# Chi-Square Distribution for Evaluating Test of Independence

There are two kinds of chi-square tests, and both tests rely on the chi-square statistic and distribution for unique objectives: one is called chi-square goodness of fit test, and the other is a chi-square test for independence. A chi-square test of independence compares two categorical variables in a contingency table to determine the relationship. A small chi-square test statistic indicates that the observed data aligns well with the expected data, whereas a large chi-square test indicates that the data doesn’t align well, and they're likely isn’t a correlation.

Chi-square distribution is often used in market research and analyzing survey response data. For example, a business can test how customers react to packaging designs by testing which colors are popular. It’s also used to test consumer reactions to brands, products, or features, such as testing age demographics and how that impacts purchasing decisions of laptops.

## Chi-square and Its Applicability in the Statistical Context

Chi-square is most commonly used in cross-tabulation, which examines the distribution of two distinct categorical variables at the same time, as well as the convergence of the variables that are displayed in a table’s cells. A test of independence checks if there is a relationship between two variables through comparison of the perceived pattern of responses within the cells against the expected patterns if the variables are independent.

In simplest terms, a chi-square statistic calculates how anticipated outcomes compare against real, observable model outcomes or data. In order for a chi-square test to be measured correctly, the data has to be derived from independent variables, taken from a substantial sized sample, raw, arbitrary, and mutually exclusive.

In a chi-square test, there’s both a null hypothesis and an expected hypothesis, similar to most statistical tests. This means that the categorical variables have no relationship, and the categorical variables do have a relationship, respectively.

The chi-square statistical formula includes:

Given that fo = the observed frequency, or the noticeable counts within the cells, and fe = is the expected frequency if there’s no current relationship between the variables.

However, the chi-square has some limitations as it is sensitive to small frequencies in cell tables, which can potentially create errors.

## Chi-square Test of Independence in R

The chi-square test of independence figures out if the values of one or two qualitative variables are dependent upon the other qualitative variable values. It basically tests if two qualitative variables are independent, or if there’s a connection between the two variables.

Below are the steps involved in implementing the chi-square test of independence in R:

• Define both the null and the alternative hypotheses.
• Import the relevant data.
• Perform validation testing in R to ensure the data is accurate.
• Prepare a contingency table, then measure the chi-square value.

## Chi-square Test for Machine Learning

In predictive modeling, statistical modeling, like the chi-square test, is used to forecast response variables depending on one or more predictors, which usually attributes that influence the response. Predictive and machine learning models are optimal when the attributes are valuable and have a significant connection to the response. However, typically it’s not previously known if the response is or is not dependent on a specific attribute, so the chi-square test evaluates if the attributes are dependent or not given that the predictors and response are categorical variables.

Below are a few scenarios for the chi-square test in machine learning:

• Attribute selection Involves selecting the ideal attributes with which to build the machine learning model in order to optimize model performance, considering that there are numerous attributes or features to choose from. The chi-square test addresses this challenge by calculating the correlation between the different features.
• Model training The chi-square test ensures that machine learning models can be developed with higher accuracy levels by testing for relevancy across potential features or attributes, and ensure the data being used for the model can produce the expected and desired results.
• Disease classification The chi-square test is often used for disease classification, such as cancer, to measure individual genes in relation to both multicategory and binary classification. This type of supervised attribute selection can reduce many of the attributes without compromising the accuracy of the test, leading to more precise classification.

## Chi-square Distribution in the Business Context

Chi-square distribution is also a popular statistical method used in business research to identify, measure, and understand the differences (if any) between variables within a given population. If a company is testing customer behaviors or products, both common in numerous business scenarios, they can determine how customers or products in any suggested category are different than what might have originally been expected. Essentially, they can find out if the outcome is because of some kind of random error, or if the difference is real and can be calculated as such.

For example, a business can collect random samples from 400 different customers, conduct a survey related to these samples, and observe the actual distribution and categorical difference across their samples. Then a chi-square test can be conducted to either validate or offer further context for the observed frequencies. The business can thereby understand if there’s a statistical difference between how the different categories replied to any given question.

It’s important to note that this doesn’t offer any insights into the exact degree of distinction between the categorical responses. Also, this test requires numerical values rather than ratios or percentages, which may be limiting to the versatility of the types of processes that can be used to perform the test. This is why this test is usually left to professional statisticians or analysts with experience and access to supportive software programs.

• ### Provides robust context

Necessary test calculations offer robust information regarding group performance in any given study, enabling deeper research understanding.

• ### Application for complex studies and tests

It’s an ideal test for research projects or studies in which parametric assumptions can’t be accounted for or met.

• ### Versatile data handling

It’s a versatile technique for evaluating data from either a two-group or a multiple-group study with equal success.

• ### Customer associations

It can determine potential associations between customer demographics, like gender or income bracket, and their brand preferences.

• ### Financial service personalization

It can be used to identify whether or not demographics play a role in the financial channel or product partiality, to help personalize services.

## Competitive Research and Statistical Analysis for Every Industry Vertical

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Contact ROP to discover how our chi-square and other statistical tests and analysis solutions can support you in your research, classification, customer behavior, product marketing, or other endeavors.

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