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As far as the background of my research is concerned, I developed a framework for sustainability management in organizations through a systematic review of literature and sustainability reports. That framework was translated into a set of 52 statements. A couple of example of these statements are as follows:

  • The leadership should inspire other organizational actors through their commitment towards sustainability management.
  • The organization should define its sustainability vision and mission.

The statements were posted online as a survey and experts in the field were invited to rate each statement on a five-point Likert scale (1 strongly disagree, 5 strongly agree). The survey was sent to 460 experts, 259 of which participated and 131 provided complete responses.

All of the survey statements were considered as hypothesis (the null was to reject the statement, which goes against the literature review) and the primary idea was to run a 1 sample t-test to compare the means against a (self-defined) test score of 3.5. If the results turn out to be significant, the null would have been rejected and the framework (in the form of the statement) would have been validated.

While studying for the 1 sample t-test, I came across its assumptions of normality and equal variance - which led me to its non-parametric alternative, one sample Wilcoxon's sign test. However, the literature suggests that the latter also makes assumptions about the symmetry of data. Since most of the survey respondents (experts) strongly-agree/agree with the statements of the survey (and rightly so, because these are representative of the literature), the data is skewed and has high kurtosis. For example, for the statements I have provided above, the skewness and kurtosis are -2.712 (st. err 0.212) and 5.438 (st. err 0.420) for the first statement and -2.9 (st. err 0.212) and 8.306 (st. err 0.420) for the second statement, respectively. Since my data is non-normal and asymmetric and does not have equal variances what should I do?

I have a few other questions as well:

1- I read in 'discovering statistics through SPSS' that if the sample size is greater than 25, one can go ahead with the one sample t-test. Would that be applicable to my case?

2- Is the Wilcoxon's sign test robust to the violation of the assumption of symmetry? If so, is there any reference I can use? Please note that I am referring to one sample Wilcoxon's sign test only.

3- If the answer to 2 is no, which other nonparametric tests can I use - which should have good power and can be performed in SPSS?

4- In addition to the hypothesis testing, I will be doing a test for non-respondents bais, the reliability of the data, and the effect size. Will these tests be sufficient to achieve the goal of my study and to develop a story for the readers/audience or should I look for some other measures too?

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  • $\begingroup$ The Wilcoxon signed-rank test makes no assumptions about the shape of the data or homogeneity. $\endgroup$ – user2974951 Feb 4 at 7:26
  • $\begingroup$ but the literature says that the test assumes symmetry of data (not normality though). Can you pease share any scientific reference which says that the Wilcoxon sign test does not make any assumption at all? Would certainly be a great help. $\endgroup$ – wnawaz Feb 5 at 8:46
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I think you misunderstood your literature. Similar questions:

Wilcoxon Signed Rank Symmetry Assumption

Appropriateness of Wilcoxon signed rank test

Is Wilcoxon Signed-Rank Test the right test to use?

TL:DR regarding Wilcoxon signed rank test

If you are testing the null hypothesis that the mean (= median) of the paired differences is zero, then the paired differences must all come from a continuous symmetrical distribution. Note that we do not have to assume that the distributions of the original populations are symmetrical - two very positively skewed distributions that differ only by location will produce a set of paired differences that are symmetrical. We also assume that the paired differences all have the same mean (= median). If you are testing the null hypothesis that the Hodges-Lehmann estimate of the median difference is zero, then the assumption of symmetry is not required.

And if this does not hold either, you can use a Sign test.

And before all this, I would check whether a regular t-test is appropriate. The assumptions here are similar in that they refer to the residuals, not the distribution of the data.

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  • $\begingroup$ Thank you for your prompt response and I completely agree with what you have said. However, you have mentioned about two distributions, which give me an impression that we are comparing two samples (regardless - dependent or independent), whereas my original question is about one sample. I have a fixed median value (test score) which I want to compare with all statements (data points) to see if the median is statistically higher (than the fixed median). In such a case, how do I find the pairwise difference? If I treat the fixed value as a pair against each data point, its still skewed. $\endgroup$ – wnawaz Feb 5 at 11:20
  • $\begingroup$ @wnawaz My bad, for a one-sample Wilcoxon test, to be a test of the median, the data need to be relatively symmetrical about their median. If this is not true then a Sign test can save you. $\endgroup$ – user2974951 Feb 7 at 8:20

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