Chi-Square Test ssumptions and Example

The Chi-Square Test of Independence is a statistical method used to determine whether there is evidence of an association between two categorical variables.

It is widely used in survey research, healthcare, social sciences, marketing, quality control, and other areas where data are organized into categories.

However, before performing a Chi-Square test, it is important to verify that the study design and data satisfy the conditions required for the test. In this guide, we will explain the key assumptions of the Chi-Square Test of Independence and demonstrate them with a practical example.

What Is the Chi-Square Test of Independence?

The Chi-Square Test of Independence evaluates whether two categorical variables are statistically associated.

For example, a researcher might want to determine whether:

• Gender is associated with political preference.
• Smoking status is associated with disease status.
• Education level is associated with employment status.
• Product preference is associated with customer age group.

The null and alternative hypotheses are:

Null hypothesis (H₀): The two categorical variables are independent.

Alternative hypothesis (H₁): The two categorical variables are associated.

The test compares the observed frequencies in a contingency table with the expected frequencies that would be obtained if the variables were independent.

Key Assumptions of the Chi-Square Test

There are several important conditions to consider before applying the Chi-Square Test of Independence.

Assumption 1: Both Variables Are Categorical

The Chi-Square Test of Independence is designed for two categorical variables.

Categorical variables consist of distinct groups or categories rather than continuous numerical measurements.

Examples include:

• Marital status: Married, Single, Divorced
• Smoking status: Smoker, Non-smoker
• Gender: Male, Female
• Political preference: Republican, Democrat, Independent
• Product preference: Product A, Product B, Product C

A numerical variable such as income or blood pressure would generally need to be categorized before using a Chi-Square test, although categorization can result in loss of information.

Assumption 2: Observations Should Be Independent

Each observation should be independent of the others.

In practical terms, one participant should contribute only one observation to the contingency table, and the classification of one participant should not determine the classification of another.

For example, if a survey examines gender and political preference, each person should normally be counted once.

Independence is primarily a study-design assumption. It cannot be established simply by looking at the contingency table.

Random sampling and appropriate study design can help support this assumption.

Repeated measurements, matched subjects, or clustered observations may violate the independence assumption. In such situations, alternative methods may be more appropriate.

Assumption 3: Categories Should Be Mutually Exclusive

Each observation should belong to exactly one category of each variable.

For example, if political preference categories are:

• Republican
• Democrat
• Independent

a participant should not simultaneously be counted in multiple categories.

Similarly, if gender is recorded as Male or Female for a particular analysis, each observation must have one corresponding category.

The categories should also be defined clearly so that observations cannot be ambiguously classified.

Assumption 4: Expected Cell Frequencies Should Be Adequate

The Chi-Square approximation works best when expected frequencies are sufficiently large.

A commonly used practical guideline is:

• No expected cell frequency should be less than 1.
• No more than 20% of expected cell frequencies should be less than 5.

Some introductory treatments simplify this to requiring expected frequencies of at least 5 in every cell. However, the more commonly cited rule for the Pearson Chi-Square approximation is the 20% / minimum 1 guideline.

If expected frequencies are too small, consider an exact test such as Fisher’s exact test for appropriate small contingency tables, or another suitable method.

Example: Gender and Political Preference

Suppose a researcher wants to determine whether gender is associated with political party preference.

A survey of 500 voters produces the following results:

We can use a Chi-Square Test of Independence to determine whether the observed differences provide sufficient evidence of an association between gender and political preference.

Checking the Assumptions

Assumption 1: Both Variables Are Categorical

The two variables are:

• Gender: Male, Female
• Political preference: Republican, Democrat, Independent

Both are categorical variables.

Therefore, this assumption is satisfied.

Assumption 2: Observations Are Independent

If each voter participated once and the sample was collected using an appropriate sampling procedure, the observations can reasonably be treated as independent.

However, this assumption depends on how the data were collected, not simply on the numbers in the table.

Therefore, the researcher must verify the study design.

Assumption 3: Categories Are Mutually Exclusive

Each voter is counted in exactly one gender category and one political preference category.

For example, a voter cannot simultaneously contribute to both the Male and Female rows.

Therefore, the categories are mutually exclusive for this analysis.

