Chi-square test

 A chi-square test is a valuable tool that aids the investigator in determining the goodness of fit. The test takes into account the size of the sample and the deviations from the expected ratio. It not only can be used for samples of different sizes but can be adapted to ratios with a different number of classes, like two classes for monohybrid crosses and four classes for dihybrid crosses. Essentially, the chi-square test is a mechanism by which deviations from a hypothetical ratio are reduced to a single value based on the size of the sample. This allows the investigator to determine the probability that a given sum of deviations will occur by chance. Expected values are obtained from the total size of the sample. If the hypothesis is that a 1:1 ratio results from a cross, the total is divided into two equal parts. For any other expected ratio, the total is divided into appropriate proportions.

Chi-Square Formula

Degrees of freedom (df) = n-1 where n is the number of classes

Steps in the chi-square test

A chi-squared test can be completed by following five simple steps:

• Identify hypotheses (null versus alternative)

• Construct a table of frequencies (observed versus expected)

• Apply the chi-squared formula

• Determine the degree of freedom (df)

• Identify the p-value (should be <0.05)

Example

1. Identify hypotheses

A chi-squared test seeks to distinguish between two distinct possibilities and hence requires two contrasting hypotheses:

Null hypothesis (H0): There is no significant difference between observed and expected frequencies.

Alternative hypothesis (H1): There is a significant difference between observed and expected frequencies.

2. Construct a table of frequencies

A table must be constructed that compares observed and expected frequencies for each possible phenotype. Expected frequencies are calculated by first determining the expected ratios and then multiplying against the observed total.

Let's test the following data to determine if it fits a 9:3:3:1 ratio.

3. Apply the chi-squared formula

Number of classes (n) = 4

Calculated Chi-square value = 0.47

4. Determine the degree of freedom (df)

Degrees of freedom represent the number of ways in which the observed outcome categories are free to vary. For Pearson's chi-square test, the degrees of freedom are equal to n - 1, where n represents the number of different phenotypic classes. The degree of freedom in this case is

(n-1) = (4-1) = 3

5. Identify the p-value

The final step is to apply the value generated to a chi-squared distribution table to determine if the results are statistically significant.

Chi-square table value

Compare the calculated chi-square value with the table chi-square value at desired degrees of freedom and probability. In this example check the table chi-square value at df = 3 and we see the probability of calculated chi-square value is greater than 0.90. By statistical convention, we use the 0.05 probability level as our critical value. If the calculated chi-square value is less than the table value at 0.05, we accept the null hypothesis, i.e. the difference between observed and expected values in each class is not real, they are the same. Since the calculated chi-square value is less than the table chi-square value, we accept the null hypothesis, that the data fits a 9:3:3:1 ratio.

The Chi-square test works well with genetic data as long as there are enough expected values in each group. In the case of small samples (less than 10 in any category) that have 1 degree of freedom, the test is not reliable. However, in such cases, the test can be corrected by using the Yates correction for continuity, which reduces the absolute value of each difference between observed and expected frequencies by 0.5 before squaring. Additionally, it is important to remember that the chi-square test can only be applied to numbers of progenies (real values), not to proportions (ratios) or percentages.