# Random Permutation Tests

At the 2014 Strata + Hadoop conference John Rauser gave a great keynote title “Statistics Without the Agonizing Pain“.  It is probably worth watching before reading the rest of this article, in it he introduces the concept of Random Permutation Tests.

“Classic” statistical tests usually make some sort of assumption about the distribution of the data e.g. normally distribution data . Are these assumptions always true? Probably not, but they are often approximately close enough to give you a useful result. By making these assumptions, these tests are called parametric.

Random Permutation Tests make no assumptions on the underlying distribution of the data. They are considered non-parametric tests. This can be extremely useful when:

• Your data just doesn’t seem to fit the distribution the classic statistical test assumes. For instance, perhaps it is bi-modal and the test assumes normality.
• You have outliers e.g. users who spend significantly more than others.
• You have a small sample size.

Random Permutation Tests can be used in almost any setting where you would compute a p-value. In this article I will focus on there use in experimental studies, you want to see if there is a difference between two treatment groups (A/B Tests, medical studies, etc.)

# Overview

The essential idea behind random permutation tests is:

1. Compute a test statistic between two (or more) groups. This could be the difference between two proportions, the difference between the means of the two groups etc.
2. Now randomly shuffle the data assigned to each group.
3. Measure the test statistic again on the shuffled data.
4. Repeat 2 and 3 many times
5. Look at where the test statistic from 1 falls in the distribution of test statistics from 2-4.

We have used steps 2-4 to empirically estimate the sampling distribution of the test statistic. From this distribution you can compute the p-value for your observed test statistic.

# Example

Let’s imagine we want to add a new widget on our checkout page of our e-commerce site to upsell products to a user.

The question we want to answer is, does adding the widget increase our revenue?

We run an A/B test with:

• Original checkout page
• Checkout page with widget

We know how much each user spent and what variant they have been given.

Lets generate some example transaction data in R:

We have randomly sampled the data from a log-normal distribution with equal mean and variance. We set the seed to ensure the results are repeatable. So in this case, we are looking to find there is no significant difference between the two datasets.

The classic statistical approach here would be to use a t-test. Let’s instead apply our random permutation test.

First, let’s compute the difference between the means of the two groups:

This gives a difference of 193.47. How likely is this to have happened by chance?

What we want to do is randomly shuffle our data between the two groups. If we were to sample (without replacement) once and compute the difference using the randomly shuffled version of groups:

This gave me a difference of 45.69. Now we will repeat this many times:

If we plot the examples, along with where our observed difference falls:

Straight away, it is fairly clear this observation could  just be due to random chance. It is not on the very extreme of the distribution. However let’s compute a p-value

On the two-tailed test, we get a p-value of 0.102, so we would accept the null hypothesis of there being no difference. Notice the add one here on both the denominator and numerator. Essentially we are adding our original measured test statistic to the random permutations. This ensures we never get a zero probability.

The standard t-test would give a p-value of 0.0985, so roughly similar. However, what if we didn’t care about the mean, but the median transaction value? Using random permutation tests, this is very simple to compute (a simple change to our code). Under classic statistical tests, we would have to go off and find the exact test we need to use under those conditions.

## Speed Improvements

On simple speed improvement is to parallelise the loop to compute the re-samples:

# Coin Package

As usual, R already has a package to help us do all of this. Using the same data as before we would run:

Generally you will find this is much faster for running large numbers of iterations.

One downside is you don’t get the visualisation of how extreme the observed data is compared to the empirical sampled histogram (Figure 2). I find this graph extremely useful when explaining how extreme a result appears to be, extremely to a non-statistical audience

# Summary

Random permutation tests are a nice alternative to classic hypothesis tests. In many cases they will give you almost exactly the same results. Being able to visualise the distribution (Figure 2) can be a massive assistance in explaining the p-value.

• Almost no assumptions on the underlying dataset being analysed
• Can be used for any test statistic (either it is implemented in coin or can be programmed yourself).
• Can be applied to all sorts of data types (numerical, ordinal, categorical) without having to remember the exact parametric test you should use.

• Computing large number of re-samples is potentially slow. Although on modern computers this is less of a concern
• Relies on the null hypothesis, that there is no association between the dataset and so the group labels are interchangeable under the null hypothesis.

Personally, I like to use both classic statistical tests and random permutation tests, even if all they do is validate one another.

