1) The chi-square test of independence is used to determine if there is a relationship between two categorical variables. It compares observed frequencies to expected frequencies if the null hypothesis of independence is true. 2) A contingency table is constructed with the observed frequencies. Expected frequencies are calculated for each cell based on row and column totals and the grand total. 3) The chi-square statistic is calculated by summing the squared differences between observed and expected frequencies divided by the expected frequency for each cell. This value is then compared to a critical value from the chi-square distribution to determine if the null hypothesis should be rejected.

1. Chi-Square Test of Independence SHAMEER P.H dept. of futures studies 2010-'12

3. normal question that intends to resolve

4. tentative formulated for empirical testing

5. tentative answer to research question

6. point to start a researchdept. of futures studies 2010-'12

7. Research Questions and Hypotheses Research question: Non-directional: No stated expectation about outcome Example: Do men and women differ in terms of conversational memory? Hypothesis: Statement of expected relationship Directionality of relationship Example: Women will have greater conversational memory than men dept. of futures studies 2010-'12

8. The Null Hypothesis Null Hypothesis - the absence of a relationship E..g., There is no difference between men’s and women’s with regards to conversational memories Compare observed results to Null Hypothesis How different are the results from the null hypothesis? We do not propose a null hypothesis as research hypothesis - need very large sample size / power Used as point of contrast for testing dept. of futures studies 2010-'12

9. Hypotheses testing When we test observed results against null: We can make two decisions: 1. Accept the null No significant relationship Observed results similar to the Null Hypothesis 2. Reject the null Significant relationship Observed results different from the Null Hypothesis Whichever decision, we risk making an error dept. of futures studies 2010-'12

10. Type I and Type II Error 1. Type I Error Reality: No relationship Decision: Reject the null Believe your research hypothesis have received support when in fact you should have disconfirmed it Analogy: Find an innocent man guilty of a crime 2. Type II Error Reality: Relationship Decision: Accept the null Believe your research hypothesis has not received support when in fact you should have rejected the null. Analogy: Find a guilty man innocent of a crime dept. of futures studies 2010-'12

11. Potential outcomes of testing Decision Accept NullReject Null R No E Relationship A L I T Y Relationship Correct decision Type I Error Correct decision Type II Error dept. of futures studies 2010-'12

12. Start by setting level of risk of making a Type I Error How dangerous is it to make a Type I Error: What risk is acceptable?: 5%? 1%? .1%? Smaller percentages are more conservative in guarding against a Type I Error Level of acceptable risk is called “Significance level” : Usually the cutoff - <.05 dept. of futures studies 2010-'12

13. Steps in Hypothesis Testing State research hypothesis State null hypothesis Decide the appropriate test criterion( eg. t test, χ2 test, F test etc.) Set significance level (e.g., .05 level) Observe results Statistics calculate probability of results if null hypothesis were true If probability of observed results is less than significance level, then reject the null dept. of futures studies 2010-'12

14. Guarding against Errors Significance level regulates Type I Error Conservative standards reduce Type I Error: .01 instead of .05, especially with large sample Reducing the probability of Type I Error: Increases the probability of Type II Error Sample size regulates Type II Error The larger the sample, the lower the probability of Type II Error occurring in conservative testing dept. of futures studies 2010-'12

16. Z test

17. F test

18. χ2test

19. ……..dept. of futures studies 2010-'12

20. Testing hypothesis for two nominal variables Variables Null hypothesis Procedure Gender Passing is not Chi-square related to gender Pass/Fail dept. of futures studies 2010-'12

21. Testing hypothesis for one nominal and one ratio variable Variables Null hypothesis Procedure Gender Score is not T-test related to gender Test score dept. of futures studies 2010-'12

22. Testing hypothesis for one nominal and one ratio variable Variable Null hypothesis Procedure Year in school Score is not related to year in ANOVA school Test score Can be used when nominal variable has more than two categories and can include more than one independent variable dept. of futures studies 2010-'12

23. Testing hypothesis for two ratio variables Variable Null hypothesis Procedure Hours spent studying Score is not related to hours Correlation spent studying Test score dept. of futures studies 2010-'12

24. Testing hypothesis for more than two ratio variables Variable Null hypothesis Procedure Hours spent studying Score is positively related to hours Classes spent studying and Multiple missed negatively related regression to classes missed Test score dept. of futures studies 2010-'12

25. Chi square (χ2 ) test dept. of futures studies 2010-'12

26. Used to: Test for goodness of fit Test for independence of attributes Testing homogeneity Testing given population variance dept. of futures studies 2010-'12

27. Chi-Square Test of Independence dept. of futures studies 2010-'12

28. Introduction (1) We often have occasions to make comparisons between two characteristics of something to see if they are linked or related to each other. One way to do this is to work out what we would expect to find if there was no relationship between them (the usual null hypothesis) and what we actually observe. dept. of futures studies 2010-'12

29. Introduction (2) The test we use to measure the differences between what is observed and what is expected according to an assumed hypothesis is called the chi-square test. dept. of futures studies 2010-'12

30. For Example Some null hypotheses may be: ‘there is no relationship between the subject of first period and the number of students absent in our class’. ‘there is no relationship between the height of the land and the vegetation cover’. ‘there is no connection between the size of farm and the type of farm’ dept. of futures studies 2010-'12

31. Important The chi square test can only be used on data that has the following characteristics: The frequency data must have a precise numerical value and must be organised into categories or groups. The data must be in the form of frequencies The expected frequency in any one cell of the table must be greater than 5. The total number of observations must be greater than 20. dept. of futures studies 2010-'12

32. Contingency table Frequency table in which a sample from a population is classified according to two attributes, which are divided in to two or more classes dept. of futures studies 2010-'12

34. Number of cells – no. of constraints dept. of futures studies 2010-'12

35. Formula χ2 = ∑ (O – E)2 E χ2 = The value of chi square O = The observed value E = The expected value ∑ (O – E)2 = all the values of (O – E) squared then added together dept. of futures studies 2010-'12

36. Critical region: dept. of futures studies 2010-'12

37. dept. of futures studies 2010-'12

38. Construct a table with the information you have observed or obtained. Observed Frequencies (O) dept. of futures studies 2010-'12

39. Expected frequency = row total x column total Grand total Now Work out the expected frequency. dept. of futures studies 2010-'12

40. (O – E)2 E Now For each of the cells calculate. dept. of futures studies 2010-'12

41. χ2Calc. = sum of all ( O-E)2/ E values in the cells. Here χ 2Calc.=36.873 Find χ 2critical From the table with degree of freedom 2 and level of significance 0.05 χ 2Critical =5.99 dept. of futures studies 2010-'12

42. Χ2table dept. of futures studies 2010-'12

43. Conclusion Compareχ2Calc.and Χ2critical obtained from the table Ifχ2Calc. Is larger thanχ2Critical.then reject null hypothesis and accept the alternative Here since χ 2Calc.is much greater than χ 2Critical, we can easily reject null hypothesis that is ; there lies a relation between the gender and choice of selection. dept. of futures studies 2010-'12

44. Reference RESEARCH METHODOLGIES - L R Potti dept. of futures studies 2010-'12