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Chi²-Test for r x s contingency tables

Chi²-Test for r x s contingency tables. PD Dr. Tim Becker Institute for Medical Biometry, Informatics and Epidemiology Bonn. Question. Does smoking status depend on home country?. Null hypothesis H 0 : smoking and home country are independent. Does smoking status depend on home country?.

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Chi²-Test for r x s contingency tables

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  1. Chi²-Test for r x s contingency tables PD Dr. Tim Becker Institute for Medical Biometry, Informatics and Epidemiology Bonn

  2. Question Does smoking status depend on home country? Null hypothesis H0: smoking and home country are independent

  3. Does smoking status depend on home country? Getting closer: Ad hoc comparison: compute frequencies per group

  4. Does smoking status depend on home country? Impression: Greek and Indonesian people seem to smoke more. Which absolute values would be expected under the null hypothesis?

  5. Which absolute values would be expected under the null hypothesis? Under H0 , the portion of Italians (Germans, Greek, Indonesian) shoulb be equal within the group of smokers and non-smokers! Portion of Italians in complete sample: (27+73)/(158+232)=0.2564 -> Expected number of smokers among Italians: 158*0.2564 (Here the portion of Italians in smokers and non-smokers are set equal to the portion in the complete sample.)

  6. Which absolute values would be expected under the null hypothesis? Task: fill the table!

  7. Which absolute values would be expected under the null hypothesis? Solution In general, expected values are computed as follows: Oij the number of observations in cell (i,j). Oi. Are the row sums and O.j the column sums. The expected Eij for cell (i,j) is Eij=O.j*Oi./N

  8. Computation of the test statistic Rule: „(Observed –Expected)²/Expected“ T=ij(Oij-Eij)²/Eij Task: Compute T for the table!

  9. Computation of the test statistic T is Chi-Square (²) distributed with (r-1)*(s-1) degrees of freedom, where r is the number of rows and s is the number of columns. In the example: 3 degrees of freedom (3 df).

  10. Test statistic: 2 – Test Simplification for 2-by-2 tables Degrees of freedom: f = 1  Quantile2 1;0.95 = 3,814 Decision: 2  3,814: H0 can not be rejected. 2 > 3,814: H0 can be rejected.

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