IB Biology Topic C4.1.15: Use of the Chi-squared Test for Association Between Two Species Notes

The Chi-Squared Test Checks Whether Two Species Are Found Together by Chance

1. The test compares observed counts with the counts expected if the two species were distributed independently.
2. In ecology it tells us whether two species are independent or associated.
   1. Independently distributed: no ecological relationship between them.
   2. Associated: found together more or less often than chance, hinting at competition or mutualism.

The Two Hypotheses You Test

1. Null hypothesis (H₀): the two species are distributed independently, with no association.
2. Alternative hypothesis (H₁): there is an association, so the species are found together more or less often than expected by chance.

The Five Steps of the Test

1. Collect data with quadrat sampling
   1. Place random quadrats across the study site.
   2. Record the presence or absence of both species in each quadrat.
2. Build a 2 × 2 contingency table
   1. Sort the quadrats into the four categories and add the row and column totals.

1. Calculate the expected value (E) for each cell

\[
E = \frac{\text{row total} \times \text{column total}}{\text{grand total}}
\]

1. Apply the chi-squared formula

\[
\chi^2 = \sum \frac{(O - E)^2}{E}
\]

• O is the observed value in each cell.
• E is the expected value for that cell.

1. Compare with the critical value
   1. For a 2 × 2 table the degrees of freedom = (rows − 1)(columns − 1) = 1.
   2. Look up the critical value at p = 0.05 with 1 degree of freedom (3.84).
   3. If the calculated χ² is greater than the critical value, reject H₀: there is a significant association.
   4. If χ² is less than or equal to the critical value, do not reject H₀: the species are distributed independently.

Reading the Result: Positive, Negative, or No Association

1. Reject H₀ (an association exists)
   1. Positive association: the species occur together more often than expected, which can suggest mutualism or shared habitat.
   2. Negative association: the species occur together less often than expected, which can suggest competition.
2. Fail to reject H₀ (no association)
   1. The species are distributed independently, so there is no significant relationship.

What the Test Tells Ecologists

1. Competition: a negative association can point to competitive exclusion between the species.
2. Mutualism: a positive association can point to a beneficial relationship between the species.

A Theory of Knowledge Angle