Nonparametric methods

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. 

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. 

With smaller sample sizes, the normality of the data becomes increasingly uncertain and nonparametric methods such as Kruskal-Wallis may be more appropriate (Zar, 1974). 

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). 

Current methods for supervised pattern recognition are numerous. Typical linear methods are linear discriminant analysis (LDA) based on distance calculation, soft independent modeling of class analogy (SIMCA), which emphasizes similarities within a class, and PLS discriminant analysis (PLS-DA), which performs regression between spectra and class memberships. More advanced methods are based on nonlinear techniques, such as neural networks. Parametric versus nonparametric computations is a further distinction. In parametric techniques such as LDA, statistical parameters of normal sample distribution are used in the decision rules. Such restrictions do not influence nonparametric methods such as SIMCA, which perform more efficiently on NIR data collections. 

The statistical software systems used for analysis of clitucal trial data can range from custom programs for specific statistical techniques to COTS packages. Such packages (e g, the SAS system, SPSS, S-Plus) provide the user with a library of statistical procedures (e.g., analysis of variance, regression, generahzed linear modelling, nonparametric methods) which can be accessed either by... 

In some instances the interest may not be in predicting the dependent variable y from the independent variable x but in determining whether they are associated. In these cases the correlation coefficient (the degree of dependence between two variables) may be calculated by either parametric or nonparametric methods, depending on the data. ... 

The nonparametric method makes no assumptions concerning the type of distribution and does not use estimates of distribution parameters. The percentiles may simply be determined by cutting off the required percentage of values in each tail of the subset reference distribution. 

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. 

The parametric method is much more complicated than the simple nonparametric method and requires computer software. The method is presented here under separate headings for testing of type of distribution, transformation of data, and the estimation of percentiles and their confidence intervals. 

The table shows an example using the 500 serum triglyceride concentrations displayed in Figure 16-3. See the text for a description of the nonparametric method. The unit of all concentrations in the table is imnol/L. 

Transform data by the logarithmic function y = ln(x) or by the square root function y = Vx and then test the fit to the Gaussian distribution. If the transformed distribution is significantly different from Gaussian shape, try another transformation or estimate the percentiles by the nonparametric method (see earlier in this chapter). Continue with the next step if the transformation to Gaussianity was successful. 

Bootstrap-based methods are reliable for estimating reference intervals. The following version using the rank-based nonparametric method is simple and reliable ... 

The presently used statistical method is the confidence interval approach. The main concern is to rule out the possibility that the test product is inferior to the comparator pharmaceutical product by more than the specified amount. Hence a one-sided confidence interval (for efficacy and/or safety) may be appropriate. The confidence intervals can be derived from either parametric or nonparametric methods. 

Van Bergeijk, K. E., Noordijk, H., Lembrechts, J., and Frissel, M. J. (1992). Influence of pH, soil type and soil organic matter content on soil-to-plant transfer of radiocesium and strontium as analyzed by a nonparametric method. J. Environ. Radioact. 15, 265-276. 

The Kaplan-Meier estimate is a nonparametric method that requires no distributional assumptions. The only assumption required is that the observations are independent. In the case of this example, the observations are event times (or censoring times) for each individual. Observations on unique study participants can be considered independent. The confidence interval approach described here is consistent with the stated preference for estimation and description of risks associated with new treatments. A method for testing the equality of survival distributions is discussed in Chapter 11. 

The sample mean is a poor measure of central tendency when the distribution is heavily skewed. Despite our best efforts at designing well-controlled clinical trials, the data that are generated do not always compare with the (deliberately chosen) tidy examples featured in this book. When we wish to make an inference about the difference in typical values among two or more independent populations, but the distributions of the random variables or outcomes are not reasonably symmetric, nonparametric methods are more appropriate. Unlike parametric methods such as the two-sample t test, nonparametric methods do not require any assumption about the shape of a distribution for them to be used in a valid manner. As the next analysis method illustrates, nonparametric methods do not rely directly on the value of the random variable. Rather, they make use of the rank order of the value of the random variable. 

In our opinion, therefore, nonparametric methods should be chosen when assumptions (such as normality for the t test) are clearly not met and the sample sizes are so small that there is very little confidence about the properties of the underlying distribution. The nonparametric method discussed in this section is a test of a shift in the distribution between two populations with a common variance represented by two samples, and it will always be valid when comparing two independent groups. 

Multiple imputations are generated by assuming a particular imputation model. Therefore, the success or failure of MI depends on the propriety of the assumed imputation model. Assumptions required in MI are (a) a model for the data values, (b) a prior distribution for parameters of the data model, and (c) the nonresponse mechanism. However, with nonparametric methods of MI, minimal distributional assumptions are required (see Section 9.6.6). 

O. G. Troyanskaya, M. E. Garber, P. O. Brown, D. Botstein, and R. B. Altman, Nonparametric methods for identifying differentially expressed genes in microarray data. Bioinformatics 18(11) 1454-1461 (2002). 

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. 

Martinez M, Friedlander L, Condon R, Meneses J, O Rangers J, Weber N, Miller M, Response to criticisms of the US FDA parametric approach for withdrawal time estimation Rebuttal and comparison to the nonparametric method proposed by Concorde and Toutain, J. Vet. Pharmacol. Ther. 2000 23 21-35. 

JD Gibbons. Nonparametric Methods for Quantitative Analysis. New York Holt, Rinehart and Winston, 1976. 

---