# Multiple Statistical Tests

A/B testing is common place in marketing and website optimisation. But why just stop at running two variations? Why not three or eight? I have certainly seen A/B tests suddenly grow to A/B/C/D… and so on. In statical literature, this is called multiple testing.

The short answer is there is nothing stopping you from doing this, however you do need to adjust your techniques for interpreting the test results or risk interpreting noise results as signal.

In fact, many A/B testing tools now offer you “multivariate testing” (Optimizely, maxymiser, etc.), which is just a fancy term for running factorial experiments and you will be running multiple tests at the same time.

• The common pitfalls when running multiple statistical tests
• Techniques to adjust for multiple comparisons
• How this affects sample size when planning experiments

# The dangers of multiple comparisons

One of my favourite examples of the danger of multiple comparisons is a study by Benett et al. 1

The very brief summary of the paper is:

• A very much dead Atlantic Salmon was placed in an fMRI machine.
• The fMRI measures the blood oxygen levels in units called voxels (think image pixels). Each voxel is measuring the change in blood oxygen levels. Typically fMRI machines will measure 130,000 voxels.
• The (dead) salmon was then shown a series of emotionally charged images and the difference between blood oxygen levels was measured.
• Funnily enough, with that many comparisons, you do find a statistically significant result. Can we conclude the dead’s salmon brain was responding to the images? Well no, obviously, it’s dead. If you do enough comparisons without any correction, you will end up with something significant.

Let’s see this in action. We will randomly sample from a binomial distribution, with p=0.5, 100 times. Each time we will test if the observed proportion is significantly different from 0.5:

Run it a few times, most times you should see one one of the p-values come out as below the alpha value. For example:

We need to control for the fact we are doing a large number of tests. Next we will look at ways to deal with these issues.

# Techniques

## Family Wise Error Rate

The Family Wise Error Rate (FWER) is the probability of making at least one type I error. If V is the number of type I errors:

$$FWER=P(V \gt0)$$

We will looking at techniques that control the FWER to ensure $$FWER \le \alpha$$. We will only look at methods that control this in the strong sense, that it is valid for all configurations of true and false hypotheses.

### Bonferroni

The Bonferroni correction is the simplest method for controlling the FWER.  Simply you now reject the null hypothesis if:

$$p_i \le \frac{\alpha}{m}$$

where $$p_i$$ is the p-value of the i-th test and$$m$$ is the number of tests.

This is a very conservative correction and if you have large number of tests, you need very small p-values to find anything significant.

To adjust p-values based on Bonferroni correction:

$$\hat{p}_i =m p_i$$

To show this in R, let’s go back to our previous example:

We should now, hopefully, see no significant results.

The p.adjust method is what we use to obtain the adjusted p-values. It is also possible to use pairwise.prop.test to do this all in one go, but I personally prefer to keep them separate (e.g. compute p-values, then adjust them).

### Holm

In Holm’s method you order the p-values in increasing order. It is called a step down procedure. You compute a critical value for each p-value:

$$a_i = \frac{\alpha}{m – i – 1}$$

Starting with the smallest p-value, keeping rejecting hypotheses until the first where $$p_i > \alpha_i$$, at which point accept $$H_i$$ and all remaining hypothesises.

$$\hat{p}_i= \left\{\begin{array}{l l}mp_i & \quad \text{if i=1}\\ \max(\hat{p}_{i-1},(m -i + 1)p_i) & \quad \text{if i=2,\dots,m}\end{array} \right.$$

In R we can simply use:

Holm’s method is equally as powerful in terms of assumptions as Bonferroni, and it less conservative. Therefore it is always suggested to use Holm’s method over Bonferroni.

### Hochberg

Holm’s method was known as a step down procedure, as you start with the smallest p-value and work upwards.

Hochbeg’s method is a step up method. You start with the largest p-value , until you find the first $$p_i \le \alpha_i$$. It uses the same critical values as Holm’s method.

The adjusted p-values are computed using:

$$\hat{p}_i= \left\{\begin{array}{l l}p_m & \quad \text{if i=m}\\ \min(\hat{p}_{i+1},(m -i + 1)p_i) & \quad \text{if i=m-1,\dots,1}\end{array} \right.$$

In R we can use:

An important consideration Hochberg’s method is that each p-value must be independent or positively dependent.  If you are running A/B tests and ensured users have only ever seen one variant, then this assumption is valid.

