1. Chi-squared Association Index
The other variants of chi-squared (looking for a difference and goodness of fit ) are covered separately

2. What does it do? There are two ways to use this test:
Looking for association between two factors
Eg: Are snail shell colour and habitat choice associated?
Looking for a difference in population/employment structures
Eg: Is the population structure the same in two villages?
You do the calculations the same way in both cases!
The structures version is strictly for comparing two sets of data that are “on a level” – this method should not be used to compare, say, local and national data – that’s covered in Chi-squared goodness of fit.

3. Planning to use it? Make sure that…
You are working with numbers of things, not, eg area, weight, length, %…
You have an average of at least 5 things (people/plants/species…) in each category

4. How does it work?
For Association, you assume (null hypothesis) there is no association
For Difference in Structures, you assume (null hypothesis) there is no difference in structures
It compares
observed values
the data you collected
expected values
what you’d get if there was really no association or no difference in structures

5. Doing the test These are the stages in doing the test:
Write down your hypotheses
Work out the expected values
Use the chi-squared formula to get a chi-squared value
Work out your degrees of freedom
Look at the tables
Make a decision
The underlined terms are hyperlinks to the appropriate slide
Examples
Association
Difference in structures

6. Hypotheses Association: H0: There is no association
H1: There is some association
Difference in structures
H0: There is no difference in structures
H1: There is some difference in structures
This is the standard form for hypotheses for this type of chi-squared test – the others are not the same

7. Expected Values Your data here will be in a table.
To work out the expected values:
Add up the totals of all the rows, and the totals of all the columns. Also find the overall total of all the data
Work out expected values using
Eg, to work out the expected value for something in 2nd row, 3rd column, multiply total of 2nd row by total of 3rd column and divide by overall total

8. Chi-Squared Formula For each cell in your table, work out
O = Observed value – your data
E = Expected value – which you’ve calculated
Then add all your values up. This gives the chi-squared value
S = “Sum of”

9. degrees of freedom = (rows – 1)(columns – 1)
The formula here for degrees of freedom is
degrees of freedom = (rows – 1)(columns – 1)
You do not need to worry about what this means –just make sure you know the formula!
But in case you’re interested – the larger your table, the more likely you are to get a “strange” result in one or more cells. The degrees of freedom is a way of allowing for this in the test.
Worth emphasising that this is a different formula to the other types of chi-squared

10. Tables This is a chi-squared table These are your significance levels
eg 0.05 = 5%
These are your degrees of freedom (df)
Perhaps a good time to check they’re OK on reading tables?

11. Make a decision
If the value you calculated is bigger than the tables, you reject your null hypothesis
If the value you calculated is smaller than the tables, you accept your null hypothesis.
Remember in each case to refer back to the actual example!
In all tests except MannWhitney & Wilcoxon, they’ll be rejecting if their value is bigger

12. Example: Snail shell colour & habitat
Samples were taken from limestone woodland and limestone pavement, and the numbers of light- and dark-shelled snails noted.
Hypotheses:
H0:Shell colour and habitat preference are not associated
H1 Shell colour and habitat preference are associated

13. The data
Light Dark
Pavement
Woodland

14. Totals Light Dark Totals Pavement 115 76 191 Woodland 69 106 175
If you have any keen mathematicians, they could also look up Yates’ correction (for 2 x 2 tables)

15. Row Total  Column Total
Expected Values
Row Total  Column Total
Overall Total
Expected value =
Expected values:
Light Dark
Pavement
Woodland
As a check, they could see that the row & column totals still add up to the same thing

16. The calculations: (O-E)2/E
Light Dark
Pavement
Woodland

17. Tables This is a chi-squared table These are your significance levels
eg 0.05 = 5%
These are your degrees of freedom (df)
Perhaps a good time to check they’re OK on reading tables?

18. The test c2 = 15.776 Degrees of freedom = (2 – 1)(2 – 1) = 1
Critical value (5%) = 3.841
Reject H0 – there is some association between snail shell colour and habitat preference

19. Tables This is a chi-squared table These are your significance levels
eg 0.05 = 5%
These are your degrees of freedom (df)
Perhaps a good time to check they’re OK on reading tables?

20. Example: Comparing population structures
Data on the population age structures of two villages were obtained. The aim is to assess whether there is any difference in the age structures.
Hypotheses:
H0:There is no difference in the villages’ population structures
H1 There is a difference in the villages’ population structures

21. The data Age Village A Village B 0 -10 16 25 11-20 12 32 21-30 32 50

22. Totals
Age A B Total
Total

23. Row Total  Column Total
Expected Values
Row Total  Column Total
Overall Total
Expected Value =
Age Village A Village B
0 -10
10-20
20-30
30-40
40-50
50+
At this stage, check whether any expected < 5.
If so, amalgamate some age categories

24. The calculations: (O-E)2/E
Age Village A Village B
0 -10
10-20
20-30
30-40
40-50
50+
0.089

25. Tables This is a chi-squared table These are your significance levels
eg 0.05 = 5%
These are your degrees of freedom (df)
Perhaps a good time to check they’re OK on reading tables?

26. The test
c2 =
c2 =
Degrees of freedom = (6 – 1)(2 – 1) = 5
Critical value (5%) =
Reject H0 – the population structures of the two villages are different

27. Tables This is a chi-squared table These are your significance levels
eg 0.05 = 5%
These are your degrees of freedom (df)
Perhaps a good time to check they’re OK on reading tables?