Non-parametric Tests
So far, all of our tests have been parametric, meaning they assume that they make assumptions about the sample distribution. We assumed that the sample followed a normal distribution in the , , and tests. In the proportions and tests, we assume that the samples follow a binomial or multinomial distribution, respectively.
But sometimes, particularly with small samples, it is not far to make these assumptions. For this reason, we have non-parametric tests that do not make any assumptions about the distribution of the sample.
While there are many, many non-parametric tests. We will go over two of the most popular ones: test and the paired signed-rank test.
Please note that there are ways to calculate -values for the test statistics of non-parametric tests, but we don’t describe how to do it here due to their esoteric nature.
Test
Cheat Sheet
- Description: Tests if the median of two independent samples (say and ) are different/more than/less than the median of another sample
- Statistic: (difference of medians)
- Sidedness: Either
- Null Hypothesis: (one-sided), (two-sided)
- Alternative Hypothesis: (one-sided), (two-sided)
- Test Statistic: where
Description
The idea behind this test is that and are proxies for . In fact, the hypotheses of the -test can be restated as Since the median is just defined as such that
As stated before, there is a way to calculate a cdf for and test it against a significance level , but it is beyond this course’s scope and better left to software.
Paired Signed-ranked Test
Cheat Sheet
Description: Tests if the sample median of a sample at one point in time () is different/more than/less than the median of a sample at a different point in time ()
Statistic: (difference of medians)
Sidedness: Either
Null Hypothesis: (one-sided), (two-sided)
Alternative Hypothesis: (one-sided), (two-sided)
Test Statistic:
Description
The function in the -statistic is called the rank function. It returns the index of when is sorted in ascending order. For example, if , then sorted that would be , so .
The function (read “sign”) in takes the “sign” of its input. It is defined as:
So contains information about the relative ranks of the difference between the observations at the time of and .
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