## Nonparametric statistics

Section 1.2 has explained that statistical testing often refers to the concepts of parameter and reference population. Section 4.3 has shown that the mean, standard deviation and correlation are parameters of the multinormal distribution, so that this distribution and others play a key role in testing quantitative data. When the data are semiquantitative, however, it does not make sense to compute statistics such as the mean or the standard deviation. In that case, hypothesis testing must be conducted with nonparametric statistics. Nonparametric tests are distribution-free, i.e. they do not assume that the samples were drawn from a population with a specified distribution (e.g. multinormal). Because of this, nonparametric statistics are useful not only when descriptors are semiquantitative, but also when quantitative descriptors do not conform to the multinormal distribution and researchers do not wish, or succeed, to normalize them. Many of the nonparametric tests are called ranking tests, because they are based on ranks of observations instead of actual quantitative values. Another advantage of nonparametric statistics is computational simplicity. Last but not least, nonparametric tests may be used with small samples, a situation that frequently occurs with ecological data. Nonparametric measures corresponding to the mean and variance (Section 4.1) are the median and range, respectively.

Nonparametric statistics cover all statistical methods developed for analysing either semiquantitative (rank statistics; Sections 5.2) or qualitative (Chapter 6) data. Rank statistics should always be used in the following situations:

yi yi yi

Figure 5.1 Three types of monotonic relationships between two descriptors: (a) linear (increasing and decreasing); (b) logistic (increasing monotonic); (c) atypical (decreasing monotonic).

1) One or several descriptors among those to be compared are semiquantitative.

Monotonic 2) The purpose of the study is to evidence monotonic relationships between quantitative descriptors. In a monotonic relationship, one of the descriptors keeps increasing or decreasing as the other increases (Fig. 5.1); the increase (or decrease) is not necessarily linear or smoothly curvilinear.

3) One or several (quantitative) descriptors are not normally distributed (see Section 4.7 for tests of normality and multinormality) and researchers do not wish to normalize them or do not succeed in doing so. Normalizing transformations are described in Subsection 1.5.6.

4) The number of observations is small.

The present Chapter first summarizes the methods available in the nonparametric approach, with reference to the corresponding parametric methods (Section 5.1). Ranking tests are then described for analysing relationships (Section 5.2) among groups of qualitative, semiquantitative, or quantitative descriptors and (Section 5.3) among groups of descriptors with mixed levels of precision (R analysis: Section 7.1). Most statistical computer packages include nonparametric procedures.