Chi square test: How to Test the Association between Two Categorical Variables with Chi square Test

1. Introduction to Chi-square Test

In this section, we delve into the fundamental concepts and principles of the chi-square test. The chi-square test is a statistical method used to determine the association between two categorical variables. It is widely employed in various fields, including social sciences, biology, and market research.

1. The Purpose of the Chi-square Test:

The Chi-square test allows us to assess whether there is a significant relationship or association between two categorical variables. It helps us understand if the observed frequencies in different categories deviate significantly from the expected frequencies.

2. Hypotheses in the Chi-square Test:

When conducting a Chi-square test, we formulate two hypotheses: the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis assumes that there is no association between the variables, while the alternative hypothesis suggests the presence of an association.

3. Calculation of the chi-square statistic:

To calculate the Chi-square statistic, we compare the observed frequencies in each category with the expected frequencies. The expected frequencies are derived from the assumption of independence between the variables. The Chi-square statistic is then calculated as the sum of the squared differences between observed and expected frequencies, divided by the expected frequencies.

4. Degrees of Freedom:

The degrees of freedom in the Chi-square test depend on the number of categories in each variable. For a 2x2 contingency table, the degrees of freedom would be 1. In general, the degrees of freedom can be calculated as (r - 1) x (c - 1), where r is the number of rows and c is the number of columns in the contingency table.

5. Interpreting the Chi-square Test:

To determine the significance of the Chi-square statistic, we compare it with the critical value from the Chi-square distribution. If the calculated Chi-square value exceeds the critical value, we reject the null hypothesis and conclude that there is a significant association between the variables.

6. Limitations of the Chi-square Test:

While the Chi-square test is a valuable tool, it does have some limitations. It assumes that the observations are independent and that the expected frequencies are not too small. Additionally, it is not suitable for analyzing continuous or ordinal variables.

7. Example Application:

Let's consider an example to illustrate the Chi-square test. Suppose we want to examine the relationship between gender (male or female) and voting preference (A, B, or C). We collect data from a sample of individuals and construct a contingency table. By applying the Chi-square test, we can determine if there is a significant association between gender and voting preference.

Remember, this is just a brief overview of the "Introduction to Chi-square Test." For a more detailed understanding, it is recommended to refer to reliable sources and consult statistical textbooks.

2. Understanding Categorical Variables

Understanding Categorical Variables is a crucial aspect when conducting a chi-square test to test the association between two categorical variables. In this section, we will delve into the intricacies of categorical variables and explore various perspectives to gain a comprehensive understanding.

1. Definition and Types:

Categorical variables, also known as qualitative variables, represent data that can be divided into distinct categories or groups. There are two main types of categorical variables: nominal and ordinal. Nominal variables have categories with no inherent order, such as colors or genders. On the other hand, ordinal variables have categories with a specific order, like education levels or satisfaction ratings.

2. Importance of Categorical Variables:

Categorical variables play a crucial role in statistical analysis as they provide valuable insights into relationships and patterns within data. By understanding the nature of these variables, we can uncover meaningful associations and make informed decisions based on the results of the Chi-square test.

3. chi-square Test and categorical Variables:

The Chi-square test is a statistical method used to determine if there is a significant association between two categorical variables. It compares the observed frequencies of each category with the expected frequencies, allowing us to assess the independence or dependence of the variables.

4. Example Scenario:

To illustrate the concept, let's consider a hypothetical study investigating the relationship between smoking habits and lung cancer. The categorical variables in this case would be "smoking status" (categories: non-smoker, occasional smoker, regular smoker) and "lung cancer diagnosis" (categories: diagnosed, not diagnosed). By applying the Chi-square test, we can analyze whether there is a significant association between these variables.

5. Interpreting chi-square Test results:

After performing the Chi-square test, we obtain a test statistic and a p-value. The test statistic measures the discrepancy between the observed and expected frequencies, while the p-value indicates the probability of obtaining such results by chance alone. If the p-value is below a predetermined significance level (e.g., 0.05), we can conclude that there is a significant association between the variables.

6. Limitations and Considerations:

While the Chi-square test is a powerful tool, it is important to acknowledge its limitations. For instance, it assumes that the observations are independent and that the expected frequencies are not too small. Additionally, the test does not provide information about the strength or direction of the association, only its presence or absence.

In summary, understanding categorical variables is essential when conducting a Chi-square test. By comprehending the types, significance, and interpretation of these variables, we can effectively analyze associations and draw meaningful conclusions. Remember, the examples provided here are for illustrative purposes, and real-world applications may vary.

