Violating the assumptions of one sample t-test and wilcoxon's sign test

Question

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?

Answer 1

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.