The choice between parametric and non-parametric tests largely depends on the distribution of your data. Non-parametric tests are commonly used when the data does not meet the assumptions of parametric tests, such as normality, or when the distribution is unknown. These tests are advantageous because they do not require the data to fit specific distribution patterns, making them ideal when dealing with irregular data. Additionally, non-parametric methods are robust and less affected by outliers or deviations from normality, offering flexibility for a wider range of data types, including ordinal and nominal data.

Parametric tests, however, may be preferable in certain situations, particularly when you have a large sample size or when the data distribution is known. The Central Limit Theorem ensures that with large samples, the sampling distribution approximates normality, justifying the use of parametric tests. When there is strong evidence that the data follows a normal distribution, parametric tests tend to be more powerful and yield more precise results. Additionally, transforming data—using logarithms or square roots—can sometimes help align it with normal distribution requirements, making parametric tests a valid option.

In conclusion, while non-parametric tests provide flexibility and robustness, parametric tests can offer more statistical power and precision when the underlying assumptions are met. Each method has advantages depending on the nature and structure of the data being analyzed.

Additionally, it is appropriate to employ a variety of statistical tests within a single study, provided that each test is harmoniously aligned with the unique properties of its corresponding variable. The nature of the data—whether nominal, ordinal, interval, or ratio—its distribution, and the sample size can vary significantly between variables, demanding distinct analytical strategies.

Lorenzo Cioni your tests for normality are basically useless, sorry to say, because if you want to conduct a paired t-test (I simply assume that this is appropriate to answer your research question, in the first place), then the normality assumption is not about the distribution of the two time point measures, but about the differences between the time points. You cannot simply derive the distribution of the differences from the original distributions (only indirectly, if both original measures are normally distributed you can expect the differences to also normally distributed, but two non-normal distributions may give [approximately] normally distributed differences)

Therefore, you should check the differences, BUT please do not solely rely on statistical tests to check normality! This is seldom helpful. It is much more important to visually inspect the distribution, via QQ-plots and histrograms etc, to get an impression WHY they are not normally distributed. For example:

1) your distribution might really be not normal, but are counts for example, then you might think about a generalized linear model with a Poisson distribution.

2) You have some outliers, but the rest of the distribution looks ok, then you might think about robust parametric approaches like trimmed means or bootstrapping (please see publications by Rand R. Wilcox on this topic).

I would generally not advise to switch to nonparametric tests, since they a) test totally different hypotheses [no they do not test median differences without strict other assumptions!], b) therefore typically do not answer the question you are originally try to answer, and c) have other assumptions, which are even harder to test imho. And this also why I would contradict Hakan Duman . Nonparametric tests are not ideal in this situations, since parametric alternatives are available and it is a myth that nonparametric tests work good in small samples, for example.

If your variables have changing distributions, it's usually better to go with non-parametric tests. These tests don't assume a specific distribution for your data, so they're more flexible when the data isn't following a normal pattern or is unpredictable.

Abdul Kuddus as already explained above, no it is not really more flexible..... only in textbook cookbooks.......

Elizabeth Asmus

Just to give an example of "changing distributions", imagine you want to compare 2 groups, one which seems to be quite normally distributed and the other heavily skewed. Now, instead of using the typical Gaussian distribution as the likelihood, you may switch to a skewed_normal likelihood, which has one additional parameter α the skewness parameter. In Bayesian distributional modeling you are not only able to predict the mean (μ) values, as in typical frequentist OLS regression, but also auxiliary parameters, like σ, to account for heteroscedasticity, and in this case α, to account for different skews. Thus you can have different skewness parameters for each group, but you still get estimates for μ instead of some rank based measures.

I think such an approach, where you account for the specifics of each group, but you are still able to test for differences in the mean values, is much more useful and better to interpret than nonparametric tests, which doesn't even test the original hypothesis.

I would love to hear your thoughts about it.

Thank you very much for your contributions!