1. Test of significance for proportions FETP India
Chi-Square
Test of significance for proportions
FETP India

2. Competency to be gained from this lecture
Test the statistical significance of proportions using the relevant Chi-square test

3. Key elements Principles of the Chi-square
Comparison of a proportion with an hypothesized value
Chi-square for 2x2 tables
Chi-square for m x n tables
Testing dose-response with Chi-square

4. Analyzing quantitative and qualitative data
Quantitative data
Qualitative data
Normal
Non-normal
Summary statistics
Mean
Median
Proportions
Statistical tests
T-test
F-test
Non-parametric test
Chi-square
Fisher exact test

5. Chi-square: Principle
The Chi–square test examines whether a series of observed (O) numbers in various categories are consistent with the numbers expected (E) in those categories on some specific hypothesis (Null hypothesis)
O= Observed value
E= Expected value
Principle

6. How the Chi-square works in practice
X2 = 0 when every observed value is equal to the expected value
As soon as an observed value differs from the expected value, the X2 exceeds zero
The value of the X2 is compared with a tabulated value
If the calculated value of X2 exceeds the tabulated value under the column p = 0.05, the null hypothesis is rejected
Principle

7. Chi-square table: Percentage points of X2 distribution
Principle

8. Use of Chi-square to compare a proportion with a hypothesized value
The reported coverage for measles in a sub-center is 80%
The chief medical officer of the district suspects that this coverage could be overestimated
Validation survey with 80 children selected using simple random sampling
56/80 (70%) vaccinated
Hypothesized value

9. Chi-square calculations
Vaccinated
Non-vaccinated
Total
Expected 64 16 80
Observed 56 24

10. Interpretation of the Chi-square
The calculated value of X2 (i.e., 5) with 1 degree of freedom exceeds the table value (3.84) at 5% level
Hence, the medical officer rejects the null hypothesis that the coverage is 80%
Hypothesized value

11. Use of Chi-square to compare proportions between two samples
Cholera outbreak affecting a village
Cases clustered around a pond
Hypothesis generating interviews suggest that many case-patients washed their utensils in the pond
The investigator compares those who washed their utensils with the others in terms of cholera incidence
2x2 tables

12. Incidence of diarrhea (cholera) among persons who washed utensils in a pond and others, South 24 Parganas, West Bengal, India, 2006
Ill
Non-ill
Total
Washed utensils in pond
50 (89%) 6 56
Did not wash utensils in pond
8 (12%) 49 57
58 (51%) 55
113
2x2 tables

13. Chi-square to test the difference in the two proportions
The proportion of persons affected by cholera in the exposed and unexposed groups differ
Three steps to test whether this difference is significant:
Calculate expected values
Compare observed and expected to calculate the Chi-square
Compare the Chi-square with tabulated value
2x2 tables

14. Step 1: Calculate the expected values (1/2)
Ill
Non-ill
Total
Washed utensils in pond 50 6 56
Did not wash utensils in pond 8 49 57
58 (51%) 55
113
51% of the population became sick
If the cholera occurred at random, these proportions apply to the two groups, exposed and unexposed
2x2 tables

15. Step 1: Calculate the expected values (2/2)
Ill
Non-ill
Total
Washed utensils in pond
50 (28)
6 (28)
56)
Did not wash utensils in pond
8 (30)
49 (27) 57
58 (51%) 55
113
51% of the 56 who washed utensils (=28) should have been sick
All other numbers can de deducted by subtraction (one degree of freedom)
2x2 tables

16. Step 2: Compare observed and expected values
Ill
Non-ill
Total
Washed utensils
50 (28)
6 (28) 56
Did not wash utensils
8 (30)
49 (27) 57 58 55
113
2x2 tables

17. Step 3: Interpretation of the Chi-square
The calculated value of X2 (i.e., 68.6) with 1 degree of freedom exceeds the table value (3.84) at 5% level
Hence, we reject the Null hypothesis that the attack rate of cholera is equal in the exposed and unexposed group
This may suggest washing utensils in the pond is a source of infection if other elements of the investigation also support the hypothesis
2x2 tables

