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Wilcoxon Test | Different test of two paired samples for non-parametric variables

Differences test is one of the most commonly used associative tests by researchers. Differences test can be conducted on both parametric and non-parametric variables. For parametric variables, the differences test can utilize the t-test assuming normally distributed data.

However, for non-parametric variables that are not normally distributed, the t-test cannot be used since the required assumptions are not met. As an alternative, researchers can use the differences test for non-parametric variables.

In this tutorial, Kanda Data will write a tutorial on the differences test for non-parametric variables, especially those using an ordinal measurement scale. The differences test for ordinal-scale variables with two paired sample groups can employ the Wilcoxon test.

Assumptions of the Wilcoxon Test

When researchers choose the Wilcoxon test to examine the differences between two variables, they need to consider the prerequisite assumptions that must be met. The first assumption is that this test is performed for two sample groups. If there are more than two sample groups being tested, this test cannot be used.

The next prerequisite assumption is that the measurement scale is ordinal. In addition, for measurement scales of interval and ratio that are not normally distributed, the Wilcoxon test can also be employed.

Furthermore, the two sample groups being tested must be paired. This means that the data comes from the same individual subjects or data population. For paired data, it should be symmetric, with an equal number of samples used. For example, a pretest and posttest are conducted on the same sample.

Case Study of the Wilcoxon Test

To provide a better understanding of how the Wilcoxon test is conducted, in this case study, Kanda Data presents a research study aimed at examining the difference in sales performance of car units before and after training among the marketing staff.

Fifteen marketing staff members from ABC car dealership were observed in this study. Each marketing staff member received training four times a month for a duration of six months. The researcher measured the sales performance using a Likert scale (an ordinal measurement scale).

Based on the inputted data, the researcher then calculated the sum of scores for each item statement on the Likert scale. The detailed total scores for the pretest and posttest can be seen in the table below:

Formulating the Statistical Hypotheses for the Research

Before conducting the Wilcoxon test, it is necessary to establish the statistical hypotheses. In line with the research objective to examine the difference in car sales performance before and after training.

The arrangement of the statistical hypotheses is as follows:

Ho: The sales performance of car units before training is not significantly different from after training.

H1: The sales performance of car units before training is significantly different from after training.

After formulating the null and alternative hypotheses, the next step is to determine the confidence interval for the research. In the above study, a confidence interval of 95% or a significance level (alpha) of 5% is applied. Therefore, the statistical hypotheses can be defined as follows:

p-value > 0.05 = Accepted Ho

p-value < 0.05 = Rejected Ho (accepted the alternative hypothesis)

Performing the Wilcoxon Test in SPSS

In this case, the Wilcoxon test is analyzed using SPSS. The first step researchers need to take is to set up the variable view in SPSS. Researchers should enter the variable names in the Name column, fill in the Labels, and specify the Measure.

Once the variable settings are completed, researchers can proceed to input the data into SPSS. Since the analysis results and corresponding scores have been previously entered in Excel, the data can be directly copied and pasted from Excel to SPSS.

The variable copying and pasting should be done in the columns that correspond to the variable names set up in the SPSS variable view. The next step is to perform the Wilcoxon test using SPSS.

The procedure involves clicking on “Analyze,” then selecting “Nonparametric Tests,” followed by clicking on “Legacy Dialog” and then “2-Related Samples.” This will bring up the “Two-Related Samples” window. A detailed outline of these analysis steps can be seen in the image below:

Please move the pretest and posttest variables to the “Test Pairs” box. Next, click on “Options,” enable “Descriptive,” and click “Continue.” The next step is to select “Wilcoxon” for the “Test Type” option.

Output of the Wilcoxon Test and Interpretation of Results

After completing all the analysis steps following the tutorial above, the output of the analysis will appear in SPSS. The analysis results include descriptive analysis and the results of the Wilcoxon test. The detailed results of the Wilcoxon test can be seen in the image below:

Based on the image above, the Wilcoxon test results (point 3) indicate a Z value of -2.815 with an Asymp Sig (2-tailed) value of 0.005. Let’s recall the hypothesis testing criteria as follows:

p-value > 0.05 = Accepted Ho

p-value < 0.05 = Rejected Ho (accepted the alternative hypothesis)

Based on the hypothesis testing criteria, it is known that the significance value (p-value) of 0.005 is smaller than 0.05. Therefore, because the p-value is less than 0.05, the null hypothesis (Ho) is rejected.

Since the null hypothesis is rejected, we accept the alternative hypothesis. Thus, it can be concluded that the sales performance of car units before training is significantly different from after training.

Furthermore, based on the descriptive statistical analysis (point 1), the mean value of Sales performance after training is 66.07, which is higher than the mean value of 64.07 before training.

Additionally, in the Wilcoxon Signed Ranks Test (point 2), it is also shown that out of the 10 samples, sales performance after training is higher than sales performance before training. From this, it can be inferred that the training for the marketing staff effectively improves sales performance.

This concludes the tutorial article that Kanda Data has written for this occasion. We hope it is beneficial and provides insights for those in need. See you in the next educational article!

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