# Parametric test or nonparametric?

I have 2 groups of sample sizes 10 and 11 respectively.

Can I apply a parametric or nonparametric test to find a significant difference in means between the groups?

• difference in what? What's the null? – Glen_b Oct 7 '14 at 7:04
• Significant difference in means of two groups – user28580 Oct 7 '14 at 7:11
• I have edited your question for spelling, clarified wording and added in your mention of means. Please check it correctly expresses your intent. – Glen_b Oct 7 '14 at 8:46
• I note that in another question of yours that refers to the same sample sizes as here, you ask about correlation. If the data are paired (which you'd need to contemplate correlation), that's critical information here. Please clarify both your questions, since it will impact the answers in both cases. – Glen_b Oct 7 '14 at 8:59

1. Since you're asking about a difference in means, if you're prepared to assume the shapes are the same under the null (that is, if the means are the same the shapes of the two population distributions are the same), and that population means exist, then you could certainly apply nonparametric tests -

i) you could apply a permutation test; the sample sizes are small enough to enumerate the full distribution.

ii) You could also apply a Wilcoxon-Mann-Whitney two sample test; this will have slightly less power than the permutation test at the normal, but may have better power if distributions are heavier-tailed. With the assumptions I mentioned, this will be a test for equality of means.

My advice would be that it's pretty pointless to test the assumption of identical shapes - firstly because the sample size is small; secondly because formal testing of assumptions answers the wrong question; and thirdly because you don't actually need to assume that they differ only by shape when H0 is false unless you want a CI for the difference in means, so you may not be able to test it in any case.

2. You could make pretty much any suitable parametric assumption (normal, exponential, gamma, Poisson, binomial, Weibull, ... etc etc) and test equality of means.

You can reasonably do a diagnostic check (such as a QQ plot or similar) of the assumption, but (again) I wouldn't advise formal testing of the assumption.

• So you are telling since sample size is very less without checking any assumptions directly I can apply the nonparametric test. But my colleague is telling we can't apply inferential statistics because of very less sample size. – user28580 Oct 7 '14 at 9:03
• I don't know what you mean by 'very less' here... less than what? Of course you can apply assumptions without checking them, but I didn't advise against checking, I advised against formal testing -- the two are not synonyms. Your colleague is mistaken -- you certainly can apply inferential statistics (parametric or nonparametric). However, your ability to check assumptions is limited and power will obviously be low. (But low power shouldn't necessarily be a problem for a pilot study.) – Glen_b Oct 7 '14 at 9:10
• See the advice (and warnings) in this question, which has much smaller sample sizes than yours. – Glen_b Oct 7 '14 at 9:15
• @Glen_b "very less" from OP means just "very small", I think. – Nick Cox Oct 7 '14 at 9:31
• @Nick Yes, I think you're right; I came to that (tentative) conclusion eventually. – Glen_b Oct 7 '14 at 10:01

The answer depends whether you can assume any parametric distribution on the two groups. E.g., if you know that the two groups are normally distributed (either by statistical hypothesis testing - see Kolmogorov Smirnov, Anderson Darling or Shapiro Wilk tests - or theoretically) then you can apply a t test to check whether the means are statistically equal... In theory, if you can assume a specific distribution on the two groups you can perform any tests based on log-likelihood. You cannot perform a test based on sample's normality distribution if you cannot assume it on both samples, also taking into account that your sample size does not allow you to rely on asymptotic results. In such case you have to consider the use of non-parametric tests (e.g. median test, Wilcoxon or Mann Whitney test) that allows you to take inference on the parameters not relying on distributional assumptions.

• Here sample size is very less. This is actually a pilot study – user28580 Oct 7 '14 at 8:57