Part 4: Parametric vs. nonparametric tests

For practical purposes, we think of parametric tests as referring to tests that assume the underlying data group(s) to be normally distributed (see part 2); they generally also assume that one’s measures derive from an equally interval scale. We think of nonparametric tests as referring to tests that do not based on these particular assumptions. Nonparametric tests are also often said to be „distribution-free“.

A potential source of confusion in working out what statistics to use in analyzing data is whether our data allows for parametric or nonparametric statistics. If we get it wrong we risk using an incorrect statistical procedure or we may use a less powerful procedure. Nonparametric tests are less powerful because they use less information in their calculation. For example, a parametric correlation uses information about the mean and deviation from the mean while a nonparametric correlation will use only the ordinal position of pairs of scores.

The basic distinction for parametric versus nonparametric is:

• If our measurement scale is nominal or ordinal then we use nonparametric statistics
• If we are using interval or ratio scales we use parametric statistics.

There are other considerations which have to be taken into account:

We have to look at the distribution of our data. Before applying parametric tests to our data, we should check that the distributions are approximately normal, i.e. follow a normal distribution. For example, MaxStat offers several normality test to check if data follow a normal distribution. We can also check the skew and kurtosis measures from the data (both should be close to zero in normally distributed data), and MaxStat offers their computation. The normality tests take into account both the skewness and kurtosis of the data, and, therefore, the application of normality tests is recommended.

If normality tests do not provide evidences for normal distribution, we have another options before selecting less powerful nonparametric tests. We can transform our data to make them more normally distributed. In some cases, transforming the data will make it fit the assumptions better. To transform data, we perform a mathematical operation on each observation, then use these transformed numbers in our statistical test. The most popular transformations are the log and square-root transformation. If transformation of data can not make the data more normally distributed, we will select an equivalent nonparametric test. The following table summarizes the differences between parametric and nonparametric tests. MaxStat does not require you to know which test is parametric or nonparametric; simply provide the information if your data follow a normal distribution (to be tested with normality tests) or not, and MaxStat choose between parametric or corresponding non-parametric test.

	Parametric	Nonparametric
Assumed distribution	normal	any
Assumed variance	homogenous	any
Type of variables	interval or ratio	nominal or ordinal
Data set relationships	independent	any
Central measure	mean	median
Advantage	more powerful	simple, less affected by outliers