [128]. These constrained chiral catalysts were then tested for the asymmetric hydrogenation of methyl ben-zoylformate to its corresponding methyl mandelate (40°C, methanol, 2 MPa H2). In the homogenous form, only the catalysts [Rh(COD) fSj-(-i-)-PMP ], [Pd(allyl) (Sj-(-i-)-PMP ] exhibit any signiflcant e.e.s under the reaction conditions (53%... 

About 20 different carboxylic acids have been tested by Dale and Mosher for comparison with MTPA 5. Some of them (mandelic and O-methylmandelic acid) have been partly described in the previous section together with MTPA. Some new and important acids have been added since. Figure 1 shows MPTA and a number of carboxylic acids which can be used in a similar way to Mosher s reagent. 

We are grateful to Mary Mandels and Raymond Andreotti of the U.S. Army Natick Laboratories for testing the mutants under controlled fermentor conditions (Figure 2) and to Goran Pettersson of the University of Uppsala, Sweden, for the antibody determinations. 

S. S. Wang, J. F. Mandell, and F. J. McGarry/Elfects of crack elevation in TDCB adhesive fracture test. Research Report R 76-3, Department of Materials Science, School of Engineering, MIT, Cambridge, MA, 1976. 

The equation indicates that a direct comparison between s yjxMn and s /x,non is not carried out as the summary above suggested. Instead, the difference between the residual variances of both models is compared to an estimation of the pure random error. There is a misconception here because according to the original formulation by Mandel the numerator of the F test should not be the difference among the residual variances. The lUPAC s Gold Book and other online resources (see www.iupac.org) were searched, but to the best of the authors knowledge there were no updates of the equations of the 1998 publication and some notes on its application were presented recently. ... 

Z. Leifer, T. Kada, M. Mandel, E. Zeiger, R. Stafford and H. S. Rosenkranz, An evaluation of the tests using DNA-repair-deficient bacteria for predicting genotoxicity and carcinogenicity, Mutat. Res., 1981, 87, 211-297. 

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