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Parametric & Nonparametric Equivalent Tests
Parametric tests assume that the variables you are testing or comparing are distributed in a particular way, typically normally distributed. This is a reasonable assumption for most physical measurements such as width or depth of a stream, it is also a reasonable assumption for multimetric indexes with 8-10 metrics. For other variables, such as conductivity or abundance, it is not a reasonable assumption because these type of variables can take on extreme values. In this situation, nonparametric tests are better. Rather than comparing the raw values, they rank the values for each variable from smallest to largest. Then the tests and comparisons are done on the ranks.
In many cases, parametric and nonparametric tests will give the same answer regarding statistical significance. When 1 or all the variables have extreme values, the results are more likely to be different. When choosing, you must decide whether extreme values are meaningful even if unusual, or if the magnitude of extreme values is simply arbitrary and doesn't provide any meaning beyond "very high (or low)."
| Type of test | Parametric test | Nonparametric test |
|---|---|---|
| 2-sample | t-test | Mann-Whitney U-test |
| Paired sample | paired t-test | Wilcoxon |
| Distribution | Chi-square | Kolmogorov-Smirnov |
| > 2 samples | 1-way Anova | Kruskal-Wallis |
| Correlation | Pearsons's r | Spearman's r |
| Crossed comparisons | factorial Anova | Friedman's; Quade |
The primary advantage of parametric testing is that you get slightly more statistical power to detect differences, but not much. If the relationship is strong, the loss of power associated with nonparametric tests is typically very small.
Another approach involves transformation of the variables to make them look more "normal" in their distribution. The advantage to this approach is that parametric statistics can be used on the transformed variables. The disadvantage is that the statistical conclusions apply to the transformed variables rather than the actual measurements taken. In addition, it may be difficult to obtain a truly normal distribution.