# Test of significance for proportions FETP India

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

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

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

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

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

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

Chi-square table: Percentage points of X2 distribution
Principle

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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