Chi-Squared Test: Skills (SL IB Biology)

Looking for associations between species

• The distribution of species in a habitat is rarely random; it usually depends on factors such as soil type, water availability, and competition
• It is sometimes possible to observe an association between the distributions of different species within a habitat, e.g.
  • Species that are in a symbiotic relationship are likely to be found next to each other; we would say that there is a positive association between the distributions of these two species
  • Species that are in direct competition for the same resources will exclude each other from their immediate surroundings, and so are likely to be found in different parts of a habitat; there might be a negative association between the distributions of these two species
• If species have no interaction with each other, then there will be no association between their distributions, and any that appears to occur will be due to chance
  • We would say that such species have distributions that are independent of each other
• Random sampling with quadrats, along with a statistical test called the chi-squared test, can be used to test for an association between two species

Using a quadrat diagram

Random sampling with quadrats can be used to study the distribution of organisms

The 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
  • Its purpose is to assess whether any difference in these results is due to chance, or due to an association between the variables being tested
• A chi-squared test can be used to analyse data from quadrat sampling to determine whether or not there is a statistically significant association between the distributions of two species
  • To the eye there may appear to be an association between the two species, but if it is not statistically significant then researchers can conclude that species distributions are independent of each other, and any appearance of association is due to chance
  • If an association is statistically significant then it must be due to an important factor, such as a symbiotic relationship
• A chi-squared test enables scientists to test hypotheses
  • A hypothesis is a testable statement about the expected outcome of an experiment
  • There are two types of hypothesis:
    • A null hypothesis states that there is no significant difference, or association, between data sets e.g. that there is no association between the distributions of two species
    • An alternative hypothesis states that there is a significant difference, or association, between data sets e.g. that there is an association (either positive or negative) between the distributions of two species
• The result of a chi-squared test enables scientists to either accept or reject a null hypothesis

Using the chi-squared test to test for association

• Step 1: Construct a contingency table for your results
  • This allows the number of quadrats that contain one, both, or neither species to be recorded
• Step 2: Calculate the row, column, and overall totals for your contingency table
• Step 3: Calculate the expected values (E) for your table
  • The results recorded in the contingency table are the observed values (O); to calculate the chi-squared value we need to calculate the expected values for each data point.
  • The expected values are what we would expect to see if the null hypothesis were correct
  • Note that this is the first step towards calculating the chi-squared value, the equation for which is:

Σ = sum of          O = observed value          E = expected value

• Step 4: Calculate the difference between the observed and expected values
  • Subtract the expected values from the observed values (O - E); some of the resulting values will be negative
• Step 5: Square each difference
  • This eliminates negative values
• Step 6: Divide each squared difference by the expected value
• Step 7: Add all of the results from step 6 together
  • This gives the chi-squared value
• Step 8: Calculate the degrees of freedom
• Step 9: Establish a probability level or p-value
  • As biologists, we work with a probability level of 0.05, or 5 %
  • This means that we can be 95 % certain that any significant difference or association is not due to chance
  • Some studies require a higher level of certainty than this, e.g. medical researchers may use a smaller p-value
• Step 10: Use a critical values table and the results of steps 8-9 to find the critical value
  • In order to understand what the chi-squared value says about the data, a table relating chi-squared values to probability is needed; this critical values table displays the probabilities that the differences between expected and observed values are due to chance
• Step 11: Compare the chi-squared value with the critical value to assess the significance

Statistical significance

• The chi-squared value, once calculated, can be compared to a critical value; this allows statistical significance to be assessed
• If the chi-squared value is larger than the critical value, there is a statistically significant difference between observed and expected values, or a statistically significant association between two sets of results
  • In this case, the null hypothesis can be rejected
• If the chi-squared value is equal to or smaller than the critical value, there is no statistically significant difference between observed and expected values, or no statistically significant association between two sets of results
  • In this case, the null hypothesis can be accepted
• To determine the critical value biologists generally use a probability level, or p-value, of 0.05, or 5 %
  • This means that if a difference or association is shown to be statistically significant at this level, there is only a 5 % probability (i.e. probability = 0.05) that this result might be due to chance