Mendelian Genetics and Chi-Square Test

Comparing Observed and Predicted Ratios

  • To evaluate if genetic traits follow Mendel's principles, observed offspring ratios are compared against predicted ratios based on Mendelian inheritance.
  • Statistical tests are used to determine if any observed differences from predictions are significant or merely due to random chance.
  • Importance of Random Chance: In any statistical analysis, it's crucial to acknowledge and account for the effects of random chance. Small deviations from expected values might not be meaningful differences.
  • Quantifying Differences: Statistics help quantify the spectrum of difference, distinguishing between minor, non-meaningful variations and significant, actual deviations.
  • Statistical Framework: This involves formulating a null hypothesis and using critical values of test statistics to determine the probability that observed differences are not due to chance.

The Chi-Square Goodness-of-Fit Test

  • Purpose: The chi-square ($\chi^2$) test is used to assess if a trait's inheritance pattern conforms to a simple Mendelian model.
  • Type of Test: It is a "goodness-of-fit" test, which evaluates how well observed data aligns with expected values.
  • Data Requirements: The test uses actual counts of individuals for each phenotypic class (observed numbers and expected numbers), not ratios, fractions, or percentages.
  • Chi-Square Formula: The chi-square statistic is calculated as the sum of the squared difference between observed and expected counts, divided by the expected count for each phenotypic class: \chi^2 = \sum \frac{(Oi - Ei)^2}{E_i} where:
    • O_i represents the observed number of individuals in phenotypic class i.
    • E_i represents the expected number of individuals in phenotypic class i.
    • \sum indicates the summation across all phenotypic classes.

Interpreting Chi-Square Results

  • Comparison to Critical Value: The calculated $\chi^2$ value is compared against a critical value obtained from a chi-square distribution table.
  • Degrees of Freedom (df): To find the correct critical value, the degrees of freedom must be determined.
    • df = (\text{Number of phenotypic classes}) - 1
    • For example, a one-locus trait with simple dominance has two phenotypic classes (dominant and recessive), so df = 2 - 1 = 1.
  • Chi-Square Distribution: This statistical distribution indicates the probability of obtaining a particular $\chi^2$ value purely by chance.
    • If the calculated $\chi^2$ value falls within the main, larger part of the distribution, the observed differences are likely due to random chance (i.e., not statistically significant).
    • If the calculated $\chi^2$ value falls in the extreme tail (typically the top 5%) of the distribution, it is highly unlikely to have occurred by chance. This suggests a real difference between observed and expected values, implying the trait is not behaving according to Mendel's principles.
  • Significance Level (p-value): A common cutoff for significance is p = 0.05 (or 5%).
    • If the calculated $\chi^2$ value exceeds the critical value for p = 0.05 (at the given degrees of freedom), we conclude that the differences are statistically significant and reject the null hypothesis. This means there's a strong likelihood that the observed inheritance pattern deviates from Mendelian principles.
    • If the calculated $\chi^2$ value is less than the critical value, the differences are not considered statistically significant, and we fail to reject the null hypothesis. This implies the observed data is consistent with Mendelian inheritance.
  • Chi-Square Table: This table is always provided and is not expected to be memorized. It lists critical values for different degrees of freedom and p-values.

Example: One-Locus Chi-Square Calculation

  • Scenario: Two heterozygous, deep-resistant individuals are crossed, producing 150 progeny.
  • Null Hypothesis (based on Mendelian principles):
    • Deep resistant phenotype: 3/4 \times 150 = 112.5 individuals (Expected)
    • Non-deep resistant phenotype: 1/4 \times 150 = 37.5 individuals (Expected)
  • Observed Data (Hypothetical for calculation):
    • Deep resistant phenotype: 100 individuals (Observed)
    • Non-deep resistant phenotype: 50 individuals (Observed, calculated as 150 - 100)
  • Calculating the $\chi^2$ Value:
    1. For Deep Resistant Class:
      (O - E)^2 / E = (100 - 112.5)^2 / 112.5 = (-12.5)^2 / 112.5 = 156.25 / 112.5 \approx 1.389
    2. For Non-Deep Resistant Class:
      (O - E)^2 / E = (50 - 37.5)^2 / 37.5 = (12.5)^2 / 37.5 = 156.25 / 37.5 \approx 4.167
    3. Total $\chi^2$ Value:
      \chi^2 = 1.389 (\text{resistant}) + 4.167 (\text{non-resistant}) \approx 5.556
  • Degrees of Freedom (df):
    • There are two phenotypic classes (resistant, non-resistant).
    • df = 2 - 1 = 1
  • Interpretation: For df = 1 and a significance level of p = 0.05, the critical value from a chi-square table is approximately 3.841. Since the calculated $\chi^2 value (approx. 5.556) is greater than the critical value (3.841), the observed differences are statistically significant. This suggests that the trait's inheritance pattern in this example is likely not following simple Mendelian principles.
  • A \chi^2$$ value less than the critical value signifies that differences are not statistically significant, meaning the data is consistent with the null hypothesis.