M09.02.15 Chi-square

The Chi-square test is a statistical method used to determine if there is a significant association between two categorical (nominal) variables. It is commonly applied in fields such as biostatistics, public health, and clinical research to assess the independence of variables or to test hypotheses about relationships in population data.

Purpose

The primary objective of the Chi-square test is to evaluate whether observed differences between groups are due to chance or if they reflect an actual association. For example, in clinical research, the test can help assess the effectiveness of a new drug by comparing recovery rates among patients who received the drug and those who did not.

Key Characteristics

• Data Type: Only nominal (categorical) data is used.
• Group Comparisons: Any number of groups can be compared, commonly formatted as 2×2, 2×3, or 3×3 tables.

Example of Chi-Square Analysis: Drug Efficacy Test

Consider a scenario where a new drug is tested for its efficacy in patient recovery:

In this example:

• Recovered: Patients who recovered after treatment
• Not Recovered: Patients who did not recover
• New Drug: Patients who received the new drug
• Placebo: Patients who received a placebo

This table forms the basis for calculating the Chi-square statistic, which will determine if the observed recovery rates are independent of the treatment type or if there is a significant association.

Steps for Conducting a Chi-Square Test

1. Define Hypotheses:
   • Null Hypothesis (H₀): There is no association between the treatment (drug vs. placebo) and recovery status (recovered vs. not recovered).
   • Alternative Hypothesis (H₁): There is an association between the treatment and recovery status.
2. Calculate Expected Frequencies: Determine what the frequencies would be if the variables were independent.
3. Compute Chi-Square Statistic (χ²): Compare observed and expected frequencies using the formula:

​where OOO is the observed frequency, and EEE is the expected frequency.

Determine Significance Level (p-value): Compare the computed χ² value to a critical value from the Chi-square distribution table.

Interpret Results: If the p-value is below the chosen significance level (e.g., 0.05), reject the null hypothesis, indicating a statistically significant association between treatment and recovery.

Points to Remember

• The Chi-square test is suitable only for nominal data.
• Expected frequencies in each cell should ideally be 5 or more to ensure validity.
• This test cannot establish causation, only association or independence.

Bibliography

• McDonald, J. H. (2014). Handbook of Biological Statistics (3rd ed.). Baltimore, Maryland: Sparky House Publishing.
• Sokal, R. R., & Rohlf, F. J. (1995). Biometry: The Principles and Practice of Statistics in Biological Research (3rd ed.). New York: W. H. Freeman.
• Altman, D. G. (1991). Practical Statistics for Medical Research. CRC Press.