5.1. How to choose a simple statistical test
Before addressing the question of how to choose a test, we describe differences between parametric and nonparametric statistics. As stated in the introduction, one has to know what kind of data one has or will obtain. In the discussion below, we use a traditional definition of “parametric” versus “nonparametric tests”. In all statistical tests, parameters of one kind or another (means, medians, etc.) are estimated. The distinction has grown murkier over the years as more and more statistical distributions become available for use in contexts where previously only the normal distribution was allowed (e.g. regression, ANOVA). “Parametric” tests assume (1) models where the residuals (the variation that is not explained by the explanatory variables one is testing, i.e. inherent biological variation of the experimental units), following fitting a linear predictor of some kind, are normally distributed, or that the data follow a (2) Poisson, multinomial, or hypergeometric distribution. This definition holds for simple models only, parametric models are actually a large class of models where all essential attributes of the data can be captured by a finite number of parameters (estimated from the data), so include many distributions and both linear and nonlinear models, but the distribution(s) must be specified when analysing the data. The complete definition is quite mathematical. A nonparametric test does not require that the data be samples from any particular distribution (i.e. they are distributionfree). This is the feature that makes them so popular.
For models based on the normal distribution, this does not mean that the dependent variable is normally distributed; in fact one hopes it is multimodal, with a different mode for each different treatment. However, if one subtracts (or conditions on) the linear predictor (e.g. subtract each treatment mean from its group of observations), the distribution of each resulting group (and all groups combined) follows the same normal distribution. Also, the discussion below pertains only to “simple” statistical tests and where observations are independent.
Note that chisquare and related tests are often considered “nonparametric” tests. This is incorrect; they are very distribution dependent (data must be drawn from Poisson, multinomial, or hypergeometric distributions), and observations must be independent. While “nonparametric” tests may not require that one samples from a particular distribution, they do require that each set of samples come from the same general distribution. That is, one sample cannot come from a rightskewed distribution and the other from a leftskewed distribution; both must have the same degree of skew and in the same direction. Note that when one has dichotomous (Yes/No) or categorical data, nonparametric tests will be required if we stay in the realm of “simple” statistical tests (Fig. 4). For parametric statistics based on the normal distribution, an important second assumption is that the variance among groups of residuals is similar (homogeneous variances, also called homoscedasticity) (as shown in Fig. 5a) and not heterogeneous variances (heteroscedasticity, Fig. 5b). If only one assumption is violated, a parametric statistic is not applicable. The alternative in such a case would be to either transform the data (see Table 4 and section 5.2.), so that the transformed data no longer violate assumptions, or to conduct nonparametric statistics. The advantage of nonparametric statistics is that they do not assume a specific distribution of the data; the disadvantage is that the power (1ß, see section 1.) is lower compared to their parametric counterparts (Wasserman, 2006), though the differences may not be great. Power itself is not of such great concern because biologically relevant effects shall be detected with a large enough effect size in a welldesigned experiment. Table 3 provides a comparison between parametric and nonparametric statistics.
Fig. 5a. Two similar distributions with different means, where variances of the two groups are homogeneous; b. shows three different distributions where the means are the same but the variances of three groups are heterogeneous.
Table 3. Comparison
between parametric and nonparametric statistics.

Parametric 
Nonparametric 
Distribution 
Normal 
Any 
Variance 
Homogenous 
Any 
General data type 
Interval or ratio (continuous) 
Interval, ratio, ordinal or nominal 
Power 
Higher 
Lower 
Example Tests 

Correlation 
Pearson 
Spearman 
Independent data 
ttest for independent samples 
MannWhitney U test 
Independent data more than 2 groups 
One way ANOVA 
Kruskal Wallis ANOVA 
Two repeated measures, 2 groups 
Matched pair ttest 
Wilcoxon paired test 
Two repeated measures, >2 groups 
Repeated measures ANOVA 
Friedman ANOVA 
Table 4. Links for GLMM models for analyses of data from cage experiments.
Distribution 
Canonical Link 
Gaussian 
identity (no transformation) 
Poisson 
log 
Binomial 
logit 
Gamma 
inverse 