3. Principles of the Chi-square Test

The chi-square test is a statistical method that can be used to test the association between two categorical variables. The chi-square test compares the observed frequencies of the categories in a contingency table with the expected frequencies that would be obtained if there was no association between the variables. The chi-square test can be used to answer questions such as:

- Is there a relationship between gender and political preference?

- Does the type of treatment affect the outcome of a disease?

- Are the preferences for different brands of products independent of age group?

In this section, we will discuss the principles of the chi-square test, such as:

1. The null and alternative hypotheses of the chi-square test.

2. The calculation of the chi-square statistic and the degrees of freedom.

3. The interpretation of the p-value and the conclusion of the test.

4. The assumptions and limitations of the chi-square test.

5. The use of the chi-square test in different scenarios.

Let's start with the first principle: the null and alternative hypotheses of the chi-square test.

1. The null and alternative hypotheses of the chi-square test.

The null hypothesis of the chi-square test is that there is no association between the two categorical variables, or that the variables are independent. The alternative hypothesis is that there is an association between the two categorical variables, or that the variables are dependent.

For example, suppose we want to test whether there is a relationship between gender and political preference. We can collect data from a random sample of voters and construct a contingency table as follows:

| Gender | Conservative | Liberal | Other |

| Male | 120 | 80 | 40 |

| Female | 90 | 110 | 60 |

| Total | 210 | 190 | 100 |

The null hypothesis is that gender and political preference are independent, meaning that the proportion of voters who prefer a certain political party is the same for both males and females. The alternative hypothesis is that gender and political preference are dependent, meaning that the proportion of voters who prefer a certain political party differs for males and females.

To test the null hypothesis, we need to calculate the expected frequencies for each cell of the contingency table, assuming that the null hypothesis is true. The expected frequency for a cell is obtained by multiplying the row total and the column total, and dividing by the grand total. For example, the expected frequency for the cell corresponding to male and conservative is:

$$\frac{120 \times 210}{500} = 50.4$$

We can do the same for the other cells and obtain the following table of expected frequencies:

| Gender | Conservative | Liberal | Other |

| Male | 50.4 | 47.6 | 22 |

| Female | 39.6 | 42.4 | 18 |

| Total | 90 | 90 | 40 |

The next step is to compare the observed frequencies with the expected frequencies using the chi-square statistic.

4. Hypothesis Formulation for Chi-square Test

One of the most important steps in performing a chi-square test is to formulate the hypothesis that will be tested. A hypothesis is a statement or claim about the relationship between two or more variables. In the context of a chi-square test, the hypothesis is usually about the association or independence between two categorical variables. For example, we might want to test whether the gender of a person is associated with their preference for a certain type of music. In this section, we will discuss how to formulate the hypothesis for a chi-square test, and what are the different types of hypotheses that can be tested. We will also provide some examples to illustrate the process of hypothesis formulation.

To formulate the hypothesis for a chi-square test, we need to follow these steps:

1. Identify the two categorical variables that we want to test. These variables are also called the row variable and the column variable, because they will form the rows and columns of a contingency table. For example, if we want to test the association between gender and music preference, the row variable could be gender and the column variable could be music preference.

2. Define the categories or levels of each variable. These categories are also called the observed frequencies, because they represent the number of observations in each cell of the contingency table. For example, if the gender variable has two categories (male and female), and the music preference variable has four categories (rock, pop, classical, and jazz), then we have eight observed frequencies in total.

3. State the null hypothesis and the alternative hypothesis. The null hypothesis is the statement that there is no association or no difference between the two categorical variables. The alternative hypothesis is the statement that there is some association or some difference between the two categorical variables. For example, the null hypothesis could be: "There is no association between gender and music preference." The alternative hypothesis could be: "There is some association between gender and music preference."

4. Choose the level of significance and the type of test. The level of significance is the probability of rejecting the null hypothesis when it is true. It is usually denoted by $\alpha$ and is often set at 0.05 or 0.01. The type of test is either one-tailed or two-tailed, depending on whether we want to test a specific direction of the association or not. For example, if we want to test whether males prefer rock music more than females, we would use a one-tailed test. If we want to test whether there is any difference in music preference between males and females, we would use a two-tailed test.

Here are some examples of hypothesis formulation for a chi-square test:

- Example 1: We want to test whether the type of pet (dog or cat) is associated with the marital status (single or married) of the owners. The row variable is pet type and the column variable is marital status. The null hypothesis is: "There is no association between pet type and marital status." The alternative hypothesis is: "There is some association between pet type and marital status." We choose a level of significance of 0.05 and a two-tailed test.