18. Simpler Chi-square formula
Ill
Non-ill
Total
Exposed a b
a+b
Unexposed c d
c+d
a+c
b+d
a+b+c+d (N)
2x2 tables

19. Application of simpler Chi-square formula to the cholera example
Ill
Non-ill
Total
Washed utensils 50 6 56
Did not wash utensils 8 49 57 58 55
113
2x2 tables

20. Corrected Chi-square formula
Ill
Non-ill
Total
Exposed a b
a+b
Unexposed c d
c+d
a+c
b+d
a+b+c+d
Note that the corrected value will always be smaller than the uncorrected which tends to exaggerate the significance of a difference
2x2 tables

21. Example of a 4x2 table Cataract Present Absent Total Hindu 10 90 100
Muslim 4 46 50
Christian 3 22 25
Others 1 9 18
167
185
Degrees of freedom = (Nbr of rows-1) x (Nbr of columns-1) =(4-1)x(2-1)= 3x1=3
N x N tables

22. Calculation of the Chi-square for a 4x2 table
Cataract
Present
Absent
Total
Hindu 10 90
100
Muslim 4 46 50
Christian 3 22 25
Others 1 9 18
167
185
Overall prevalence of cataract = 9.6%
Apply 9.6% proportion to all groups to calculate expected values
Use generic formula
N x N tables

23. Interpretation of a Chi-square for a m x n table
The Chi-square tests the overall Null hypothesis that all frequencies are distributed at random
If the Null hypothesis is rejected, it means the distribution is heterogeneous
It is not possible to:
“Attribute” the difference to a particular group
Regroup categories according to differences observed and test with a 2x2 table (i.e., post-hoc analysis)
Test with multiple 2x2 tables (i.e., multiple comparisons)
N x N tables

24. Key rule about Chi-square
Chi- square test should be applied on qualitative data set out in the form of frequencies
Chi– square test should not be done on:
Percentages
Rates
Ratios
Mean values
N x N tables

25. Limitations to the use of Chi-square
When sample size is small, other exact tests (e.g., Fisher exact test) are preferred and calculated with computer
N < 30
Expected value < 5
When several expected cell frequencies are less than one, it is better to amalgamate rows / columns
N x N tables

26. Testing a dose-response relationship with a Chi-square
Overall m x n Chi-square
Tests the null hypothesis that the odds ratios do not differ
No particular conditions needed
Overall test, easy to compute
Rough conclusions
Chi-square for trend
Dose-reponse

27. Exposure to injections with reusable needles and acute hepatitis B, Thiruvananthapuram, Kerala, India, 1992
Potential risk factors
Cases (N=160)
Controls (N=160)
Odds ratio
95% confidence interval
None 51
120
1.0
N/A 
Single injection 41 25
3.9
Multiple injections 29 7
9.8
3.8-26
Heterogeneous exposures categories
Overall 3x2 Chi-square : 42, 2 degrees of freedom, p<
Dose-reponse

28. Testing a dose-response relationship with a Chi-square
Overall m x n Chi-square
Chi-square for trend
Tests for a linear trend for the increase of the odds ratios with increased levels of exposure
Requires equal interval in the exposure categories
Can be calculated on a computer
Refined conclusions
Dose-reponse

29. Odds of typhoid according to raw onion consumption, Darjeeling, West Bengal, India, 2005-2006
Weekly raw onion servings
Cases (n=123)
Controls (n=123)
Odds ratio # % OR
95% CI
1-2 19 23 28 34 1
N/A
3-4 25 31 27 61
1.4
0.6 – 3.3 5+ 38 46 8 5 7
2.5 – 21
Homogeneous exposures categories
Chi-square for trend: 16.8; P-value:
Dose-reponse

30. Take home messages
The Chi-square compares expected and observed counts
The Chi-square can compare an actual proportion with a theoretical one
The Chi-square is the basic test to compare proportions in epidemiological 2x2 tables
The Chi-square can be used also as a global test for m x n tables
Specific Chi-squares can be used to test for dose-response effect