What is a Chi-Square Test and Why Do We use it?

Overview

• What is the chi-square test? How does it work?
• Learn about the different types of Chi-Square tests and where and when you should apply them
• We will also implement a chi-square test in R in this article

Introduction

"Science is advanced by proposing and testing a hypothesis, not by declaring questions unsolvable" - Nick Matzke

Let's start with a case study. I want you to think of your favorite restaurant right now. Let's say you can predict a certain number of people arriving for lunch five days a week. At the end of the week, you observe that the expected footfall was different from the actual footfall.

Sounds like a prime statistics problem? That's the idea!

So, how will you check the statistical significance between the observed and the expected footfall values? Remember this is a categorical variable - 'Days of the week' - with 5 categories [Monday, Tuesday, Wednesday, Thursday, Friday].

One of the best ways to deal with this is by using the Chi-Square test.

We can always opt for z-tests, t-tests or ANOVA when we're dealing with continuous variables. But the situation becomes tricky when working with categorical features (as most data scientists will attest to!). I've found the chi-square test to be quite helpful in my own projects.

So let's dive into the article to understand all about the chi-square test, what it is, how it works and how we can implement it in R.

If you are new statistics and data science, I would recommend the below resources to get a comprehensive overview of the two broad topics:

• Comprehensive & Practical Guide to Inferential Statistics
• Introduction to Data Science Course

Table of Contents

• What are Categorical Variables?
• What is a Chi-Square Test and Why Do We Use It?
• Assumptions of the Chi-Square Test
• Types of Chi-Square Tests (With implementation in R)
  1. Chi-Square Goodness of Fit Test
  2. Chi-Square Test of Association between Two Variables

What are Categorical Variables?

I'm sure you've encountered categorial variables before, even if you might not have intuitively recognized them. They can be tricky to deal with in the data science world so let's first define them.

Categorical variables fall into a particular category of those variables that can be divided into finite categories. These categories are generally names or labels. These variables are also called qualitative variables as they depict the quality or characteristics of that particular variable.

For example, the category “Movie Genre” in a list of movies could contain the categorical variables - "Action", "Fantasy", "Comedy", "Romance", etc.

There are broadly two types of categorical variables:

1. Nominal Variable: A nominal variable has no natural ordering to its categories. They have two or more categories. For example, Marital Status (Single, Married, Divorcee); Gender (Male, Female, Transgender), etc.
2. Ordinal Variable: A variable for which the categories can be placed in an order. For example, Customer Satisfaction (Excellent, Very Good, Good, Average, Bad), and so on

When the data we want to analyze contains this type of variable, we turn to the chi-square test, denoted by χ², to test our hypothesis.

What is a Chi-Square Test and Why Do We use it?

A Chi-Square test is a test of statistical significance for categorical variables.

Let's learn the use of chi-square with an intuitive example.

A research scholar is interested in the relationship between the placement of students in the statistics department of a reputed University and their C.G.P.A (their final assessment score).

He obtains the placement records of the past five years from the placement cell database (at random). He records how many students who got placed fell into each of the following C.G.P.A. categories - 9-10, 8-9, 7-8, 6-7, and below 6.

If there is no relationship between the placement rate and the C.G.P.A., then the placed students should be equally spread across the different C.G.P.A. categories (i.e. there should be similar numbers of placed students in each category).

However, if students having C.G.P.A more than 8 are more likely to get placed, then there would be a large number of placed students in the higher C.G.P.A. categories as compared to the lower C.G.P.A. categories. In this case, the data collected would make up the observed frequencies.

So the question is, are these frequencies being observed by chance or do they follow some pattern?

Here enters the chi-square test! The chi-square test helps us answer the above question by comparing the observed frequencies to the frequencies that we might expect to obtain purely by chance.

Chi-square test in hypothesis testing is used to test the hypothesis about the distribution of observations/frequencies in different categories.

We are almost at the implementing aspect of chi-square tests but there's one more thing we need to learn before we get there.

Assumptions of the Chi-Square Test

Just like any other statistical test, the chi-square test comes with a few assumptions of its own:

1. The χ2 assumes that the data for the study is obtained through random selection, i.e. they are randomly picked from the population
2. The categories are mutually exclusive i.e. each subject fits in only one category. For e.g.- from our above example - the number of people who lunched in your restaurant on Monday can't be filled in the Tuesday category
3. The data should be in the form of frequencies or counts of a particular category and not in percentages
4. The data should not consist of paired samples or groups or we can say the observations should be independent of each other
5. When more than 20% of the expected frequencies have a value of less than 5 then Chi-square cannot be used.  To tackle this problem: Either one should combine the categories only if it is relevant or obtain more data

Types of Chi-Square Tests (By manual calculations and with implementation in R)

Chi-Square Goodness of Fit Test

This is a non-parametric test. We typically use it to find how the observed value of a given event is significantly different from the expected value. In this case, we have categorical data for one independent variable, and we want to check whether the distribution of the data is similar or different from that of the expected distribution.

