Chi-Square (X2) Test of Independence 

The chi-square test of independence is an inferential statistical test commonly used to determine if there is a significant relationship between two categorical variables or if the variables are "independent" of each other. For example, a biologist might want to determine if two species of organisms associate (are found together) in a community. 

Does Species A associate with Species B?

Species A

Species B

Just like other statistical tests, the Chi-Square Test for Independence tests two hypotheses:

Null Hypothesis (Ho):
"There is not a significant relationship between variables, the variables are independent of each other; any association between variables is likely due to chance and sampling error."

For example:  

• There is no significant association between Species A and Species B; the species are independent of each other.  The location of Species A has no effect on the location of Species B.

Alternative Hypothesis (H1):
"There is a significant (positive or negative) relationship between variables; the association between variables is likely not due to chance or sampling error."

For example:  

• There is a significant association between Species A and Species B; the species are dependent.  Either Species A significantly associates with Species B or Species A does not significantly associate with Species B.

How to Calculate a Chi-Square Test of Independence

The first step is to collect raw data for the occurrence of each variable.  In ecology studies, tcv  gfbhis is often done via random sampling using a quadrant.  In our example, there are five quadrants.  Determine: 

• The number of quadrants with both species present
• The number of quadrants with Species A but not Species B
• The number of quadrants with Species B but not Species A
• The number of quadrants with neither species 

Then create a "contingency table" to display the results. In a contingency table, rows represent one categorical variable and columns represent the other, with each cell containing the observed frequency for that combination. Always include row totals, column totals, and a grand total.

Next you need to determine what would be EXPECTED assuming the variables are independent of each other.  
​Expected frequencies = (row total X column total) / grand total

Now that you have OBSERVED and EXPECTED values, the chi-square statistic can be calculated.  Chi-square statistic computation involves comparing observed and expected frequencies using the formula:
χ² = Σ[(Observed - Expected)²/Expected].

The final calculated chi-square value is determined by summing the values across all cells in the contingency table. The larger the chi-square value, the greater the difference between your observed data and what you'd expect under independence.

χ² = ​ 0.0 + 0.1 = 0.1 + 0.2 = 0.4

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The calculated χ²  value is than compared to the “critical value  χ²” found in an χ² distribution table.  The χ² distribution table represents a theoretical curve of  expected results. The expected results are based on DEGREES OF  FREEDOM. 

Degrees of Freedom = (number of rows - 1) X (number of columns - 1)
In our example, DF = (2-1) X (2-1) = 1 X 1 = 1
*the row and column for the total in the contingency table are not included
 
The χ² distribution table is organized by the Level of  Significance.  The level of significance is the maximum tolerable probability of accepting a false null hypothesis.  We use 0.05.  ​

• If the calculated value is lower than the critical value in the table at the 0.05 level of significance, accept the null hypothesis and conclude that there is NO significant dependency between the variables.   

• If the calculated value is higher than the critical value in the table at the 0.05 level of significance, reject the null hypothesis and conclude that there IS a significant dependency between the variables.  ​

​
For example, with a DF=1, a value greater than 3.841 is required to be considered statistically significant (at p = 0.05). Since the X2 we calculated (0.4) is less than 3.841, there is NOT a significant association between Species A and Species B.  The location of Species A has no significant effect on the location of Species B, any association between species is likely due to chance and sampling error.

Even statistically significant results may have limited biological relevance, so always interpret your statistical conclusions within the broader context of biological principles and real-world applications. Consider factors like effect size, sample representativeness, and the biological mechanisms that might explain any observed relationships.

Sample size requirements must be met before applying this test. Each expected frequency in your contingency table should be at least 5, and your total sample size should be sufficiently large to ensure reliable results. If these conditions aren't met, you'll need to either collect more data or consider alternative statistical approaches.