Chi-Square Part II Fenster Chi-Square Part II Let us see how this works in another example. Attitudes towards Research Attitudes Towards Statistics Favorable Neither favorable nor unfavorable Favorable Unfavorable Row Totals Neither favorable nor unfavorable Unfavorable Col.
Download ReportTranscript Chi-Square Part II Fenster Chi-Square Part II Let us see how this works in another example. Attitudes towards Research Attitudes Towards Statistics Favorable Neither favorable nor unfavorable Favorable Unfavorable Row Totals Neither favorable nor unfavorable Unfavorable Col.
Chi-Square Part II Fenster Chi-Square Part II Let us see how this works in another example. Attitudes towards Research Attitudes Towards Statistics Favorable Neither favorable nor unfavorable Favorable Unfavorable Row Totals 9 26 13 48 Neither favorable nor unfavorable 19 75 83 177 Unfavorable 16 56 110 182 Col. Totals 44 157 206 407 Chi-Square Part II It has been argued that people with favorable attitudes towards research tend to have favorable attitudes towards statistics. Question: If we knew the attitudes towards research of a respondent, can we predict the attitude toward statistics? Chi-Square Part II Step 2 H1: Knowledge of attitudes toward research does help us predict attitudes towards statistics. Step 1 HO: Knowledge of attitudes toward research does not help us predict attitudes towards statistics. Chi-Square Part II Selecting a significance level: Let’s use =.05. This gives us a χ2 critical of 9.488. Your book says the χ2 critical of 9.5. Step 4: Collect and summarize sample data. We will use the chi-square test with 4 degrees of freedom. Why four? df=(r-1) X (c-1) We have 3 rows and 3 columns. so we get df= (3-1) X (3-1)= 2 X 2=4 Chi-Square Part II If we find a χ2 greater than or equal to 9.5 we reject the null hypothesis and conclude that attitudes towards research can predict attitudes towards statistics. If we find a χ2 less than 9.5 we fail to reject the null hypothesis and conclude attitudes towards research cannot predict attitudes towards statistics. Calculation of Expected Frequencies Expected frequencies= (Row total) X (Column Total) Grand Total Calculation of Expected Frequencies Cell a – Favorable attitudes towards both research and statistics. (44) X (48) = 5.18 407 Calculation of Expected Frequencies Cell b – Neither favorable or unfavorable attitudes towards research, favorable attitudes towards statistics. (157) X (48) = 18.51 407 Calculation of Expected Frequencies Cell c –Unfavorable attitudes towards research, favorable attitudes towards statistics (206) X (48) = 24.29 407 Calculation of Expected Frequencies Cell d – Favorable attitudes towards research, neither favorable or unfavorable attitudes towards statistics (44) X (177) = 19.13 407 Calculation of Expected Frequencies Cell e - Neither favorable or unfavorable attitudes towards both statistics and research (157) X (177) = 68.27 407 Calculation of Expected Frequencies Cell f – Unfavorable attitudes towards research, neither favorable or unfavorable attitudes towards statistics (206) X (177) = 89.58 407 Calculation of Expected Frequencies Cell g – Favorable attitudes towards research, unfavorable attitudes towards statistics (44) X (182) = 19.67 407 Calculation of Expected Frequencies Cell h - Neither favorable or unfavorable attitudes towards research, unfavorable attitudes towards statistics (157) X (182) = 70.20 407 Calculation of Expected Frequencies Cell i – Unfavorable attitudes towards both research and statistics (206) X (182) = 92.11 407 So we set up our chi-square table Cell f observed f expected f observed-f expected (i.e., RESIDUAL S) (f observed-f expected)2 (f observed-f expected)2/f expected a 9 5.18 3.82 14.59 2.81 b 26 18.51 7.49 56.1 3.03 c 13 24.29 -11.29 127.46 5.25 d 19 19.13 -0.13 0.17 0.008 e 75 68.27 6.73 45.29 0.6 f 83 89.58 -6.58 43.29 0.5 g 16 19.67 -3.67 13.46 0.67 h 56 70.20 -14.2 201.64 2.87 i 110 92.11 17.89 320.05 3.5 Total 407 407.00 0.00 20.2 Hypothesis Testing with Chi-Square Step 5: Making a decision 2 Χ observed= 20.2 χ2 critical= 9.488. Decision: REJECT HO, and conclude that attitudes towards research allow us to predict attitudes towards statistics. Hypothesis Testing with Chi-Square Notes about chi-square: (1) Σ (f observed - f expected)=0. The RESIDUALS ALWAYS SUM TO ZERO. If Σ (f observed - f expected) does not equal zero (within rounding error), you have made a calculation error. Recheck your work. Hypothesis Testing with Chi-Square The chi-square test itself cannot tell us anything about directionality. One way to get directionality in the chisquare is to look at the (f observed- f expected) column. We see that certain cells occur much less frequently than we would expect. Hypothesis Testing with Chi-Square For example cell c (unfavorable attitudes towards research but favorable attitudes towards statistics) occurs much less frequently than we would expect on the basis of chance. Analysis of Residuals Cell f observed f expected f observed-f expected (i.e., RESIDUAL S) (f observed-f expected)2 (f observed-f expected)2/f expected a 9 5.18 3.82 14.59 2.81 b 26 18.51 7.49 56.1 3.03 c 13 24.29 -11.29 127.46 5.25 d 