# Paired or not paired t-test in untreated/treated experimental setup

I am analysing the experiment, where the cells isolated from 4 different healthy donors were treated with a certain substance or left untreated (cells from each donor were divided into 2 parts: 1 part was treated, another part was not). I am a bit confused, which type of statistical test I should choose, if I want to answer a question, whether the treatment influences the behaviour of these cells or not.

Untreated: 1.5 1.21 2.13 1.11

Treated: 3.95 1.85 4.1 2.54;

I would rather treat these data as paired, as the untreated and treated groups are not fully independent, because they come from the same donors. However, when I searched for the papers with similar experimental setup the unpaired t-test is most often used for the analysis. Are my consideration of using a paired test correct in this case?

Another moment is the choice between parametric and non-parametric test. Is this possible to conclude about distribution type if n=4? Are there any rules for choosing a statistical test with such a small sample size?

Thank you!

• Since the cells from each donor was divided into a control and treated, I would consider that a matched pair test. Without any information on the characterization of the output, it is impossible to provide advice between a parametric or non parametric test. Jan 10, 2021 at 2:17
• Thank you for your answer! Untreated: 1.5 1.21 2.13 1.11; Treated: 3.95 1.85 4.1 2.54; Shapiro-Wilk test shows normal distribution and QQplot looks normal as well; Difference untreated/treated is also normally distributed. I suppose paired t-test will work here best then Jan 10, 2021 at 13:27
• With only 4 points it is difficult to test for normality. This is where experience comes into play, Is there an expectation of a normality. I would guess yes and use a paired sample t.test. Jan 12, 2021 at 4:02

A nonparametric Wilcoxon signed-rank test will not reject at the 5% level with $$n=4$$ pairs. Even for a one-sided test, the smallest possible P-value is $$1/2^4 = 1/16 > 0.05.$$

wilcox.test(c(1,10,100,1000), alt="g")

Wilcoxon signed rank test

data:  c(1, 10, 100, 1000)
V = 10, p-value = 0.0625
alternative hypothesis: true location is greater than 0

• Thank you for your answer! Yes, this was also my concern, as I have read that Wilcoxon signed-rank test requires min n=6. So basically if I choose to do a paired test here the paired t-test will be the only choice Jan 10, 2021 at 13:33
• Another question is what would be an option, when the variable is not normally distributed with n=4, probably then only transformation to normalise the data. But if data can not be normalised.. Jan 10, 2021 at 13:39
• If you have data, then please edit them into your question. Not possible to speculate usefully on appropriate analysis without seeing data. With only four pairs it may not be possible to determine normality. The fundamental difficulty is small sample size. // In R, 90% of the time t.test(rnorm(4, 10, 5), alt="g")\$p.val returns a P-value less than 0.05, but I'm not sure I see the relevance of that just from what you have said about your experiment and data. Jan 10, 2021 at 17:54