D3.2.21 Use of a chi-squared test on data from dihybrid crosses (HL) Notes

Using the Chi-Squared Test on Data from Dihybrid Crosses

1. Consider a scientist studying inheritance patterns in pea plants.
2. They've performed a dihybrid cross and predicted a 9:3:3:1 phenotypic ratio in the F2 generation.
3. But when they count the offspring, the numbers don't match perfectly.
4. The scientist now needs to determine if this deviation is due to chance or if something else is at play.
5. The chi-squared test helps you answer this question by comparing your observed results to the expected ones.

Why Use the Chi-Squared Test?

1. In genetics, the chi-squared test is used to:
   1. Compare observed results (actual data) to expected results (based on predictions).
   2. Determine if deviations are due to chance or if they suggest other factors, like gene linkage.

Observed vs. Expected Results

1. In a dihybrid cross, the expected phenotypic ratio for unlinked genes is 9:3:3:1.
2. This means:
   1. 9/16 of the offspring should show both dominant traits.
   2. 3/16 should show one dominant and one recessive trait.
   3. 3/16 should show the other dominant and recessive trait.
   4. 1/16 should show both recessive traits.

Calculating Chi-Squared ($\chi^2$)

The chi-squared formula is:

$$\chi^2 = \sum \frac{(O - E)^2}{E}$$

where $O$ is the observed frequency and $E$ is the expected frequency.

Statistical Significance and $p = 0.05$

1. The significance level ($p = 0.05$) means there is a 5% chance that the observed differences are due to random variation.
2. If $p < 0.05$, the results are considered statistically significant, suggesting that the null hypothesis should be rejected.

Step-by-Step Guide to the Chi-Squared Test

1. State the Hypotheses:
   1. $H_0$​: The traits fit the expected Mendelian ratio.
   2. $H_1$: The traits deviate from the expected Mendelian ratio.
2. Set Up a Contingency Table:
   1. Create a table to organize observed and expected values for each phenotype.
3. Calculate Expected Frequencies:
   1. Use the predicted ratio (e.g., 9:3:3:1) to calculate expected frequencies.
   2. Example: If there are 160 offspring, $E$ for each phenotype is:
      1. $9/16 \times 160 = 90$(dominant traits)
      2. $3/16 \times 160 = 30$ (mixed traits)
      3. $1/16 \times 160 = 10$ (recessive traits)
4. Apply the Chi-Squared Formula:
   1. Calculate $\chi^2$ for each phenotype, then sum the values.
5. Determine Degrees of Freedom:
   1. Degrees of freedom ($df$) = Number of phenotypic categories $- 1$.
   2. For a 9:3:3:1 ratio: $df = 4 - 1 = 3$.
6. Compare with Critical Values:
   1. Use a chi-squared table to find the critical value for $df = 3$ at $p = 0.05$ (significance level).
   2. If $\chi^2$ is larger than the critical value, reject $H_0$​.