Hochberg’s method is considered more powerful than Holm’s as it will reject more hypotheses.

### Hommel

Hommel’s procedure is more powerful than Hochberg’s but slightly harder to understand. It has exactly the assumptions at Hochberg’s (independent or positively dependent p-values) and being more powerful it should be preferred over Hochberg’s.

Hommel’s procure rejects all p-values that are $$\le \frac{\alpha}{j}$$

The value for $$j$$ is found by:

$$j = \max_{i=1,\dots,m}\{ p_{m-i+k}\gt\frac{k\alpha}{i}\textrm{for }k=1,\dots,i\}$$

Think of this as us looking at all the sets of hypotheses. We are trying to find the largest $$j$$ where the condition $$p_{n-i+k}\gt\frac{k\alpha}{i}$$ is true for all the values of $$k$$.

Let’s look at this in practice 2. Suppose we have 3 hypotheses with p-values $$p_1=0.024$$,$$p_2=0.030$$ and$$p_3=0.073$$. To find $$j$$ we compute:

For $$i=1$$, $$p_3 = 0.073 > \alpha = 0.05$$

For $$i=2$$, $$p_3 = 0.073 > \alpha = 0.05, p_2 = 0.030 >\frac{ \alpha}{2}=0.025$$

For $$i=3$$, $$p_3 = 0.073 > \alpha = 0.05, p_2 = 0.030 <\frac{ 2\alpha}{3}=0.033$$,

$$p_1=0.024 > \frac{\alpha}{3} = 0.0167$$

We find two sets of hypotheses where the statement is true for all $$k$$. The values of $$i$$ for these two sets were $$\{1,2\}$$, so $$j=\max\{1,2\}=2$$. We use $$j$$ and reject any p-values less than $$\frac{\alpha}{2}$$. Which in this case is $$p_1=0.024$$.

In the following R code, I have provided code for calculating $$j$$ and the adjusted p-values. The adjusted p-value calculation followed the algorithm in the Appendix of Wright’s paper:

As with the other methods, R provides a pre-built function for calculating the adjusted p-values:

## False Discovery Rate

Up till now we have focused on methods to control the Family Wise Error Rate (FWER). Let V be the number of false positives in hypotheses we reject and P be the number of hypotheses rejected. In FWER methods we are ensuring:

$$P(V \ge 0) \le\alpha$$

We have applied various method (Holm, Hochberg, etc.) in order to ensure this. If we want to control False Discovery Rate (FDR), we will ensure:

$$FDR =\mathbb{E}(\frac{V}{P}) \le \alpha$$

That is the expected false positive rate will be below $$\alpha$$. Many FWER methods can be seen as too conservative and often suffer from low power. FDR methods offer a way to increase power but maintain a principled bound on the error.

The FDR methods were developed on the observation

4 false discoveries out of 10 rejected null hypotheses

is much more serious than

20 false discoveries out of 100 rejected null hypotheses

That is when we are running a large number of tests, we are possibly willing to accept a percentage of false discoveries if we still find something interesting.

Christopher Genovese has an extremely good tutorial on FDR and worth looking over for much more detail.

R provides two methods for calculating adjusted p-values (sometimes called q-values in the FDR context) based on controlling the FDR:

BH is the original Benjamini-Hochberg procedure where the concept of FDR was introduced. It has the same assumptions as the Hochberg procedure (e.g. independent or positively dependent p-values). BY is Benjamini–Yekutieli procedure, it has no assumptions on the dependency of the p-values (but as such it is more conservative than BH).

FDR methods have become popular in genomic studies, neuroscience, biochemistry, oncology and plant sciences.  Particularly where there is a large number (thousands often) of tests to be performed, but you don’t want to miss interesting interactions.

## Choosing a method

We have covered a variety of techniques to handle multiple tests, but what one should you use? This is my very crude suggestions for choosing which method to use:

1. I have less than or equal to 50 tests and each test is independent or positively dependent – Use Hommel.
2. I have less than or equal to 50 tests but I do not know if the tests are independent or positively dependent – Use Holm.
3. I have more than 50 tests and each test is independent or positively dependent – Use FDR Benjamini-Hochberg.
4. I have more than 50 tests but I do not know if the tests are independent or positively dependent – Use FDR Benjamini–Yekutieli.

