1. Associations Let’s do the  2  dance It’s a question of frequency

2. Association Analysis The  2  (Chi-squared) test can be used to assess whether there is an association between sedentary organisms Asks question: do organisms tend to occur together? Sedentary organisms include plants and a few animals such as barnacles It can also look for disassociations It only uses frequency data i.e. presence/absence tallies using quadrats

3.  2 example Student asks question: does ling (Calluna vulgaris) tend to grow together with bell heather (Erica cinerea)?

4.  2 example Null hypothesis: There is no statistically significant association between ling and bell heather on an area of moorland i.e. their distributions are independent of each other Alternative hypothesis: There is a statistically significant association between ling and bell heather on an area of moorland i.e. the distributions of the two species are not independent of each other

5.  2 observed values Data: 200 quadrats placed randomly in an area of moorland The presence/ absence of ling and bell heather in each one was recorded in a table

6.  2 expected values

8. Step 2 Put expected value into 2 x 2 contingency table and calculate other expected values by subtraction from totals Expected value given by 122 – 81.7 = 40.3

9.  2 values This is the test statistic

10.  2 critical values

11.  2 conclusions The  2 value of16.82 is > critical value of 3.84 for 1 degree of freedom at the 5% probability level [p<0.05] The result is also significant at the 1% level [p<0.01] Therefore, we can accept that there is a highly significant difference between the observed and expected values We reject the Null Hypothesis and accept the Alternative Hypothesis

12.  2 conclusions Both or neither species occur more often than expected by chance Just one of the species occurs less frequently than expected by chance Therefore, we conclude that there is a positive association between ling and heather.

13. Exercise:  2 example Carry out a  2 (chi-squared) test to see if there is an association or disassociation between bell heather and bilberry

14. Exercise:  2 Data Start by stating your hypotheses, then do the sums! Bell Heather No Bell Heather Total Bilberry125567 No Bilberry 8845133 Total100 200

15. Exercise:  2 Questions What is the  2 test statistic? Can you reject the Null Hypothesis? Look at the table of observed and expected values – what do you notice? What do you conclude?

16. Exercise:  2 Answers The  2 test statistic is 13.04 which is > critical values at 5% and 1% levels of significance The Null Hypothesis can be rejected There are fewer quadrats with both or neither species, and more with just one species than expected....conclude that there is a significant disassociation between bilberry and bell heather These two species both occur on the moorland, but tend to occupy different microhabitats