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## The Chi-Square Test

The χ^{2} statistic is used in genetics to illustrate if there are deviations from the expected outcomes of the alleles in a population. The general assumption of any statistical test is that there are no significant deviations between the measured results and the predicted ones. This lack of deviation is called the** null hypothesis** (**H _{0}**). χ

^{2}statistic uses a distribution table to compare results against at varying levels of probabilities or

**critical values**. If the χ

^{2}value is greater than the value at a specific probability, then the null hypothesis has been rejected and a significant deviation from predicted values was observed. Using Mendel’s laws, we can count phenotypes after a cross to compare against those predicted by probabilities (or a Punnett Square).

In order to use the table, one must determine the stringency of the test. The lower the p-value, the more stringent the statistics. **Degrees of Freedom** (**DF**) are also calculated to determine which value on the table to use. Degrees of Freedom are the number of classes or categories there are in the observations minus 1. DF=n-1

In the example of corn kernel color and texture, there are 4 classes: Purple & Smooth, Purple & Wrinkled, Yellow & Smooth, Yellow & Wrinkled. Therefore, DF = 4 – 1 = 3 and choosing p < 0.05 to be the threshold for significance (rejection of the null hypothesis), the χ^{2} must be greater than 7.82 in order to be significantly deviating from what is expected. With this dihybrid cross example, we expect a ratio of 9:3:3:1 in phenotypes where 1/16th of the population are recessive for both texture and color while 9/16th of the population display both color and texture as the dominant. 3/16th will be dominant for one phenotype while recessive for the other and the remaining 3/16th will be the opposite combination.

With this in mind, we can predict or have expected outcomes using these ratios. Taking a total count of 200 events in a population, 9/16(200)=112.5 and so forth. Formally, the χ^{2} value is generated by summing all combinations of:

** (observed-expected) ^{2}/expected**

### Chi-Square Test: Is this coin fair or weighted? (activity)

- Everyone in the class should flip a coin 2x and record the result (assumes class is 24)
- Fair coins are expected to land 50% heads and 50% tails
- 50% of 48 results should be 24
- 24 heads and 24 tails is already written in the “Expected” column

- As a class, compile the results in the “Observed” column (total of 48 coin flips)
- In the last column, subtract the expected heads from the observed heads and square it, then divide by the number of expected heads
- In the last column, subtract the expected tails from the observed tails and square it, then divide by the number of expected tails
- Add the values together from the last column to generate the χ
^{2}value - Compare the value with the value at 0.05 with DF=1
- there are 2 classes or categories (head or tail), so DF = 2 – 1 = 1
- Were the coin flips fair (not significantly deviating from 50:50)?

Let’s say that the coin tosses yielded 26 Heads and 22 Tails. Can we assume that the coin was unfair? If we toss a coin an odd number of times (eg. 51), then we would expect that the results would yield 25.5 (50%) Heads and 25.5 (50%) Tails. But this isn’t a possibility. This is when the χ^{2} test is important as it delineates whether 26:25 or 30:21 etc. are within the probability for a fair coin.

### Chi-Square Test of Kernel Coloration and Texture in an F_{2} Population (activity)

- From the counts, one can assume which phenotypes are dominant and recessive
- Fill in the “Observed” category with the appropriate counts
- Fill in the “Expected Ratio” with either 9/16, 3/16 or 1/16
- The total number of counted event was 200, so multiply the “Expected Ratio” x 200 to generate the “Expected Number” fields
- Calculate the (Observed-Expected)
^{2}/Expected for each phenotype combination - Add all (Observed-Expected)
^{2}/Expected values together to generate the χ^{2}value and compare with the value on the table where DF=3 - Do we reject the Null Hypothesis or were the observed numbers as we expected as roughly 9:3:3:1?
- What would it mean if the Null Hypothesis was rejected? Can you explain a case in which we have observed values that are significantly altered from what is expected?

Tags: quantitative reasoning, integration of knowledge, guided inquiry