Nonparametric

Often the goal of a data analysis problem requites more than simple classification of samples into known categories. It is very often desirable to have a means to detect oudiers and to derive an estimate of the level of confidence in a classification result. These ate things that go beyond sttictiy nonparametric pattern recognition procedures. Also of interest is the abiUty to empirically model each category so that it is possible to make quantitative correlations and predictions with external continuous properties. As a result, a modeling and classification method called SIMCA has been developed to provide these capabihties (29—31). 

It should be emphasized that the estimation methods presented previously apply to any hazard paper and, in addition, to a nonparametric fit to the data obtained by drawing a smooth curve through data on any hazard paper. 

When a set of data does not plot as a straight line on any of the available papers, then one may wish to draw a smooth curve through the data points on one of the plotting papers, and use the curve to obtain estimates of distribution percentiles and probabilities of failure for various given times. With such a nonparametric fit to the data, it is usually unsatisfactory to extrapolate beyond the data because it is difficult to determine how to extend how to extend the curve. Nonparametric fitting is best used only if the data contain a reasonably large number of failures. 

A first evaluation of the data can be done by running nonparametric statistical estimation techniques like, for example, the Nadaraya-Watson kernel regression estimate [2]. These techniques have the advantage of being relatively cost-free in terms of assumptions, but they do not provide any possibility of interpreting the outcome and are not at all reliable when extrapolating. The fact that these techniques do not require a lot of assumptions makes them... 

Figure 3.1 Time course of implanted tumor volume for one experimental subject (Control) and associated fitted model curves (solid line, exponential model dashed line, nonparametric kernel estimate).

All conventional approaches (mathematical and stochastic programming, parametric and nonparametric regression analysis) adopt as a common solution format real vectors, x and as performance criterion,... 

To address the modified problem statements and uncover final solutions with the desired alternative formats, data-driven nonparametric learning methodologies, based on direct sampling approaches, were described. They require far fewer assumptions and a priori decisions on the part of the user than most conventional techniques. These practical frameworks for extracting knowledge from operating data present the final uncovered solutions to the decisionmaker in formats that are both easy to understand and implement. 

S. Siegel. Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill, Tokyo, 1956. 

Porter PS, Rao ST, Ku J-Y, Poirot RL, Dakins M (1997) Small sample properties of nonparametric bootstrap t confidence intervals. J Air Waste Management Assoc 47 1197-1203 Powell R, Hergt J, Woodhead J (2002) Improving isochron calcttlatiorrs with robust statistics and the bootstrap. Chem Geol 185 191- 204... 

Our goal is to estimate the function P(r) from the set of discrete observations Y(tj). We use a nonparametric approach, whereby we seek to estimate the function without supposing a particular functional form or parameterization. We require that our estimated function be relatively smooth, yet consistent with the measured data. These competing properties are satisfied by selecting the function that minimizes, for an appropriate value of the regularization parameter X, the performance index ... 

R. L. Eubank 1999, Nonparametric Regression and Spline Smoothing, Marcel Dekker, New York. 

If the normal approximation to the binomial distribution is not valid (that is, more than 20% of expected cell counts are less than 5) for drug therapy and symptom of headache, then you can use Fisher s exact test, which is a nonparametric test, to test for a difference in proportions. To get the p-value using Fisher s exact test, you run the following SAS code ... 

For continuous variables you may be required to provide inferential statistics along with the descriptive statistics that you generate from PROC UNIVARIATE. The inferential statistics discussed here are all focused on two-sided tests of mean values and whether they differ significantly in either direction from a specified value or another population mean. Many of these tests of the mean are parametric tests that assume the variable being tested is normally distributed. Because this is often not the case with clinical trial data, we discuss substitute nonparametric tests of the population means as well. Here are some common continuous variable inferential tests and how to get the inferential statistics you need out of SAS. 

Finally, if the ldl change variable is not normally distributed, then you can run a nonparametric test on the change-from-baseline LDL value, like this ... 

If the two sample populations are not normally distributed, then you can use the nonparametric Wilcoxon rank sum test to compare the population means. The following SAS code compares the ldl change change-from-baseline means for active drug and placebo ... 

