Parametric Tests vs. Non-Parametric Tests
Inferential statistics fall into two categories, namely; parametric and non-parametric tests. The statistical approach to use depends on the level of data that you wish to examine. Generally, parametric tests are suitable for normally distributed data while non-parametric tests are applied in cases where the assumptions of parametric tests cannot be met. Parametric tests assume a normal distribution of values or a “bell-shaped” curve. However, it is rare to have such data and it’s safe to say that parametric tests are designed for non-real or idealized data. On the other hand, non-parametric tests are designed for real data; data that is skewed, lumpy, and have outliers and gaps scattered about. These tests are also called distribution-free tests because they don’t have strict assumptions to check with regard to data distribution. Hence, non-parametric tests don’t assume that your data follow a specific distribution (normal distribution). In addition, the choice between parametric tests is influenced by the level of measurement of the dependent variable. As a general rule of thumb, parametric tests should be selected when the dependent variable is measured on a continuous scale (continuous variable). In contrast, non-parametric tests are typically selected when the level of measurement of the dependent variable is nominal or ordinal and with discrete variables. Also, as aforementioned, non-parametric tests are applied when the dependent variable is continuous but the data is not normally distributed. Read more on different levels of measurement.
When to Use Parametric or Non-Parametric Tests?
The decision on whether to use a parametric test or a non-parametric test depends on whether it is the mean or median that is more accurately representing the center of your data distribution. Generally, parametric tests assess group means while non-parametric tests assess group medians. If your sample size is large enough and the mean more accurately represents the center of your data set’s distribution, use a parametric test. However, if you have a small sample size (n<30), you may be forced to use a non-parametric test. This is because checking for normality of data produced by a small sample can be difficult to tell whether the data are normal or not. For instance, the histogram may not be smooth even if the data are normal. Also, the data displayed in a histogram with a small sample is most of the time asymmetrical but there are occasions when there is no significant evidence of symmetry or asymmetry, and thus impossible to tell. On the other hand, if the median more accurately represents the center of your data set’s distribution, use a non-parametric test even if the sample size is large. In addition, non-parametric tests are used when the levels of measures being used do not lend themselves to a normal distribution or the distribution has no meaning, such as the color of eyes. In other words, nominal and ordinal measures require a non-parametric test in most cases.
Common Parametric and Non-Parametric Tests
The first step in deciding whether to use a parametric or a non-parametric test is to check normality. The easiest option is to perform a simple check using a histogram. If your histogram is roughly symmetrical, you can safely assume the data is normally distributed and a parametric test can be used. However, if the histogram is not symmetrical, then a non-parametric test is more appropriate. The following statistical analyses can be applied when your data is normal;
- T-tests
- ANOVA
- Regression Analysis
- Pearson’s Rank Correlation
If the data does not have a normal distribution or the normality of data is in doubt, you can use the following statistical analyses;
- One sample sign test
- One sample Wilcoxon Signed Rank test
- Friedman test
- Kruskal-Wallis test
- Mann-Whitney test (Wilcoxon rank sum test)
- Mood’s Median test
- Spearman Rank Correlation
You may take the following steps if the data is not normal before performing a non-parametric test on any statistical software, such as STATA and SAS;
- You can consider a transformation of data if it has a generally skewed distribution. Sometimes, there are patterns that can be observed when data is significantly skewed in one direction or the other. From the observed patterns, the histogram can be reframed so that the patterns are accounted for and the histogram displays more normality
- If the non-normality is a result of outliers, you can consider whether or not to delete them. If the outliers are significantly distant from the mean, you can investigate them further to establish if they resulted from errors in data collection. If not, you can check if they hold much context to your research. You can safely remove outliers from the dataset if they don’t hold much context so that the histogram displays more normality.
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Choosing the Most Appropriate Test
When selecting a statistical test, the most important thing to consider is what is the main hypothesis- are testing some presupposed relationship between variables or differences between groups? In some cases, there are no hypotheses and the researcher only wants to see what is there. If there are no hypotheses, then there is no statistical test. The statistical test to use is determined by your data. Luckily, most of the frequently used parametric tests have non-parametric counterparts.
Differences between Discrete Groups
If you are interested in examining differences between discrete groups, the most prevalent parametric tests used are the independent samples t-test and ANOVA (analysis of variance). An independent samples t-test is used to assess for differences in a continuous dependent variable between two groups. On the other hand, ANOVA is used to assess for differences in a continuous dependent variable when there are more than two groups to be compared. The non-parametric alternatives to these tests are the Mann-Whitney U test and the Kruskal-Wallis test, respectively. These tests are applied when the parametric assumptions are not met or when the dependent variable is measured on an ordinal scale.
Differences between Paired/ Matched Groups or in One Group
One may also be interested in examining for differences in a dependent variable among one group over a period of time, for instance, pre-test and post-test or paired groups, such as experimental and control groups. The most applicable parametric tests to use in such a scenario are the dependent samples t-test and the repeated measures ANOVA. The dependent samples t-test is used to compare scores of a continuous variable at two different points in time or under two different conditions. On the other hand, the repeated measures ANOVA is used to compare scores of a continuous variable over three or more time points OR under three or more different conditions. The non-parametric versions of these two tests are the Wilcoxon-Signed Rank test and the Friedman test (Friedman ANOVA), respectively.
Association between Variables
When examining the strength of association between two variables, the most frequent parametric test used is the Pearson rank correlation (r). It is used when one wishes to assess the relationship between two continuous variables. The non-parametric alternative to the Pearson correlation is the Spearman rank correlation (ρ). It is used when one or both variables are measured on an ordinal scale.
The table below shows related pairs of parametric and non-parametric hypothesis tests.
| Parametric Tests | Non-Parametric Tests |
| One sample t-test | One sample Sign, one sample Wilcoxon |
| Two sample t-test | Mann-Whitney test (ordinal), Chi-square (nominal) |
| One-Way ANOVA | Kruskal-Wallis, Mood’s median test |
| Repeated measures ANOVA | Friedman test |
| Pearson Correlation | Spearman correlation |