Chi-Square Test

The Chi-Squared test is used to compare what you have measured (observed) against what may be anticipated (expected).

We establish a hypothesis for the feature under investigation and then convert it to a null hypothesis. The null hypothesis states that no relationship between the two population parameters exists. We use it because it helps us see if our hypothesis has validity. It is impossible to prove something with absolute certainty. However, we can disprove a null hypothesis, which allows us to accept that our hypothesis is valid, and we use ‘confidence levels’ and ‘critical values’ to do this.

The difference between expected and observed results in experiments can be described in two ways:

• Statistically significant
• Statistically insignificant (happened by chance)

Chi-squared tests should only be done using categorical data and if specific criteria are met.

What are the criteria for performing a chi-squared test?

• Only raw counts can be used – not ratios, rates, fractions or percentages.
• The comparison between theoretical (expected) and experimental (observed) results is being made.

What are the assumptions of the chi-squared test?

Additionally, the chi-squared test makes several assumptions:

• The comparisons are made on random samples.
• The expected count of each cell is greater than one (>1).
• No more than 20% of the cells have expected counts less than 5 (<5).

How do you calculate chi-squared?

The formula looks very scary - but don’t panic! We can break it down into steps.

What is the formula for chi-squared?

$$\begin{gathered} X^2=\sum _{}\frac{{(O-E)}^2}{E} \\ {X}^2=Theteststatistic \\ \sum _{}=Thesumof \\ O=Observedfrequency \\ E=Expectedfrequency \end{gathered}$$

In other words, chi-squared $X^2$ is the sum of the square of the difference between the observed values and expected values ${(O-E)}^2$, divided by the expected values (E).

To help you understand how we would calculate the chi-squared, we will use flower phenotype as an example.

To calculate:

• Obtain the expected and observed results for the experiment (as shown in the table below)
• Calculate the difference between each set of results
• Square each difference
• Divide each squared difference by the expected value
• Use the sum of these answers to obtain the chi-squared value

Table 1. Example of a table to find values for Chi-Squared calculation.

How do you calculate degrees of freedom and use a chi-squared distribution table?

The Chi-Squared test has little meaning on its own – it needs to be compared to ‘critical values’, which are found in tables or on graphs as calculated by statistical experts.

First, you must decide the confidence level you want to use. The most common is generally 95% and/or 99%, meaning for every 100 times you carried out the test, you would get chance results on five occasions or one occasion.

Table 2. Confidence, uncertainty and probability levels.

We then use the value we have obtained in the Chi-Squared test to see if the data is statistically significant. A distribution table is used for this. The distribution table relates the chi-squared value with probabilities. We also use degrees of freedom to determine the number of comparisons made.

For a chi-squared test, the degrees of freedom equal the number of categories minus one (n-1). You will also need to determine your p-value.

Here is an example of a standard chi-squared table. You read the table by looking at the row corresponding to the degrees of freedom used in your experiment and the column corresponding to your p-value. You will find your critical value at the intersection of these rows and columns.

A single gene determines seed type with a dominant allele that produces smooth seeds and a recessive allele that produces wrinkled seeds. Mendel’s experiment resulted in 5474 smooth and 1850 wrinkled seeds. Allowing for some statistical error, how can we tell if this result fits our expected ratio?

When following these steps, it is useful to summarise your calculations in a table like so:

Table 4. Another example of how to obtain the values for the equation.

What if the observed values are significantly different from the expected values?

Suppose the observed values are significantly different to expected values. If the result is smaller than or equal to the stated p-value, there are some things we may want to consider. As discussed in the article on sex-linkage, autosomal linkage, and epistasis, there are several reasons why we might observe patterns of inheritance that do not fit Mendelian ratios.

• The trait might be sex-linked, meaning the sex of the individual affects whether they can inherit the trait.
• Two genes might be found on the same chromosome and thus exhibit linkage.
• Epistasis might also affect the phenotypes expressed by the individual.