19 19.13 -0.13 0.17 0.008 e 75 68.27 6.73 45.29 0.6 f 83 89.58 -6.58 43.29 0.5 g 16 19.67 -3.67 13.46 0.67 h 56 70.20 -14.2 201.64 2.87 i 110 92.11 17.89 320.05 3.5 Total 407 407.00 0.00 20.2 Hypothesis Testing with Chi-Square We can also see that three cells that capture consistency of attitudes between research and statistics (cell a favorable attitudes for both, cell e neither favorable or unfavorable attitudes towards both, cell i unfavorable attitudes for both) all have a positive values for (f observed- f expected). Those three cells are consistent with the (unstated and untested) hypothesis that individuals tend to have similar attitudes for both research and statistics Hypothesis Testing with Chi-Square Only by examining the (f observed- f expected) can we give any statement on the directionality of the relationship. [We could also analyze the column percentages as we move across categories of the independent variable to give us insight on directionality.] Hypothesis Testing with Chi-Square 3) In this example, why do we get statistical significance? We can say that the cells d, e, f and g do not contribute to the statistical significance of the overall relationship. The individual chisquare values for these four cells are all very small. The overall relationship is significant because of the other cells. Analysis of Residuals Cell f observed f expected f observed-f expected (i.e., RESIDUAL S) (f observed-f expected)2 (f observed-f expected)2/f expected a 9 5.18 3.82 14.59 2.81 b 26 18.51 7.49 56.1 3.03 c 13 24.29 -11.29 127.46 5.25 d 19 19.13 -0.13 0.17 0.008 e 75 68.27 6.73 45.29 0.6 f 83 89.58 -6.58 43.29 0.5 g 16 19.67 -3.67 13.46 0.67 h 56 70.20 -14.2 201.64 2.87 i 110 92.11 17.89 320.05 3.5 Total 407 407.00 0.00 20.2 Hypothesis Testing with Chi-Square Chi-square allows us to decompose the overall relationship into its component parts. This decomposition allows us to assess whether all categories contribute to the significance of the overall relationship. Hypothesis Testing with Chi-Square Limitations for χ2 So far we have stressed the virtues for χ2 such as weak assumptions, and a statistical significance test appropriate for nominal level data. This is why chi-square is so popular. There are two limitations for χ2, one minor and one major. Hypothesis Testing with Chi-Square Minor Limitation When the expected cell frequency is less than 5, χ2 rejects the null hypothesis too easily. (Note: this means the EXPECTED frequency and NOT the OBSERVED frequency). Solution: Use Yates' correction Yates’ correction Take the | (f observed- f expected) | -0.5 Hypothesis Testing with Chi-Square Major Limitation We have set up a null hypothesis that there is no relationship between two variables and have tried to reject this hypothesis. We refer to a relationship as being statistically significant when we have established, subject to the risk of type I error, that there is a relationship between two variables. But does rejecting the null hypothesis mean the relationship is significant in the sense of being a strong or an important Hypothesis Testing with Chi-Square Remember significance levels are dependent upon sample size. Let us say that you wanted to investigate the relationship between gender and level of tolerance. You had no money to investigate this relationship, so you handed out questionnaires around UML and found the following: Hypothesis Testing with Chi-Square Gender Attitudes towards racial tolerance High Males Females Row Totals 24 26 50 Low 26 24 50 Column Totals 50 50 100 Hypothesis Testing with Chi-Square Is there a significant relationship between gender and attitudes towards racial tolerance? Let us use α=.05. We have one degree of freedom. χ2 critical=3.8. χ2 observed=0.16. Since χ2 observed (0.16) < χ2 critical (3.8), we FAIL to reject the null hypothesis and conclude that gender does not help us predict to attitudes towards racial tolerance. Now let us say you had an extremely ambitious study and you found the following relationship Gender Attitudes towards racial tolerance Males Females Row Totals High 2400 2600 5000 Low 2600 2400 5000 Column Totals 5000 5000 10000 Hypothesis Testing with Chi-Square Is there a significant relationship between gender and attitudes towards racial tolerance? Let us use α=.05. We have one degree of freedom. χ2 critical=3.8, χ2 observed=16.0. Since χ2 observed (16.0) > χ2 critical (3.8), we easily reject the null hypothesis and conclude that gender does help us predict to attitudes towards racial tolerance. Hypothesis Testing with Chi-Square χ2 is sensitive to the number of cases in the sample. Even though the proportions in the cells remain unchanged, the new χ2 is 100 times the old chi-square because we have 100 times the number of cases. Hypothesis Testing with Chi-Square Corrections for the sample size problem Pearson's contingency coefficient (You can ask for the Contingency Coefficient with SPSS CROSSTABS’ output). Hypothesis Testing with Chi-Square χ2 χ2 + N where N=total number of cases in sample Problem with C: Cannot attain 1.0 in perfect relationship. As the syllabus says, there is no ideal solution to the sample size problem with chi-square. C=