Differences in Parametric and Non Parametric Tests
Parametric tests are those that make assumptions about the parameters of the population distribution from which the sample is drawn. This is often the assumption that population data are normally distributed. Nonparametric tests are "nonparametric" and can therefore be used for non-normal variables.
Nonparametric tests are valid for both non normal and normal data, so why not always use them?
It seems prudent to use nonparametric tests in all cases, which would save the trouble of testing for normality. However, parametric testing is preferred for the following reasons:
1. We are seldom only interested in a significance test; We want to say something about the population the samples came from, and it's best for you to do that with it
Parameter estimates and confidence intervals.
2. It is difficult to build flexible models with nonparametric tests that allow for multiple regression confounders, for example.
3. Parametric tests typically have more statistical power than their nonparametric counterparts. In other words, significant differences are more likely to be detected when they really exist.
Do nonparametric tests compare medians?
It is a widely held belief that a Mann-Whitney U test is actually a test for differences in the medians. However, two groups could have the same median and still have a significant Mann-Whitney U test. Consider the following data for two groups of 100 observations each. Group 1: 98 (0), 1, 2; Group 2: 51(0), 1, 48(2).
The median is 0 in both cases, but from the Mann-Whitney test P<0.0001. Only if we are willing to make the additional assumption that the difference in the two groups is simply a change in location (i.e. the distribution of the data in one group is simply changed by a fixed amount from the other) can we say that the proof is a median difference test.
However, if the groups have the same distribution, a change in location shifts the medians and means by the same amount, and therefore the difference in medians is equal to the difference in means. Therefore, the Mann-Whitney U test is also a test for the difference in means. How is the Mann-Whitney U test related to the t-test? If the ranges of the data were entered into a 2-sample t-test program instead of the data themselves, the resulting P-value would be very close to that of a Mann-Whitney U-test.
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