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Chi-Square Test of significance for proportions FETP India

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

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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

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

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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

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How the Chi-square works in practice X 2 = 0 when every observed value is equal to the expected value As soon as an observed value differs from the expected value, the X 2 exceeds zero The value of the X 2 is compared with a tabulated value If the calculated value of X 2 exceeds the tabulated value under the column p = 0.05, the null hypothesis is rejected Principle

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Chi-square table: Percentage points of X 2 distribution Principle

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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

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Chi-square calculations Vaccinated Non- vaccinatedTotal Expected Observed562480

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Interpretation of the Chi-square The calculated value of X 2 (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

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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

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Incidence of diarrhea (cholera) among persons who washed utensils in a pond and others, South 24 Parganas, West Bengal, India, 2006 IllNon-illTotal Washed utensils in pond50 (89%)656 Did not wash utensils in pond8 (12%)4957 Total58 (51%) x2 tables

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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: 1.Calculate expected values 2.Compare observed and expected to calculate the Chi-square 3.Compare the Chi-square with tabulated value 2x2 tables

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Step 1: Calculate the expected values (1/2) IllNon-illTotal Washed utensils in pond Did not wash utensils in pond Total58 (51%) % of the population became sick If the cholera occurred at random, these proportions apply to the two groups, exposed and unexposed 2x2 tables

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Step 1: Calculate the expected values (2/2) IllNon-illTotal Washed utensils in pond 50 (28)6 (28)56) Did not wash utensils in pond 8 (30)49 (27)57 Total58 (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

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Step 2: Compare observed and expected values IllNon-illTotal Washed utensils50 (28)6 (28)56 Did not wash utensils8 (30)49 (27)57 Total x2 tables

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Step 3: Interpretation of the Chi-square The calculated value of X 2 (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

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Simpler Chi-square formula IllNon-illTotal Exposedaba+b Unexposedcdc+d Totala+cb+da+b+c+d (N) 2x2 tables

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Application of simpler Chi-square formula to the cholera example IllNon-illTotal Washed utensils50656 Did not wash utensils84957 Total x2 tables

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Corrected Chi-square formula IllNon-illTotal Exposedaba+b Unexposedcdc+d Totala+cb+da+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

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Example of a 4x2 table Cataract PresentAbsentTotal Hindu Muslim44650 Christian32225 Others1910 Total Degrees of freedom = (Nbr of rows-1) x (Nbr of columns-1) =(4-1)x(2-1)= 3x1=3 N x N tables

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Calculation of the Chi-square for a 4x2 table Cataract PresentAbsentTotal Hindu Muslim44650 Christian32225 Others1910 Total Overall prevalence of cataract = 9.6% Apply 9.6% proportion to all groups to calculate expected values Use generic formula N x N tables

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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

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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

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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

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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

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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 N/A Single injection Multiple injections Overall 3x2 Chi-square : 42, 2 degrees of freedom, p< Heterogeneous exposures categories Dose-reponse

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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

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Odds of typhoid according to raw onion consumption, Darjeeling, West Bengal, India, Weekly raw onion servings Cases (n=123) Controls (n=123) Odds ratio #%#%OR95% CI N/A – – 21 Chi-square for trend: 16.8; P-value: Homogeneous exposures categories Dose-reponse

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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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