Chi-squared test

What Is Chi-squared test?

The Chi-squared test (often written as $\chi^2$ test) is a non-parametric statistical hypothesis test used to determine if there is a statistically significant association between two categorical data variables or if an observed distribution of a single categorical variable differs from an expected frequency distribution. It is a fundamental tool in statistical analysis, particularly when dealing with count data rather than continuous measurements. The Chi-squared test helps evaluate how likely any observed difference between sets of data arose by chance.

History and Origin

The Chi-squared test was first introduced by statistician Karl Pearson in 1900.²⁴ Pearson's paper laid the groundwork for what became known as the Chi-squared test of goodness-of-fit.²³ His work allowed statisticians to interpret findings using methods that did not depend on the assumption of a normal distribution.²² This development was a significant contribution to the modern theory of statistics in the early 20th century, particularly for hypothesis testing with nominal variables.²¹

Key Takeaways

• The Chi-squared test evaluates whether observed frequencies significantly differ from expected frequencies in categorical data.
• It is widely used for tests of independence (association between two categorical variables) and goodness-of-fit (comparing observed distribution to a theoretical one).
• A key output is the Chi-squared statistic, which is compared against a Chi-squared distribution to derive a p-value.²⁰
• The test is sensitive to sample size and requires sufficient expected frequency in each category to ensure reliable results.¹⁹
• The Chi-squared test is non-parametric, meaning it does not assume a specific underlying data distribution.¹⁸

Formula and Calculation

The Chi-squared test statistic is calculated using the following formula:

$\chi^2 = \sum_{i=1}^{n} \frac{(O_i - E_i)^2}{E_i}$

Where:

• $O_i$ = the observed frequency (actual count) for each category.
• $E_i$ = the expected frequency for each category under the null hypothesis.
• $\sum$ = sum across all categories.
• $n$ = the number of categories.

To determine the expected frequency ($E_i$) in a test of independence using a contingency table, the formula is:

$E_{i,j} = \frac{(\text{Row Total}) \times (\text{Column Total})}{\text{Grand Total}}$

The calculated Chi-squared value is then compared to a critical value from the Chi-squared distribution table, based on the chosen significance level (alpha) and the degrees of freedom.

Interpreting the Chi-squared test

Interpreting the Chi-squared test involves comparing the calculated Chi-squared statistic to a critical value or, more commonly, evaluating its associated p-value. The p-value represents the probability of observing a Chi-squared statistic as extreme as, or more extreme than, the one calculated, assuming that the null hypothesis is true.¹⁷

If the p-value is less than or equal to the predetermined significance level (commonly 0.05), the result is considered to be statistical significance, and the null hypothesis is rejected. This suggests that there is sufficient evidence to conclude that the observed differences are not due to random chance, and there is a relationship between the categorical variables (for a test of independence) or a significant difference from the expected distribution (for a goodness-of-fit test).¹⁶ Conversely, if the p-value is greater than the significance level, one fails to reject the null hypothesis, indicating that there is not enough evidence to claim a significant difference or association.¹⁵

Hypothetical Example

Imagine a market research firm wants to determine if customer preference for a new financial app's interface (Option A vs. Option B) is independent of their age group (Under 30, 30-50, Over 50). They survey 200 randomly selected users, recording their preferred interface and age group.

The null hypothesis ($H_0$) is that interface preference is independent of age group. The alternative hypothesis ($H_a$) is that interface preference is dependent on age group.

Observed Frequencies ($O_i$):

Expected Frequencies ($E_i$) (calculated as (Row Total * Column Total) / Grand Total):

Now, calculate the Chi-squared statistic:
$\chi^2 = \frac{(40-30)^2}{30} + \frac{(20-30)^2}{30} + \frac{(35-40)^2}{40} + \frac{(45-40)^2}{40} + \frac{(25-30)^2}{30} + \frac{(35-30)^2}{30}$
$\chi^2 = \frac{100}{30} + \frac{100}{30} + \frac{25}{40} + \frac{25}{40} + \frac{25}{30} + \frac{25}{30}$
$\chi^2 = 3.33 + 3.33 + 0.625 + 0.625 + 0.833 + 0.833 = 9.58$

With (Rows - 1) * (Columns - 1) = (3-1) * (2-1) = 2 degrees of freedom, and assuming a 0.05 statistical significance level, the critical Chi-squared value is approximately 5.991. Since 9.58 > 5.991, the null hypothesis is rejected. This suggests that there is a statistically significant relationship between age group and interface preference. This analysis helps inform product development and user experience design.

Practical Applications

The Chi-squared test finds numerous practical applications across various fields, including finance and economics, for performing data analysis on categorical data.

