Genetics Chi-Square Test: Epistasis and Hypothesis Testing

Genetics Chi-Square Test Walkthrough

Overview of the Chi-Square (χ²) Test

The Chi-Square Goodness-of-Fit Test is a statistical method used in genetics to determine if observed offspring counts match the expected genetic ratios derived from Mendelian principles and epistasis. This test helps evaluate whether deviations from expected ratios are due to chance or indicate a significant difference.

• Formula: $\chi^2 = \sum \frac{(O - E)^2}{E}$ where O = observed value, E = expected value.

Dominant Epistasis (12:3:1 Ratio)

Dominant epistasis occurs when a dominant allele at one locus masks the expression of alleles at another locus, resulting in a modified phenotypic ratio.

• Hypothesis: The data follows a 12:3:1 ratio.

• Observed Values: 720 (white), 140 (yellow), 60 (purple)

• Expected Values:

  • White: $\frac{12}{16} \times 920 = 690$

  • Yellow: $\frac{3}{16} \times 920 = 172.5$

  • Purple: $\frac{1}{16} \times 920 = 57.5$

• Calculation: $\chi^2 = \frac{(720-690)^2}{690} + \frac{(140-172.5)^2}{172.5} + \frac{(60-57.5)^2}{57.5}$ $= \frac{900}{690} + \frac{1056.25}{172.5} + \frac{6.25}{57.5}$ $= 1.304 + 6.123 + 0.109 = 7.536$

• Interpretation: Degrees of freedom (df) = 2. Critical value (p=0.05) = 5.99. Since 7.536 > 5.99, we reject the null hypothesis (the observed data does not fit the expected ratio).

Duplicate Recessive Epistasis (9:7 Ratio)

Duplicate recessive epistasis occurs when two genes can compensate for each other, and both must have at least one dominant allele for the phenotype to be expressed, resulting in a 9:7 ratio.

• Hypothesis: The data follows a 9:7 ratio.

• Observed Values: 306 (white), 280 (purple)

• Expected Values:

  • White: $\frac{9}{16} \times 586 = 329.6$

  • Purple: $\frac{7}{16} \times 586 = 256.4$

• Calculation: $\chi^2 = \frac{(306-329.6)^2}{329.6} + \frac{(280-256.4)^2}{256.4}$ $= \frac{552.16}{329.6} + \frac{561.76}{256.4}$ $= 1.675 + 2.192 = 3.867$

• Interpretation: Degrees of freedom (df) = 1. Critical value (p=0.05) = 3.84. Since 3.867 > 3.84, we reject the null hypothesis (the observed data does not fit the expected ratio).

Summary Table

The following table summarizes the critical values for different types of epistasis and their degrees of freedom:

Interaction	Ratio	df	Critical Value
Recessive Epistasis	9:3:4	2	5.99
Dominant Epistasis	12:3:1	2	5.99
Duplicate Recessive	9:7	1	3.84
Dominant Inhibitor	13:3	1	3.84
Duplicate Dominant	15:1	1	3.84

Key Terms and Concepts

• Epistasis: Interaction between genes where one gene masks or modifies the expression of another.

• Degrees of Freedom (df): Number of categories minus one; used to determine the critical value in chi-square tests.

• Critical Value: The threshold value for rejecting the null hypothesis at a given significance level (commonly p=0.05).

• Null Hypothesis: Assumes no significant difference between observed and expected values.

Example Application

Chi-square tests are widely used in genetics to validate Mendelian ratios, analyze gene interactions, and confirm inheritance patterns in experimental crosses.

Additional info: The notes provide a focused walkthrough on epistasis ratios and hypothesis testing using the chi-square method, relevant to Ch.5 (Extensions of Mendelian Inheritance) and Ch.24 (Quantitative Genetics).