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Contingency Tables (cross tabs) Generally used when variables are nominal and/or ordinal Even here, should have a limited number of variable attributes (categories) Even here, should have a limited number of variable attributes (categories) Some find these very intuitiveothers struggle It is very easy to misinterpret these critters It is very easy to misinterpret these critters

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Interpreting a Contingency Table WHAT IS IN THE INDIVIDUAL CELLS? The number of cases that fit in that particular cell The number of cases that fit in that particular cell In other words, frequencies (number of cases that fit criteria)In other words, frequencies (number of cases that fit criteria) For small tables, and/or small sample sizes, it may be possible to detect relationships by eyeballing frequencies. For most.. For small tables, and/or small sample sizes, it may be possible to detect relationships by eyeballing frequencies. For most.. Convert to Percentages: a way to standardize cells and make relationships more apparentConvert to Percentages: a way to standardize cells and make relationships more apparent

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Example 1 Random sample of UMD students to examine political party membership Are UMD students more likely to belong to particular political parties? Are UMD students more likely to belong to particular political parties? DemocratDemocrat RepublicanRepublican IndependentIndependent GreenGreen (N=40 UMD students) (N=40 UMD students)

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Univariate Table Univariate Table Null: The distribution is even across all categories. (N=40 UMD students) CategoriesF% Republican1230% Democrat1435% Independent923% Green510%

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Example 2 A survey of 10,000 U.S. residents Is ones political view related to attitudes towards police? What are the DV and IV? What are the DV and IV? Convention for bivariate tables The IV is on the top of the table (dictates columns) The IV is on the top of the table (dictates columns) The DV is on the side (dictates rows). The DV is on the side (dictates rows).

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Bivariate Table Attitude Towards Police Political Party Total RepubDemocratLibertarianSocialist Favorable29002100180305210 Unfav.19001800160283888 Total48003900340589098

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The Percentages of Interest Attitude Towards Police Political Party Total RepubDemocratLibertarianSocialist Favorable2900 (60%) 2100 (54%) 180 (53%) 30 (52%) 5210 Unfav19001800160283888 Total48003900340589098

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The Test Statistic for Contingency Tables Chi Square, or 2 Calculation Calculation Observed frequencies (your sample data)Observed frequencies (your sample data) Expected frequencies (UNDER NULL)Expected frequencies (UNDER NULL) Intuitive: how different are the observed cell frequencies from the expected cell frequencies Intuitive: how different are the observed cell frequencies from the expected cell frequencies Degrees of Freedom: Degrees of Freedom: 1-way = K-11-way = K-1 2-way = (# of Rows -1) (# of Columns -1)2-way = (# of Rows -1) (# of Columns -1)

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One-Way CHI SQUARE The most simple form of the Chi square is the one-way Chi square test For Univariate Table Do frequencies observed differ significantly from an even distribution?Do frequencies observed differ significantly from an even distribution? Null hypothesis: there are no differences across the categories in the populationNull hypothesis: there are no differences across the categories in the population

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Chi Square: Steps 1. Find the expected (under null hypothesis) cell frequencies 2. Compare expected & observed frequencies cell by cell 3. If null hypothesis is true, expected and observed frequencies should be close in value 4. Greater the difference between the observed and expected frequencies, the greater the possibility of rejecting the null

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Calculating 2 2 = [(f o - f e ) 2 /f e ] Where F e = Row Marginal X Column Marginal Where F e = Row Marginal X Column MarginalN So, for each cell, calculate the difference between the actual frequencies (observed) and what frequencies would be expected if the null was true (expected). Square, and divide by the expected frequency. Add the results from each cell.

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1-WAY CHI SQUARE 1-way Chi Square Example: There is an even distribution of membership across 4 political parties (N=40 UMD students) Find the expected cell frequencies ( F e = N / K) Find the expected cell frequencies ( F e = N / K) Categories FoFoFoFo FeFeFeFe Republican1210 Democrat1410 Independ.910 Green510

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1-WAY CHI SQUARE 1-way Chi Square Example: There is an even distribution of membership across 4 political parties (N=40 UMD students) Compare observed & expected frequencies cell-by-cell Compare observed & expected frequencies cell-by-cell Categories FoFoFoFo FeFeFeFe f o - f e Republican12102 Democrat14104 Independ.910 Green510-5

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1-WAY CHI SQUARE 1-way Chi Square Example: There is an even distribution of membership across 4 political parties (N=40 UMD students) Square the difference between observed & expected frequencies Square the difference between observed & expected frequencies Categories FoFoFoFo FeFeFeFe f o - f e (f o - f e ) 2 Republican121024 Democrat1410416 Independ.9101 Green510-525

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1-WAY CHI SQUARE 1-way Chi Square Example: There is an even distribution of membership across 4 political parties (N=40 UMD students) Divide that difference by expected frequency Divide that difference by expected frequency Categories FoFoFoFo FeFeFeFe f o - f e (f o - f e ) 2 (f o - f e ) 2 /f e Republican1210240.4 Democrat14104161.6 Independ.91010.1 Green510-5252.5 ====4.6

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Interpreting Chi-Square Chi-square has no intuitive meaning, it can range from zero to very large As with other test statistics, the real interest is the p value associated with the calculated chi-square value As with other test statistics, the real interest is the p value associated with the calculated chi-square value Conventional testing = find 2 (critical) for stated alpha (.05,.01, etc.)Conventional testing = find 2 (critical) for stated alpha (.05,.01, etc.) Reject if 2 (observed) is greater than 2 (critical) Reject if 2 (observed) is greater than 2 (critical) SPSS: find the exact probability of obtaining the 2 under the null (reject if less than alpha)SPSS: find the exact probability of obtaining the 2 under the null (reject if less than alpha)

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Example of Chi-Square Sampling Distributions (Assuming Null is True)

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Interpreting 2 the old fashioned way: The UMD political party example Chi square = 4.6 df (1-way Chi square) = K-1 = 3 X 2 (critical) (p

2-WAY CHI SQUARE Compare expected & observed frequencies cell by cellCompare expected & observed frequencies cell by cell X 2 (obtained) = 4.920X 2 (obtained) = 4.920 df= (r-1)(c-1) = 1 X 1 = 1df= (r-1)(c-1) = 1 X 1 = 1 X 2 (critical) for alpha of.05 is 3.841 (Healey Appendix C)X 2 (critical) for alpha of.05 is 3.841 (Healey Appendix C) Obtained > CriticalObtained > Critical CONCLUSION:CONCLUSION: Reject the null: There is a relationship between the team that students root for and their opinion of Brett Favre (p