The 50 tests is an arbitrary choice but in general if you are using a lot of tests you might want to consider FDR methods.

## Sample Size

Intuitively, the more tests you run, the larger sample size you will need. While there a certainly more complicated methods for determining sample size, I will describe one simple method:

• Apply the Bonferroni correction $$\alpha’=\frac{\alpha}{m}$$
• Plug this $$\alpha’$$ into the sample size calculations discussed previously.

This is likely to be an over-estimate of the sample size required, but will give you a good indication of how many more samples will be needed.

### Alternatives

This is by no means an exhaustive description of how to deal with multiple tests. In this article I have focused on ways to adjust existing p-values. Other methods it may be worth exploring:

• ANOVA and Tukey HSD tests (Post-hoc analysis in general). One issue here is that it assumes normality of the data. This may mean you need to perform transformations like Arcsine Square Root 3 on proportion data.
• Bootstrap methods – Simulating how unlikely a sample is to occur if they were not independent.
• Bayesian methods

Potentially in later articles I will try to explore some of these methods further.

1. See  http://prefrontal.org/files/posters/Bennett-Salmon-2009.pdf
2. Example courtesy of http://www2.math.uu.se/research/pub/Ekenstierna.pdf
3. https://www.biostars.org/p/6189/

# Significance Testing and Sample Size

A common question when analysing any sort of online data:

Is this result statistically significant?

• A/B Testing
• Survey Analysis

In this post, we will explore ways to answer this question and determine the sample sizes we need.

Note: This post was heavily influenced by Marketing Distillery’s excellent  A/B Tests in Marketing.

## Terminology

First let us define some standard statistical terminology:

• We are deciding between two hypotheses – the null hypothesis and the alternative hypothesis.
• The null hypothesis is the default assumption that there is no relationship between two values.  For instance, if comparing conversion rates of two landing pages, the null hypothesis is that the conversion rates are the same.
• The alternative hypothesis is what we assume if the null hypothesis is rejected. It can be two-tailed, the conversion rates are not the same, or one-tailed, where we state the direction of the inequality.
• A result is determined to be statistically significant if the observed effect was unlikely to have been seen by random chance.
• How unlikely is determined by the significance level. Commonly it is set to 5%. It is the probability of rejecting the null hypothesis when it is true. This probability is denoted by $$\alpha$$

$$\alpha = P(rejecting\ null\ hypothesis\ |\ null\ hypothesis\ is\ true)$$

• Rejecting the null hypothesis when the null hypothesis is true, is called a type I error.
• A type II error occurs when the null hypothesis is false,  but is erroneously failed to be rejected. This probability of this is denoted by $$\beta$$

$$\beta = P(fail\ to\ reject\ null\ hypothesis\ |\ null\ hypothesis\ is\ false)$$

• The power of a test is $$1-\beta$$. Commonly this is set to 90%.
• The p-value is calculated from the test statistic. It is the probability that the observed result would have been seen by chance assuming the null hypothesis is true.
• To reject the null hypothesis we need a p-value that is lower than the selected significance level.

## Types of Data

Usually we have two types of data we want to perform significance tests with:

1. Proportions e.g. Conversion Rates, Percentages
2. Real valued numbers e.g. Average Order Values, Average Time on Page

In this post, we will look at both.

## Tests for Proportions

A common scenario is we have run an A/B test of two landing pages and we wish to test if the conversion rate is significantly different between the two.

An important assumption here is that the two groups are mutually exclusive e.g. you can only have been shown one of the landing pages.

Null hypothesis is that the proportions are equal in both groups.

The test can be performed in R using:

Under the hood this is performing a Pearson’s Chi-Squared Test.

Regarding the parameters:

x – Is usually a 2×2 matrix giving the counts of successes and failures in each group (conversions and non-conversions for instance).

n – Number of trials performed in each group. Can leave null if you provide x as a matrix.

p – Only if you want to test for equality against a specific proportion (e.g. 0.5).

alternative – Generally only used when testing against a specific p. Changes the underlying test to use a z-test. See Two-Tailed Test of Population Proportion for more details. The z-test may not be a good assumption with small sample sizes.

conf.level – Used only in the calculation of a confidence interval. Not used as part of the actual test.

correct – Usually safe to apply continuity correction. See my previous post for more details on the correction.

