"Validity" and "importance" are two different aspects, and I really like that you mention both!

Distributional assumptions are a statistical, procedural, logical neccessity to evaluate empirical data. They exist in and come from the realm of logic; they do not represent anything from the realm of our observations, and they exists without need of any observations. The cruicial aspect is that such assumptions can be more or less useful - and how useful they can be depends on the factual long-run behaviour of the data.

Distributional assumptions are idealizations (logical constructs) that can be more or less well met by actual data, but strictly mathematically no real data will ever perfecly meet any assumption. I like to compare this with the shape of a circle. A circle is a logical construct and we can use this as a model in great many very useful calculations. But: there is nothing like a "real circle" in the whole (known) universe. Sometimes an egg is "circular enough", sometimes a milled metal coin may not be "circular enough". It depends on the purpose and on the fact if the existing difference between the model (circle) and the acual shape is important for what should be done with this object regarding its "circularity". This is similar with observations/data and the statistical assumptions.

In statistics, many assumptions actually refer to the sampling distribution or the likelihood function. The shapes of these depend on the distribution of the data and on the sample size. For increasing sample sizes they usually approximate the shape of the normal probability density. Therefor, for large samples, methods based on the assumption of a "normal-shaped" sampling distribution or likelihood function work generally quite well, no matter what the distribution of the data actually is. In your specific example the sample size was 450, what is rather large. For such large samples will the sampling distribution of the rank sums (used in KW-test) approximate the normal distribution, and it is likely that your spftware will have used this normal approximation in the KW-test to get the p-value.

So:

"Valitity": Assumptions are never valid (mathematically). [Therefore formal tests on such assumptions are logically flawed, because the null hypothesis is known to be false a priori, and a failure to reject it is a known false-negative result]

"Importance": Severe violations of assumptions can give severely biased results. How much bias and which direction must be regarded as "important" depends on the particular case and on the aims of the analysis.

Final note:

Another assumption that is common to ALL analyses is that the sampling was done appropriately [most methods assume random samples] and that the underlying model (grouping structure, monotone or linear relationships, etc.) is correct. We can never know whether or not the model is correct (models are actually also idealizations!) - we must assume it, but we may check if the data we observed seems to "disagree" (severely enough to be regarded "important").

Thank you Dr.  Wilhelm, however, the assumptions and the meaning of valid and importance are known.

The question was different from the answers!!

"The question, how much being strict to those statistical condition is valid and important?"

This depends on the circumstances of the study, the kind of data, the sample size, the aim of the study, and the actual consequences that may follow from the presumes insights. 

I think this follows from my post above.

That'a s bit like asking what kind of weighing maschine one should use. And the answer is: it depends on what you want to weigh. Gravity-based instuments will be biased on the moon and won't work in space. A high-precision instrument will not be useful to weigh cattles, weighing the masses of individual molecue fragments requires a different technique (MALDI-TOF) than weighing floor for a cake (kitchen scale), etc.

Dear Muayyad and Jochen,

I dare write something more about hypotheses and assumptions.

In a statistical test one has:

assumptions A1 to An  and hypotheses H1 to Hm.

We, conceptually, discriminate between these but the test does not. So when a significant result is found and the assumptions are not verified we do not know what is tested: an assumption or a hypothesis. That is why assumptions have to be tested/ verified beforehand (when possible).

In the examples that you describe, Muayyad, something special is going on: you compare parametric and non parametric statistics.

Under certain cicumstances ( for instance linearity when comparing pearson's r and spearmans rho) it is logical that the results are similar.

But: when data that meet the assumptions of parametric test are analyzed with a non parametric test, the non parametric test will have less power (because it uses less information from the data). This will be visible, particularly, in smaller samples where power is an issue.

Greetings,

Pierre

based on what is mentioned above and my observation: can we say that any test in parametric statistics can be done with nonparametric and it gives same results?

No, not any test. For instance, there is no way to use "nonparametric tests" to test a specific functional relationship other than a "monotone relationship". There is further no way to test interactions (multiplicative effects of factor-combinations), and it is also not possible to test effects that are intermingled with a covariance structure of other fixed or random factors.

For very simple cases there are pairs of "parametric" and "nonparametric" tests that will approch similar results as the sample size goes to infinity.

Dear Muayyad,

I agree with Jochen.

I'll give an example:

When the relation between two variables is non-linear (but monotone) a non-parametric measure of correlation (say Kendall's tau) is larger than a parametric measure (Pearsons r).

And also: when applying a non-parametric method on data that satisfy the assumptions of a parametric method, p-values will be larger than for the parametric method. The difference in p-value will become smaller as sample size increases; like Jochen writes:"that will approch similar results as the sample size goes to infinity."

Greetings,

Pierre