课 33, 主题 6

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7-1-6-the-chi-squared-test

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Predicting inheritance: chi-squared test

A statistical test called the chi-squared test determines whether or not there is a significant difference between the observed and expected results in an experiment
- If the difference between results is statistically significant this suggests the presence of a factor that isn’t being accounted for
  - E.g. linkage between genes
- When a difference is not significant, any differences that are observed can be said to be due to chance alone
The chi-squared test is carried out when the data is categorical, i.e. falls into distinct groups

Calculating chi-squared values

Obtain the expected (E) and observed (O) results for the experiment
Calculate the difference between each set of results
Square each difference
- It is irrelevant whether the difference is positive or negative
Divide each squared difference by the expected value
Add the resulting values together to get a sum of these answers to obtain the chi-squared value

Analysing chi-squared values

To work out what the chi-squared value means, we need to compare the chi-squared value to a critical value
The critical value is read from a table of critical values and depends on the probability level used and the degrees of freedom
- Biologists generally use a probability level of 0.05 or 5 %
  - This means that there is only a 5 % probability that any difference between O and E has occurred by chance
- The degrees of freedom takes into account the number of comparisons made, and is calculated as follows:

degrees of freedom = number of classes – 1

Making a Conclusion

Comparison	Conclusion
χ² ≥ critical value	Significant difference – not due to chance → Reject the null hypothesis
χ² ≥ critical value
χ² < critical value	No significant difference – due to chance → Accept the null hypothesis
χ² < critical value

Interpreting significance

It is possible to use the critical values table to assess the probability level at which any difference between observed and expected values becomes significant, e.g.
- If χ² falls between critical values for different p-levels:
  - Estimate the probability range
  - E.g. χ² between p = 0.05 and p = 0.10 → the probability that any difference is due to chance is 5–10%
- A very large χ² (e.g. greater than the critical value at p = 0.001)
  - Indicates less than 0.1% probability that any difference between the O and E is due to chance
  - This provides strong evidence against the null hypothesis

Worked Example

An experiment was carried out on the inheritance of two genes in rabbits; one for coat colour and one for ear length. In this dihybrid cross, the expected ratio of phenotypes was 9 : 3 : 3 : 1.

Rabbits with the heterozygous genotype were bred together, and the phenotypes of all the offspring were recorded.

Complete a chi-squared test to determine whether the difference between observed and expected offspring ratios is significant.

Step 1: complete a table like the one below

Table showing offspring phenotypes: brown coat long/short ears, black coat long/short ears. Includes observed, expected numbers, ratios, and calculations.

Note that the expected values can be calculated as follows:

9 + 3 + 3 + 1 = 16

128 (total number of rabbits) ÷ 16 = 8

3 x 8 = 24

9 x 8 = 72

Step 2: use the table contents to calculate the chi-squared value

1 ÷ 72 = 0.014

9 ÷ 24 = 0.375

4 ÷ 24 = 0.167

chi-squared value = ∑(O – E)² ÷ E

= 0.014 + 0.375 + 0.167 + 0

= 0.56

Step 3: compare the chi-squared value to the critical value

Table showing critical values based on degrees of freedom (1-4) and significance levels (0.1, 0.05, 0.01, 0.001) for hypothesis testing.

The degrees of freedom can be calculated as follows:

In this example there are 4 phenotypes:

4 – 1 = 3 degrees of freedom

We are biologists so we work at a probability level of 0.05

The critical value is therefore 7.82

Step 4: draw conclusions

The chi-squared value of 0.56 is smaller than the critical value of 7.82
This means that there is no significant difference between the expected and observed results and any differences that do occur are due to chance
- 0.56 would be located somewhere to the left-hand side of the table, indicating that there is a higher than 10 % probability that the difference between O and E is due to chance
A null hypothesis can be accepted

Examiner Tips and Tricks

When calculating a chi-squared value, it is very helpful to create a table like the one seen in the worked example. This will help you with your calculations and ensure you don’t get confused.

You should also be prepared to suggest reasons why results might be significantly different. For example, there could be linkage between the genes being analysed.

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