Non-parametric method

Ideally, to characterize the spatial distribution of pollution, one would like to know at each location x within the site the probability distribution of the unknown concentration p(x). These distributions need to be conditional to the surrounding available information in terms of density, data configuration, and data values. Most traditional estimation techniques, including ordinary kriging, do not provide such probability distributions or "likelihood of the unknown values pC c). Utilization of these likelihood functions towards assessment of the spatial distribution of pollutants is presented first then a non-parametric method for deriving these likelihood functions is proposed. 

Non-linear models, such as described by the Michaelis-Menten equation, can sometimes be linearized by a suitable transformation of the variables. In that case they are called intrinsically linear (Section 11.2.1) and are amenable to ordinary linear regression. This way, the use of non-linear regression can be obviated. As we have pointed out, the price for this convenience may have to be paid in the form of a serious violation of the requirement for homoscedasticity, in which case one must resort to non-parametric methods of regression (Section 12.1.5). 

It is fair to say that statisticians tend to disagree somewhat regarding the value of non-parametric methods. Some statisticians view them very favourably while others are reluctant to use them unless there is no other alternative. 

Clearly the main advantage of a non-parametric method is that it makes essentially no assumptions about the underlying distribution of the data. In contrast, the corresponding parametric method makes specific assumptions, for example, that the data are normally distributed. Does this matter Well, as mentioned earlier, the t-tests, even though in a strict sense they assume normality, are quite robust against departures from normality. In other words you have to be some way off normality for the p-values and associated confidence intervals to be become invalid, especially with the kinds of moderate to large sample sizes that we see in our trials. Most of the time in clinical studies, we are within those boundaries, particularly when we are also able to transform data to conform more closely to normality. 

Further, there are a number of disadvantages of non-parametric methods ... 

With parametric methods confidence intervals can be calculated which link directly with the p-values recall the discussion in Section 9.1. With non-parametric methods the p-values are based directly on the calculated ranks and it is not easy to obtain a confidence interval in relation to parameters that have a clinical meaning that link with this. This compromises our ability to provide an assessment of clinical benefit. 

Non-parametric methods reduce power. Therefore if the data are normally distributed, either on the original scale or following a transformation, the non-parametric test will be less able to detect differences should they exist. 

For these reasons non-parametric methods are used infrequently within the context of clinical trials and they tend only to be considered if it is clear that a corresponding parametric approach, either directly or following a data transformation, is unsuitable. 

Non-parametric methods statistical tests which make no assumptions about the distributions from which the data are obtained. These can be used to show iifferences, relationships, or association even when the characteristic observed can not be measured numerically. 

Signal processing may also involve parameter estimation methods (e.g., resolution of a spectral curve into the sum of Gaussian functions). Even in such cases, however, we may need non-parametric methods to approximate the position, height and half-width of the peaks, used as initial estimates in the parameter estimation procedure. 

The most commonly employed univariate statistical methods are analysis of variance (ANOVA) and Student s r-test [8]. These methods are parametric, that is, they require that the populations studied be approximately normally distributed. Some non-parametric methods are also popular, as, f r example, Kruskal-Wallis ANOVA and Mann-Whitney s U-test [9]. A key feature of univariate statistical methods is that data are analysed one variable at a rime (OVAT). This means that any information contained in the relation between the variables is not included in the OVAT analysis. Univariate methods are the most commonly used methods, irrespective of the nature of the data. Thus, in a recent issue of the European Journal of Pharmacology (Vol. 137), 20 out of 23 research reports used multivariate measurement. However, all of them were analysed by univariate methods. 

Two non-parametric methods for hypothesis testing with PCA and PLS are cross-validation and the jackknife estimate of variance. Both methods are described in some detail in the sections describing the PCA and PLS algorithms. Cross-validation is used to assess the predictive property of a PCA or a PLS model. The distribution function of the cross-validation test-statistic cvd-sd under the null-hypothesis is not well known. However, for PLS, the distribution of cvd-sd has been empirically determined by computer simulation technique [24] for some particular types of experimental designs. In particular, the discriminant analysis (or ANOVA-like) PLS analysis has been investigated in some detail as well as the situation with Y one-dimensional. This simulation study is referred to for detailed information. However, some tables of the critical values of cvd-sd at the 5 % level are given in Appendix C. 

