Introduction to Chi-Square Test in Genetics

The chi-square test is a statistical method used to determine whether there is a significant difference between the expected and observed frequencies in categorical data. In genetics, it is particularly useful for analyzing the inheritance patterns of traits and alleles. By applying the chi-square test, researchers can evaluate hypotheses about genetic variations and assess whether certain traits are inherited according to Mendelian principles.

Example 1: Mendelian Inheritance of Pea Plants

In an experiment studying the inheritance of a particular trait in pea plants—specifically, flower color—researchers cross two heterozygous plants (Pp) where ‘P’ represents purple flowers (dominant) and ‘p’ represents white flowers (recessive). The expected ratio for the offspring is 3 purple to 1 white.

In a sample of 160 offspring, the observed results were:

• 120 purple flowers
• 40 white flowers

To conduct the chi-square test, we first calculate the expected frequencies:

• Expected purple = (3/4) * 160 = 120
• Expected white = (1/4) * 160 = 40

Next, we calculate the chi-square statistic using the formula:

egin{align}
ext{Chi-square} =
rac{(O - E)^2}{E}
ext{where} O = ext{Observed}, E = ext{Expected}
egin{align}
ext{Chi-square} =
rac{(120 - 120)^2}{120} +
rac{(40 - 40)^2}{40} = 0 + 0 = 0
ext{Degrees of freedom} = 2 - 1 = 1
ext{Critical value at 0.05 significance level} = 3.841

Since 0 < 3.841, we fail to reject the null hypothesis. The flower color inheritance follows Mendelian ratios.

Example 2: Blood Type Distribution in a Population

A research team is interested in the distribution of blood types (A, B, AB, O) in a specific population. The expected distribution based on known genetic frequencies is:

• Type A: 26%
• Type B: 20%
• Type AB: 4%
• Type O: 50%

In a sample of 200 individuals, the observed counts were:

• Type A: 48
• Type B: 40
• Type AB: 8
• Type O: 104

We calculate the expected counts for each blood type:

• Expected Type A = 0.26 * 200 = 52
• Expected Type B = 0.20 * 200 = 40
• Expected Type AB = 0.04 * 200 = 8
• Expected Type O = 0.50 * 200 = 100

Using the chi-square formula:

egin{align*}
ext{Chi-square} =
rac{(48 - 52)^2}{52} +
rac{(40 - 40)^2}{40} +
rac{(8 - 8)^2}{8} +
rac{(104 - 100)^2}{100}
\ ext{Degrees of freedom} = 4 - 1 = 3
ext{Critical value at 0.05 significance level} = 7.815

Calculating:
ext{Chi-square} =
rac{(4)^2}{52} + 0 + 0 +
rac{(4)^2}{100} = 0.3077 + 0.16 = 0.4677

Since 0.4677 < 7.815, we fail to reject the null hypothesis. The blood type distribution in this population is consistent with expected frequencies.

Example 3: Trait Distribution in Fruit Flies

A geneticist studying Drosophila melanogaster (fruit flies) investigates the inheritance of eye color, where red eyes (R) are dominant over white (r). A cross between two heterozygous flies (Rr) is performed, and the expected phenotypic ratio is 3 red: 1 white.

In a sample of 120 offspring, the observed counts were:

• Red eyes: 90
• White eyes: 30

Calculating expected counts:

• Expected red = (3/4) * 120 = 90
• Expected white = (1/4) * 120 = 30

Using the chi-square formula:

egin{align*}
ext{Chi-square} =
rac{(90 - 90)^2}{90} +
rac{(30 - 30)^2}{30} = 0 + 0 = 0
\ ext{Degrees of freedom} = 2 - 1 = 1
ext{Critical value at 0.05 significance level} = 3.841

Since 0 < 3.841, we fail to reject the null hypothesis. The eye color inheritance follows the expected Mendelian ratio.

Conclusion

These examples demonstrate how the chi-square test serves as a valuable tool in genetics to analyze inheritance patterns and validate genetic hypotheses. By understanding these applications, researchers can gain insights into the inheritance of traits and the underlying genetic mechanisms.