Assumption 4: Expected Frequencies Are Adequate

Expected frequencies are calculated using: Eij​=Grand Total(Row Total)(Column Total)​

For example, the expected frequency for Male Republicans is: E=500250×230​ E=115

The complete expected-frequency table is:

Every expected frequency is greater than 5.

Therefore, the expected-frequency condition is satisfied.

How the Chi-Square Statistic Is Calculated

The Pearson Chi-Square statistic is: χ2=∑E(O−E)2​

where:

• O = observed frequency
• E = expected frequency

The statistic measures how different the observed frequencies are from the frequencies expected under the assumption of independence.

A larger Chi-Square statistic generally indicates a greater discrepancy between the observed and expected frequencies.

Degrees of Freedom

For a contingency table, the degrees of freedom are: df=(r−1)(c−1)

where:

• r = number of rows
• c = number of columns

For our 2 × 3 table: df=(2−1)(3−1) df=2

Interpreting the Test

For the example above, the Pearson Chi-Square test produces a p-value of approximately 0.649.

Using a significance level of: α=0.05

we compare: p=0.649>0.05

Therefore, we fail to reject the null hypothesis.

The appropriate conclusion is:

There is insufficient statistical evidence of an association between gender and political preference in this sample.

This does not prove that gender and political preference are independent in the population. It means that the observed differences are not statistically significant at the 5% significance level.

Chi-Square Test in R

The test can be performed easily in R using chisq.test().

data<-matrix(c(120, 90, 40,110, 95, 45),nrow=2,byrow=TRUE)dimnames(data) <-list(Gender=c("Male", "Female"),Preference=c("Republican", "Democrat", "Independent"))chisq.test(data)

You can examine the expected frequencies using:

test<-chisq.test(data)test$expected

This is useful for checking whether the expected-frequency condition is satisfied.

What If the Expected Frequencies Are Too Small?

When expected frequencies are very small, the standard Pearson Chi-Square approximation may not be reliable.

Depending on the table and study design, alternatives include:

Fisher’s Exact Test

For small contingency tables, Fisher’s exact test can provide an exact p-value.

In R:

fisher.test(data)

For larger tables, computational considerations may affect which exact or approximate method is practical.

Combining Categories

If categories are substantively similar and there is a valid scientific reason to combine them, researchers may sometimes collapse sparse categories.

However, categories should not be combined simply to obtain a significant result.

Chi-Square Test vs Fisher’s Exact Test

Important: Statistical Significance Is Not Effect Size

A significant Chi-Square test tells you that there is evidence of an association, but it does not tell you how strong that association is.

For a 2 × 2 table, Phi (φ) can be used as an effect-size measure.

For larger contingency tables, Cramér’s V is commonly used.

This distinction is important because a very large sample can produce a statistically significant association even when the practical association is small.

Common Mistakes

Mistake 1: Treating a Non-Significant Result as Proof of No Association

A p-value greater than 0.05 does not prove independence.

It means there is insufficient evidence to reject independence under the specified test and assumptions.

Mistake 2: Ignoring the Study Design

The independence assumption depends heavily on how observations were collected.

Repeated measurements or clustered data require methods that account for the dependence.

Mistake 3: Using Chi-Square for Continuous Variables

The standard Chi-Square Test of Independence is designed for categorical data.

For continuous variables, methods such as correlation, regression, t-tests, or ANOVA may be more appropriate depending on the research question.

Mistake 4: Looking Only at the P-Value

A complete analysis should consider:

• Chi-Square statistic
• Degrees of freedom
• P-value
• Effect size
• Observed and expected frequencies
• Study design

Conclusion

The Chi-Square Test of Independence is a powerful and widely used method for investigating associations between categorical variables.

Before performing the test, researchers should verify that:

1. The variables are categorical.
2. Observations are appropriately independent.
3. Categories are mutually exclusive.
4. Expected cell frequencies are adequate for the Chi-Square approximation.

In the gender and political preference example, all expected frequencies are sufficiently large, and the test produces a p-value of approximately 0.649. Because this is greater than 0.05, there is insufficient evidence to conclude that gender and political preference are associated in the sample.

However, a Chi-Square test should not be interpreted in isolation. Effect size, study design, sample size, and the practical importance of the association should also be considered when drawing conclusions.

Antecedent Variables in Statistics – FINNSTATS