Let's consider the above example where the research scholar was interested in the relationship between the placement of students in the statistics department of a reputed University and their C.G.P.A.

In this case, the independent variable is C.G.P.A with the categories 9-10, 8-9, 7-8, 6-7, and below 6.

The statistical question here is: whether or not the observed frequencies of placed students are equally distributed for different C.G.P.A categories (so that our theoretical frequency distribution contains the same number of students in each of the C.G.P.A categories).

We will arrange this data by using the contingency table which will consist of both the observed and expected values as below:

C.G.P.A.	10-9	9-8	8-7	7-6	Below 6	Total
Observed Frequency	30	35	20	10	5	100
Expected Frequency	20	20	20	20	20	100

After constructing the contingency table, the next task is to compute the value of the chi-square statistic. The formula for chi-square is given as:

where,

χ2 = Chi-Square value
Oi = Observed frequency
Ei = Expected frequency

Let us look at the step-by-step approach to calculate the chi-square value:

Step 1: Subtract each expected frequency from the related observed frequency. For example, for the C.G.P.A category 10-9, it will be "30-20 = 10". Apply similar operation for all the categories
Step 2: Square each value obtained in step 1, i.e. (O-E)2. For example: for the C.G.P.A category 10-9, the value obtained in step 1 is 10. It becomes 100 on squaring. Apply similar operation for all the categories
Step 3: Divide all the values obtained in step 2 by the related expected frequencies i.e. (O-E)2/E. For example: for the C.G.P.A category 10-9, the value obtained in step 2 is 100. On dividing it with the related expected frequency which is 20, it becomes 5. Apply similar operation for all the categories
Step 4: Add all the values obtained in step 3 to get the chi-square value. In this case, the chi-square value comes out to be 32.5
Step 5: Once we have calculated the chi-square value, the next task is to compare it with the critical chi-square value. We can find this in the below chi-square table against the degrees of freedom (number of categories - 1) and the level of significance:

In this case, the degrees of freedom are 5-1 = 4. So, the critical value at 5% level of significance is 9.49.

Our obtained value of 32.5 is much larger than the critical value of 9.49. Therefore, we can say that the observed frequencies are significantly different from the expected frequencies. In other words, C.G.P.A is related to the number of placements that occur in the department of statistics.

Let's further solidify our understanding by performing the Chi-Square test in R.

The Chi-Square Goodness of Fit Test in R

Let's implement the chi-square goodness of fit test in R. Time to fire up RStudio!

Problem Statement

Let's understand the problem statement before we dive into R.

An organization claims that the experience of the employees of different departments is distributed in the following categories:

11 - 20 Years = 20%
21 - 40 Years = 17%
6 - 10 Years = 41% and
Up to 5 Years = 22%

A random sample of 1470 employees is collected. Does this random sample provide evidence against the organization's claim?

You can download the data here.

Setting up the hypothesis

• Null hypothesis: The true proportions of the experience of the employees of different departments are distributed in the following categories: 11 - 20 Years = 20%, 21 - 40 Years = 17%, 6 - 10 Years = 41% and up to 5 Years = 22%
• Alternative hypothesis: The distribution of experience of the employees of different departments differs from what the organization states

Let's begin!

Step 1: First, import the data
Step 2: Validate it for correctness in R:

#Step 1 - Importing Data
#_______________________________________________________

#Importing the csv data 
data<-read.csv(file.choose())

#Step 2 - Validate data for correctness
#______________________________________________________

#Count of Rows and columns
dim(data)

#View top 10 rows of the dataset
head(data,10)

Output:

#Count of Rows and columns 

[1] 1470 2 
#View top 10 rows of the dataset 

age.intervals Experience.intervals 
1 41 - 50 6 - 10 Years 
2 41 - 50 6 - 10 Years 
3 31 - 40 6 - 10 Years 
4 31 - 40 6 - 10 Years 
5 18 - 30 6 - 10 Years 
6 31 - 40 6 - 10 Years 
7 51 - 60 11 - 20 Years 
8 18 - 30 Upto 5 Years 
9 31 - 40 6 - 10 Years 
10 31 - 40 11 - 20 Years

Step 3: Create a proportion table for expected frequencies:

#Step 3 - Calculate the proportion of experience of employees

#Proportion table for observed frequencies
prop.table((table(data$Experience.intervals)))

Output:

11 - 20 Years 21 - 40 Years 6 - 10 Years Upto 5 Years 
0.2312925 0.1408163 0.4129252 0.2149660

Step 4: Calculate the chi-square value:

#Step 4 - Calculate the chi-square value
#_______________________________________________________

chisq.test(x = table(data$Experience.intervals), 
           p = c(0.2, 0.17, 0.41, 0.22))

Output:

Chi-square test for given probabilities

data: table(data$Experience.intervals)
X-squared = 14.762, df = 3, p-value = 0.002032

The p-value here is less than 0.05. Therefore, we will reject our null hypothesis. Hence, the distribution of experience of the employees of different departments differs from what the organization states.

