**Chi-Square Part II** Fenster

**Chi-Square Part II** • Let us see how this works in another example.

**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**

**Hypothesis Testing with Chi-Square** • Step 5: Making a decision • Χ2 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 chi-square 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**

**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 chi-square values for these four cells are all very small. The overall relationship is significant because of the other cells.

**Analysis of Residuals**

**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 one? • Not necessarily.

**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**

**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

**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** • C= 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.