Testing Genetic Ratios Help (page 2)
Sampling Theory and Size
The application of statistics to genetic problems can help predict and evaluate outcomes of crosses and experiments.
If we toss a coin, we expect that half of the time it will land heads up and half of the time tails up. This hypothesized probability is based upon an infinite number of coin tossings wherein the effects of chance deviations from 0.5 in favor of either heads or tails cancel one another. All actual experiments, however, involve finite numbers of observations and therefore some deviation from the expected numbers (sampling error) is to be anticipated. Let us assume that there is no difference between the observed results of a coin-tossing experiment and the expected results that cannot be accounted for by chance alone (null hypothesis). How great a deviation from the expected 50 : 50 ratio in a given experiment should be allowed before the null hypothesis is rejected? Conventionally, the null hypothesis in most biological experiments is rejected when the deviation is so large that it could be accounted for by chance less than 5%of the time. Such results are said to be significant. When the null hypothesis is rejected at the 5% level, we take 1 chance in 20 of discarding a valid hypothesis. It must be remembered that statistics can never render absolute proof of the hypothesis, but merely sets limits to our uncertainty. If we wish to be even more certain that the rejection of the hypothesis is warranted we could use the1% level, often called highly significant, in which case the experimenter would be taking only 1 chance in 100 of rejecting a valid hypothesis.
If our coin-tossing experiment is based on small numbers, we might anticipate relatively large deviations from the expected values to occur quite often by chance alone. However, as the sample size increases, the deviation should become proportionately less, so that in a sample of infinite size the plus and minus chance deviations cancel each other completely to produce the 50 : 50 ratio.
Degrees of Freedom
Assume a coin is tossed 100 times. We may arbitrarily assign any number of heads from 0 to 100 as appearing in this hypothetical experiment. However, once the number of heads is established, the remainder is tails and must add to 100. In other words, we have n – 1 degrees of freedom(df) in assigning numbers at random to the n classes within an experiment.
EXAMPLE 2.37 In an experiment involving three phenotypes (n = 3), we can fill two of the classes at random, but the number in the third class must constitute the remainder of the total number of individuals observed. Therefore, we have 3 – 1 = 2 degrees of freedom.
EXAMPLE 2.38 A 9 : 3 : 3 : 1 dihybrid ratio has four phenotypes (n = 4). There are 4 – 1 = 3 degrees of freedom. The number of degrees of freedom in these kinds of problems is the number of variables (n) under consideration minus 1. For most genetic problems, the degrees of freedom will be 1 less than the number of phenotypic classes. Obviously, the more variables involved in an experiment the greater the total deviation may be by chance
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.
- 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.
- 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 |oi 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:
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