- Example 2: We want to test whether the blood type (A, B, AB, or O) is independent of the eye color (blue, brown, green, or hazel) of the students. The row variable is blood type and the column variable is eye color. The null hypothesis is: "Blood type and eye color are independent." The alternative hypothesis is: "Blood type and eye color are not independent." We choose a level of significance of 0.01 and a two-tailed test.

- Example 3: We want to test whether the smoking status (smoker or non-smoker) is related to the lung cancer risk (high or low) of the patients. The row variable is smoking status and the column variable is lung cancer risk. The null hypothesis is: "Smoking status and lung cancer risk are not related." The alternative hypothesis is: "Smoking status and lung cancer risk are related." We choose a level of significance of 0.05 and a one-tailed test.

5. Calculation of Chi-square Statistic

The chi-square statistic is a measure of how well the observed frequencies of a categorical variable match the expected frequencies under a certain hypothesis. It is calculated by summing up the squared differences between the observed and expected frequencies, divided by the expected frequencies. The larger the chi-square statistic, the more the observed data deviate from the expected data. The chi-square statistic can be used to test the association between two categorical variables by comparing the observed frequencies in a contingency table with the expected frequencies under the assumption of independence. The null hypothesis is that there is no association between the two variables, and the alternative hypothesis is that there is some association. The chi-square test can be performed using the following steps:

1. Construct a contingency table that shows the observed frequencies of the two categorical variables. For example, suppose we want to test the association between gender and eye color. We can collect data from a random sample of 100 people and record their gender and eye color. The contingency table might look like this:

| | Blue | Brown | Green | Total |

| Male | 12 | 34 | 4 | 50 |

| Female| 18 | 22 | 10 | 50 |

| Total | 30 | 56 | 14 | 100 |

2. Calculate the expected frequencies for each cell of the contingency table under the assumption of independence. This can be done by multiplying the row total and the column total, and dividing by the grand total. For example, the expected frequency for the cell corresponding to male and blue eyes is $$\frac{50 \times 30}{100} = 15$$. The expected frequencies for the other cells can be calculated similarly. The contingency table with the expected frequencies in parentheses might look like this:

| | Blue | Brown | Green | Total |

| Male | 12 (15) | 34 (28) | 4 (7) | 50 |

| Female| 18 (15) | 22 (28) | 10 (7) | 50 |

| Total | 30 | 56 | 14 | 100 |

3. Calculate the chi-square statistic by summing up the squared differences between the observed and expected frequencies, divided by the expected frequencies. The formula is $$\chi^2 = \sum \frac{(O-E)^2}{E}$$, where O is the observed frequency and E is the expected frequency. For example, the contribution of the cell corresponding to male and blue eyes to the chi-square statistic is $$\frac{(12-15)^2}{15} = 0.6$$. The contributions of the other cells can be calculated similarly. The chi-square statistic is the sum of all these contributions, which is $$\chi^2 = 0.6 + 1.29 + 1.29 + 0.6 + 1.29 + 1.29 + 1.29 + 0.6 + 1.29 = 9.6$$.

4. Compare the chi-square statistic with the critical value from the chi-square distribution with the appropriate degrees of freedom. The degrees of freedom are calculated by multiplying the number of rows minus one and the number of columns minus one. For example, in this case, the degrees of freedom are $$(2-1) \times (3-1) = 2$$. The critical value can be obtained from a chi-square table or a calculator. For a significance level of 0.05, the critical value for 2 degrees of freedom is 5.991. Since the chi-square statistic is larger than the critical value, we reject the null hypothesis and conclude that there is a significant association between gender and eye color.

6. Determining Degrees of Freedom

One of the most important concepts in statistical inference is the degrees of freedom. The degrees of freedom are a measure of how much information we have in our data to estimate a parameter or test a hypothesis. In this section, we will learn how to determine the degrees of freedom for a chi-square test, which is a common method to test the association between two categorical variables. We will also see how the degrees of freedom affect the shape and critical values of the chi-square distribution, and how to interpret the results of a chi-square test.

To determine the degrees of freedom for a chi-square test, we need to consider the following steps:

1. Create a contingency table that summarizes the frequencies of the two categorical variables. For example, suppose we want to test the association between gender and eye color. We can create a table that shows the number of males and females with different eye colors, as shown below.

| | Blue | Brown | Green | Total |

| Male | 20 | 30 | 10 | 60 |

| Female| 15 | 25 | 15 | 55 |

| Total | 35 | 55 | 25 | 115 |

2. Calculate the number of rows and columns in the contingency table. In our example, we have two rows (male and female) and three columns (blue, brown, and green).

3. Use the formula: degrees of freedom = (number of rows - 1) x (number of columns - 1). In our example, the degrees of freedom are (2 - 1) x (3 - 1) = 2.

