Question

explain condition under which you would use a nonparametric test

Answer 1

Solution:

Reasons to Use Nonparametric Tests

Reason 1: Your area of study is better represented by the median

This is my favourite reason to use a nonparametric test and the one that isn’t mentioned often enough! The fact that you can perform a parametric test with non-normal data doesn’t imply that the mean is the best measure of the central tendency for your data.

For example, the centre of a skewed distribution, like income, can be better measured by the median where 50% are above the median and 50% are below. If you add a few billionaires to a sample, the mathematical mean increases greatly even though the income for the typical person doesn’t change.

When your distribution is skewed enough, the mean is strongly affected by changes far out in the distribution’s tail whereas the median continues to more closely reflect the centre of the distribution. For these two distributions, a random sample of 100 from each distribution produces means that are significantly different, but medians that are not significantly different.

Reason 2: You have a very small sample size

If you don’t meet the sample size guidelines for the parametric tests and you are not confident that you have normally distributed data, you should use a nonparametric test. When you have a really small sample, you might not even be able to ascertain the distribution of your data because the distribution tests will lack sufficient power to provide meaningful results.

In this scenario, you’re in a tough spot with no valid alternative. Nonparametric tests have less power to begin with and it’s a double whammy when you add a small sample size on top of that!

Reason 3: You have ordinal data, ranked data, or outliers that you can’t remove

Typical parametric tests can only assess continuous data and the results can be significantly affected by outliers. Conversely, some nonparametric tests can handle ordinal data, ranked data, and not be seriously affected by outliers. Be sure to check the assumptions for the nonparametric test because each one has its own data requirements.

Nonparametric tests are sometimes called distribution-free tests because they are based on fewer assumptions (e.g., they do not assume that the outcome is approximately normally distributed). Parametric tests involve specific probability distributions (e.g., the normal distribution) and the tests involve estimation of the key parameters of that distribution (e.g., the mean or difference in means) from the sample data. The cost of fewer assumptions is that nonparametric tests are generally less powerful than their parametric counterparts (i.e., when the alternative is true, they may be less likely to reject H₀).

It can sometimes be difficult to assess whether a continuous outcome follows a normal distribution and, thus, whether a parametric or nonparametric test is appropriate. There are several statistical tests that can be used to assess whether data are likely from a normal distribution. The most popular are the Kolmogorov-Smirnov test, the Anderson-Darling test, and the Shapiro-Wilk test¹. Each test is essentially a goodness of fit test and compares observed data to quantiles of the normal (or other specified) distribution. The null hypothesis for each test is H₀: Data follow a normal distribution versus H₁: Data do not follow a normal distribution. If the test is statistically significant (e.g., p<0.05), then data do not follow a normal distribution, and a nonparametric test is warranted. It should be noted that these tests for normality can be subject to low power. Specifically, the tests may fail to reject H₀: Data follow a normal distribution when in fact the data do not follow a normal distribution. Low power is a major issue when the sample size is small - which unfortunately is often when we wish to employ these tests. The most practical approach to assessing normality involves investigating the distributional form of the outcome in the sample using a histogram and to augment that with data from other studies, if available, that may indicate the likely distribution of the outcome in the population.

There are some situations when it is clear that the outcome does not follow a normal distribution. These include situations:

• when the outcome is an ordinal variable or a rank,
• when there are definite outliers or
• When the outcome has clear limits of detection.