In general, the power of parametric tests are greater than the power of the alternative nonparametric test when assumptions are met. As the sample size increases and becomes larger, the power of the nonparametric test approaches it parametric alternative.
In other words, when using a nonparametric test more data is needed to detect the same size difference as a parametric equivalent.
Nonparametric tests also assume that the underlying distributions are symmetric but not necessarily normal. The assumption is these test statistics are "distribution-free. In other words, there aren't any assumptions made about the population parameters.
When the choice exist on whether to use the parametric or nonparametric test, if the distribution is fairly symmetric, the standard parametric test are better choices than the nonparametric alternatives.
For example, if you are not sure if two data sets are normally distributed it may be safer to substitute the Mann-Whitney test to reduce the risk of drawing a wrong conclusion when testing two means.
Nonparametric tests are used when:
Advantages of nonparametric tests:
Disadvantages of nonparametric tests:
These are the most commonly used nonparametric equivalents. There are many more nonparametric tests but these are most likely to arise on a Six Sigma project and a certification exam.
A sample and a target (or given value)
Parametric - One Sample t test (testing means)
Nonparametric - One Sample Wilcoxon or One Sample Sign (testing medians)
Two independent samples
Parametric - Two Sample t test (testing means)
Nonparametric - Mann-Whitney (testing medians)
>2 independent samples
Parametric - ANOVA (testing means)
Nonparametric - Mood's median or Kruskal-Wallis test (testing medians)
Three of more matched (dependent) samples
Parametric - Two way ANOVA for matched samples
Nonparametric - Friedman Test
Use the Runs Test to examine the randomness of the data and whether a sequence of samples are randomly distributed.
Use Spearman Rank Correlation to examine the correlation between two data sets.
Use Goodman Kruska's Gamma testing the association for ranked variables.
Testing Variances - Levene's test is alternate if F and t test assumptions are not met and is used to check the homogeneity of variances across samples.
Follow these steps when you believe the data does not meet normality assumptions:
There are other transformation methods such as the Yeo-Johnson which is similar to the Box-Cox but can be used with negative values and zero.
The word "transformation" implies that each data point has the same calculation applied to it (including any specification limits when analyzing capability). Data that is meets the assumption of normality allows a wider variety of test to be used.
Other transformations are called Squared, Reciprocal, Exponential, Log (x), Lambert Gaussian, and Tukey's Ladder of Powers.
These are more advance topics of which you should consult a Master Black Belt to ensure the most appropriate transformation is used depending on the distribution of the data.
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