4. Use the degrees of freedom to find the critical value of the chi-square distribution for a given significance level. The critical value is the point on the chi-square distribution that separates the rejection and non-rejection regions of the hypothesis test. For example, if we use a significance level of 0.05, we can find the critical value from a chi-square table or a calculator. The critical value for 2 degrees of freedom and 0.05 significance level is 5.991.

5. Compare the observed chi-square statistic with the critical value to make a decision about the hypothesis test. The observed chi-square statistic is calculated from the contingency table using the formula: $$\chi^2 = \sum \frac{(O - E)^2}{E}$$ where O is the observed frequency and E is the expected frequency under the null hypothesis of no association. In our example, the observed chi-square statistic is 2.667. Since this is less than the critical value of 5.991, we fail to reject the null hypothesis and conclude that there is no evidence of an association between gender and eye color.

7. Interpreting the Chi-square Test Results

One of the most important steps in performing a chi-square test is interpreting the results. The chi-square test can tell us whether there is a significant association between two categorical variables, but it cannot tell us the nature or strength of that association. To understand the meaning and implications of the chi-square test results, we need to look at some additional information, such as the contingency table, the expected frequencies, the effect size, and the post-hoc tests. In this section, we will discuss how to interpret the chi-square test results from different perspectives and provide some examples to illustrate the concepts.

Some of the points that we need to consider when interpreting the chi-square test results are:

1. The p-value: The p-value is the probability of obtaining a chi-square statistic as extreme or more extreme than the one observed, assuming that the null hypothesis of no association is true. A small p-value (usually less than 0.05) indicates that the observed association is unlikely to be due to chance and that we can reject the null hypothesis. A large p-value (usually greater than 0.05) indicates that the observed association is likely to be due to chance and that we cannot reject the null hypothesis. For example, if we perform a chi-square test on the data below and obtain a p-value of 0.001, we can conclude that there is a significant association between gender and color preference.

| Gender | Blue | Red | Green | Yellow |

| Male | 20 | 15 | 10 | 5 |

| Female | 10 | 20 | 15 | 5 |

2. The contingency table: The contingency table shows the observed frequencies of the two categorical variables in each cell. The contingency table can help us visualize the pattern of the association and identify which cells have the largest or smallest differences between the observed and expected frequencies. For example, in the table above, we can see that males tend to prefer blue more than females, while females tend to prefer red more than males. We can also see that both genders have similar preferences for green and yellow.

3. The expected frequencies: The expected frequencies are the frequencies that we would expect to observe in each cell if the null hypothesis of no association is true. They are calculated by multiplying the row total and the column total and dividing by the grand total. For example, the expected frequency for the cell of male and blue is (30 x 30) / 75 = 12. The expected frequencies are used to calculate the chi-square statistic and measure the discrepancy between the observed and expected frequencies. The larger the discrepancy, the more likely the null hypothesis is false.

4. The effect size: The effect size is a measure of the strength or magnitude of the association between the two categorical variables. There are different ways to calculate the effect size for the chi-square test, such as phi, Cramer's V, or odds ratio. The effect size can range from 0 to 1, where 0 means no association and 1 means a perfect association. The effect size can help us evaluate the practical significance of the chi-square test results, beyond the statistical significance. For example, a chi-square test may yield a significant p-value, but the effect size may be very small, indicating that the association is weak and may not have much practical relevance.

5. The post-hoc tests: The post-hoc tests are additional tests that can be performed after the chi-square test to compare the observed frequencies of specific cells or groups of cells. The post-hoc tests can help us identify which pairs of categories have a significant difference in their frequencies and how large that difference is. There are different types of post-hoc tests for the chi-square test, such as the standardized residuals, the adjusted standardized residuals, the Bonferroni correction, or the Marascuilo procedure. The post-hoc tests can provide more detailed and specific information about the nature of the association between the two categorical variables. For example, a post-hoc test may reveal that the difference between males and females in their preference for blue is significant and large, while the difference between males and females in their preference for green is not significant and small.

8. Limitations of the Chi-square Test

The chi-square test is a widely used statistical method for testing the association between two categorical variables. However, like any other statistical test, it has some limitations that need to be considered before applying it to real-world data. In this section, we will discuss some of the common limitations of the chi-square test and how to deal with them or avoid them.

Some of the limitations of the chi-square test are:

1. The chi-square test assumes that the observations are independent. This means that the outcome of one observation does not affect the outcome of another observation. For example, if we want to test the association between gender and smoking status, we need to make sure that the gender of one person does not influence the smoking status of another person. However, in some cases, this assumption may not hold. For example, if we sample people from the same household, family, or social group, their gender and smoking status may be correlated. In such cases, the chi-square test may give misleading results. To avoid this problem, we need to ensure that the sampling method is random and that the sample size is large enough to represent the population of interest.

