Statistical F test

Malinowski, E.R., "Statistical F-Tests for Abstract Factor Analysis and Target Testing 1,/. Chemo. 1987 (1) 49-60... 

E.R. Malinowski, Statistical F-tests for abstract factor analysis and target testing. J. Chemom., 3 (1988)49-60. 

When the uncertainty in the parameter values becomes too large, the analyst should consider reducing the model. The correlation matrix between parameters can be useful in selecting the parameters that can be removed to make the model smaller. There are statistical criteria that can be used to select the better model. These include the Akaike Information Criterion (AIC) value and the F-test. The AIC value is calculated using the WSS, the number of parameters in the model, and the number of data points. The model with the lower AIC values is usually selected as the better model. The statistical F-test involves the calculation of an F value from the WSS and degrees of freedom from two analyses. The calculated F value is compared with the tabled values and a decision can be made whether the more complex model provides a significant improvement in the fit to the data. The analyst using a combination of subjective and objective criteria can make an educated decision about the best model. 

The "whiteness" of the residuals can be analyzed by plotting their cumulative periodogram and observing closeness to the ideal ramp line. Another common diagnostic technique is to determine the statistical efficiency of adding one or two parameters (and terms) to the model. This will check if the minimal parametric model chosen is truly parsimonious. The statistical F-test based on the extra-sum-of-squares principle is useful for this efficiency test. 

There are several cases where NMR spectroscopy has been used to investigate copolymers which deviate from the terminal model for copolymerisation (see also chapter 3). For example, Hill and co-workers [23, 24] have examined sequence distributions in a number of low conversion styrene/acrylonitrile (S/A) copolymers using carbon-13 NMR spectroscopy. Previous studies on this copolymer system, based on examination of the variation of copolymer composition with monomer feed ratio, indicated significant deviation from the terminal model. In order to explain this deviation, propagation conforming to the penultimate (second-order Markov) and antepenultimate (third-order Markov) models had been proposed [25-27]. Others had invoked the complex participation model as the cause of deviation [28]. From their own copolymer/comonomer composition data. Hill et al [23] obtained best-fit reactivity ratios for the terminal, penultimate, and the complex participation models using non-linear methods. After application of the statistical F-test, they rejected the terminal model as an inadequate description of the data in comparison to the other two models. However, they were unable to discriminate between the penultimate and complex participation models. Attention was therefore turned to the sequence distribution of the polymer. 

If several properties must be predicted from a single NIR speetrum, this method should be applied for each property, in order to correct each calibration model independently. Moreover, the slope/bias correction requires the computation of y-values for the standardization samples, which makes the use of generic standards impossible. Finally, this simple approach can only be used for instrument standardization when the differences between calibration and prediction steps are rather simple. When more complex instrumental differences occur, the predictions cannot be corrected by a simple univariate correction. To decide whether the slope/bias correction method can be successfully applied, a procedure based on a statistical F-test was proposed [41]. 

We have examined the copolymer composition and triad fraction data in terms of the various polymerization models. Mathematical analysis of the composition and triad fraction data using a non-linear least squares method, followed by application of the statistical F-test to discriminate between the fit of each of the models to the data (1),... 

J.M.G. Cowie (University of Stirling, Stirling, Scotland) I think if you use the statistical F-test you can also determine whether you are actually improving the fit or not. 

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