• Market Research: Companies use the Chi-squared test to analyze survey data, for instance, to determine if customer demographics (e.g., age, income bracket) are independent of their preference for certain investment products or financial services. This can inform targeted marketing strategies.
• A/B Testing in Fintech: In the development of financial applications, A/B tests often compare two versions of a feature (e.g., different layouts for a trading dashboard). The Chi-squared test can assess if observed differences in user behavior, such as conversion rates or click-through rates, are statistically significant and not just due to random chance.¹³, ¹⁴
• Fraud Detection: Financial institutions might use the Chi-squared test to assess if certain categorical patterns of transactions (e.g., type of transaction vs. time of day) are significantly different from expected patterns, potentially flagging unusual activities indicative of fraud.
• Risk Management: In assessing credit risk, institutions might categorize loan applicants by various factors (e.g., credit score range, employment status) and loan outcomes (default/non-default). A Chi-squared test can determine if these categorical factors are independently associated with default rates.
• Economic Analysis: Economists might apply the Chi-squared test to analyze categorical economic data, such as whether unemployment rates differ significantly across different industrial sectors or if consumer spending habits are associated with different economic policy regimes.
• Compliance and Regulation: Regulators might use the Chi-squared test to check if the distribution of certain financial products or services among different demographic groups aligns with non-discriminatory practices or if reported incidents align with expected frequencies.

Limitations and Criticisms

While a widely used tool for inferential statistics, the Chi-squared test has several limitations that users should consider to avoid misinterpretations.

One significant limitation is its sensitivity to sample size. With a very large sample, even a trivial relationship or a small deviation from the expected frequency can appear statistically significant, even if it lacks practical meaning.¹¹, ¹² Conversely, with very small sample sizes, the test's reliability can be compromised, particularly if the expected frequency in any cell of the contingency table falls below a certain threshold (typically five).⁹, ¹⁰ In such cases, alternative tests like Fisher's exact test might be more appropriate.⁸

Another criticism is that the Chi-squared test only indicates whether a relationship exists between categorical variables; it does not imply causation or the strength of the association.⁷ It also does not provide insights into the nature or direction of the relationship. Furthermore, the test assumes that observations are independent and randomly sampled.⁵, ⁶ Violations of these assumptions, such as dependent observations (e.g., repeated measurements from the same individuals), can lead to inaccurate results and inflated Type I error rates (false positives).⁴ The Chi-squared test also performs best with categorical data and is not designed for quantitative data or continuous variables, where other statistical tests (like t-tests or ANOVA) would be more suitable.

Chi-squared test vs. t-test

Both the Chi-squared test and the t-test are fundamental tools in hypothesis testing, but they are applied in different scenarios based on the type of data being analyzed and the research question.

The core distinction lies in the nature of the data. The Chi-squared test examines relationships within categorical data, such as nominal or ordinal scales, by comparing observed frequency counts against expected counts. For instance, it could assess if there's a relationship between investment style and investor age group. In contrast, the t-test is used for quantitative data and determines if there is a statistically significant difference between the means of two groups. An example would be comparing the average returns of two different investment portfolios. Both tests ultimately lead to a p-value to help researchers make decisions about their null hypothesis, but they address different types of comparisons.

FAQs

When should I use a Chi-squared test?

You should use a Chi-squared test when you want to analyze categorical data to see if there's a relationship between two such variables (test of independence) or if an observed set of frequencies for a single categorical variable differs significantly from a hypothesized or expected set of frequencies (goodness-of-fit test). For example, it can determine if there's an association between gender and preferred investment platform.

What is the null hypothesis for a Chi-squared test?

For a Chi-squared test of independence, the null hypothesis typically states that there is no association or relationship between the two categorical variables being examined. For a Chi-squared goodness-of-fit test, the null hypothesis states that the observed frequency distribution does not differ significantly from the expected or theoretical distribution.

##³# What do "observed" and "expected" frequencies mean?
Observed frequency ($O_i$) refers to the actual counts or number of occurrences recorded in each category from your collected data. Expected frequency ($E_i$) refers to the counts that would be anticipated in each category if the null hypothesis were true (i.e., if there were no relationship between variables or if the data perfectly matched a theoretical distribution). The Chi-squared test quantifies the difference between these observed and expected values.

Can the Chi-squared test be used for small samples?

The Chi-squared test's reliability decreases with small sample sizes, particularly if the expected frequency in any individual cell of the data table falls below five. In such cases, the approximation to the Chi-squared distribution may not be accurate. For very small samples, alternative exact tests, such as Fisher's exact test, are generally recommended.

##²# Does a significant Chi-squared result imply causation?
No, a statistically significant result from a Chi-squared test indicates only that there is a non-random association or relationship between the categorical variables. It does not imply a cause-and-effect relationship. Establishing causation typically requires experimental design and consideration of other factors.¹