1. Two-dimensional Chi-square
Sometimes, we want to classify cases on two dimensions at the same time – for example, we might want to classify newly-qualified physicians on the basis of their choice of type of practice and their sex.
If we did this, we could ask whether there is any relationship between the two – that is, are women and men equally likely to choose each type?

2. Two-dimensional Chi-square
If we classify a set of cases on two dimensions, and the two dimensions are independent of each other, then the proportions of events in the categories on one dimension should be the same in all the categories on the other dimension:
Thus, if choice of type of medical practice is independent of sex, then the proportions of men choosing various types of practice should be the same as the proportions of women…

3. Two-dimensional Chi-square
Specialty
Sex Rural GP City GP Specialist Σ
Male
Female
In this data set, there are four times as many women in the sample as men. There are also four times as many women in each specialty – thus, choice of specialty appears to be independent of sex.

4. Two-dimensional Chi-square
Specialty
Sex Rural GP City GP Specialist Σ
Male
Female
In this data set, there are four times as many men as women – but again the proportions are constant across specialties. Again, choice of specialty appears to be independent of sex.

5. Two-dimensional Chi-square
Specialty
Sex Rural GP City GP Specialist Σ
Male
Female
In this data set, there are equal numbers of women and men. But the proportions vary across specialties – thus, choice of specialty appears to be dependent on sex.

6. Two-dimensional Chi-square
The null hypothesis in the two-dimensional chi-square test is that the two dimensions are not related (that is, they are independent). To test this hypothesis, we need to compute expected values for each of the cells defined by the two dimensions.
In there were 25 rural GPs in our sample, and if type of practice were independent of sex, then half of the rural GPs should be men and half women.

7. Two-dimensional Chi-square
Our expected values reflect two proportions: the proportion of the sample in each sex category and the proportion in each practice category:
Specialty
Sex Rural GP City GP Specialist Σ
Male
Female Σ

8. Two-dimensional Chi-square
We’ll step through the calculations:
Specialty
Sex Rural GP City GP Specialist Σ
Male
Female Σ

9. Sex Rural GP City GP Specialist Σ Male 12.5 55 32.5 100
Specialty
Sex Rural GP City GP Specialist Σ
Male
Female Σ
First, notice this number – the sum of all the observations
Then note this number – the number of males

10. Sex Rural GP City GP Specialist Σ Male 12.5 55 32.5 100
Specialty
Sex Rural GP City GP Specialist Σ
Male
Female Σ
Then note this number – the number of rural GPs
This number is calculated as:
100 * 25 = 12.5
200

11. Two-dimensional Chi-square
Thus, expected values are computed as
Expected value = (Row total * column total)
sum of observations.
If you can do that, you can do the 2-dimensional chi-square.

12. Two-dimensional Chi-square
For the physicians example, we compute:
Χ2 = [ ]2 + [5-12.5]2 + [45-55]2 + [65-55]2
+ [ ]2 + [ ]2 =

13. Two-dimensional Chi-square
For the 2-D chi-square, degrees of freedom are:
(r-1)(c-1)
where r = # of rows and c = # of columns. Here, r = 2, c = 3, so d.f. = 1 * 2 = 2.
Thus, Χ2crit = Χ2(.05,2) = Our decision is to reject the null hypothesis (that the two dimensions are independent).

14. Formula for compute expected values
More generally, the rule for working out expected values in two dimensional classifications is:
Ê(nij) = ri * cj n
where n = total number of observations (cases in the sample)

15. Chi-square – Example 1 (from last week)
At a recent meeting of the Coin Flippers Society, each member flipped three coins simultaneously and the number of tails occurring was recorded. 1b. Subsequently, the number of tails each member flipped was determined for different value coins. The data are shown on the next slide as the number of members throwing different numbers of tails with different value coins.

16. Chi-square – Example 1b Coin Number of Tails Value 0 1 2 3
Is there evidence that the number of tails is affected by coin value? (α = .05)

17. Σ Chi-square – Example 1b HO: The two classifications are independent
HA: The two classifications are dependent
Test statistic: Χ2 = [nij – Ê(nij)]2
Ê(nij)
Rejection region: Χ2obt > Χ2crit = Χ2(.05, 6) = Σ

18. Chi-square – Example 1b
The first step is to compute the expected values for each cell, using the formula:
Ê(nij) = ri * cj n
For the top left cell, we get: (65) (162) = 21.06
500

19. Chi-square – Example 1b
Using the formula for all the other cells gives:
We are now ready to compute Χ2 obtained.

20. Chi-square – Example 1b Χ2obt = [20-21.06]2 + … + [20-17.7]2
= 4.032
Decision: do not reject HO - there is no evidence that the number of tails is affected by coin value.

21. Chi-square – Example 2b
There is an “old wives’ tale” that babies don’t tend to be born randomly during the day but tend more to be born in the middle of the night, specifically between the hours of 1 AM and 5 AM. To investigate this, a researcher collects birth-time data from a large maternity hospital. The day was broken into 4 parts: Morning (5 AM to 1 PM), Mid-day (1 PM to 5 PM), Evening (5 PM to 1 AM), and Mid-night (1 AM to 5 AM).

22. Chi-square – Example 2b
The numbers of births at these times for the last three months (January to March) are shown below:
Morning Mid-day Evening Mid-night

23. Chi-square – Example 2b
A question can certainly be raised as to whether the pattern reported above is peculiar to births in the winter months or reflects births at other times of the year as well.
The data obtained from the same hospital during the hottest summer months last year are shown on the next slide, along with the original data.

24. Chi-square – Example 2b Morn Midd Even Mid-night Σ 110 50 100 100 360 Σ
Are the two patterns different? (α = .05)

25. Σ Chi-square – Example 2b HO: The two classifications are independent
HA: The two classifications are dependent
Test statistic: Χ2 = [nij – Ê(nij)]2
Ê(nij)
Rejection region: Χ2obt > Χ2crit = Χ2(.05, 3) = 7.81 Σ

26. Chi-square – Example 2b
The first step is to compute the expected values for each cell, using the formula:
Ê(nij) = ri * cj n
For the top left cell, we get: (200) (360) = 112.5
640

27. Chi-square – Example 2b Using the formula for the other cells we get:
Morn Midd Even Midn
Cold
Hot

28. Chi-square – Example 1b Χ2obt = [110-112.5]2 + … + [70-74.375]2 =
Decision: do not reject HO - there is no evidence that the pattern of births is different in the hot months compared to the rest of the year.