If the populations to be compared across treatments are not normally distributed, you can use the nonparametric Kruskal-Wallis test of the distributions by running PROC NPAR1WAY as follows ... 

Time-to-event analysis in clinical trials is concerned with comparing the distributions of time to some event for various treatment regimens. The two nonparametric tests used to compare distributions are the log-rank test and the Cox proportional hazards model. The Cox proportional hazards model is more useful when you need to adjust your model for covariates. 

The basic wood densities (dry) for different species were obtained from Ref. [36]. A basic density value obtained from the weighted average of the densities of each site s species was used for the species that for various reasons could not be identified. For estimation of SOC (soil organic carbon), equation (6) was used [30]. For data analysis, the nonparametric type test was chosen. We used the INFOSTAT software, and a value of 0.05 was considered significant. 

Neural networks are extensively used to develop nonparametric models and are now the method of choice when electronic noses are used to analyze complex mixtures, such as wines and oils.5 Judgments made by the neural network cannot rely on a parametric model that the user has supplied because no model is available that correlates chemical composition of a wine to the wine s taste. Fortunately, the network can build its own model from scratch, and such models often outperform humans in determining the composition of oils, perfumes, and wines. 

Lucy, D., Aykroyd, R.G. and Pollard, A.M. (2002). Nonparametric calibration for age estimation. Journal of the Royal Statistical Society C, Applied Statistics 51 183-196. 

Wu, X., and Cinar, A. (1996). An adaptive robust M-estimator for nonparametric nonlinear system identification. J. Proc. Control 6, 233-239. 

Three methods of analysis—linear regression (Gad, 1999 Steel and Torrie, 1960) a multiple comparison analysis, Dunnett s method (Dunnett, 1955) and a nonparametric analysis, such as Kruskal-Wallis (Gad, 1999)—can all be recommended. Each has its strengths and weaknesses, and other methods are not excluded. 

Kruskal-Wallis s nonparametric method avoids the complications of transformations of weighting and is about as powerful as any other method. However, it is inappropriate when the response declines markedly at high dose. 

Gehan-Breslow Modification of Kruskal-Wallis Test is a nonparametric test on censored observations. It assigns more weight to early incidences compared to Cox-Tarone test. 

Statistical methods are based on specific assumptions. Parametric statistics, those most familiar to the majority of scientists, have more stringent underlying assumptions than do nonparametric statistics. Among the underlying assumptions for many parametric statistical methods (such as the analysis of variance) is that the data are continuous. The nature of the data associated with a variable (as described previously) imparts a value to that data, the value being the power of the statistical tests which can be employed. 

Rank transformations Depends on nature of samples As a bridge between parametric and nonparametric statistics... 

The analysis of rank data, what is generally called nonparametric statistical analysis, is an exact parallel of the more traditional (and familiar) parametric methods. There are methods for the single comparison case (just as Student s t-test is used) and for the multiple comparison case (just as analysis of variance is used) with appropriate post hoc tests for exact identification of the significance with a set of groups. Four tests are presented for evaluating statistical significance in rank data the Wilcoxon Rank Sum Test, distribution-free multiple comparisons, Mann-Whitney U Test, and the Kruskall-Wallis nonparametric analysis of variance. For each of these tests, tables of distribution values for the evaluations of results can be found in any of a number of reference volumes (Gad, 1998). 

It should be clearly understood that for data that do not fulfill the necessary assumptions for parametric analysis, these nonparametric methods are either as powerful or in fact, more powerful than the equivalent parametric test. 

The Wilcoxon Rank-Sum test is commonly used for the comparison of two groups of nonparametric (inteval or not normally distributed) data, such as those which are not measured exactly but rather as falling within certain limits (for example, how many animals died during each hour of an acute study.) The test is also used when there is no variability (variance = 0) within one or more of the groups we wish to compare (Sokal and Rohlf, 1994). 

The distribution-free multiple comparison test should be used to compare three or more groups of nonparametric data. These groups are then analyzed two at a time for any significant differences (Hollander and Wolfe, 1973, pp. 124-129). The test can be used for data similar to those compared by the rank-sum test. We often employ this test for reproduction and mutagenicity studies (such as comparing survival rates of offspring of rats fed various amounts of test materials in the diet). 

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