16.12 Chi-Squared (χ²)
Overview of the Chi-Squared (χ²) Test
- Purpose: The chi-squared test (χ² test) is a statistical method used to compare observed results with expected results. It helps determine if differences are due to random chance or if they indicate a significant deviation from expected patterns.
- Application in Genetics: Often used to evaluate inheritance patterns, such as expected Mendelian ratios, to see if experimental results align with theoretical predictions.
Steps in Conducting a Chi-Squared Test
- Calculate Expected Numbers:
- Based on the total number of offspring and the predicted phenotypic ratios.
- Example: In a dihybrid cross expecting a 9:3:3:1 ratio and 144 offspring:
- Record Observed and Expected Values:
- Enter data in a table format with columns for observed numbers (O), expected numbers (E), and the calculations.
- Calculate χ² Value Using the Formula:
- Steps:
- Degrees of Freedom (df):
- Calculated as number of phenotypic categories minus 1.
- Example: For a 9:3:3:1 ratio with four categories, ( df = 4 – 1 = 3 ).
- Interpret χ² Value Using a χ² Table:
- Compare the χ² value with critical values at various probability levels (e.g., 0.1, 0.05, 0.01).
- The critical probability (p-value) is usually set at 0.05:
- If χ² is less than the critical value: Accept the null hypothesis (differences are likely due to chance).
- If χ² is greater than the critical value: Reject the null hypothesis (differences are significant).
Example Calculation
Example Observed Data for 144 Offspring (from a 9:3:3:1 Expected Ratio):
| Phenotype | Observed (O) | Expected (E) | ( O – E ) | (O – E)2 | (O – E)2/E |
|---|---|---|---|---|---|
| Purple, cut | 86 | 81 | +5 | 25 | 25/81 = 0.31 |
| Purple, potato | 26 | 27 | -1 | 1 | 1/27 = 0.04 |
| Green, cut | 24 | 27 | -3 | 9 | 9/27 = 0.33 |
| Green, potato | 8 | 9 | -1 | 1 | 1/9 = 0.11 |
- Total χ² Value: ( 0.31 + 0.04 + 0.33 + 0.11 = 0.79 )
Interpreting the χ² Value
- Degrees of Freedom:
- ( df = 4 – 1 = 3 )
- Critical Value from χ² Table (for df = 3, p = 0.05): 7.82
- Comparison and Conclusion:
- Calculated χ² (0.79) < Critical χ² (7.82):
- The χ² value is well below 7.82, indicating a high probability that any difference is due to chance. The results support the null hypothesis that observed differences are not significant.
Using the Null Hypothesis
- Null Hypothesis: Assumes no significant difference between observed and expected results (any observed deviations are due to chance).
- Decision: Based on the χ² test result, either accept or reject the null hypothesis:
- Accept Null Hypothesis (if χ² < critical value): Differences are due to chance.
- Reject Null Hypothesis (if χ² > critical value): Differences are likely significant; assumptions may need reconsideration.
Practice Question
- Question 23: In a genetic cross, observed offspring are:
- Grey body, dark eyes: 54
- Grey body, pale eyes: 4
- White body, dark eyes: 4
- White body, pale eyes: 18
- Expected Ratio: Calculate based on Mendelian predictions, then use χ² to determine if differences are significant.
Key Points to Remember
- Interpretation of χ²: A value below the critical threshold supports the null hypothesis; a value above suggests a significant difference.
- Expected Ratios: Based on Mendelian inheritance patterns.
- Degrees of Freedom: Equals number of categories minus 1.