Typical data sets in chemistry contain 20 to 100 objects with 3 to 20 features. This small number of objects is not sufficient for a reasonably secure estimation of probability densities. Hence the application of parametric methods is not possible. The use of non parametric methods that make no assumptions about the underlying statistical distribution of data is necessary. These methods, however, do not allow for statements about the confidence of the results. 

Initial experiments used principal component analysis (PCA) to investigate the multivariate response. PCA is a non-parametric method which outputs linear combinations of the input values (the principal components ), such that the majority of variation is concentrated in the first few components. 

Introduce non-parametric methods where data is converted to rankings, so they become distribution-free tests ... 

Describe four widely applicable non-parametric methods (Mann-Whitney, Wilcoxon paired samples, Krukal-Wallisand Spearman correlation)... 

When non-parametric methods are applied to data that is normally distributed, they are slightly less powerful than their parametric equivalents, although the difference is not great. For the tests covered in this chapter, the non-parametric test has about 95 per cent of the power of its parametric equivalent. In other words, if our data is normally distributed then a sample of 19 tested by a parametric method would provide about the same power as a sample of 20 tested by the non-parametric equivalent. Since the power of the two types of test is so similar, it is not surprising that the P values generated [0.034 for the f-test (when applied to the transformed data) and 0.036 for the Mann-Whitney test] are barely different. 

A common view is that, when planning any experiment where data will be collected on an ordinal scale, we may as well reconcile ourselves to the use of non-parametric methods from the outset. 

Unless there is specific evidence that the data is likely to behave itself unusually well, just accept that non-parametric methods will have to be used. Power loss will, at worst, be very slight. 

Where data are reasonably normally distributed, non-parametric methods are a little less powerful than their parametric equivalents, but where the data are severely nonnormal, the non-parametric test may be much more powerful. With non-normally distributed interval scale data, the best solution is transformation to normality, but failing that, non-parametric methods can be used with only modest loss of power. 

Ordinal data can potentially approximate to a normal distribution, but tends to be severely non-normal. The use of non-parametric methods is not obligatory with such data, but is common practice. 

Quantitative continuous data may be evaluated by standard statistical methods. It is inappropriate to use parametric statistical methods on semiquantitative data (i.e., renal injury light miscroscopic assessment scores), although appropriate non-parametric methods (e.g., Duncan s rank-sum procedure) may be used. 

Parametric data were presented as mean SD. To determine differences in glutamate concentrations, a repeated-measures analysis of variance was performed. The cutaneous sensation, hind-limb motor function, and morphological changes of the spinal cord were analyzed with a non-parametric method (Kruskal-Wallis test) followed by the Mann-Whitney U-test. 

This is a non-parametric method that calculates the distances matrix between all n observations and uses the following assignation rule sample is assigned to the group most represented among the nearest k observations . Generally k is odd, and the size of the groups is also taken into account. 

The discriminant analysis techniques discussed above rely for their effective use on a priori knowledge of the underlying parent distribution function of the variates. In analytical chemistry, the assumption of multivariate normal distribution may not be valid. A wide variety of techniques for pattern recognition not requiring any assumption regarding the distribution of the data have been proposed and employed in analytical spectroscopy. These methods are referred to as non-parametric methods. Most of these schemes are based on attempts to estimate P(x g > and include histogram techniques, kernel estimates and expansion methods. One of the most common techniques is that of K-nearest neighbours. 

For t descriptive statistics should be given. If t jj is to be subjected to a statistical analysis this should be based on non-parametric methods and should be applied to untransformed data. A sufficient number of samples around predicted maximal concentrations should have been taken to improve the accuracy of the t jj estimate. For parameters describing the elimination phase (Tj/j) only descriptive statistics should be given. 

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