Chi-Square Test for Association/Independence

The second type of chi-square test is the Pearson’s chi-square test of association. This test is used when we have categorical data for two independent variables and we want to see if there is any relationship between the variables.

Let's take another example to understand this. A teacher wants to know the answer to whether the outcome of a mathematics test is related to the gender of the person taking the test. Or in other words, she wants to know if males show a different pattern of pass/fail rates than females.

So, here are two categorical variables: Gender (Male and Female) and mathematics test outcome (Pass or Fail). Let us now look at the contingency table:

	Boys	Girls
Pass	17	20
Fail	8	5

By looking at the above contingency table, we can see that the girls have a comparatively higher pass rate than boys. However, to test whether this observed difference is significant or not, we will carry out the chi-square test.

The steps to calculate the chi-square value are as follows:

Step 1: Calculate the row and column total of the above contingency table:

	Boys	Girls	Total
Pass	17	20	37
Fail	8	5	13
Total	25	25	50

Step 2: Calculate the expected frequency for each individual cell by multiplying row sum by column sum and dividing by total number:

Expected Frequency = (Row Total x Column Total)/Grand Total

For the first cell, the expected frequency would be (37*25)/50 = 18.5. Now, write them below the observed frequencies in brackets:

	Boys	Girls	Total
Pass	17(18.5)	20(18.5)	37
Fail	8(6.5)	5(6.5)	13
Total	25	25	50

Step 3: Calculate the value of chi-square using the formula:

Calculate the right-hand side part of each cell. For example, for the first cell, ((17-18.5)^2)/18.5 = 0.1216.

Step 4: Then, add all the values obtained for each cell. In this case, the values are:

0.1216+0.1216+0.3461+0.3461 = 0.9354

Step 5: Calculate the degrees of freedom, i.e. (Number of rows-1)(Number of columns-1) = 11 = 1

The next task is to compare it with the critical chi-square value from the table we saw above.

The Chi-Square calculated value is 0.9354 which is less than the critical value of 3.84. So in this case, we fail to reject the null hypothesis. This means there is no significant association between the two variables, i.e, boys and girls have a statistically similar pattern of pass/fail rates on their mathematics tests.

Let's further solidify our understanding by performing the chi-square test in R.

Test for Independence in R

Problem statement

A Human Resources department of an organization wants to check whether age and experience of the employees are dependent on each other. For this purpose, a random sample of 1470 employees is collected with their age and experience. You can download the data here.

Setting up the hypothesis

Null hypothesis: Age and Experience are two independent variables

Alternative hypothesis: Age and Experience are two dependent variables

Let's begin!

Step 1: First, import the data

Step 2: Validate it for correctness in R:

Output:

#Count of Rows and columns

[1] 1470 2 
> #View top 10 rows of the dataset 

age.intervals Experience.intervals 
1 41 - 50 6- 10 Years 
2 41 - 50 6 - 10 Years 
3 31 - 40 6 - 10 Years 
4 31 - 40 6 - 10 Years 
5 18 - 30 6 - 10 Years 
6 31 - 40 6 - 10 Years 
7 51 - 60 11 - 20 Years 
8 18 - 30 Upto 5 Years 
9 31 - 40 6 - 10 Years 
10 31 - 40 11 - 20 Years

Step 3: Construct a contingency table and calculate the chi-square value:

#Step 3 - Make a table and Calculate the chi-square value
#_______________________________________________________

ct<-table(data$age.intervals,data$Experience.intervals)
ct
chisq.test(ct)

Output:

ct<-table(data$age.intervals,data$Experience.intervals)
> ct

11 - 20 Years 21 - 40 Years 6 - 10 Years Upto 5 Years
18 - 30 22 0 172 192
31 - 40 190 20 308 101
41 - 50 85 112 110 15
51 - 60 43 75 17 8
> chisq.test(ct) 

Pearson's Chi-squared test 

data: ct 
X-squared = 679.97, df = 9, p-value < 2.2e-16

The p-value here is less than 0.05. Therefore, we will reject our null hypothesis. We can conclude that age and experience are two dependent variables, aka as the experience increases, the age also increases (and vice versa).

END NOTES

In this article, we learned how to analyze the significant difference between data that contains categorical measures in it with the help of chi-square tests. We enhanced our knowledge on the use of chi-square, assumptions involved in carrying out the test, and how to conduct different types of chi-square tests both manually and in R.

This article was originally published on my blog - What is the Chi-Square Test and How Does it Work? An Intuitive Explanation with R Code .

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