The important part of the output of prop.test is the p-value. If this is less than your desired significance level (say 0.05) you reject the null hypothesis.

In this example, the p-value is not less than our desired significance level (0.05) so we cannot reject the null hypothesis.

Before running your test, you should fix your sample size in each group in advance. The R library pwr has various functions for helping you do this. The function pwr.2p.test can be used for this:

Any one of the parameters can be left blank and the function will estimate its value. For instance, leaving n, the sample size, blank will mean the function will compute the desired sample size.

The only new parameter here is h. This is the minimum effect size you wish to be able to detect. h is calculated as the difference of the arcsine transformation of two proportions:

$$h = 2 \arcsin(\sqrt{p_1}) – 2\arcsin(\sqrt{p_2})$$

Assuming you have an idea of the base rate proportion (e.g. your current conversion rate) and the minimum change you want to detect, you can use the follow R code to calculate h:

# Tests for real-values

Now imagine we want to test if a change to a eCommerce web-page increases the average order value.

The major assumption we are going to make is that the data we are analysing is normally distributed. See my previous post on how to check if the data is normally distributed.

It may be possible to transform the data to a normal distribution, for instance if the data is log-normal. Time on page often looks to fit a log-normal distribution, in this case you can just take the log of the times on page.

The test we will run is the two-sample t-test. We are testing if the means of the two groups are significantly different.

The parameters:

x – The samples in one group

y – The samples in the other group. Leave NULL for a single sample test

alternative – Perform two-tailed or one-tailed test?

mu – Use this if you know the true mean

paired – TRUE for the paired t-test, used for data where the two samples have a natural partner in each set, for instance comparing the weights of people before and after a diet.

var.equal – Assume equal variance in the two groups or not

conf.level – Used in calculating the confidence interval around the means. Not part of the actual test.

The test can be run using:

In this example, as our p-value is greater than 0.05 significance level, so we cannot reject the null hypothesis.

As before, before running the experiment we should set the sample size required. Using the pwr library we can use:

d is the effect size. For t-tests the affect size is assessed as:

$$d = \frac{|\mu_1 – \mu_2|}{\sigma}$$

where $$\mu_1$$ is the mean of group 1, $$\mu_2$$ is the mean of group 2 and $$\sigma$$ is the pooled standard deviation.

Ideally you should set d based on your problem domain, quantifying the effect size you expect to see. If this is not possible, Cohen’s book on power analysis “Statistical Power Analysis for the Behavioral Sciences”, suggests setting d to be 0.2, 0.5 and 0.8 for small, medium and large effect sizes respectively.

# Testing for Normality

The most widely used distribution in statistical analysis is the normal distribution.  Often when performing some sort of data analysis, we need to answer the question:

Does this data sample look normally distributed?

There are a variety of statistical tests that can help us answer this question, however first we should try to visualise the  data.

# Visualisation

Below is a simple piece of R code to create a histogram and normal Q-Q plot:

On top of the histogram we have have overlayed the normal probability density distribution using the sample mean and standard deviation.

In the Q-Q plot, we are plotting the quantiles of the sample data against the quantiles of a standard normal distribution. We are looking to see a roughly linear relationship between the sample and the theoretical quantiles.

In this example, we would probably conclude, based on the graphs, that the data is roughly normally distributed (which is true in this case as we randomly generated it from a normal distribution).

It is worth bearing in mind, many data analysis techniques assume normality (linear regression, PCA, etc.) but are still useful even with some deviations from normality.

# Statistical Tests for Normality

We can also use several tests for normality. The two most common are the Anderson-Darling test and the Shapiro-Wilk test.

The Null Hypothesis of both these tests is that the data is normally distributed.

## Anderson-Daring Test

The test is based on the distance between the empirical distribution function (EDF) and the cumulative distribution function (CDF) of the underlying distribution (e.g. Normal). The statistic is the weighted sum of the difference between the EDF and CDF, with more weight applied at the tails (making it better at detecting non-normality at the tails).

Below is the R code to perform this test:

As the p-value is above the significance level, we would accept the null hypothesis (normality).

## Shapiro-Wilk Test

The Shapiro-Wilk test is a regression-type test that uses the correlation of sample order statistics (the sample values arranged in ascending order) with those of a normal distribution. Below is the R code to perform the test:

Again, as the p-value is above the significance level, we would accept the null hypothesis (normality).

