Nonparametric results

Parametric results are numerical, whereas nonparametric results often aim for yes-or-no answers and often require no assumption about the underlying distribution of observations. The use of a median instead of the mean to estimate the location of a distribution is an example of the use of distribution-free statistics. The median is more robust than the mean in that it is more distribution-free, but the median is less efficient (requires more observations to achieve the same precision) than the mean for... 

Nonparametric analysis provides powerful results since the rehahility calculation is unconstrained to fit any particular pre-defined lifetime distribution. However, this flexibility makes nonparametric results neither easy nor convenient to use for different purposes as often encountered in engineering design (e.g., optimization). In addition, some trends and patterns are more clearly identified and recognizable with parametric analysis. Several possible methods can be used to fit a parametric distribution to the nonparametric estimated rehability functions (as provided by the Kaplan-Meier estimator), such as graphical procedures or inference procedures. See Lawless (2003) for details. We choose in this paper the maximum likelihood estimation (MLE) technique, assuming that the sateUite subsystems failure data are arising from a WeibuU piobabihly distribution, as expressed in Equations 1,2. 

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). 

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). 

We will follow the guidance of Albert Einstein to make everything as simple as possible, but not simpler. The reader will find practical formulae to compute results like the correlation matrix, but will also be reminded that there exist other possibilities of parameter estimation, like the robust or nonparametric estimation of a correlation. 

Quantitative methodology uses large or relatively large samples of subjects (as a rule students) and tests or questionnaires to which the subjects answer. Results are treated by statistical analysis, by means of a variety of parametric methods (when we have continuous data at the interval or at the ratio scale) or nonparametric methods (when we have categorical data at the nominal or at the ordinal scale) (30). Data are usually treated by standard commercial statistical packages. Tests and questionnaires have to satisfy the criteria for content and construct validity (this is analogous to lack of systematic errors in measurement), and for reliability (this controls for random errors) (31). 

Optimization techniques may be classified as parametric statistical methods and nonparametric search methods. Parametric statistical methods, usually employed for optimization, are full factorial designs, half factorial designs, simplex designs, and Lagrangian multiple regression analysis [21]. Parametric methods are best suited for formula optimization in the early stages of product development. Constraint analysis, described previously, is used to simplify the testing protocol and the analysis of experimental results. 

The online statistical calculations can be performed at http //members.aol.com/ johnp71/javastat.html. To carry out linear regression analysis as an example, select Regression, correlation, least squares curve-fitting, nonparametric correlation, and then select any one of the methods (e.g., Least squares regression line, Least squares straight line). Enter number of data points to be analyzed, then data, x and y . Click the Calculate Now button. The analytical results, a (intercept), b (slope), f (degrees of freedom), and r (correlation coefficient) are returned. 

The results are expressed as the number of cells per high power field for each 10 pm distance. Statistical analysis for thick filter assays can be performed using many methods including a twotailed student to test or a KruskalWallis nonparametric analysis of variance on averaged triplicate values. 

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

Originally a simple nonparametric method for determination of percentiles was recommended by the IFCC. However, the newer bootstrap method is currently the best method available for determination of reference limits. The more complex parametric method is seldom necessary, but it will also be presented here owing to its popularity and frequent misapplication. When we compare the results obtained by these methods, we usually find that the estimates of the percentiles are very similar. Detailed descriptions of these methods are given later in this chapter. 

When a model is used for descriptive purposes, goodness-of-ht, reliability, and stability, the components of model evaluation must be assessed. Model evaluation should be done in a manner consistent with the intended application of the PM model. The reliability of the analysis results can be checked by carefully examining diagnostic plots, key parameter estimates, standard errors, case deletion diagnostics (7-9), and/or sensitivity analysis as may seem appropriate. Conhdence intervals (standard errors) for parameters may be checked using nonparametric techniques, such as the jackknife and bootstrapping, or the prohle likelihood method. Model stability to determine whether the covariates in the PM model are those that should be tested for inclusion in the model can be checked using the bootstrap (9). 

A study is considered valid if the results obtained with positive and negative controls are consistent with the laboratory s historical data and with the literature. Statistical analysis is usually applied to compare treated and negative control groups. Both pairwise and linear trend tests can be used. Because of the low background and Poisson distribution, data transformation (e.g., log transformation) is sometimes needed before using tests applicable to normally distributed data. Otherwise, nonparametric analyses should be preferred. 

The aim of supervised classification is to create rules based on a set of training samples belonging to a priori known classes. Then the resulting rules are used to classify new samples in none, one, or several of the classes. Supervised pattern recognition methods can be classified as parametric or nonparametric and linear or nonlinear. The term parametric means that the method makes an assumption about the distribution of the data, for instance, a Gaussian distribution. Frequently used parametric methods are EDA, QDA, PLSDA, and SIMCA. On the contrary, kNN and CART make no assumption about the distribution of the data, so these procedures are considered as nonparametric. Another distinction between the classification techniques concerns the... 

So for example, if p = 0.60 and B = 1000 then the lower and upper 95% Cl shifts from the 25th and 975th observation to the 73rd and 993rd observation, respectively. The nonlinear transformation of the Z-distribution affects the upper and lower values differentially. The bias-corrected method offers the same advantages as the percentile method but offers better coverage if the bootstrap distribution is asymmetrical. The bias-corrected method is not a true nonparametric method because it makes use of a monotonic transformation that results in a normal distribution centered on f(x). If b = 0 then the bias-corrected method results are the same as the percentile method. 

For several years, it has been acknowledged that whenever possible bioassay data should be corrected for early death. In both CDC s and EPA s analysis of Kociba et al (6), the individual animal pathology results were not available. Recently, these data were made available and the analysis showed some interesting results (76-77). The nonparametric Kaplan-Meier estimates of the probability of a female rat developing a hepatocellular neoplastic nodule or carcinoma by the end of 25 months, the duration of the Kociba study, are shown in Figure 5. These estimates are computed separately for each dose level and take into account the observation times of each rat. The values of these estimates for the two highest dose levels are 0.81 at... 

The biomass growth and the lipase activity results were statistically evaluated using the software Statistica (Windows release 6.0). As the data did not present normal distribution, nonparametric statistic was used. Therefore, the Kmskal-Wallis test was performed to... 

For the pre-intervention surveys logarithmic transformation of the UIC produced symmetric and approximately normal distributions and where possible analysis was performed using the transformed iodine concentration. For many of the results, however, medians and results of nonparametric tests are reported to enable comparison with the post-intervention surveys. In examining the differences between years some subjects had been sampled on both occasions. To remove the effect of the assumed correlation between subjects sampled on multiple occasions, a linear mixed model analysis including subjects as random effects was used to test the difference between log iodine in the pre-intervention years. 

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