Question

• In your own words, explain why you would use a nonparametric test instead of a parametric test. How are they different? How are they the same?
• Come up with a research study example for each of the different kinds of Chi square tests: no difference, goodness of fit, and test of independence.

Answer 1

It’s safe to say that most people who use statistics are more familiar with parametric analyses than nonparametric analyses. Nonparametric tests are also called distribution-free tests because they don’t assume that your data follow a specific distribution.

You may have heard that you should use nonparametric tests when your data don’t meet the assumptions of the parametric test, especially the assumption about normally distributed data. That sounds like a nice and straightforward way to choose, but there are additional considerations.

Reasons to Use Parametric Tests

Reason 1: Parametric tests can perform well with skewed and nonnormal distributions

This may be a surprise but parametric tests can perform well with continuous data that are nonnormal if you satisfy the sample size guidelines in the table below.

1-sample t test	Greater than 20
2-sample t test	Each group should be greater than 15
One-Way ANOVA	If you have 2-9 groups, each group should be greater than 15. If you have 10-12 groups, each group should be greater than 20.

Reason 2: Parametric tests can perform well when the spread of each group is different

While nonparametric tests don’t assume that your data follow a normal distribution, they do have other assumptions that can be hard to meet. For nonparametric tests that compare groups, a common assumption is that the data for all groups must have the same spread (dispersion). If your groups have a different spread, the nonparametric tests might not provide valid results.

On the other hand, if you use the 2-sample t test or One-Way ANOVA, you can simply go to the Options subdialog and uncheck Assume equal variances. Voilà, you’re good to go even when the groups have different spreads!

Reason 3: Statistical power

Parametric tests usually have more statistical power than nonparametric tests. Thus, you are more likely to detect a significant effect when one truly exists.

Reasons to Use Nonparametric Tests

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

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

For example, the center 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 center 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.