Permutation hypothesis test in R. Exploring a powerful simulation... | by Serafim Petrov | Towards Data Science

To compare outcomes in experiments, we often use Student’s t-test. It assumes that data are randomly selected from the population, arrived in large samples (>30), or normally distributed with equal variances between groups.If we do not happen to meet these assumptions, we may use one of the simulation tests. In this article, we will introduce the Permutation Test.
Rather than assuming underlying distribution, the permutation test builds its distribution, breaking up the associations between or among groups. Often we are interested in the difference of means or medians between the groups, and the null hypothesis is that there is no difference there. We may ask the question: from all the possible permutations, how extreme our data would look like? All possible permutations would represent a theoretical distribution. In practice, there is no need to perform ALL permutations to build the theoretical distribution, but run a reasonable number of simulations to take a sample from that distribution. Usually, there are 10k or 100k simulations.
Suppose we have a chain of 12 retail stores, and we want to test a new sales technique. We may assign or just pick, for instance, 5 stores and try it there. Then compare average sales after a certain period with sales in the rest 7 stores.
The data are not randomly collected, presumably not normally distributed; and the sample itself has fewer than 30 observations.In our dataset, we have two columns: treatment — a binary variable indicating whether the store implemented the new approach; outcome — a numeric variable recoding a sales amount at the end of a period.
From the boxplot representation, we may see that the groups are fairly different and the variance is approximately equal.
The histogram of outcome does not appear normally distributed; however, the Shapiro-Wilk normality test returns a not significant p-value = 0.3654
To find the difference in means between groups, we use build-in commands and save it to a variable original:
original <- diff(tapply(outcome, treatment, mean))
The difference is ~37.8 points. Now we ask the question: if we shuffle the data ten thousand times, how often we would observe this or more extreme difference?
We run a simulation drawing 10k shaffles without replacement, and record the difference in means for each permutation. Then we perform a two-sided test computing the number of records the absolute value of whose exceeded absolute values of original difference.
The function returns the simulated distribution and the p-value.
If we plot the distribution, we may observe that our original difference is not particularly extreme, with an exact p-value of 0.1822818
In many research projects, this would indicate that there is no statistically significant difference between groups: however, in this business case, I would pick the option that there is some evidence that the sales approach affects the outcome.
If we run the Welch t-test, we would get a similar p-value of 0.1815. The Shapiro-Wilk normality test indicated that the data were presumably obtained from the Normal Distribution, but it is not a general case.
R also has several libraries to run permutation tests. One of them is library(coin), which performs an independence test. It also returns a similar p-value of 0.1858.
The Permutation test is a powerful tool in measuring effects in experiments. It is easy to implement, and it does not rely on many assumptions as other tests do. It has not been widely popular until the simulation on computers became routinely implemented.
It is closely related to the other simulation test: the bootstrapping hypothesis test, where the samples are drawn with replacement.
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