2. The chi-square test is sensitive to the sample size. The larger the sample size, the more likely the chi-square test will detect a significant association, even if the association is very weak or trivial. On the other hand, the smaller the sample size, the less likely the chi-square test will detect a significant association, even if the association is strong or meaningful. For example, if we have a sample of 10 people and we want to test the association between gender and eye color, we may not find a significant result, even if there is a strong association in the population. However, if we have a sample of 1000 people and we want to test the same association, we may find a significant result, even if there is no association in the population. To avoid this problem, we need to choose an appropriate sample size that is neither too small nor too large, and that is based on the expected effect size and the desired power of the test.

3. The chi-square test requires that the expected frequencies are not too small. The expected frequency is the number of observations that we would expect to see in each cell of the contingency table under the null hypothesis of no association. For example, if we have a sample of 100 people and we want to test the association between gender and smoking status, and we observe that 40 people are male and 60 people are female, and that 20 people are smokers and 80 people are non-smokers, then the expected frequency for the cell of male smokers is $$\frac{40 \times 20}{100} = 8$$. The chi-square test assumes that the expected frequencies are large enough to approximate a normal distribution. However, if the expected frequencies are too small, the chi-square test may not be valid. A common rule of thumb is that the expected frequencies should be at least 5 in each cell. If this condition is not met, we may need to use a different test, such as Fisher's exact test, or combine some categories to increase the expected frequencies.

9. Conclusion and Further Applications

The chi-square test is a powerful and versatile statistical tool that can be used to test the association between two categorical variables. It can help us answer questions such as: Is there a difference in the preferences of customers based on their age group? Is there a relationship between the type of treatment and the outcome of a disease? Is there a bias in the selection of candidates based on their gender? In this blog, we have learned how to perform the chi-square test using a contingency table, how to interpret the results using the p-value and the effect size, and how to check the assumptions and limitations of the test.

The chi-square test has many applications in various fields of study, such as:

1. Biology and Medicine: The chi-square test can be used to test the hypothesis of genetic inheritance, such as Mendel's laws of segregation and independent assortment. For example, we can use the chi-square test to determine if the observed ratios of phenotypes in a cross between two plants are consistent with the expected ratios based on the genotypes. The chi-square test can also be used to compare the frequencies of different types of diseases or outcomes among different groups of patients or treatments. For example, we can use the chi-square test to evaluate if a new drug is effective in reducing the mortality rate of a certain disease compared to a placebo or a standard treatment.

2. Psychology and Education: The chi-square test can be used to test the hypothesis of the independence of two psychological or educational variables, such as personality traits, learning styles, attitudes, or behaviors. For example, we can use the chi-square test to examine if there is a difference in the frequency of extraversion or introversion among students who choose different majors. The chi-square test can also be used to compare the performance or achievement of different groups of students or learners based on different factors, such as gender, age, or teaching method. For example, we can use the chi-square test to assess if there is a difference in the pass rate of an exam among students who received online or face-to-face instruction.

3. Sociology and Politics: The chi-square test can be used to test the hypothesis of the association between two social or political variables, such as demographic characteristics, opinions, or preferences. For example, we can use the chi-square test to investigate if there is a relationship between the gender and the voting behavior of a population. The chi-square test can also be used to compare the distribution or proportion of different categories or groups among different populations or samples. For example, we can use the chi-square test to compare the ethnic diversity of two cities or countries.

The chi-square test is not without its limitations, however. Some of the limitations are:

- The chi-square test requires that the data are in the form of counts or frequencies, not proportions or percentages.

- The chi-square test assumes that the observations are independent, meaning that each observation belongs to only one category of each variable and that the categories of each variable are mutually exclusive.

- The chi-square test assumes that the expected frequencies are sufficiently large, usually at least 5, to ensure the validity of the approximation of the chi-square distribution. If the expected frequencies are too small, the chi-square test may not be accurate or reliable.

- The chi-square test does not provide information about the direction or the strength of the association between the variables, only whether there is a significant association or not. To measure the direction or the strength of the association, we need to use other statistics, such as the phi coefficient, Cramer's V, or the odds ratio.

The chi-square test is a useful and flexible method to test the association between two categorical variables. It can help us answer many interesting and important questions in various domains of knowledge. However, we need to be aware of the assumptions and limitations of the test and use it appropriately and cautiously. We also need to supplement the chi-square test with other statistics or methods to gain a deeper and richer understanding of the data and the phenomena we are studying.

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