WK11: Statistical inference: Two categorical variables: Chi-square test for independence

Chi-Square Test for Independence

Overview

The Chi-square test for independence is used to determine if there is a relationship between two categorical variables. It assesses whether the counts for specific categories in one variable are dependent on the categories of the other variable.

Example: Sickle Cell Trait and Malaria

Background

• Question: Is there a relationship between being a sickle cell carrier and protection against malaria?
• Study: 543 African children were checked for sickle cell trait and malaria infection.
• Variables:
  • Sickle cell trait (presence or absence)
  • Malaria (presence or absence of heavy infection)

Data

• 36 children with sickle cell trait, 36 were heavily infected with malaria.
• 407 children without sickle cell trait, 152 were heavily infected with malaria.

Contingency Table

Steps for Chi-Square Test

1. Hypotheses and Significance Level:

   • Null Hypothesis ($H_0$): There is no relationship between the two categorical variables (they are independent).
     • Example: There is no relationship between the presence of sickle cell trait and malaria.
   • Alternative Hypothesis ($H_a$): The two categorical variables are dependent.
     • Example: There is a relationship between the presence of sickle cell trait and malaria.
   • Significance Level ($\alpha$): 5% (0.05) in this example.

2. Check Conditions for Use of the Test:

   • Random Sample: The data should come from a random sample of the population.
   • Expected Counts:
     • All expected counts should be at least 1.
     • At least 80% of the cells in the two-way table should have an expected count of at least 5.

3. Calculate the Test Statistic:

   • Observed Counts: The actual counts from the sample data.
   • Expected Counts: The counts expected under the assumption of no relationship between the variables.
     • Calculated as: $\frac{\text{Row Total} \times \text{Column Total}}{\text{Table Total}}$
     • For example, if $T<em>A$ is the total for category A of variable X, $T</em>B$ is the total for category B of variable Y, and $T$ is the table total, the expected count is: $\frac{T<em>A \times T</em>B}{T}$
   • Example calculation for sickle cell trait and malaria:
     • Proportion of children with sickle cell trait: $\frac{136}{543} \approx 0.2505$ (25.05%)
     • Expected count for children with malaria and sickle cell trait: $188 \times \frac{136}{543}$
     • Expected count for children without malaria but with sickle cell trait: $355 \times \frac{136}{543}$
     • Expected count for children without sickle cell trait and without malaria: 266.09
   • Chi-square Statistic ($\chi^2$):
     • Formula: $\chi^2 = \sum \frac{(\text{Observed Count} - \text{Expected Count})^2}{\text{Expected Count}}$
     • The sum is calculated across all cells in the contingency table.
     • Example result: $\chi^2 = 5.33$

4. Find the P-value:

   • The p-value is the probability of obtaining a test statistic as large or larger than the calculated one, assuming the null hypothesis is true.
   • Degrees of Freedom (df): $(\text{Number of Rows} - 1) \times (\text{Number of Columns} - 1)$
     • In the sickle cell example: $(2 - 1) \times (2 - 1) = 1$
   • Use a Chi-square distribution table to find the p-value corresponding to the calculated $\chi^2$ statistic and degrees of freedom.
   • Example: For $\chi^2 = 5.33$ and df = 1, the p-value is between 0.02 and 0.025.

5. Make a Statistical Decision and Conclusion:

   • If the p-value is less than or equal to the significance level ($\alpha$), reject the null hypothesis. This suggests there is a statistically significant relationship between the variables.
   • If the p-value is greater than $\alpha$, fail to reject the null hypothesis. This suggests there is not enough evidence to support a relationship between the variables.
   • Example Conclusion: Since the p-value range (0.02 - 0.025) is below the significance level of 0.05, we reject the null hypothesis. The sample provides statistically significant evidence that there is a relationship between the presence of sickle cell trait and the presence of malaria.