Quotes from Siegel & Castellan (Nonparametric statistics) :

1. Nonparametric tests are called 'distribution-free', they do not assume that the scores are drawn from a population distributed in a certain way.

2. Many of these tests are identified as 'ranking tests', which suggests their other principal merits: these tests may be used with scores which are not exact in any numerical sense.

3. Another advantage is their computational simplicity.

4. These tests are most usefull with small sample sizes.

Prof Feys raises some good points. Further, the central limit theorem is not as powerful as many people believe. Best wishes David Booth

I don't agree with the last two points given by Jos:

3. Nonparametric (distribution-free) methods are not computationally simple. They are conceptually simple, but they are often computationally demanding.

4. These tests are least useful with small samples - exactly because small samples do not provide much information about the actual distribution. This is typically associated more with a loss of power than with bias, so this problem is typically not harmful, so to say.

And there are some other points unmentioned:

- Nonparametric (distribution-free) methods do not generalize to more complicated models including covariables

- Nonparametric (distribution-free) methods do not allow modelling explicit non-linearity nor non-additive effects (interactions)

- It is typically unclear how the tested statistical hypothesis of nonparametric (distribution-free) methods is to be interpreted (typically, the tested statistical hypothesis is not even understood by people applying such tests).

- Nonparametric (distribution-free) methods can (and usually do) address different statistical hypotheses than parametric tests do; sometimes these hypotheses are of scientific interest, in which case these methods should be used.

The Chi-square test is a non-parametric statistic, also called a distribution free test. Non-parametric tests should be used when any one of the following conditions pertains to the data: The level of measurement of all the variables is nominal or ordinal.

That's also not that simple. A Chi-square test is a test that uses a (approximately) Chi-squared distributed statisic to evaluate the "p-value". If you do a z-test using z², you do a Chi-squared test, just like a t-test using t² is an F-test, and the Chi-squared test is a large-sample approximation of the F-test. The Chi-quared statistic/distribution is used for approximate likelihood ratio tests also for models with ratio scaled response variables.

Hello Chetna Chauhan

In my opinion, read a little about non-parametric tests with the names McNemar's Test, Cochran's Q, Marginal Homogeneity, Kendall's coefficient of concordance, Wald-Wolfowitz, Jonckheere-Terpstra, Moses extreme reaction.

After that, you will realize that non-parametric tests are good and useful. In some cases, they do things that parametric tests cannot.

I think there are two common mistakes about when and how to use non-parametric tests.

Studying the non-parametric tests that I mentioned some of above will show you that the two main reasons for using non-parametric tests are:

There's a lot that could said here, but I'll just make a few points.

• As suggested by some of the comments, the distinction between parametric and nonparametric tests or models is not always clear. (But I think it's a useful distinction for discussion and learning.)
• Here, I'll assume you're talking about certain rank-based tests that we might call traditional nonparametric tests like Wilcoxon-Mann-Whitney and Kruskal-Wallis. That are in some sense analogous to parametric tests like t-test and one-way anova.
• We may not need nonparametric tests. Often there is a modern method that could replace these tests, for example ordinal regression in some cases or generalized linear models. Or simulation-based methods like permutation tests and bootstrapping.
• Rank-based tests address hypotheses that may be of interest. For example, if one group tends to have higher values than another group (stochastic dominance). If that's what you want to know, then a nonparametric test is the test you may want.
• They may be useful with small sample sizes. (Note disagreement by Jochen Wilhelm above.)
• There may be a pedagogical reason for teaching about these tests. They're (kind of) simple to understand and calculate by hand. (Note partial disagreement by Jochen Wilhelm above.) And they apply to a very specific situation, which may be easier for a student to understand than approaches that apply in more general circumstances.
• On the other hand, some people think we should teach the most general methods --- say, general linear models, generalized linear models, ordinal regression --- and not bother students with these specific tests that have difficult names like Wilcoxon, Mann-Whitney, Kruskal-Wallis, Friedman.
• The CLT doesn't work miracles. In some cases quite large sample sizes are needed to get reasonable results.
• Personally, I think it's a bad idea to rely on the CLT without looking at and understanding your data. I wish we wouldn't teach this idea. Often, a different model is more appropriate, depending on the nature of the data.

As others have noted, the distinction between parametric and non-parametric is not very clear. See this comment in the VassarStats online textbook, for example:

• http://vassarstats.net/textbook/parametric.html

Note as well that the Wilcoxon-Mann-Whitney test, which is typically described as non-parametric, estimates a parameter. See Ronán Michael Conroy's nice Stata Journal article for more info. ;-)

• https://www.stata-journal.com/article.html?article=st0253

I think when someone says "nonparametric" they should state if they are not interested in any parameters, and say why, or state which they are not. It seems the odd definition of non-parametric as related to not assuming the normal distribution got popularized in the 1950s with:

"the first techniques of inference which appeared were those which made a good many assumptions about the nature of the population from which the scores were drawn, Since population values are `parameters', these statistical techniques are called parametric." \citep[p. ~2]{Siegel1956}

While that book was useful as a how-to in the days before computers, its explanations have not been as useful imo. It leads to people saying things like a chi-sq test is non-parametric, but presumably when expressed as a regression it might not be. But I think the questionner's main question is how quickly does the central limit theorem come into help, which depends on the distribution and what one means by "help", but the following is a good place to start:

@incollection{Tukey1960,

author = {John W. Tukey},

year = {1960},

title = {A survey of sampling from contaminated distributions},

booktitle = {Contributions to Probability and Statistics: Essays in Honor of {H}arold {H}otelling},

editor = {Ingram Olkin and Sudhist G. Ghurye and Wassily Hoeffding and William G. Madow and Henry B. Mann},

pages = {448-485},

publisher = {Stanford Univ. Press},

address = {Stanford, CA},

}

If the small samples are not normally distributed, non-parametric tests should be performed instead of the Student's t-test since they do not require assumptions about the distribution of population.

Chuck A Arize , did you read Jochen Wilhelm 's post, #4?

To expand on some earlier points. As Sal Mangiafico notes the CLT does not work miracles.

1) It doesn't apply to all cases (notably for some relatively common cases such as ratios of variables where the variance is undefined). It applies to averages of distributions with defined mean and variance.

2) The CLT relies on asymptotic convergence. So if it applies and you have infinite n everything is fine. In practice averages from different distributions converge at different rates. Generally very skewed and very heavy tailed distributions converge slower.

For example Poisson distribution with lambda = 10 and lambda = 1/1000 converge at very different rates. It isn't safe to assume that a large sample allows you to invoke the CLT. Though if your original distribution is fairly symmetrical and has light tails (excess kurtosis close 0) then samples of n =20 or n=30 might be fine. Likewise with heavy tails and asymmetry a sample of 10,000 might be insufficient.

For those interested in a more detailed understanding of the CLT I recommend this excellent paper:

Zhang, X., Astivia, O. L. O., Kroc, E., & Zumbo, B. D. (2022). How to think clearly about the central limit theorem. Psychological Methods. Advance online publication.

To answer your question, the use of non-parametric tests arises when the assumptions of parametric tests are not met or when the data is not normally distributed.

Non-parametric tests are particularly useful when dealing with ordinal data, small sample sizes, or outliers.

Now how a large sample can be useful from clt?

Non-parametric tests are used when the data does not meet the assumptions of parametric tests. Non-parametric tests do not require the data to be normally distributed and are more robust to outliers. They are also used when the sample size is small.