How to Analyze Wilcoxon signed-rank test using Minitab

Introduction

The Wilcoxon signed-rank test determines whether there is a statistically significant difference in the median of a dependent variable between two related groups. It is the nonparametric equivalent to the paired t-test. As the Wilcoxon signed-ranks test does not assume normality of the differences of the two related groups, it can be used when this assumption has been violated and the use of the paired t-test is inappropriate.

For example, you could use a Wilcoxon signed-rank test to determine whether there is a difference in the number of customer complaints before and after a training course for sales reps designed to improve their customer interactions (i.e., the dependent variable would be “customer complaints”, and the two related groups would be the two different “time points”; that is, the number of customer complaints “before” the training course and the number of customer complaints “after“ the training course. Alternately, you could use a Wilcoxon signed-rank test to understand whether there is a difference in productivity based on whether packers in a factory listen to background music (i.e., the dependent variable would be "productivity", measured in terms of the number of parcels packers processed per day, and the two related groups would be the two different "conditions" participants were exposed to; that is, the number of parcels processed when packers listen to background music (condition A) compared to the number of parcels processed when packers do not listen to background music (condition B)).

In this guide, we show you how to carry out a Wilcoxon signed-rank test using Minitab, as well as interpret and report the results from this test. However, before we introduce you to this procedure, you need to understand the different assumptions that your data must meet in order for a Wilcoxon signed-rank test to give you a valid result. We discuss these assumptions next.

Assumptions

The Wilcoxon signed-rank test has three "assumptions". You cannot test the first two of these assumptions with Minitab because they relate to your study design and choice of variables. However, you should check whether your study meets these two assumptions before moving on. If these assumptions are not met, there is likely to be a different statistical test that you can use instead. Assumptions #1 and #2 are explained below:

• Assumption #1: Your dependent variable should be measured at a continuous level (i.e., they are interval or ratio variables) or ordinal level. Examples of such continuous variables include height (measured in feet and inches), temperature (measured in ^oC), salary (measured in US dollars), revision time (measured in hours), intelligence (measured using IQ score), firm size (measured in terms of the number of employees), age (measured in years), reaction time (measured in milliseconds), grip strength (measured in kg), power output (measured in watts), test performance (measured from 0 to 100), sales (measured in number of transactions per month), academic achievement (measured in terms of GMAT score), and so forth. Examples of ordinal variables include Likert items (e.g., a 7-point scale from "strongly agree" through to "strongly disagree"), amongst other ways of ranking categories (e.g., a 5-point scale explaining how much a customer liked a product, ranging from "Not very much" to "Yes, a lot"). If you are unsure whether your dependent variable is continuous (i.e., measured at the interval or ratio level) or ordinal, see our Types of Variable guide.
• Assumption #2: Your independent variable should consist of two categorical, "related groups" or "matched pairs". "Related groups" indicates that the same participants are present in both groups. The reason that it is possible to have the same participants in each group is because each participant has been measured on two occasions on the same dependent variable. For example, you might have measured 100 participants' salary in US dollars (i.e., the dependent variable) before and after they took an MBA to improve their salary (i.e., the two "time points" where participants' salary was measured – "before" and "after" the MBA course – reflect the two "related groups" of the independent variable). Since the same participants were measured at these two time points, the groups are related. It is also common for related groups to reflect two different conditions that all participants undergo (i.e., these conditions are sometimes called interventions, treatments or trials). For example, you might have measured 50 participants' test anxiety (i.e., the dependent variable) when they underwent a hypnotherapy programme (condition A) compared to undergoing a counselling session (condition B) designed to reduce such anxiety (i.e., the two "conditions" where participants' test anxiety was measured – "condition A" and "condition B" – reflect the two "related groups" of the independent variable).

Assumption #3 relates to the nature of your data and can be checked using Minitab. You have to check that your data meets this assumption because if it does not, the results you get when running a Wilcoxon signed-rank test might not be valid. In fact, do not be surprised if your data violates this assumption. This is not uncommon. However, there are possible solutions to correct such violations (e.g., transforming your data) such that you can still use a Wilcoxon signed-rank test. Assumption #3 is explained below:

• Assumption #3: The distribution of the differences between the two related groups are symmetrical in shape (i.e., the distribution of differences between the scores of both groups of the independent variable – for example, the difference scores of "carbohydrate-only drink" and "carbohydrate-protein drink" trials – are symmetrical in shape). If the distribution of differences is symmetrically shaped, you can analyse your study using the Wilcoxon signed-rank test. If not, it may be possible to transform your data to achieve a symmetrically-shaped distribution of differences. However, if this option is not possible or desirable (Altman, 1999), you might run a sign test instead of the Wilcoxon signed-rank test since the sign test does not require this assumption to be met.

In practice, checking for assumption #3 will probably take up most of your time when carrying out a Wilcoxon signed-rank test. However, it is not a difficult task, and Minitab provides all the tools you need to do this.
In the section, Test Procedure in Minitab, we illustrate the Minitab procedure required to perform a Wilcoxon signed-rank test assuming that no assumptions have been violated. First, we set out the example we use to explain the Wilcoxon signed-rank test procedure in Minitab.