The Shapiro-Wilk test is slightly more powerful than the Anderson test, but is limited to 5000 samples.

# Test Caution

While these tests can be useful, they are not infallible. I would recommend looking at the histogram and Q-Q plots first, and use the tests as another check.

In particular small and large samples sizes can cause problems for the tests.

With a sample size of 10 or less, it is unlikely the test will detect non-normality (e.g. reject the null) even if the distribution is truly non-normal.

With a large sample size of 1000 samples or more, a small deviation from normality (some noise in the sample) may be concluded as significant and reject the null.

Overall your two best tools are the histogram and Q-Q plot. Potentially using the tests as additional indicators.

# Pearson’s Chi-Squared Test

The Pearson’s Chi-Squared Test is a commonly used statistical test applied to categorical data. It has two major use-cases:

• A goodness of fit test measuring how well a observed data fits a theoretical distribution.
• A test of independence assessing how likely two categorical variables are independent of one another.

## Procedure

The procedure for performing a Chi-Squared Test is as follows:

1. Compute the Chi-Squared statistic $$\chi^2$$.
2. Compute the degrees of freedom.
3. Use the Chi-Squared distribution to compute the p-value.

## Goodness of fit

Imagine we have been rolling a dice and want to test if it is fair. We collect a table of rolls:

Our test statistic is calculated as:

$$\chi^2 = \sum^n_{i=1}\frac{(O_i – E_i)^2}{E_i}$$

where

$$O_i$$ is the observed count

$$E_i$$ is the expected count under the theoretical distribution

$$n$$ is the number of categories

$$N$$ is total observations

In the dice case,$$n=6$$ as we have six sides. As our theoretical distribution is that the dice is fair, $$E_i$$ is simply $$\frac{N}{6}$$. A more complicated case might be if the theoretical distribution was a normal distribution. Here we would need to compute the probability density over various intervals (e.g. $$P(a \lt x \leq b)$$ and multiply by$$N$$.

The degrees of freedom is $$n – (s + 1)$$ where $$s$$ is the number of estimated parameters. For the dice example the degrees of freedom is 5, as there are no free parameters needed to be estimated. If the theoretical distribution was normal with unknown mean and variance, the degrees of freedom would be $$n-(2 + 1)$$, $$s$$ is 2 as we would need to compute the sample mean and variance.

## Test of Independence

The most common use-case of the Chi-Squared test is for testing if their is an association between two categorical variables. Some examples are:

• Is there a relationship between gender and the party you vote for?
• Is there a relationship between a landing page version and conversions (A/B testing)

One important assumption is that they must be independent groups e.g. you can only vote for one party, you are only shown one landing page.

To begin we form a contingency table, such as:

This example is a 2×2 table, but we can also have tables of 3×2, 3×3 etc.

If we have $$r$$ rows and $$c$$ columns the expected frequency is:

$$E_{i,j}= \frac{(\sum_{k=1}^cO_{i,k})(\sum_{k=1}^rO_{k,j})}{N}$$

e.g. Row Total * Column Total / N.

The test statistic is then:

$$\chi^2=\sum^r_{i=1}\sum^c_{j=1}\frac{(O_{i,j} – E_{i,j})^2}{E_{i,j}}$$

The number of degrees of freedom is $$(r – 1)(c – 1)$$.

Our null hypothesis is the variables (gender and party for instance) are independent. Our alternative hypothesis is that they have an association (but not the direction of the association). If the p-value is below the significance level, we reject the null hypothesis and the evidence suggests there is a relationship between the two variables.

## Small Cell Counts

The assumption made is that test statistic is chi-squared distributed. The assumption is true in the limit.

The assumption can break down when a cell has a very small count. Often if the expected cell count is less than 5 (sometimes 10), the Yate’s correction for continuity is applied. The effect of this is to make a more conservative estimate, but does also increasing the likelihood of a type II error.

If the cell counts are less than 10 and it is a 2×2 contingency table, an alternative is to apply Fisher’s exact test. The test can be applied on general r x c contingency tables using monte-carlo simulation.

## Implementation

The chisq.test function in R implements the Chi-Squared test including the ability for use Yates’ correction.

The fisher.test function implements Fisher’s exact test.