Testing Genetic Ratios Help

Chi-Square Test

In order to evaluate a genetic hypothesis, we need a test that can convert deviations from expected values into the probability of such inequalities occurring by chance. Furthermore, this test must also take into consideration the size of the sample and the number of variables (degrees of freedom). The chi-square test (pronounced ki-square; symbolized x²) includes all of these factors.

where means to sum what follows it as the i classes increase from 1 to n, o represents the number of observations within a class, e represents the number expected in the class according to the hypothesis under test, and n is the number of classes. The value of chi-square may then be converted into the probability that the deviation is due to chance by entering Table 2-3 at the appropriate number of degrees of freedom.

An alternative method for computing chi-square in problems involving only two phenotypes will give the same result as the conventional method and often makes computation easier:

where a and b are the numbers in the two phenotypic classes and r is the expected ratio of a to b.

1. Chi-Square Limitations. The chi-square test as used for analyzing the results of genetic experiments has two important limitations: (1) it must be used only on the numerical or raw data itself, never on percentages or ratios derived from the data; (2) it cannot properly be used for experiments wherein the expected frequency within any phenotypic class is less than 5.
2. Corrections for Small Samples. The formula from which the chi-square table is derived is based upon a continuous distribution, namely, that of the "normal" curve. Such a distribution might be expected when we plot the heights of a group of people. The most frequent class would be the average height and successively fewer people would be in the taller or shorter phenotypes. All sizes are possible from the shortest to the tallest, i.e., heights form a continuous distribution. However, the kinds of genetic problems in the previous chapters of this book involve separate or discrete phenotypic classes such as blue eyes vs. brown eyes. A correction should therefore be applied in the calculation of chi-square to correct for this lack of continuity. The "Yates Correction for Continuity" is applied as follows, where |o_i e _i| is an absolute (positive) value:

This correction usually makes little difference in the chi-square of most problems, but may become an important factor near the critical values. The Yates correction should be routinely applied whenever only 1 degree of freedom exists, or in small samples where each expected frequency is between 5 and 10. If the corrected and uncorrected methods each lead to the same conclusion, there is no difficulty. However, if these methods do not lead to the same conclusion, then either more data need to be collected or a more sophisticated statistical test should be employed. Do not apply the Yates correction for problems in this book unless requested to do so.

Practice problems for these concepts can be found at: 

• Patterns of Inheritance Practice Problems
• Patterns of Inheritance Practice Test