Nonparametric analysis

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. 

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). 

Statistical analysis for these assays can be performed using many statistical programs including a twotailed student t-test or a KruskalWallis nonparametric analysis of variance on averaged triplicate values. 

Figure 14-9 DoD plot for comparison of two drug assays nonparametric analysis. A histogram shows the relative frequency of N = 65 differences with demarcated 2.5 and 97.5 percentiles determined nonparametrleally.The 90% CIs of the percentiles are shown. These were derived by the bootstrap technique.

For the sake of this example, we use the data from the parametric ANOVA example to illustrate the Kruskal-Wallis test. If it seems at all strange to use the same data for both examples, a parametric analysis and a nonparametric analysis, it is worth noting that a nonparametric analysis is always appropriate for a given dataset meeting the requirements at the start of the chapter. Parametric analyses are not always appropriate for all datasets. 

There are a number of nonparametric analysis methods dealing with continuous data. The last statistical method included in this chapter is to be used when the continuous outcome is time to an event. 

A. Hillis, M. Maguire, B. S. Hawkins, and M. M. Newhouse, The Markov process as a general method for nonparametric analysis of right-censored medical data. / Chron Disease 39 595-604 (1986). 

If, on the other hand, it is feared that the outcome itself will not be Normally distributed conditional on the fitted model and a suitable generalized linear model cannot be found, then various mixed strategies permit one to perform a partial adjustment and then proceed to a nonparametric analysis (Koch et al., 1998 Lesaffre and Senn, 2003). 

Nonparametric analysis A term used rather loosely with various meanings but which may be defined here as a method of analysis which makes rather fewer assumptions about the shape and form of the data than are required for a parametric analysis. 

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. 

Data taken from Tables 1 and 2 of Laber el al. (1992). Differences in means tested by Kmskall-Wallace nonparametric analysis of variance, H statistic with 1 df, uncorrected p values presented. 

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